Physics Reports 423 (2006) 1 – 48 www.elsevier.com/locate/physrep
The phenomenology of Dvali–Gabadadze–Porrati cosmologies Arthur Lue Department of Physics and Astronomy, University of Texas at San Antonio, 6900 North Loop 1604 West, San Antonio, TX 78249, USA Accepted 3 October 2005 Available online 5 December 2005 editor: M. P. Kamionkowski
Abstract Cosmologists today are confronted with the perplexing reality that the universe is currently accelerating in its expansion. Nevertheless, the nature of the fuel that drives today’s cosmic acceleration is an open and tantalizing mystery. There exists the intriguing possibility that the acceleration is not the manifestation of yet another mysterious ingredient in the cosmic gas tank (dark energy), but rather our first real lack of understanding of gravity itself, and even possibly a signal that there might exist dimensions beyond that which we can currently observe. The braneworld model of Dvali, Gabadadze and Porrati (DGP) is a theory where gravity is altered at immense distances by the excruciatingly slow leakage of gravity off our three-dimensional Universe and, as a modifiedgravity theory, has pioneered this line of investigation. I review the underlying structure of DGP gravity and those phenomenological developments relevant to cosmologists interested in a pedagogical treatment of this intriguing model. © 2005 Published by Elsevier B.V. PACS: 04.50.+h; 98.80.−k
Contents 1. The contemporary universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Gravitational leakage into extra dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The formal arena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Preliminary features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Cosmology and global structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. The modified Friedmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The Brane worldsheet in a Minkowski bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Luminosity distances and other observational constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Recovery of Einstein gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The van Dan–Veltman–Zakharov discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Case study: cosmic strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. The Einstein solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. DGP cosmic strings: the weak-Brane limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. The r/r0 → 0 limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4. The picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. The Schwarzschild-like solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. The field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-mail address:
[email protected]. 0370-1573/$ - see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.physrep.2005.10.007
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4.3.2. The weak-Brane limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3. The r/r0 → 0 limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Modified gravitational forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Background cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Metric potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Gravitational regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Bulk solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1. Weak-field analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2. A note on bulk boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Anomalous orbit precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Nearly circular orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Solar system tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Large-scale structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Linear growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Nonlinear growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Observational consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Gravitational lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Localized sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. The late-time ISW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Leakage and depletion of anisotropic power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Prospects and complications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Spherical symmetry and Birkhoff’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Beyond isolated spherical perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Exotic phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4. Ghosts and instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Gravity’s future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. The contemporary universe Cosmology in this decade is said to be thriving in a golden age. With the cornucopia of observational data from both satellite and ground-based surveys, an increasingly coherent phenomenological and theoretical picture is now emerging. And while our understanding of cosmic expansion, primordial nucleosynthesis, the microwave background and other phenomena allows particle physics and cosmology to use the very vastness of our Universe to probe the most incomprehensibly high energies, this golden age of cosmology is offering the first new data regarding the physics on immense scales in of themselves. In other words, while modern cosmology is a ratification of the notion of the deep connection between the very small and the very large, it offers also the opportunity to challenge fundamental physics itself at the lowest of energies, an unexplored infrared domain. A central example highlighting this theme is that physicists are currently faced with the perplexing reality that the Universe is accelerating in its expansion [1,2]. That startling reality is only driven home with the observation of the onset of this acceleration [3]. The acceleration represents, in essence, a new imbalance in the governing gravitational equations: a universe filled only with ordinary matter and dark matter (ingredients for which we have independent corroboration) should decelerate in its expansion. What drives the acceleration thus remains an open and tantalizing question. Instructively, physics historically has addressed such imbalances in the governing gravitational equation in either one of two ways: either by identifying sources that were previously unaccounted for (e.g., Neptune and dark matter) or by altering the governing equations (e.g., general relativity). Standard cosmology has favored the first route to addressing the imbalance: a missing energy-momentum component. Indeed, a “conventional” explanation exists for the cause of that acceleration—in general relativity, vacuum energy provides the repulsive gravity necessary to drive accelerated cosmological expansion. Variations on this vacuum-energy theme, such as quintessence, promote the energy density to the potential energy density of a dynamical field. Such additions to the roster of cosmic sources of energy-momentum are collectively referred to as dark energy. If it exists, this mysterious dark energy would constitute the majority of the energy density of the universe today.
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However, one may also entertain the alternative viewpoint. Can cosmic acceleration be the first new signal of a lack of understanding of gravitational interactions? That is the cosmic acceleration the result, not of the contents of the cosmic gas tank, as it were, but a consequence of the engine itself. This is the question that intrigues and excites us, and more importantly, how we can definitively answer that question. How can one definitively differentiate this modification of the theory of gravity from dark energy? Cosmology can offer a fresh opportunity to uncover new fundamental physics at the most immense of scales.1 Understanding cosmic acceleration and whether it indicates new fundamental physics serves as a first concrete step in the program of exploiting cosmology as a tool for understanding new infrared physics, i.e., physics on these immense scales. In 2000, Dvali, Gabadadze and Porrati (DGP) set forth a braneworld model of gravity by which our observed four-dimensional Universe resides in a larger, five-dimensional space. However, unlike popular braneworld theories at the time, the extra dimension featured in this theory was astrophysically large and flat, rather than large compared to particle-physics scales but otherwise pathologically small compared to those we observe. In DGP braneworlds, gravity is modified at large (rather than at short) distances through the excruciatingly slow evaporation of gravitational degrees of freedom off of the brane Universe. It was soon shown by Deffayet that just such a model exhibits cosmological solutions that approach empty universes that nevertheless accelerate themselves at late times. Having pioneered the paradigm of self-acceleration, DGP braneworld gravity remains a leading candidate for understanding gravity modified at ultralarge distances; nevertheless, much work remains to be done to understand its far-reaching consequences. This article is intended to be a coherent and instructive review of the material for those interested in carrying on the intriguing phenomenological work, rather than an exhaustive account of the rather dissonant, confusing and sometimes mistaken literature. In particular, we focus on the simple cases and scenarios that best illuminate the pertinent properties of DGP gravity as well as those of highest observational relevance, rather than enumerate the many intriguing variations which may be played out in this theory. We begin by setting out the governing equations and the environment in which this model exists, while giving a broad picture of how its gross features manifest themselves. We then provide a detailed view of how cosmology arises in this model, including the emergence of the celebrated self-accelerating phase. At the same time, a geometric picture of how such cosmologies evolve is presented, from the perspective of both observers in our Universe as well as hypothetical observers existing in the larger bulk space. We touch on observational constraints for this specific cosmology. We then address an important problem/subtlety regarding the recovery of four-dimensional Einstein gravity. It is this peculiar story that leads to powerful and accessible observable consequences for this theory, and is the key to differentiating a modified-gravity scenario such as DGP gravity from dark-energy scenarios. We then illuminate the interplay of cosmology with the modification of the gravitational potentials and spend the next several sections discussing DGP gravity’s astronomical and cosmological consequences. Finally, we finish with the prospects of future work and some potential problems with DGP gravity. We will see that DGP gravity provides a rich and unique environment for potentially shedding light on new cosmology and physics. 2. Gravitational leakage into extra dimensions We have set ourselves the task of determining whether there is more to gravitational physics than is revealed by general relativity. Extra dimension theories in general, and braneworld models in particular, are an indispensable avenue with which to approach understand gravity, post-Einstein. Extra dimensions provide an approach to modifying gravity with out abandoning Einstein’s general relativity altogether as a basis for understanding the fundamental gravitational 1 There is a bit of a semantic point about what one means by dark energy versus modified gravity, i.e., altering the energy-momentum content of a theory versus altering the field equations themselves. For our qualitative discussion here, what I mean by dark energy is some (possibly new) field or particle that is minimally coupled to the metric, meaning that its constituents follow geodesics of the metric. An alternative statement of this condition is that the new field is covariantly conserved in the background of the metric. I am presuming that the metric alone mediates the gravitational interaction. Thus, a modified-gravity theory would still be a metric theory, minimally coupled to whatever energy-momentum exists in that paradigm, but whose governing equations are not the Einstein equations. We also wish to emphasize the point that we are not addressing the cosmological constant problem here, i.e., why the vacuum energy is zero, or at least much smaller than the fundamental Planck scale, MP4 . When we refer to either dark energy or a modified-gravity explanation of cosmic acceleration, we do so with the understanding that a vanishing vacuum energy is explained by some other means, and that dark energy refers to whatever residual vacuum energy or potential energy of a field may be driving today’s acceleration, and that modified-gravity assumes a strictly zero vacuum energy.
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braneworld
Fig. 1. DGP gravity employs the braneworld scenario. Matter and all standard model forces and particles are pinned to a strictly four-dimensional braneworld. Gravity, however, is free to explore the full five-dimensional bulk.
interaction. Furthermore, the braneworld paradigm (Fig. 1) allows model builders a tool by which to avoid very real constraints on the number of extra dimensions coming from standard model observations. By explicitly pinning matter and standard model forces onto a (3 + 1)-dimensional brane Universe while allowing gravity to explore the larger, higher-dimensional space, all nongravitational physics follows the standard phenomenology. Ultimately, the game in braneworld theories is to find a means by which to hide the extra dimensions from gravity as well. Gravity is altered in those regimes where the extra dimensions manifest themselves. If we wish to explain today’s cosmic acceleration as a manifestation of extra dimensions, it makes sense to devise a braneworld theory where the extra dimensions are revealed at only the largest of observable distance scales. 2.1. The formal arena The braneworld theory [4] of DGP represents a leading model for understanding cosmic acceleration as a manifestation of new gravity. The bulk is this model is an empty five-dimensional Minkowski space; all energy-momentum is isolated on the four-dimensional brane Universe. The theory is described by the action [4]: 1 3 5 √ 4 S(5) = − d x −gR + d x −g (4) Lm + SGH . (2.1) M 16 M is the fundamental five-dimensional Planck scale. The first term in S(5) is the Einstein–Hilbert action in five di(4) mensions for a five-dimensional metric gAB (bulk metric) with Ricci scalar R and determinant g. The metric g is (4) 2 the induced (four-dimensional) metric on the brane, and g is its determinant. The contribution SGH to the action is a pure divergence necessary to ensure proper boundary conditions in the Euler–Lagrange equations. An intrinsic curvature term is added to the brane action [4]: 1 − (2.2) MP2 d4 x −g (4) R (4) . 16 Here, MP is the observed four-dimensional Planck scale.3 The gravitational field equations resulting from the action equations (2.1) and (2.2) are M 3 GAB + MP2 (x 5 − z(x ))GAB = 8(x 5 − z(x ))TAB (x ) , (4)
(2.3)
where GAB is the five-dimensional Einstein tensor, G(4) is the Einstein tensor of the induced metric on the brane (4) g , and where x 5 is the extra spatial coordinate and z(x ) represents the location of the brane as a function of the 2 Throughout this paper, we use A, B, . . .={0, 1, 2, 3, 5} as bulk indices, , , . . .={0, 1, 2, 3} as brane space–time indices, and i, j, . . .={1, 2, 3} as brane spatial indices. 3 Where would such a term come from? The intrinsic curvature term may be viewed as coming from effective-action terms induced by quantum matter fluctuations that live exclusively on the brane Universe (see [4–6] for details). There is an ongoing discussion as to whether this theory is unstable to further quantum gravity corrections on the brane that reveal themselves at phenomenologically important scales [7–14]. While several topics covered here are indeed relevant to that discussion (particularly Section 4), rather than becoming embroiled in this technical issue, we studiously avoid quantum gravity issues here and treat gravity as given by Eqs. (2.1) and (2.2) for our discussion, and that for cosmological applications, classical gravity physics is sufficient.
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four-dimensional coordinates of our brane Universe, {x }. Note that the energy-momentum tensor only resides on the brane surface, as we have constructed. While the braneworld paradigm has often been referred to as “string-inspired,” we are not necessarily wedded to that premise. One can imagine a more conventional scenario where physics is still driven by field theory, where the brane is some sort of solitonic domain wall and conventional particles and standard model forces are states bound to the domain wall using usual quantum arguments. This approach does require that DGP gravity still exhibits the same properties as described in this review when the brane has a nonzero thickness [15–18,9]. While the situation seems to depend on the specifics of the brane’s substructure, there exist specific scenarios in which it is possible to enjoy the features of DGP gravity with a thick, soliton-like brane. Unlike other braneworld theories, DGP gravity has a fanciful sort of appeal; it uncannily resembles the Flatland-like world one habitually envisions when extra dimensions are invoked. The bulk is large and relatively flat enjoying the properties usually associated with a Minkowski space–time. Bulk observers may look down from on high upon the brane Universe, which may be perceived as being an imbedded surface in this larger bulk. It is important to note that the brane position remains fully dynamical and is determined by the field equations, Eq. (2.3). While a coordinate system may devised in which the brane appears flat, the brane’s distortion and motion are, in that situation, registered through metric variables that represent the brane’s extrinsic curvature. This is a technique used repeatedly in order to ease the mathematical description of the brane dynamics. Nevertheless, we will often refer to a brane in this review as being warped or deformed or the like. This terminology is just shorthand for the brane exhibiting a nonzero extrinsic curvature while imagining a different coordinate system in which the brane is nontrivially imbedded. 2.2. Preliminary features In order to get a qualitative picture of how gravity works for DGP braneworlds, let us take small metric fluctuations around flat, empty space and look at gravitational perturbations, hAB , where gAB = AB + hAB ,
(2.4)
where AB is the five-dimensional Minkowski metric. Choosing the harmonic gauge in the bulk jA hAB = 21 jB hA A ,
(2.5)
where the 5-components of this gauge condition leads to h5 = 0 so that the surviving components are h and h55 . The latter component is solved by the following:
(5) h55 = (5) h ,
(2.6)
where (5) is the five-dimensional d’Alembertian. The -component of the field equations (2.3) become, after a little manipulation [6], M 3 (5) h + MP2 (x 5 )((4) h − j j h55 ) = 8(T − 13 T )(x 5 ) ,
(2.7)
where (4) is the four-dimensional (brane) d’Alembertian, and where we take the brane to be located at x 5 = 0. Fourier transforming just the four-dimensional space–time x to corresponding momentum coordinates p , and applying boundary conditions that force gravitational fluctuations to vanish one approaches spatial infinity, then gravitational fluctuations on the brane take the form [6] 8 1 ˜ ˜h (p, x 5 = 0) = ˜ T (p ) − T (p ) . (2.8) 3 MP2 p 2 + 2M 3 p We may recover the behavior of the gravitational potentials from this expression. There exists a new physical scale, the crossover scale r0 =
MP2 , 2M 3
(2.9)
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that governs the transition between four- and five-dimensional behavior. Ignoring the tensor structure of Eq. (2.8) until future sections, the gravitational potential of a source of mass m is Vgrav ∼ −
Gbrane m , r
(2.10)
when r>r0 . When r?r0 Vgrav ∼ −
Gbulk m , r2
(2.11)
where the gravitational strengths are given by Gbulk = M −3 and Gbrane = MP2 . That is, the potential exhibits fourdimensional behavior at short distances and five-dimensional behavior (i.e., as if the brane were not there at all) at large distances. For the crossover scale to be large, we need a substantial mismatch between MP , the conventional fourdimensional Planck scale (corresponding to the usual Newton’s constant, Gbrane = G) and the fundamental, or bulk, Planck scale M. The fundamental Planck scale M has to be quite small4 in order for the energy of gravity fluctuations to be substantially smaller in the bulk versus on the brane, the energy of the latter being controlled by MP . Note that when M is small, the corresponding Newton’s constant in the bulk, Gbulk , is large. Paradoxically, for a given source mass m, gravity is much stronger in the bulk. There is a simple intuition for understanding how DGP gravity works and why the gravitational potential has its distinctive form. When MP ?M, there is a large mismatch between the energy of a gravitational fluctuation on the brane versus that in the bulk. That is, imagine a gravitational field of a given amplitude (unitless) and size (measured in distance). The corresponding energies are roughly Ebrane ∼ MP2 d3 x(jh)2 ∼ MP2 × size , (2.12) Ebrane ∼ M 3
d3 x dx 5 (jh)2 ∼ M 3 × (size)2 ∼ Ebrane ×
size . r0
(2.13)
What happens then is that while gravitational fluctuations and field are free to explore the entire five-dimensional space unfettered, they are much less substantial energetically in the bulk. Imagine an analogous situation. Consider the brane as a metal sheet immersed in air. The bulk modulus of the metal is much larger than that of air. Now imagine that sound waves represent gravity. If one strikes the metal plate, the sound wave can propagate freely along the metal sheet as well as into the air. However, the energy of the waves in the air is so much lower than that in the sheet, the wave in the sheet attenuates very slowly, and the wave propagates in the metal sheet virtually as if there were no bulk at all. Only after the wave has propagated a long distance is there a substantial amount of attenuation and an observer on the sheet can tell an “extra” dimension exists, i.e., that the sound energy must have been lost into some unknown region (Fig. 2). Thus at short distances on the sheet, sound physics appears lower dimension, but at larger distances corresponding to the distance at which substantial attenuation has occurred, sound physics appears higher dimensional. In complete accord with this analogy, what results in DGP gravity is a model where gravity deviates from conventional Einstein gravity at distances larger than r0 . While a brane observer is shielded from the presence of the extra dimensions at distance scales shorter than the crossover scale, r0 . But from the nature of the above analogy, it should be clear that the bulk is not particularly shielded from the presence of brane gravitational fields. In the bulk, the solution to Eq. (2.7) for a point mass has equipotential surfaces as depicted in Fig. 3. From the bulk perspective, the brane looks like a conductor which imperfectly repels gravitational potential lines of sources existing away from the brane, and one that imperfectly screens the gravitational potential of sources located on the brane [19].
4 To have r ∼ H −1 , today’s Hubble radius, one needs M ∼ 100 MeV. That is, bulk quantum gravity effects will come into play at this low 0 0
energy. How does a low-energy quantum gravity not have intolerable effects on standard model processes on the brane? Though one does not have a complete description of that quantum gravity, one may argue that there is a decoupling mechanism that protects brane physics from bulk quantum effects [6].
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7
r0
Fig. 2. At distances much smaller than the crossover scale r0 , gravity appears four-dimensional. As a graviton propagates, its amplitude leaks into the bulk. On scales comparable to r0 that amplitude is attenuated significantly, thus, revealing the extra dimension.
Gbulk
Gbrane
Fig. 3. A point source of mass m is located on the brane. The gravitational constant on the brane is Gbrane = MP−2 , whereas in the bulk Gbulk = M −3 (left diagram). Gravitational equipotential surfaces, however, are not particularly pathological. Near the matter source, those surfaces are lens-shaped. At distances farther from the matter source, the equipotential surfaces become increasingly spherical, asymptoting to equipotentials of a free point source in five-dimensions. That is, in that limit the brane has no influence (right diagram).
3. Cosmology and global structure Just as gravity appears four-dimensional at short distances and five-dimensional at large distances, we expect cosmology to be altered in a corresponding way. Taking the qualitative features developed in the last section, we can now construct cosmologies from the field equations, Eq. (2.3). We will find that the cosmology of DGP gravity provides an intriguing avenue by which one may explain the contemporary cosmic acceleration as the manifestation of an extra dimension, revealed at distances the size of today’s Hubble radius. 3.1. The modified Friedmann equation The first work relevant to DGP cosmology appeared even before the article by Dvali, Gabadadze and Porrati, though these studies were in the context of older braneworld theories [20–22]. We follow here the approach of Deffayet [23] who first noted how one recovers a self-accelerating solution from the DGP field equations, Eq. (2.3). The general time-dependent line element with the isometries under consideration is of the form ds 2 = N 2 (, z) d2 − A2 (, z)ij di dj − B 2 (, z) dz2 ,
(3.1)
where the coordinates we choose are , the cosmological time; i , the spatial comoving coordinates of our observable Universe; and z, the extra dimension into the bulk. The three-dimensional spatial metric is the Kronecker delta, ij , because we focus our attention on spatially flat cosmologies. While the analysis was originally done for more general homogeneous cosmologies, we restrict ourselves here to this observationally viable scenario.
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Recall that all energy-momentum resides on the brane, so that the bulk is empty. The field equations, Eq. (2.3), reduce to B˙ 3 A A A B 3 A˙ A˙ G00 = 2 + − 2 + − =0 , (3.2) A A A B N A A B B ¨ 2A B¨ 1 A˙ 2N˙ A˙ B˙ N˙ 2A˙ Gij = 2 ij + − − − − A B A N A B N A N 1 i N 2A A 2N A B N 2A − 2 j + + + − + =0 , (3.3) N A A N A B N A B A˙ N˙ A˙ 3 A N A 3 A¨ G55 = 2 − − − 2 + =0 , (3.4) A A N A N A A N B ˙ A A˙ N B˙ A − − =0 , (3.5) G05 = 3 A AN B A in the bulk. Prime denotes differentiation with respect to z and dot denotes differentiation with respect to . We take the brane to be located at z = 0. This prescription does not restrict brane surface dynamics as it is tantamount to a residual coordinate gauge choice. Using a technique first developed in Refs. [24–27], the bulk equations, remarkably, may be solved exactly given that the bulk energy-momentum content is so simple (indeed, it is empty here). Taking the bulk to be mirror (Z2 ) symmetric across the brane, we find the metric components in Eq. (3.1) are given by [23] a¨ , a˙ A = a ∓ |z|a˙ , B =1 , N = 1 ∓ |z|
(3.6)
where there remains a parameter a() that is yet to be determined. Note that a() = A(, z = 0) represents the usual scale factor of the four-dimensional cosmology in our brane Universe. Note that there are two distinct possible cosmologies associated with the choice of sign. They have very different global structures and correspondingly different phenomenologies as we will illuminate further. First, take the total energy-momentum tensor which includes matter and the cosmological constant on the brane to be TBA |brane = (z) diag(− , p, p, p, 0) .
(3.7)
In order to determine the scale factor a(), we need to employ the proper boundary condition at the brane. This can be done by taking Eq. (2.3) and integrating those equations just across the brane surface. Then, the boundary conditions at the brane requires 8G N , (3.8) =− N z=0 3 A 8G = (3p + 2 ) . (3.9) A z=0 3 Comparing this condition to the bulk solutions, Eq. (3.6), these junction conditions require a constraint on the evolution of a(). Such an evolution is tantamount to a new set of Friedmann equations [23]: H 8G = () r0 3
(3.10)
˙ + 3( + p)H = 0 ,
(3.11)
H2 ± and
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4
4d
(r0H)2
3
self-accelerating (-)
2
FLRW (+)
1
0 -2
0
2
4
6
8
10
8πr02ρ/MP2 Fig. 4. The solid curve depicts Eq. (3.10) while the dotted line represents the conventional four-dimensional Friedmann equation. Two cosmological phases clearly emerge for any given spatially homogeneous energy-momentum distribution.
where we have used the usual Hubble parameter H = a/a. ˙ The second of these equations is just the usual expression of energy-momentum conservation. The first equation, however, is indeed a new Friedmann equation that is a modification of the convential four-dimensional Friedmann equation of the standard cosmological model. Let us examine Eq. (3.10) more closely. The new contribution from the DGP braneworld scenario is the introduction of the term ±H /r0 on the left-hand side of the Friedmann equation. The choice of sign represent two distinct cosmological phases. Just as gravity is conventional four-dimensional gravity at short scales and appears five-dimensional at large distance scales, so too the Hubble scale, H (), evolves by the conventional Friedmann equation at high Hubble scales but is altered substantially as H () approaches r0−1 . Fig. 4 depicts the new Friedmann equation. Deffayet [23] first noted that there are two distinct cosmological phases. First, there exists the phase in Eq. (3.10) employing the upper sign which had already been established in Refs. [20–22], which transitions between H 2 ∼ to H 2 ∼ 2 . We refer to this phase as the Friedmann–Lemaître–Robertson–Walker (FLRW) phase. The other cosmological phase corresponds to the lower sign in Eq. (3.10). Here, cosmology at early times again behaves according to a conventional four-dimensional Friedmann equation, but at late times asymptotes to a brane self-inflationary phase (the asymptotic state was first noted by Shtanov [21]). In this latter self-accelerating phase, DGP gravity provides an alternative explanation for today’s cosmic acceleration. If one were to choose the cosmological phase associated with the lower sign in Eq. (3.10), and set the crossover distance scale to be on the order of H0−1 , where H0 is today’s Hubble scale, DGP could account for today’s cosmic acceleration in terms of the existence of extra dimensions and a modifications of the laws of gravity. The Hubble scale, H (), evolves by the conventional Friedmann equation when the universe is young and H () is large. As the universe expands and energy density is swept away in this expansion, H () decreases. When the Hubble scale approaches a critical threshold, H () stops decreasing and saturates at a constant value, even when the Universe is devoid of energy-momentum content. The Universe asymptotes to a deSitter accelerating expansion. 3.2. The Brane worldsheet in a Minkowski bulk It is instructive to understand the meaning of the two distinct cosmological phases in the new Friedmann equation, Eq. (3.10). In order to do that, one must acquire a firmer grasp on the global structure of the bulk geometry, Eq. (3.6), and its physical meaning [23,28]. Starting with Eq. (3.6) and using the technique developed by Deruelle and Dolezel [29] in a more general context, an explicit change of coordinates may be obtained to go to a canonical five-dimensional
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Minkowskian metric ds 2 = dT 2 − (dX 1 )2 − (dX 2 )2 − (dX 3 )2 − (dY 5 )2 .
(3.12)
The bulk metric in this cosmological environment is strictly flat, and whatever nontrivial space–time dynamics we experience on the brane Universe is derived from the particular imbedding of a nontrivial brane surface evolving in this trivial bulk geometry. Rather like the popular physics picture of our standard cosmology, we may think of our Universe as a balloon or some deformable surface expanding and evolving in a larger-dimensional space. Here in DGP gravity, that picture is literally true. The coordinate transformation from Eq. (3.6) to the explicitly flat metric equation (3.12) is given by 2 1 1 a 2 d a˙ T = A(z, ) +1− 2 − d 3 , 4 2 4a˙ a˙ d a 1 2 1 a 2 d a˙ 5 Y = A(z, ) −1− 2 − d 3 , 4 2 4a˙ a˙ d a Xi = A(z, )i ,
(3.13)
where 2 = ij i j . For clarity, we can focus on the early Universe of cosmologies of DGP braneworlds to get a picture of the global structure, and in particular of the four-dimensional big bang evolution at early times. Here, we restrict ourselves to radiation domination (i.e., p= 13 ) such that, using Eqs. (3.10) and (3.11), a()=1/2 when H ?r0−1 using appropriately normalized time units. A more general equation of state does not alter the qualitative picture. Eq. (3.10) shows that the early cosmological evolution on the brane is independent of the sign choice, though the bulk structure will be very different for the two different cosmological phases through the persistence of the sign in Eq. (3.6). The global configuration of the brane worldsheet is determined by setting z = 0 in the coordinate transformation equation (3.13). We get 2 4 T = 1/2 + 1 − − 3/2 , 4 3 2 4 Y 5 = 1/2 − 1 − − 3/2 , 4 3 Xi = 1/2 i .
(3.14)
The locus of points defined by these equations, for all (, i ), satisfies the relationship Y+ =
3 1 i 2 1 3 (X ) + Y− , 4Y− 3
(3.15)
i=1
where we have defined Y± = 21 (T ± Y 5 ). Note that if one keeps only the first term, the surface defined by Eq. (3.15) would simply be the light cone emerging from the origin at (T , Xi , Y 5 ) = 0. However, the second term ensures that the brane worldsheet is timelike except along the Y+ -axis. Moreover, from Eq. (3.14), we see that Y− = 1/2 ,
(3.16)
implying that Y− acts as an effective cosmological time coordinate on the brane. The Y+ -axis is a singular locus corresponding to = 0, or the big bang.5 This picture is summarized in Figs. 5 and 6. Taking Y 0 as its time coordinate, a bulk observer perceives the braneworld as a compact, hyperspherical surface expanding relativistically from an initial big bang singularity. Fig. 5 shows a space–time diagram representing this picture where the three-dimensional hypersurface of the brane Universe is 5 The big bang singularity when r < ∞ is just the origin Y = Y = X i = 0 and is strictly pointlike. The rest of the big bang singularity (i.e., − + when Y+ > 0) corresponds to the pathological case when r = ∞.
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brane worldsheet
T
Y5
big bang
Xi Fig. 5. A schematic representation of the brane worldsheet from an inertial bulk reference frame. The bulk time coordinate, T, is the vertical direction, while the other directions represent all four spatial bulk coordinates, X i and Y 5 . The big bang is located along the locus Y 5 = T , while the dotted surface is the future lightcone of the event located at the origin denoted by the solid dot. The curves on the brane worldsheet are examples of equal cosmological time, , curves and each is in a plane of constant Y 5 + T . Figure from Ref. [28].
time
Fig. 6. Taking time-slicings of the space–time diagram shown in Fig. 5 (and now only suppressing one spatial variable, rather than two) we see that the big bang starts as a pointlike singularity and the brane universe expands as a relativistic shockwave from that origin. Resulting from the peculiarities of the coordinate transformation, the big bang persists eternally as a lightlike singularity (see Fig. 5) so that for any given time slice, one point on the brane surface is singular and is moving at exactly the speed of light relative to a bulk observer.
depicted as a circle with each time slice. Note that a bulk observer views the braneworld as spatially compact, even while a cosmological brane observer does not. Simultaneously, a bulk observer sees a spatially varying energy density on the brane, whereas a brane observer perceives each time slice as spatially homogeneous. Fig. 6 depicts the same picture as Fig. 5, but with each successive image in the sequence representing a single time slice (as seen by a bulk observer). The big bang starts as a strictly pointlike singularity, and the brane surface looks like a relativistic shock originating from that point. The expansion factor evolves by Eq. (3.10) implying that at early times near the big bang, its expansion is indistinguishable from a four-dimensional FLRW big bang and the expansion of the brane bubble decelerates. However, as the size of the bubble becomes comparable to r0 , the expansion of the bubble starts to deviate significantly from four-dimensional FLRW. Though the brane cosmological evolution between the FLRW phase and the self-accelerating phase is indistinguishable at early times, the bulk metric equations (3.6) for each phase is quite distinct. That distinction has a clear geometric
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V~C
B+ Bb
Fig. 7. The brane surface at a given time instant as seen from a inertial bulk observer. While from a brane observer’s point of view (observer b), a constant-time slice of the universe is infinite in spatial extent, from a bulk observer’s point of view, the brane surface is always compact and spheroidal (imagine taking a time slice in Fig. 5). That spheroidal brane surface expands at near the speed of light from a bulk observer’s point of view. In the FLRW phase, a bulk observer exists only inside the expanding brane surface, watching that surface expand away from him/her (observer B+ ). In the self-accelerating phase, a bulk observer only exists outside the expanding brane surface, watch that surface expand towards him/her (observer B− ).
interpretation: the FLRW phase (upper sign) corresponds to that part of the bulk interior to the brane worldsheet, whereas the self-accelerating phase (lower sign) corresponds to bulk exterior to the brane worldsheet (see Fig. 7). The full bulk space is two copies of either the interior to the brane worldsheet (the FLRW phase) or the exterior (the self-accelerating phase), as imposed by Z2 -symmetry. Those two copies are then spliced across the brane, so it is clear that the full bulk space cannot really be represented as imbedded in a flat space. It is clear that the two cosmological phases really are quite distinct, particularly at early times when the universe is small. In the FLRW phase, the bulk is the tiny interior of a small brane bubble. From a bulk observer’s point of view, space is of finite volume, and he/she witnesses that bulk space grow as the brane bubble expands away from him/her. The intriguing property of this space is that there are shortcuts through the bulk one can take from any point on the brane Universe to any other point. Those shortcuts are faster than the speed of light on the brane itself, i.e., the speed of a photon stuck on the brane surface [28]. In the self-accelerating phase, the bulk is two copies of the infinite volume exterior, spliced across the tiny brane bubble. Here a bulk observer witnesses the brane bubble rapidly expanding towards him/her, and eventually when the bubble size is comparable to the crossover scale r0 , the bubble will begin to accelerate towards the observer approaching the speed of light. Because of the nature of the bulk space, one cannot take shortcuts through the bulk. The fastest way from point A to B in the brane Universe is disappointingly within the brane itself. 3.3. Luminosity distances and other observational constraints How do we connect our new understanding of cosmology in DGP gravity to the real (3 + 1)-dimensional world we actually see? Let us focus our attention on the expansion history governed by Eq. (3.10) and ask how one can understand this with an eye toward comparison with existing data. In the dark energy paradigm, one assumes that general relativity is still valid and that today’s cosmic acceleration is driven by a new smooth, cosmological component of sufficient negative pressure (referred to as dark energy) whose energy density is given by DE and so that the expansion history of the universe is driven by the usual Friedmann equation H2 =
8G ( M + DE ) , 3
(3.17)
the dark energy has an equation of state w = pDE / DE , so that DE () = 0DE a −3(1+w) ,
(3.18)
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13
if w is constant; whereas DE has more complex time dependence if w varies with redshift. Dark energy composed of just a cosmological constant (w = −1) is fully consistent with existing observational data; however, both w > − 1 and w < − 1 remain observationally viable possibilities [3]. We wish to get a more agile feel for how the modified Friedmann equation of DGP gravity, Eq. (3.10), H2 ±
H 8G () , = r0 3
behaves in of itself as an evolution equation. We are concerned with the situation where the Universe is only populated with pressureless energy-momentum constituents, M , while still accelerating in its expansion at late time. We must then focus on the self-accelerating phase (the lower sign) so that H2 −
H 8G () . = r0 3 M
(3.19)
Then the effective dark energy density of the modified Friedmann equation is then 8G eff H . = 3 DE r0
(3.20)
The expansion history of this model and its corresponding luminosity distance redshift relationship was first studied in Refs. [30,31]. By comparing this expression to Eq. (3.18), one can mimic a w-model, albeit with a time-varying w. One sees immediately that the effective dark energy density attenuates with time in the self-accelerating phase. Employing the intuition devised from Eq. (3.18), this implies that the effective w associated with this effective dark energy must always be greater than negative one.6 What are the parameters in this model of cosmic expansion? We assume that the universe is spatially flat, as we have done consistently throughout this review. Moreover, we assume that H0 is given. Then one may define the parameter
M in the conventional manner such that
M = 0M (1 + z)3 ,
(3.21)
where
0M =
8G 0M 3H02
.
(3.22)
It is imperative to remember that while M is the sole energy-momentum component in this Universe, spatial flatness does not imply M = 1. This identity crucially depends on the conventional four-dimensional Friedmann equation. One may introduce a new effective dark energy component, r0 , where
r0 =
1 , r0 H
(3.23)
to resort to analogous identity: 1 = M + r0 .
(3.24)
This tactic, or something similar, is often used in the literature. Indeed, one may even introduce an effective timeeff / eff . Using Eq. (3.11) and the time derivative of Eq. (3.19), dependent weff (z) ≡ pDE DE weff (z) = −
1 . 1 + M
(3.25)
6 It must be noted that if one were to go into the FLRW phase, rather than self-accelerating phase, and relax the presumption that the cosmological constant be zero (i.e., abandon the notion of completely replacing dark energy), then there exists the intriguing possibility of gracefully achieving weff < − 1 without violating the null-energy condition, without ghost instabilities and without a big rip [32–34].
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1
dL/dLΛCDM
0.95
0.9
0
1 z
2
Fig. 8. dLw (z)/dL CDM (z) and dLDGP (z)/dL CDM (z) for a variety of models. The reference model is the best-fit flat CDM, with 0M = 0.27. The dashed curves are for the constant—w models with ( 0M , w) = (0.2, −0.765), (0.27, −0.72), and (0.35, −0.69) from top to bottom. The solid curves are for the DGP models with the same 0M as the constant—w curves from top to bottom.
Taking Eq. (3.19), we can write the redshift dependence of H (z) in terms of the parameters of the model:
1 1 H (z) 1 = + + 4 0M (1 + z)3 . H0 2 r0 H 0 r02 H02
(3.26)
While it seems that Eq. (3.26) exhibits two independent parameters, 0M and r0 H0 , Eq. (3.24) implies that r0 H0 =
1 1 − 0M
,
(3.27)
yielding only a single free parameter in Eq. (3.26). The luminosity-distance takes the standard form in spatially flat cosmologies: z dz DGP , dL (z) = (1 + z) H (z) 0
(3.28)
using Eq. (3.26). We can compare this distance with the luminosity distance for a constant—w dark-energy model dLw (z) = (1 + z)
z 0
H0−1 dz
(3.29)
w 3 3(1+w)
w M (1 + z) + (1 − M )(1 + z)
and in Fig. 8 we compare these two luminosity distances, normalized using the best-fit CDM model for a variety of
0M values. What is clear from Fig. 8 is that for all practical purposes, the expansion history of DGP self-accelerating cosmologies are indistinguishable from constant—w dark-energy cosmologies. They would in fact be identical except for the fact that weff (z) has a clear and specific redshift dependence given by Eq. (3.25). The original analysis done in Refs. [30,31] suggests that SNIA data favors an 0M low compared to other independent measurements. Such a tendency is typical of models resembling w > − 1 dark energy. Supernova data from that period implies that the best-fit Omega 0M is
0M = 0.18+0.07 −0.06 ,
(3.30)
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at the one-sigma level resulting from chi-squared minimization. Eq. (3.24) implies that the corresponding best-fit estimation for the crossover scale is −1 r0 = 1.21+0.09 −0.09 H0 .
(3.31)
Subsequent work using supernova data [32,33,35–40] refined and generalized these results. The most recent supernova results [3] are able to probe the deceleration/acceleration transition epoch [40]. Assuming a flat universe, this data suggests a best-fit 0M
0M = 0.21 ,
(3.32)
corresponding to a best-fit crossover scale r0 = 1.26H0−1 .
(3.33)
Similar results were obtained when relaxing flatness or while applying a Gaussian prior on the matter density parameter. Another pair of interesting possibilities for probing the expansion history of the universe is using the angular size of high-z compact radio-sources [41] and using the estimated age of high-z objects [42]. Both these constraints are predicated on the premise that the only meaningful effect of DGP gravity is the alteration of the expansion history. However, if the objects are at high enough redshift, this may be a plausible scenario (see Section 5). Finally, one can combine supernova data with data from the cosmic microwave background (CMB). Again, we are presuming the only effect of DGP gravity is to alter the expansion history of the universe. While that is mostly likely a safe assumption at the last scattering surface (again see Section 5), there are O(1)-redshift effects in the CMB, such as the late-time integrate Sachs–Wolfe effect, that may be very sensitive to alterations of gravity at scales comparable to today’s Hubble radius. We pursue such issues later in this review. For now, however, we may summarize the findings on the simpler presumption [31]. Supernova data favors slightly lower values of 0M compared to CMB data for a flat universe. However, a concordance model with 0M = 0.3 provided a good fit to both sets (pre-WMAP CMB data) with 2 ≈ 140 for the full data set (135 data points) with a best-fit crossover scale r0 ∼ 1.4H0−1 .
4. Recovery of Einstein gravity Until now, we have ignored the crucial question of whether adopting DGP gravity yields anomalous phenomenology beyond the alteration of cosmic expansion history. If we then imagine that today’s cosmic acceleration were a manifestation of DGP self-acceleration, the naive expectation would be that all anomalous gravitational effects of this theory would be safely hidden at distances substantially smaller than today’s Hubble radius, H0−1 , the distance at which the extra dimension is revealed. We will see in this section that this appraisal of the observational situation in DGP gravity is too naive. DGP gravity represents an infrared modification of general relativity. Such theories often have pathologies that render them phenomenologically not viable. These pathologies are directly related to van Dam–Veltman–Zakharov (VDVZ) discontinuity found in massive gravity [43–45]. DGP does not evade such concerns: although gravity in DGP is fourdimensional at distances shorter than r0 , it is not four-dimensional Einstein gravity—it is augmented by the presence of an ultra-light gravitational scalar, roughly corresponding to the unfettered fluctuations of the braneworld on which we live. This extra scalar gravitational interaction persists even in the limit where r0−1 → 0. This is a phenomenological disaster which is only averted in a nontrivial and subtle matter [46–49]. Let us first describe the problem in detail and then proceed to understanding its resolution. 4.1. The van Dan–Veltman–Zakharov discontinuity General relativity is a theory of gravitation that supports a massless graviton with two degrees of freedom, i.e., two polarizations. However, if one were to describe gravity with a massive tensor field, general covariance is lost and the graviton would possess five degrees of freedom. The gravitational potential (represented by the quantity h =g − )
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generated by a static source T is then given by (in three-dimensional momentum space, q i ) 8 1 1 2 T (q ) = − − T hmassive 3 MP2 q 2 + m2
(4.1)
for a massive graviton of mass m around a Minkowski-flat background. While similar in form to the gravitational potential in Einstein gravity 8 1 1 massless 2 (4.2) (q ) = − 2 2 T − T h 2 MP q it nevertheless has a distinct tensor structure. In the limit of vanishing mass, these five degrees of freedom may be decomposed into a massless tensor (the graviton), a massless vector (a graviphoton which decouples from any conserved matter source) and a massless scalar. This massless scalar persists as an extra degree of freedom in all regimes of the theory. Thus, a massive gravity theory is distinct from Einstein gravity, even in the limit where the graviton mass vanishes as one can see when comparing Eqs. (4.1) and (4.2). This discrepancy is a formulation of the VDVZ discontinuity [43–45]. The most accessible physical consequence of the VDVZ discontinuity is the gravitational field of a star or other compact, spherically symmetric source. The ratio of the strength of the static (Newtonian) potential to that of the gravitomagnetic potential is different for Einstein gravity compared to massive gravity, even in the massless limit. Indeed the ratio is altered by a factor of order unity. Thus, such effects as the predicted light deflection by a star would be affected significantly if the graviton had even an infinitesimal mass. This discrepancy appears for the gravitational field of any compact object. An even more dramatic example of the VDVZ discontinuity occurs for a cosmic string. A cosmic string has no static potential in Einstein gravity; however, the same does not hold for a cosmic string in massive tensor gravity. One can see why using the potentials (Eqs. (4.1) and (4.2)). The potential between a cosmic string with T = diag(T , −T , 0, 0) and a test particle with T˜ = diag(2M˜ 2 , 0, 0, 0) is Vmassless = 0,
Vmassive ∼
T M˜ ln r , MP2
(4.3)
where the last expression is taken in the limit m → 0. Thus in a massive gravity theory, we expect a cosmic string to attract a static test particle, whereas in general relativity, no such attraction occurs. The attraction in the massive case can be attributed to the exchange of the remnant light scalar mode that comes from the decomposition of the massive graviton modes in the massless limit. The gravitational potential in DGP gravity, Eq. (2.8), has the same tensor structure as that for a massive graviton and perturbatively has the same VDVZ problem in the limit that the graviton linewidth (effectively r0−1 ) vanishes. Again, this tensor structure is the result of an effective new scalar that may be associated with a brane fluctuation mode, or more properly, the fluctuations of the extrinsic curvature of the brane. Because, in this theory, the brane is tensionless, its fluctuations represent a very light mode and one may seriously ask the question as to whether standard tests of scalar-tensor theories, such as light-deflection by the sun, already rule out DGP gravity by wide margins. It is an important and relevant question to ask. We are precisely interested in the limit when r0−1 → 0 for all intents and purposes. We want r0 to be the size of the visible Universe today, while all our reliable measurements of gravity are on much smaller scales. However, the answers to questions of observational relevance are not straightforward. Even in massive gravity, the presence of the VDVZ discontinuity is more subtle than just described. The potential, Eq. (4.1), is only derived perturbatively to lowest order in h or T . Vainshtein proposed that this discontinuity does not persist in the fully nonlinear classical theory [50]. However, doubts remain [51] since no self-consistent, fully nonlinear theory of massive tensor gravity exists (see, for example, Ref. [52]). If the corrections to Einstein gravity remain large even in limit r0 → ∞, the phenomenology of DGP gravity is not viable. The paradox in DGP gravity seems to be that while it is clear that a perturbative, VDVZ-like discontinuity occurs in the potential somewhere (i.e., Einstein gravity is not recovered at short distances), no such discontinuity appears in the cosmological solutions; at high Hubble scales, the theory on the brane appears safely like general relativity [46]. What does this mean? What is clear is that the cosmological solutions at high Hubble scales are extremely nonlinear,
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and that perhaps, just as Vainshtein suggested for massive gravity, nonlinear effects become important in resolving the DGP version of the VDVZ discontinuity. 4.2. Case study: cosmic strings We may ask the question of how nonlinear, nonperturbative effects change the potential, Eq. (2.8), per se. Indeed, as a stark and straightforward exercise, we may ask the question in DGP gravity, does a cosmic string attract a static test particle or not in the limit? We will see that corrections remain small and that the recovery of Einstein gravity is subtle and directly analogous to Vainshtein’s proposal for massive gravity. DGP cosmic strings provided the first understanding of how the recovery of Einstein gravity occurs in noncosmological solutions [47]. Cosmic strings offer a conceptually clean environment and a geometrically appealing picture for how nonperturbative effects drive the loss and recover of the Einstein limit in DGP gravity. Once it is understood how the VDVZ issue is resolved in this simpler system, understanding it for the Schwarzschild-like solution becomes a straightforward affair. 4.2.1. The Einstein solution Before we attempt to solve the full five-dimensional problem for the cosmic string in DGP gravity, it is useful to review the cosmic string solution in four-dimensional Einstein gravity [53,54]. For a cosmic string with tension T, the exact metric may be represented by the line element: ds 2 = dt 2 − dx 2 − (1 − 2GT)−2 dr 2 − r 2 d 2 .
(4.4)
This represents a flat space with a deficit angle 4GT. If one chooses, one can imagine suppressing the x-coordinate and imagining that this analysis is that for a particle in (2 + 1)-dimensional general relativity. Eq. (4.4) indicates that there is no Newtonian potential (i.e., the potential between static sources arising from g00 ) between a cosmic string and a static test particle. However, a test particle (massive or massless) suffers an azimuthal deflection of 4GT when scattered around the cosmic string, resulting from the deficit angle cut from space–time. Another way of interpreting this deflection effect may be illuminated through a different coordinate choice. The line element, Eq. (4.4), can be rewritten as ds 2 = dt 2 − dx 2 − (y 2 + z2 )−2GT [dy 2 + dz2 ] .
(4.5)
Again, there is no Newtonian gravitational potential between a cosmic string and a static test particle. There is no longer an explicit deficit angle cut from space–time; however, in this coordinate choice, the deflection of a moving test particle results rather from a gravitomagnetic force generated by the cosmic string. In the weak field limit, one may rewrite Eq. (4.5) as a perturbation around flat space, i.e., g = + h , as a series in the small parameter GT such that h00 = hxx = 0 ,
(4.6)
hyy = hzz = 4GT ln r ,
(4.7)
where r = y 2 + z2 is the radial distance from the cosmic string. So, interestingly, one does recover the logarithmic potentials that are expected for codimension-2 objects like cosmic strings in (3 + 1)-dimensions or point particles in (2 + 1)-dimensions. They appear, however, only in the gravitomagnetic potentials in Einstein gravity, rather than in the gravitoelectric (Newtonian) potential. 4.2.2. DGP cosmic strings: the weak-Brane limit We wish to fine the space–time around a perfectly straight, infinitely thin cosmic string with a tension T, located on the surface of our brane Universe (see Fig. 9). Alternatively, we can again think of suppressing the coordinate along the string so that we consider the space–time of a point particle, located on a two-dimensional brane existing in a (3 + 1)-dimensional bulk. As in the cosmological solution, we assume a mirror, Z2 -symmetry, across the brane surface at z = /2. Einstein equations (2.3) may now be solved for this system.
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z
braneworld r
A φ
Fig. 9. A schematic representation of a spatial slice through a cosmic string located at A. The coordinate x along the cosmic string is suppressed. The coordinate r represents the three-dimensional distance from the cosmic string A, while the coordinate z denotes the polar angle from the vertical axis. In the no-gravity limit, the braneworld is the horizontal plane, z = /2. The coordinate is the azimuthal coordinate. Figure from Ref. [47].
There is certainly a regime where one may take a perturbative limit when GT is small and so that given gAB = AB + hAB , the four-dimensional Fourier transform of the metric potential on the brane is given by Eq. (2.8). For a cosmic string, this implies that when r?r0 , h00 = hxx = −
1 4r0 GT , 3 r
(4.8)
hyy = hzz = −
2 4r0 GT . 3 r
(4.9)
Graviton modes localized on the brane evaporate into the bulk over distances comparable to r0 . The presence of the brane becomes increasingly irrelevant as r/r0 → ∞ and a cosmic string on the brane acts as a codimension-three object in the full bulk. When r>r0 , 1 h00 = hxx = 4GT ln r , 3
(4.10)
2 hyy = hzz = 4GT ln r . 3
(4.11)
The metric potentials when r>r0 represent a conical space with deficit angle 23 4GT. Thus in the weak field limit, we expect not only an extra light scalar field generating the Newtonian potential, but also a discrepancy in the deficit angle with respect to the Einstein solution. We can ask the domain of validity of the perturbative solution. The perturbative solution considered only terms in Eq. (2.3) linear in hAB , or correspondingly, linear in GT. When GT>1, this should be a perfectly valid approach to self-consistently solving Eq. (2.3). However, there is an important catch. While GT is indeed a small parameter, DGP gravity introduces a large parameter r0 into the field equations. Actually, since r0 is dimensionful, the large parameter is more properly r0 /r. Thus, there are distances for which nonlinear terms in Eq. (2.3) resulting from contributions from the extrinsic curvature of the brane r0 ∼ (GT)2 , (4.12) r cannot be ignored, even though they are clearly higher order in GT. Nonlinear terms such as these may only be ignored when [47] √ r?r0 4GT . (4.13) Thus, the perturbative solution given by the metric potential, Eq. (2.8), is not valid in all regions. In particular, the perturbative solution is not valid in the limit where everything is held fixed and r0 → ∞, which is precisely the limit of interest.
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απ/2 braneworld A
Fig. 10. A spatial slice through the cosmic string located at A. As in Fig. 9 the coordinate x along the cosmic string is suppressed. The solid angle wedge exterior to the cone is removed from the space, and the upper and lower branches of the cone are identified. This conical surface is the braneworld (z = /2 or sin z = ). The bulk space now exhibits a deficit polar angle (cf. Fig. 9). Note that this deficit in polar angle translates into a conical deficit in the braneworld space. Figure from Ref. [47].
4.2.3. The r/r0 → 0 limit For values of r violating Eq. (4.13), nonlinear contributions to the Einstein tensor become important and the weak √ field approximation breaks down, even when the components h >1. What happens when r>r0 4GT? We need to find a new small expansion parameter in order to find a new solution that applies for small r. Actually, the full field equations (2.3) provide a clue [47]. A solution that is five-dimensional Ricci flat in the bulk, sporting a brane surface that is four-dimensional Ricci flat, is clearly a solution. Fig. 10 is an example of such a solution (almost). The bulk is pure vanilla five-dimensional Minkowski space, clearly Ricci flat. The brane is a conical deficit space, a space whose intrinsic curvature is strictly zero. The field equations, Eq. (2.3) should be solved. Why the space depicted in Fig. 10 is not exactly a solution comes from the Z2 -symmetry of the bulk across the brane. The brane surface has nontrivial extrinsic curvature even though it has vanishing intrinsic curvature. Thus a polar deficit angle space has a residual bulk curvature that is a delta-function at the brane surface, and Eq. (2.3) are not exactly zero everywhere for that space. Fortunately, the residual curvature is subleading in r/r0 , and one may perform a new systematic perturbation in this new parameter, r/r0 , starting with the space depicted in Fig. 10 as the zeroth-order limit. The new perturbative solution on the brane is given using the line element ds 2 = N 2 (r)|sin z= (dt 2 − dx 2 ) − A2 (r)|sin z= dr 2 − 2 r 2 d 2 , where the metric components on the brane are [47] 1 − 2 r N(r)|sin z= = 1 + , 2 r0 1 − 2 r A(r)|sin z= = 1 − 2 r0
(4.14)
(4.15)
(4.16)
and the deficit polar angle in the bulk is (1 − ) where sin(/2) = , while the deficit azimuthal angle in the brane itself is 2(1 − ). The deficit angle on the brane is given by = 1 − 2GT ,
(4.17)
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g00-1
Einstein
weak-brane
0 + ...
~ T ln r 2 MP r0
0 r0
T MP2
1/2
5d
~
T r0 MP2 r
r0
r
Fig. 11. The Newtonian potential for a cosmic string has the following regimes: outside r0 , the cosmic string appears as a codimension-3 object, i.e., a Schwarzschild source, and so its potential goes as r −1 ; inside r0 , the string appears as a codimension-2 object, i.e., a true string source. Outside r0 (T /MP2 )1/2 , however, the theory appears Brans–Dicke and one generates a logarithmic scalar potential associated with codimension-2 objects. Inside the radius r0 (T /MP2 )1/2 from the string source, Einstein gravity is recovered and there is no Newtonian potential.
which is precisely equivalent to the Einstein result. The perturbative scheme is valid when 1 − 2 √ r>r0 ∼ r0 4GT ,
(4.18)
which is complementary to the regime of validity for the weak-brane perturbation. Moreover, Eq. (4.18) is the regime of interest when taking r0 → ∞ while holding everything else fixed, i.e., the one of interest in addressing the VDVZ discontinuity. What we see is that, just like for the cosmological solutions, DGP cosmic strings do no suffer a VDVZlike discontinuity. Einstein gravity is recovered in the r0 → ∞ limit, precisely because of effects nonlinear, and indeed nonperturbative, in GT. 4.2.4. The picture Fig. 11 depicts how in different parametric regimes, we find different qualitative behaviors for the brane metric around a cosmic string in DGP gravity. Though we have not set out the details here, the different perturbative solutions discussed are part of the same solution in the bulk and on the brane [47], i.e., the trunk and the tail of the elephant, as it were. For an observer at a distance r?r0 from the cosmic string, where r0−1 characterizes the graviton’s effective linewidth, the cosmic string appears as a codimension-three object in the full bulk. The metric is Schwarzschild-like in this regime. When r>r0 , brane effects become important, and the cosmic string appears as a codimension-two object on the brane. If the source √ is weak (i.e., GT is small), the Einstein solution √ with a deficit angle of 4GT holds on the brane only when r>r0 4GT. In the region on the brane when r?r0 4GT (but still where r>r0 ), the weak field approximation prevails, the cosmic string exhibits a nonvanishing Newtonian potential and space suffers a deficit angle different from 4GT. The solution presented here supports the Einstein solution near the cosmic string in the limit that r0 → ∞ and recovery of Einstein gravity proceeded precisely as Vainshtein suggested it would in the case of massive gravity: nonperturbative effects play a crucial role in suppressing the coupling the extra scalar mode. Far from the source, the gravitational field is weak, and the geometry of the brane (i.e., its extrinsic curvature with respect to the bulk) is not substantially altered by the presence of the cosmic string. The solution is a perturbation in GT around the trivial space depicted in Fig. 9. Propagation of the light scalar mode is permitted and the solution does not correspond to that from general relativity. However near the source, the gravitational fields induce a nonperturbative extrinsic curvature in the brane, in a manner reminiscent of the popular science picture used to explain how matter sources warp geometry. Here, the picture is literally true. The solution here is a perturbation in r/r0 around the space depicted in Fig. 10. The brane’s extrinsic curvature suppresses the coupling of the scalar mode to matter and only the tensor mode remains, thus Einstein gravity is recovered.
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4.3. The Schwarzschild-like solution So while four-dimensional Einstein gravity is recovered in a region near a cosmic string source, it is recovered in a region much smaller than the region where one naively expected the extra dimension to be hidden, i.e., the larger radius r0 . Einstein gravity is only recovered within a region much smaller than r0 , a region whose size is dictated by the source strength. Do the insights elucidated using cosmic string translate for a star-like object? If so, that would have fantastic observational consequences. We would have a strong handle for observing this theory in a region that is accessible in principle, i.e., at distances much smaller than today’s Hubble radius. Indeed, Gruzinov first showed how that recovery of Einstein gravity in the Schwarzschild-like solution is exactly analogous to what was found for the cosmic string, and moreover, is also in exactly the spirit of Vainshtein’s resolution of the VDVZ discontinuity for massive gravity [48]. 4.3.1. The field equations We are interested in finding the metric for a static, compact, spherical source in a Minkowski background. Under this circumstance, one can choose a coordinate system in which the metric is static (i.e., has a timelike Killing vector) while still respecting the spherical symmetry of the matter source. Let the line element be ds 2 = N 2 (r, z)dt 2 − A2 (r, z)dr 2 − B 2 (r, z)[d2 + sin2 d 2 ] − dz2 .
(4.19)
This is the most general static metric with spherical symmetry on the brane. The bulk Einstein tensor for this metric is
2 B 2 B 2B A 2A B 2B A B 1 1 zz zz z z z − + 2 − + +2 + 2 , Gtt = 2 − 2 B A B A B A B B A B B Grr
Bz2 B 2 2Bzz N z Bz N B Nzz 1 1 + 2 − + +2 + 2 , = 2− 2 2 N B N B N B B A B B
1 N B N A N B A B = − 2 + − + − N B N A N B A B A Azz Bzz N z Az N z Bz A z Bz Nzz + + + + + , − N A B N A N B A B
1 1 N 2B N A N B A B B 2 z + − +2 −2 + 2 Gz = 2 − 2 N B N A N B A B B A B
B2 Nz Az N z Bz A z Bz − +2 +2 + z2 , N A N B A B B
G
= G
(4.20)
Nz 2Bz Az N 2B + + + . Gzr = − N B A N B The prime denotes partial differentiation with respect to r, whereas the subscript z represents partial differentiation with respect to z. We wish to solve the five-dimensional field equations, Eq. (2.3). This implies that all components of the Einstein tensor, Eq. (4.20), vanish in the bulk but satisfy the following modified boundary relationships on the brane. Fixing the residual gauge B|z=0 = r and imposing Z2 -symmetry across the brane Az 2 A 2Bz r0 1 8r0 2 − + = 2 − + 2 (1 − A ) + (r) , A B r A A r MP2 Nz 8r0 2Bz r0 2 N 1 − p(r) , + = 2 + 2 (1 − A2 ) − N B A r N r MP2 Nz Az Bz r0 N N A 1 N A 8r0 − + + = 2 − + − − p(r) , (4.21) N A B N N A r N A A MP2
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when z = 0. These brane boundary relations come from Gtt , Grr and G , respectively. We have chosen a gauge in which the brane, while still dynamical, appears flat. All the important extrinsic curvature effects discussed in the last section will appear in the z-derivatives of the metric components evaluated at the brane surface, rather than through any explicit shape of the brane. We are interested in a static matter distribution (r), and we may define an effective radially dependent Schwarzschild radius 8 r 2 Rg (r) = 2 r (r) dr , (4.22) MP 0 where we will also use the true Schwarzschild radius, rg = Rg (r → ∞). We are interested only in weak matter sources, g (r). Moreover, we are most interested in those parts of space–time where deviations of the metric from Minkowski are small. Then, it is convenient to define the functions {n(r, z), a(r, z), b(r, z)} such that N(r, z) = 1 + n(r, z) ,
(4.23)
A(r, z) = 1 + a(r, z) ,
(4.24)
B(r, z) = r[1 + b(r, z)] .
(4.25)
Since we are primarily concerned with the metric on the brane, we can make a gauge choice such that b(r, z = 0) = 0 identically so that on the brane, the line element 2 2 ds 2 = 1 + n(r)|z=0 dt 2 − 1 + a(r)|z=0 dr 2 − r 2 d ,
(4.26)
takes the standard form with two potentials, n(r)|z=0 and a(r)|z=0 , the Newtonian potential and a gravitomagnetic potential. Here we use d as shorthand for the usual differential solid angle. We will be interested in small deviations from flat Minkowski space, or more properly, we are only concerned when n(r, z), a(r, z) and b(r, z)>1. We can then rewrite our field equations, Eq. (4.20), and brane boundary conditions, Eq. (4.21), in terms of these quantities and keep only leading orders. The brane boundary conditions become 2a r0 2a − (az + 2bz ) = r0 − − 2 + 2 Rg (r) , r r r 2n 2a − (nz + 2bz ) = r0 − 2 , r r n a − (nz + az + bz ) = r0 n + − . (4.27) r r Covariant conservation of the source on the brane allows one to ascertain the source pressure, p(r), given the source density (r): pg = −n g .
(4.28)
The pressure terms were dropped from Eq. (4.27) because there are subleading here. 4.3.2. The weak-Brane limit Just as for the cosmic string, there is again a regime where one may take a properly take the perturbative limit when rg is small. Again, given gAB = AB + hAB , the four-dimensional Fourier transform of the metric potential on the brane is given by Eq. (2.8) 1 8 1 h˜ (p) = 2 2 T˜ − T˜ . 3 MP p + p/r0
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For a Schwarzschild solution, this implies that when r?r0 h00 = −
4 r0 rg , 3 r2
(4.29)
hxx = hyy = hzz = −
2 r 0 rg 3 r2
(4.30)
and r>r0 h00 = −
4 rg , 3 r2
(4.31)
hxx = hyy = hzz = −
2 rg . 3 r2
(4.32)
It is convenient to write the latter in terms of our new potentials for the line element, Eq. (4.26), n(r)|z=0 = −
4 rg , 3 2r
(4.33)
a(r)|z=0 = +
2 rg . 3 2r
(4.34)
This is actually the set of potentials one expects from Brans–Dicke scalar-tensor gravity with Dicke parameter = 0. Einstein gravity would correspond to potentials whose values are −rg /2r and +rG /2r, respectively. As discussed earlier, there is an extra light scalar mode coupled to the matter source. That mode may be interpreted as the fluctuations of the free brane surface. Again, just as in the cosmic string case, we see that significant deviations from Einstein gravity yield nonzero contributions to the right-hand side of Eq. (4.21). Because these are multiplied by r0 , this implies that the extrinsic curvatures (as represented by the z-derivatives of the metric components at z = 0) can be quite large. Thus, while we neglected nonlinear contributions to the field equations, Eq. (4.20), bilinear terms in those equations of the form Az /AB z /B, for example, are only negligible when [48] r?r∗ ≡ (rg r02 )1/3 ,
(4.35)
even when rg is small and n, a>1. When r>r∗ , we need to identify a new perturbation scheme. 4.3.3. The r/r0 → 0 limit The key to identifying a solution when r>r∗ is to recognize that one only needs to keep certain nonlinear terms in Eq. (4.20). So long as n, a, b>1, or equivalently r?rg , the only nonlinear terms that need to be included are those terms bilinear in Az /A and Bz /B [48]. Consider a point mass source such that Rg (r) = rg = constant. Then, the following set of potentials on the brane are [48]: rg r rg , (4.36) n=− + 2r 2r02 a=+
rg − 2r
rg r 8r02
.
(4.37)
The full bulk solution and how one arrives at that solution will be spelled out in Section 5 when we consider the more general case of the Schwarzschild-like solution in the background of a general cosmology, a subset of which is this Minkowski background solution. That the inclusion of terms only nonlinear in az and bz was sufficient to find solutions valid when r>r∗ is indicative that the nonlinear behavior arises from purely spatial geometric factors [48]. In particular, inserting the potentials, Eqs. (4.36) and (4.37), into the expressions, Eq. (4.27), indicates that the extrinsic curvatures of the brane, i.e., az |z=0 and bz |z=0 , play a crucial role in the nonlinear nature of this solution, indeed a solution inherently nonperturbative
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weak-brane phase
braneworld Einstein phase Fig. 12. Given a mass source located on the brane, inside the radius r∗ (the green hemisphere), the brane is dimpled significantly generating a nonperturbative extrinsic curvature. Brane fluctuations are suppressed in this region (Einstein phase). Outside this radius, the brane is free to fluctuate. The bulk in this picture is above the brane. The mirror copy of the bulk space below the brane is suppressed.
V(r)
Einstein
weak-brane
4 Gmr0 3 r2
4 Gm 3 r
Gm + ... r
0
5d
1/3
r* =(r02 rg)
r0
r
Fig. 13. The Newtonian potential V (r) = g00 − 1 has the following regimes: outside r0 , the potential exhibits five-dimensional behavior (i.e., 1/r 2 ); inside r0 , the potential is indeed four-dimensional (i.e., 1/r) but with a coefficient that depends on r. Outside r∗ we have Brans–Dicke potential while inside r∗ we have a true four-dimensional Einstein potential.
in the source strength rg . This again is directly analogous to the cosmic string but rather than exhibiting a conical distortion, the brane is now cuspy. The picture of what happens physically to the brane is depicted in Fig. 12. When a mass source is introduced in the brane, its gravitational effect includes a nonperturbative dimpling of the brane surface (in direct analogy with the popular physics picture of how general relativity works). The brane is dimpled significantly in a region within a radius r∗ of the matter source. The extrinsic curvature suppresses the light brane bending mode associated with the extra scalar field inside this region, whereas outside this region, the brane bending mode is free to propagate. Thus four-dimensional Einstein gravity is recovered close the mass source, but at distances less than r∗ , not distances less than r0 . Outside r∗ , the theory appears like a four-dimensional scalar-tensor theory, in particular, four-dimensional linearized Brans–Dicke, with parameter = 0. A marked departure from Einstein gravity persists down to distances much shorter than r0 . Fig. 13 depicts the hierarchy of scales in this system.
5. Modified gravitational forces So, we expect a marked departure of the metric of a spherical, compact mass at distances comparable to r∗ and greater. The potentials, Eqs. (4.36) and (4.37), provide the form of the corrections to Einstein gravity as r approaches r∗ while the in the weak-brane phase, i.e., when r?r∗ but when r is still much smaller than r0 , the potentials are given by Eqs. (4.33) and (4.34). From our treatment of the cosmic expansion history of DGP gravity, if we are to comprehend the contemporary cosmic acceleration as a manifestation of extra-dimensional effects, wish to set r0 ∼ H0−1 . The distance r∗ clearly plays an important role in DGP phenomenology. Table 1 gives a few examples of what r∗ would be if given source masses were isolated in an empty Universe when r0 ∼ H0−1 .
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Table 1 Example values for r∗ Earth Sun Milky Way (1012 M )
1.2 pc 150 pc 1.2 Mpc
100 10-1
rg/2r
10-2 10-3
N-1
10-4 10-5
rg/r0 = 10-6
δNDGP= (rgr/2r02)1/2
10-6 10-7 10-8
δNcosmo = (r/r0)2/2
10-9 10-10 -5 10
10-4
10-3
10-2
10-1
100
101
102
103
104
105
1/3
r/(r02 rg)
Fig. 14. The corrections to the Newtonian potential become as large as the potential itself just as the contributions from cosmology become dominant. It is for this reason that we expect the background cosmology to has a significant effect on the modified potential in the weak-brane regime.
However, there is a complication when we wish to understand the new gravitational forces implied by Eqs. (4.33) and (4.34) the context of cosmology [55]. The complication arises when we consider cosmologies whose Hubble radii, H −1 , are comparable or even smaller than r0 . Take H −1 = constant = r0 for example. In such an example, one may actually regard the Hubble flow as being as making an effective contribution to the metric potentials Ncosmo = −
r2 . r02
(5.1)
Fig. 14 depicts a representative situation. The corrections computed in the Minkowski case become important just at the length scales where the cosmology dominates the gravitational potential. That is, the regime rr∗ is just that region where cosmology is more important than the localized source. Inside this radius, an observer is bound in the gravity well of the central matter source. Outside this radius, an observer is swept away into the cosmological flow. Thus, one cannot reliably apply the results from a Minkowski background under the circumstance of a nontrivial cosmology, particularly, when we are interested in DGP gravity because of its anomalous cosmological evolution. We need to redo the computation to include the background cosmology, and indeed we will find a cosmology-dependent new gravitational force [55,56,34]. This computation is more nuanced than for a static matter source in a static Minkowski background. We are still interested in finding the metric for compact, spherically symmetric overdensities, but now there is time-evolution to contend with. However, if we restrict our attention to distance scales such that rH >1 and to nonrelativistic matter sources,7 then to leading-order in r 2 H 2 and zH, the solutions to the field equations, Eq. (2.3), are also solutions to the static equations, i.e., the metric is quasistatic, where the only time dependence comes from the slow evolution of 7 There are a number of simplifications that result from these approximations. See Ref. [56] for an enumeration of these as well as the caveats concerning straying to far from the approximations.
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the extrinsic curvature of the brane. To be explicit, we are looking at the nonrelativistic limit, where the gravitational potentials of a matter source depend only on the instantaneous location of the matter’s constituents, and not on the motion of those constituents. Incidentally, we left the details of the solution to the Minkowski problem (which were first treated in Refs. [48,49]) to this section. All the arguments to be employed may be used, a fortiori, as a special case of the more general cosmological background. 5.1. Background cosmology One can choose a coordinate system in which the cosmological metric respects the spherical symmetry of the matter source. We are concerned with processes at distances, r, such that rH>1. Under that circumstance it is useful to change coordinates to a frame that surrenders explicit brane spatial homogeneity but preserves isotropy r(, i ) = a() , t (, i ) = +
2 H ()a 2 () 2
(5.2) (5.3)
for all z and where 2 = ij i j . The line element becomes ds 2 = [1 ∓ 2(H + H˙ /H )|z| − (H 2 + H˙ )r 2 ] dt 2 − [1 ∓ 2H |z|][(1 + H 2 r 2 ) dr 2 + r 2 d ] − dz2 ,
(5.4)
where here dot represents differentiation with respect to the new time coordinate, t. Moreover, H = H (t) in this coordinate system. All terms of O(r 3 H 3 ) or O(z2 H 2 , zHrH) and higher have been neglected. The key is that because we are interested primarily in phenomena whose size is much smaller than the cosmic horizon, the effect of cosmology is almost exclusively to control the extrinsic curvature, of the brane. This can be interpreted as a modulation of the brane’s stiffness or the strength of the scalar gravitational mode. In the coordinate system described by Eq. (5.4), the bulk is like a Rindler space. This has a fairly natural interpretation if one imagines the bulk picture [23,28]. One imagines riding a local patch of the brane, which appears as hyperspherical surface expanding into (or away from) a five-dimensional Minkowski bulk. This surface either accelerates or decelerates in its motion with respect to the bulk, creating a Rindler-type potential. Note that we are keeping the upper- and lowersign convention to represent the two cosmological phases. While we are nominally focussed on self-acceleration, we will see that contributions from the sign have important effects on the modified gravitational potentials. 5.2. Metric potentials We have chosen a coordinate system, Eq. (4.19), in which a compact spherical matter source may have a quasistatic metric, yet still exist within a background cosmology that is nontrivial (i.e., deSitter expansion). Let us treat the matter distribution to be that required for the background cosmology, Eq. (5.4), and add to that a compact spherically symmetric matter source, located on the brane around the origin (r = 0, z = 0) TBA |brane = (z) diag( (r) + B , −p(r) − pB , −p(r) − pB , −p(r) − pB , 0) ,
(5.5)
where B and pB are the density and pressure of the background cosmology, and where (r) is the overdensity of interest and p(r) is chosen to ensure the matter distribution and metric are quasistatic. We may define an effective Schwarzschild radius 8 r 2 r (r, t) dr . (5.6) Rg (r, t) = 2 MP 0 We solve the perturbed Einstein equations in quasistatic approximation by generalizing the method used in [55], obtaining the metric of a spherical mass overdensity (t, r) in the background of the cosmology described by Eqs. (3.10) and (3.11). Because we are interested only in weak matter sources, g (r), and since we are interested in solutions well away from the cosmic horizon, we can still exploit doing an expansion using Eq. (4.24) and keeping only leading orders. Now, we just need to take care to include leading orders in terms of H as well as other parameters of interest.
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We are particularly concerned with the evaluation of the metric on the brane. However, we need to take care that when such an evaluation is performed, proper boundary conditions in the bulk are satisfied, i.e., that there no singularities in the bulk, which is tantamount to having spurious mass sources there. In order to determine the metric on the brane, we implement the following approximation [55,56,34]:
H˙ nz |z=0 = ∓ H + . H
(5.7)
Note that this is just the value nz would take if there were only a background cosmology, but we are making an assumption that the presence of the mass source only contributes negligibly to this quantity at the brane surface. With this one specification, a complete set of equations, represented by the brane boundary conditions, Eq. (4.21), and Gzz =0, exists on the brane so that the metric functions may be solved on that surface without reference to the bulk. At the end of this section, we check that the assumption, Eq. (5.7), indeed turns out the be the correct one that ensures that no pathologies arise in the bulk. The brane boundary conditions, Eq. (4.21), now take the form 2a r0 2a −(az + 2bz ) = r0 − − 2 + 2 Rg (r) + 3H (r0 H ± 1) , r r r 2n 2 2a −2bz = r0 − 2 + r0 (3H 2 + 2H˙ ) ± (H 2 + H˙ ) , r H r a n 2 + r0 (3H 2 + 2H˙ ) ± (H 2 + H˙ ) , (5.8) −(az + bz ) = r0 n + − r r H where we have substituted Eqs. (3.10) and (3.11) for the background cosmological density and pressure and where we have neglected second-order contributions (including those from the pressure necessary to keep the compact matter source quasistatic). There are now five equations on the brane with five unknowns. The solution on the brane is given by the following. For a cosmological background with arbitrary evolution H (), we find that [55,56,34] rn (t, r)|brane = a(t, r)|brane =
Rg [1 + (r)] − (H 2 + H˙ )r 2 , 2r
Rg 1 [1 − (r)] + H 2 r 2 . 2r 2
(5.9) (5.10)
Note that the cosmological background contribution is included in these metric components. The function (r) is defined as ⎡ ⎤ 2R 3 8r 3r ⎣ g (r) = 2 1 + 02 − 1⎦ (5.11) 4r0 Rg 9 r 3 and H˙ = 1 ± 2r0 H 1 + . 3H 2
(5.12)
Just as for the modified Friedmann equation, Eq. (3.10), there is a sign degeneracy in Eq. (5.12). The lower sign corresponds to the self-accelerating cosmologies. These expressions are valid on the brane when r>r0 , H −1 . In both Eqs. (5.9) and (5.10), the first term represent the usual Schwarzschild contribution with a correction governed by (r) resulting from brane dynamics (as depicted in Fig. 15), whereas the second term represents the leading cosmological contribution. Let us try to understand the character of the corrections.
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(rgr02)1/2 r
∆(r)
0
r0H
r0H = 1 -1/3 (r3/2rgr02)1/2 Fig. 15. The function (r) represents a normalized correction to Newton’s constant, G, i.e., Geff =G[1+ (r)]. In the self-accelerating cosmological phase, for small r, (r) asymptotes to −(r 3 /2rg r02 )1/2 , i.e., a correction independent of cosmology. For large r (but also when r >H −1 ), (r) asymptotes the constant value 1/3. This value is − 13 in the saturated limit r0 H = 1, and goes like O(1/r0 H ) as r0 H → ∞. The boundary between
the two regimes is thus r∗ = (rg r02 /2 )1/3 . For the FLRW phase, the graph is just changed by a sign flip, with the exception that the most extreme curve occurs not when r0 H = 1, but rather when H = 0.
5.3. Gravitational regimes One may consolidate all our results and show from Eqs. (5.9)–(5.11) that there exists a scale [55,56,34],
1/3 r02 Rg r∗ = 2
(5.13)
with H˙ . = 1 ± 2r0 H 1 + 3H 2
Inside a radius r∗ the metric is dominated by Einstein gravity but has corrections which depend on the global cosmological phase. Outside this radius (but at distances much smaller than both the crossover scale, r0 , and the cosmological horizon, H −1 ) the metric is weak-brane and resembles a scalar-tensor gravity in the background of a deSitter expansion. This scale is modulated both by the nonperturbative extrinsic curvature effects of the source itself as well as the extrinsic curvature of the brane generated by its cosmology. The qualitative picture described in Fig. 12 is generalized to the picture shown in Fig. 16. Keeping Fig. 15 in mind, there are important asymptotic limits of physical relevance for the metric on the brane, Eqs. (5.9) and (5.10). First, when r>r∗ , the metric is close to the Schwarzschild solution of four-dimensional general relativity. Corrections to that solution are small: Rg Rg r n=− , (5.14) ± 2r 2r02 Rg ∓ a= 2r
Rg r 8r02
.
(5.15)
The background cosmological expansion becomes largely unimportant and the corrections are dominated by effects already seen in the Minkowski background. Indeed, there is no explicit dependence on the parameter governing cosmological expansion, H. However, the sign of the correction to the Schwarzschild solution is dependent on the global properties of the cosmological phase. Thus, we may ascertain information about bulk, five-dimensional cosmological
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r
braneworld
Fig. 16. The four-dimensional universe where we live is denoted by the large spherical brane. A local mass source located, for example, near its north pole dynamically dimples the brane, inducing a nonperturbative extrinsic curvature. That extrinsic curvature suppress coupling the mass source to the extra scalar mode and, within the region dictated by the radius r∗ given by Eq. (5.13), Einstein gravity is recovered. Outside r∗ , the gravitational field is still modulated by the effects of the extrinsic curvature of the brane generated be background cosmology.
behavior from testing details of the metric where naively one would not expect cosmological contributions to be important. Cosmological effects become important when r?r∗ . The metric is dominated by the cosmological flow, but there is still an attractive potential associated with the central mass source. Within the cosmological horizon, r>H −1 , this residual potential is Rg 1 n = − 1+ , (5.16) 2r 3 Rg 1 a = 1− . (5.17) 2r 3 This is the direct analog of the weak-brane phase one finds for compact sources in Minkowski space. The residual effect of the matter source is a linearized scalar-tensor gravity with Brans–Dicke parameter =
3 2
( − 1) .
(5.18)
Notice that as r0 H → ∞, we recover the Einstein solution, corroborating results found for linearized cosmological perturbations [62]. At redshifts of order unity, r0 H ∼ O(1) and corrections to four-dimensional Einstein gravity are substantial and H (z)-dependent. These results found in this analysis were also followed up in Ref. [63]. 5.4. Bulk solutions We should elaborate how the solution, Eqs. (5.9) and (5.10), were ascertained as well as explicitly write the solutions of the potentials {n, a, b} in the bulk [55,56,34]. The key to our solution is the assumption, Eq. (5.7). It allows one to solve the brane equations independently of the rest of the bulk and guarantees the asymptotic bulk boundary conditions. In order to see why Eq. (5.7) is a reasonable approximation, we need to explore the full solution to the bulk Einstein equations, GAB (r, z) = 0 ,
(5.19)
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satisfying the brane boundary conditions, Eq. (5.8), as well as specifying that the metric approach the cosmological background, Eq. (5.4), for large values of r and z, i.e., far away from the compact matter source. First, it is convenient to consider not only the components of the Einstein tensor, Eq. (4.20), but also the following components of the bulk Ricci tensor (which also vanishes in the bulk): N A N B Nzz N z Az N z Bz 1 N Rtt = 2 − +2 + + +2 , (5.20) N N A N B N N A N B A Nzz Azz 2Bzz + + . (5.21) N A B We wish to take Gzr = 0, Gzz = 0, and Rzz = 0 and derive expressions for A(r, z) and B(r, z) in terms of N (r, z). Only two of these three equations are independent, but it is useful to use all three to ascertain the desired expressions. Rzz =
5.4.1. Weak-field analysis Since we are only interested in metric when r, z>r0 , H −1 for a weak matter source, we may rewrite the necessary field equations using the expressions, Eq. (4.24). Since the functions, {n(r, z), a(r, z), b(r, z)} are small, we need only keep nonlinear terms that include z-derivatives. The brane boundary conditions, Eq. (4.21), suggest that az and bz terms may be sufficiently large to warrant inclusion of their nonlinear contributions. It is these z-derivative nonlinear terms that are crucial to the recover of Einstein gravity near the matter source. If one neglected these bilinear terms as well, one would revert to the linearized, weak-brane solution (cf. Ref. [48]). Integrating Eq. (5.21) twice with respect to the z-coordinate, we get n + a + 2b = zg 1 (r) + g2 (r) ,
(5.22)
where g1 (r) and g2 (r) are to be specified by the brane boundary conditions, Eq. (5.8), and the residual gauge freedom b(r)|z=0 = 0, respectively. Integrating the Gzr -component of the bulk Einstein tensor, Eq. (4.20), with respect to the z-coordinate yields r(n + 2b) − 2(a − b) = g3 (r) .
(5.23)
The functions g1 (r), g2 (r), and g3 (r) are not all independent, and one can ascertain their relationship with one another by substituting Eqs. (5.22) and (5.23) into the Gzz bulk equation. If one can approximate nz = ∓(H + H˙ /H ) for all z, then one can see that Gzr = 0, Gzz = 0, and Rzz = 0 are all consistently satisfied by Eqs. (5.22) and (5.23), where the functions g1 (r), g2 (r), and g3 (r) are determined at the brane using Eqs. (5.9) and (5.10) and the residual gauge freedom b(r)|z=0 = 0: r0 H˙ g1 (r) = ∓ 4H + − 2 Rg , (5.24) H r r Rg Rg 1 g2 (r) = − r 2 H˙ + (1 − ) + dr 2 (1 + ) , (5.25) 2 2r 2r 0 Rg (5.26) (1 − 3) − (2H 2 + H˙ )r 2 , 2r where we have used the function (r), defined in Eq. (5.11). Using Eqs. (5.22)–(5.26), we now have expressions for a(r, z) and b(r, z) completely in terms of n(r, z) for all (r, z). Now we wish to find n(r, z) and to confirm that nz = ∓(H + H˙ /H ) is a good approximation everywhere of interest. Eq. (5.20) becomes 2n H 2 + H˙ H 2 + H˙ n + + nzz = ± g1 (r) ± , (5.27) r H H g3 (r) =
where again we have neglected contributions if we are only concerned with r, z>r0 , H −1 . Using the expression, Eq. (5.24), we find n +
2n r0 (H 2 + H˙ ) [Rg (r)] . + nzz = −3(H 2 + H˙ ) ∓ r r 2H
(5.28)
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Then, if we let 1 2 H˙ H 2 + H˙ r 1 2 ˙ z − (H + H )r ∓ r0 n=1∓ H + dr 2 Rg (r)(r) + n(r, z) , H 2 H r 0
(5.29)
where n(r, z) satisfies the equation n +
2n + nzz = 0 , r
we can solve Eq. (5.30) by requiring that n vanish as r, z → ∞ and applying the condition Rg H 2 + H˙ rn |z=0 = 1 + 1 ± 2r0 (r) , 2r H
(5.30)
(5.31)
on the brane as an alternative to the appropriate brane boundary condition for n(r, z) coming from a linear combination of Eq. (5.8). We can write the solution explicitly ∞ n(r, z) = dkc(k)e−kz sin kr , (5.32) 0
where c(k) =
2
∞
dr r sin krn|z=0 (r) .
(5.33)
0
We can then compute nz |z=0 , arriving at the bound Rg (r) 1 r dr nz |z=0 r 0 r2 for all r>r0 , H −1 . Then, H˙ nz |z=0 = ∓ H + + nz |z=0 . H
(5.34)
(5.35)
When the first term in Eq. (5.35) is much larger than the second, Eq. (5.7) is a good approximation. When the two terms in Eq. (5.35) are comparable or when the second term is much larger than the first, neither term is important in the determination of Eqs. (5.9) and (5.10). Thus, Eq. (5.7) is still a safe approximation. One can confirm that all the components of the five-dimensional Einstein tensor, Eq. (4.20), vanish in the bulk using field variables satisfying the relationships, Eqs. (5.22), (5.23) and (5.29). The field variables a(r, z) and b(r, z) both have terms that grow with z, stemming from the presence of the matter source. However, one can see that with the following redefinition of coordinates: R = r − zr 0
Rg , r2
r
Z=z+
dr 0
Rg , r2
(5.36) (5.37)
that to leading order as z → H −1 , the desired Z-dependence is recovered for a(R, Z) and b(R, Z) (i.e., ∓H Z), and the Newtonian potential takes the form H˙ 1 n(R, Z) = ∓ H + Z − (H 2 + H˙ )R 2 + · · · . (5.38) H 2 Thus, we recover the desired asymptotic form for the metric of a static, compact matter source in the background of a cosmological expansion.
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5.4.2. A note on bulk boundary conditions A number of studies have been performed for DGP gravity that have either arrived at or used modified force laws different than those given by Eqs. (5.9)–(5.12) [57–61]. How does one understand or reconcile such a discrepancy? Remember that one may ascertain the metric on the brane without reference to the bulk because there are five unknown quantities, {N (r), A(r), Nz (r), Az (r), Bz (r)}, and there are four independent equations (the three brane boundary conditions, (5.8), and the Gzz -component of the bulk Einstein equations) on the brane in terms of only these quantities. One need only choose an additional relationship between the five quantities in order to form a closed, solvable system. We chose Eq. (5.7) and showed here that it was equivalent to choosing the bulk boundary condition that as z became large, one recovers the background cosmology. The choice made in these other analyses is tantamount to a different choice of bulk boundary conditions. One must be very careful in the analysis of DGP gravitational fields that one is properly treating the asymptotic bulk space, as it has a significant effect on the form of the metric on the brane.
6. Anomalous orbit precession We have established that DGP gravity is capable of generating a contemporary cosmic acceleration with dark energy. However, it is of utmost interest to understand how one may differentiate such a radical new theory from a more conventional dark energy model concocted to mimic a cosmic expansion history identical to that of DGP gravity. The results of the previous sections involving the nontrivial recovery of Einstein gravity and large deviations of this theory from Einstein gravity in observably accessible regimes is the key to this observational differentiation. Again, there are two regimes where one can expect to test this theory and thus, there are two clear regimes in which to observationally challenge the theory. First deep within the gravity well, where r>r∗ and where the corrections to general relativity are small, but the uncertainties are also correspondingly well-controlled. The second regime is out in the cosmological flow, where r?r∗ (but still r>r0 ∼ H0−1 ) and where corrections to general relativity are large, but our observations are also not as precise. We focus on the first possibility in this section and will go to the latter in the next section. 6.1. Nearly circular orbits Deep in the gravity well of a matter source, where the effects of cosmology are ostensibly irrelevant, the correction to the gravitational potentials may be represented by effective correction to Newton’s constant r3 (r) = ± , (6.1) 2r02 Rg that appears in the expressions, Eqs. (5.9) and (5.10). Though there is no explicit dependence on the background H () evolution, there is a residual dependence on the cosmological phase through the overall sign. In DGP gravity, tests of a source’s Newtonian force leads to discrepancies with general relativity [48,55,64]. Imagine a body orbiting a mass source where Rg (r) = rg = constant. The perihelion precession per orbit may be determined in the usual way given a metric of the form, Eqs. (4.24) and (4.26) J AN , (6.2) = dr 2 2 r E − N 2 (1 + J 2 /r 2 ) where E = N 2 dt/ds and J = r 2 d /ds are constants of motion resulting from the isometries of the metric, and ds is the differential proper time of the orbiting body. With a nearly circular orbit deep within the Einstein regime (i.e., when r>r∗ so that we may use Eqs. (5.14) and (5.15)), the above expression yields 1/2 3rg 3 r3 = 2 + . ∓ r 2 2r02 rg
(6.3)
The second term is the famous precession correction from general relativity. The last term is the new anomalous precession due to DGP brane effects. This latter correction is a purely Newtonian effect. Recall that any deviations of
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a Newtonian central potential from 1/r results in an orbit precession. The DPG correction to the precession rate is now [48,55,64] d 3 = ∓5 as/year . DGP = ∓ dt 8r0
(6.4)
Note that this result is independent of the source mass, implying that this precession rate is a universal quantity dependent only on the graviton’s effective linewidth (r0−1 ) and the overall cosmological phase. Moreover, the final result depends on the sign of the cosmological phase [55]. Thus one can tell by the sign of the precession whether self-acceleration is the correct driving force for today’s cosmic acceleration. It is extraordinary that a local measurement, e.g., in the inner solar system, can have something definitive to say about gravity on the largest observable scales. 6.2. Solar system tests Nordtvedt [65] quotes precision for perihelion precession at 430 as/year for Mercury and 10 as/year for Mars. Improvements in lunar ranging measurements [66,64] suggest that the Moon will be sensitive to the DGP correction, Eq. (6.4), in the future lunar ranging studies. Also, BepiColombo (an ESA satellite) and MESSENGER (NASA) are being sent to Mercury at the end of the decade, will also be sensitive to this correction [67]. Counterintuitively, future outer solar system probes that possess precision ranging instrumentation, such as Cassini [68], may also provide ideal tests of the anomalous precession. Unlike post-Newtonian deviations arising from Einstein corrections, this anomaly does not attenuate with distance from the sun; indeed, it amplifies. This is not surprising since we know that corrections to Einstein gravity grow as gravity weakens. More work is needed to ascertain whether important inner or outer solar system systematics allow this anomalous precession effect to manifest itself effectively. If so, we may enjoy precision tests of general relativity even in unanticipated regimes [69–73]. The solar system seems to provide a most promising means to constrain this anomalous precession from DGP gravity.8 It is also interesting to contrast solar system numbers with those for binary pulsars. The rate of periastron advance for the object PSR 1913+16 is known to a precision of 4 × 104 as/year [75]. This precision is not as good as that for the inner solar system. A tightly bound system such as a binary pulsar is better at finding general relativity corrections to Newtonian gravity because it is a stronger gravity system than the solar system. It is for precisely the same reason that binary pulsars prove to be a worse environment to test DGP effects, as these latter effects become less prominent as the gravitational strength of the sources become stronger. As a final note, one must be extremely careful about the application of Eqs. (5.9) and (5.10). Remember they were derived for a spherical source in isolation. We found that the resulting metric, while in a weak-field regime, i.e., rg /r>1, was nevertheless still very nonlinear. Thus, superpossibility of central sources is no longer possible. This introduces a major problem for the practical application of Eqs. (5.9) and (5.10) to real systems. If we take the intuition developed in the last two sections, however, we can develop a sensible picture of when these equations may be applicable. We found that being in the Einstein regime corresponds to being deep within the gravity well of a given matter source. Within that gravity well, the extrinsic curvature of the brane generated by the source suppressed coupling of the extra scalar mode. Outside a source’s gravity well, one is caught up in the cosmological flow, the brane is free to fluctuate, and the gravitational attraction is augmented by an extra scalar field. If one takes a “rubber sheet” picture of how this DGP effect works, we can imagine several mass sources placed on the brane sheet, each deforming the brane in a region whose size corresponds with the mass of the source (see Fig. 17). These deformed regions will in general overlap with each other in some nontrivial fashion. However, the DGP corrections for the orbit of a particular test object should be dominated by the gravity well in which the test object is orbiting. For example, within the solar system, we are clearly in the gravity well of the sun, even though we also exist inside a larger galaxy, which in turn is inside a cluster, etc. Nevertheless, the space–time curvature around us is dominated by the sun, and for DGP that implies the extrinsic curvature of the brane in our solar system is also dominated by the sun. This picture also implies that if we are not rooted in the gravity well of a single body, then the quantitative form of the correction given by Eqs. (5.14) and (5.15) is simply invalid. That is, three body systems need 8 One should also note that DGP gravity cannot explain the famous Pioneer anomaly [74]. First, the functional form of the anomalous acceleration in DGP is not correct: it decays as a r −1/2 power-law with distance from the sun. Moreover, the magnitude of the effect is far too small in the outer solar system.
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cosmic horizon of A
mass cluster A
Fig. 17. The universe is populated by a variety of matter sources. For sources massive enough to deform the brane substantially, Einstein gravity is recovered within the gravity well of each region (the green circles whose sizes are governed by r∗ ∼ (mass)1/3 ). Outside these gravity wells, extra scalar forces play a role.
to be completely reanalyzed. This may have relevant consequences for the moon which has substantial gravitational influences from both the earth and the sun.
7. Large-scale structure Future solar system tests have the possibility of probing the residual deviation from four-dimensional Einstein gravity at distances well below r∗ . Nevertheless, it would be ideal to test gravitational physics where dramatic differences from Einstein gravity are anticipated. Again, this is the crucial element to the program of differentiating a modified-gravity scenario such as DGP from a dark-energy explanation of today’s cosmic acceleration that has an identical expansion history. A detailed study of large-scale structure in the Universe can provide a test of gravitational physics at large distance scales where we expect anomalous effects from DGP corrections to be large. In the last section, we have seen that the modified force law is sensitive to the background cosmological expansion, since this expansion is intimately tied to the extrinsic curvature of the brane [23,28], and this curvature controls the effective Newtonian potential [55,56,34]. This gives us some measure of sensitivity to how cosmology affects the changes in the growth of structure through the modulation of a cosmology-dependent Newtonian potential. We may then proceed and compare those results to the standard cosmology, as well as to a cosmology that exactly mimics the DGP expansion history using dark energy. The description of the work presented in this section first appeared in Ref. [56]. 7.1. Linear growth The modified gravitational potentials given by Eqs. (5.9) and (5.10) indicate that substantial deviations from Einstein gravity occur when rr∗ . More generally, from our discussion of the VDVZ discontinuity, we expect large deviations from general relativity any time an analysis perturbative in the source strength is valid. This is true when we want to understand the linear growth of structure in our universe. So long as we consider early cosmological times, when perturbations around the homogeneous cosmology are small, or at later times but also on scales much larger than the clustering scale, a linear analysis should be safe. The linear regime is precisely what we have analyzed with the potential given by Eq. (2.8), or, more generally, Eqs. (5.16) and (5.17) Rg 1 n = − 1+ , 2r 3 Rg 1 , a = 1− 3 2r in a background cosmology (remember, we are still only considering nonrelativistic sources at scales smaller than the horizon size). Because we are in the linear regime, we may think of these results as Green’s function solutions to the
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35
general physical problem. Thus, these potentials are applicable beyond just spherical sources and we deduce that 1 ∇ 2 n(r, t) = 4G 1 + (x, t) , (7.1) 3 1 2 ∇ a(r, t) = 4G 1 − (x, t) (7.2) 3 for general matter distributions, so long as they are nonrelativistic. What results is a linearized scalar-tensor theory with cosmology-dependent Brans–Dicke parameter, Eq. (5.18) =
3 2
( − 1) .
Note this is only true for weak, linear perturbations around the cosmological background, and that once overdensities becomes massive enough for self-binding, these results cease to be valid and one needs to go to the nonlinear treatment. Nonrelativistic matter perturbations with () = / (t) evolve via ¨ + 2H ˙ = 4G 1 + 1 , (7.3) 3 where (t) is the background cosmological energy density, implying that self-attraction of overdensities is governed by an evolving Geff : 1 Geff = G 1 + . (7.4) 3 Here the modification manifests itself as a time-dependent effective Newton’s constant, Geff . Again, as we are focused on the self-accelerating phase, then from Eq. (5.12) H˙ = 1 − 2r0 H 1 + . 3H 2 As time evolves the effective gravitational constant decreases. For example, if 0m = 0.3, Geff /G = 0.72, 0.86, 0.92 at z = 0, 1, 2. One may best observe this anomalous repulsion (compared to general relativity) through the growth of large-scale structure in the early universe. That growth is governed not only by the expansion of the universe, but also by the gravitational self-attraction of small overdensities. Fig. 18 depicts how growth of large-scale structure is altered by DGP gravity. The results make two important points: (1) growth is suppressed compared to the standard cosmological model since the expansion history is equivalent to a w(z) dark-energy model with an effective equation of state given by Eq. (3.25) weff (z) = −
1 1 + m
and (2) growth is suppressed even compared to a dark energy model that has identical expansion history from the alteration of the self-attraction from the modified Newton’s constant, Eq. (7.4). The latter point reiterates the crucial feature that one can differentiate between this modified-gravity model and dark energy. 7.2. Nonlinear growth We certainly want to understand large-scale structure beyond the linear analysis. Unlike the standard cosmological scenario where the self-gravitation of overdensities may be treated with a linear Newtonian gravitational field, in DGP gravity the gravitational field is highly nonlinear, even though the field is weak. This nonlinearity, particularly the inability to convert the gravitational field into a superposition of point-to-point forces, poses an enormous challenge to understanding growth of structure in DGP gravity. Nevertheless, we already have the tools to offer at least a primitive preliminary analysis. We do understand the full nonlinear gravitational field around spherical, nonrelativistic sources, Eqs. (5.9)–(5.12). Consider the evolution
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1 0.98 0.96 0.94 0.92 0.9 0.88 0.86
0.8 0.6 0.4 0.2 0 0
2
4
6
8
10
z Fig. 18. The top panel shows the ratio of the growth factors D+ (dashed lines) in DGP gravity [Eq. (7.3)] and a model of dark energy (DE) with an equation of state such that it gives rise to the same expansion history (i.e., given by Eq. (3.10), but where the force law is still given by general relativity). The upper line corresponds to 0m = 0.3, the lower one to 0m = 0.2. The solid lines show the analogous result for velocity perturbations factors f ≡ d ln D+ /d ln a. The bottom panel shows the growth factors as a function of redshift for models with different expansion histories, corresponding to (from top to bottom) CDM ( 0m = 0.3), and DGP gravity with 0m = 0.3, 0.2 respectively. Figure from Ref. [56].
of a spherical top-hat perturbation (t, r) of top-hat radius Rt . At subhorizon scales (H r>1), the contribution from the Newtonian potential, n(t, r), dominates the geodesic evolution of the overdensity. The equation of motion for the perturbation is [56] √ ˙2 ¨ − 4 + 2H ˙ = 4G (1 + ) 1 + 2 1 1+−1 , 3 1+ 3
(7.5)
where ≡ 8r02 Rg /92 Rt3 . Note that for large , Eq. (7.5) reduces to the standard evolution of spherical perturbations in general relativity. Fig. 19 shows an example of a full solution of Eq. (7.5) and the corresponding solution in the cosmological constant case. Whereas such a perturbation collapses in the CDM case at z = 0.66 when its linearly extrapolated density contrast is c = 1.689, for the DGP case the collapse happens much later at z = 0.35 when its c = 1.656. In terms of the linearly extrapolated density contrasts things do not look very different, in fact, when the full solutions are expressed as a function of the linearly extrapolated density contrasts, lin = D+ i /(D+ )i they are very similar to within a few percent. This implies that all the higher-order moments of the density field are very close to those for CDM models, for example, the skewness is less than a 1% difference from CDM. This close correspondence of the higher-order moments can be useful by allowing the use of nonlinear growth to constrain the bias between galaxies and dark matter in the same way as it is done in standard case, thus inferring the linear growth factor from the normalization of the power spectrum in the linear regime. Although the result in the right panel in Fig. 19 may seem a coincidence at first sight, Eq. (7.5) says that the nontrivial correction from DGP gravity in square brackets is maximum when = 0 (which gives the renormalization of Newton’s constant). As increases
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104
1.05
103
ACDM
(z)
1.04
102
1.03 DGP / ACDM 1.02
10 1.01 DGP
1 0
1
2 z
3
0.4
0.6
0.8 lin
1
1.2
1.4
Fig. 19. Numerical solution of the spherical collapse. The left panel shows the evolution for a spherical perturbation with i = 3 × 10−3 at zi = 1000 for 0m = 0.3 in DGP gravity and in CDM. The right panel shows the ratio of the solutions once they are both expressed as a function of their linear density contrasts. Figure from Ref. [56].
the correction disappears (since DGP becomes Einstein at high-densities), so most of the difference between the two evolutions happens in the linear regime, which is encoded in the linear growth factor. 7.3. Observational consequences What are the implications of these results for testing DGP gravity using large-scale structure? A clear signature of DGP gravity is the suppressed (compared to CDM) growth of perturbations in the linear regime due to the different expansion history and the addition of a repulsive contribution to the force law. However, in order to predict the present normalization of the power spectrum at large scales, we need to know the normalization of the power spectrum at early times from the CMB. A fit of the to pre-WMAP CMB data was performed in Ref. [31] using the angular diameter distance for DGP gravity, finding a best-fit (flat) model with 0m 0.3, with a very similar CMB power spectrum to the standard cosmological constant model (with 0m 0.3 and 0 = 0.7) and other parameters kept fixed at the same value. Here we use this fact, plus the normalization obtained from the best-fit cosmological constant power-law model from WMAP [76] which has basically the same (relevant for large-scale structure) parameters as in Ref. [31], except for the normalization of the primordial fluctuations which has increased compared to pre-WMAP data (see e.g., Fig. 11 in Ref. [77]). The normalization for the cosmological constant scale-invariant model corresponds to present rms fluctuations in spheres of 8 Mpc/ h, 8 = 0.9 ± 0.1 (see Table 2 in [76]). Fig. 20 shows the present value of 8 as a function of 0m for DGP gravity, where we assume that the best-fit normalization of the primordial fluctuations stays constant as we change 0m , and recompute the transfer function and growth factor as we move away from 0m = 0.3. Since most of the contribution to 8 comes from scales r < 100h/Mpc, we can calculate the transfer function using Einstein gravity, since these modes entered the Hubble radius at redshifts high enough that they evolve in the standard fashion. The value of 8 at 0m = 0.3 is then given by 0.9 times the ratio of the DGP to CDM growth factors shown in the bottom panel of Fig. 18. The error bars in 8 reflect the uncertainty
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1 0.9 0.8 0.7
8
0.6 0.5 0.4 0.3 0.2 0.1 0 0.1
0.2
0.3 0 Ωm
0.4
0.5
Fig. 20. The linear power spectrum normalization, 8 , for DGP gravity as a function of 0m . The vertical lines denote the best-fit value and 68% +0.07 confidence level error bars from fitting to type-IA supernovae data from [31], 0m = 0.18−0.06 . The other lines correspond to 8 as a function of
0m obtained by evolving the primordial spectrum as determined by WMAP by the DGP growth factor. Figure from Ref. [56].
in the normalization of primordial fluctuations, and we keep them a constant fraction as we vary 0m away from 0.3. We see in Fig. 20 that for the lower values of 0m preferred by fitting the acceleration of the universe, the additional suppression of growth plus the change in the shape of the density power spectrum drive 8 to a rather small value. This could in part be ameliorated by increasing the Hubble constant, but not to the extent needed to keep 8 at reasonable values. The vertical lines show the best-fit and 1 error bars from fitting DGP gravity to the supernova data from Ref. [31]. This shows that fitting the acceleration of the universe requires approximately 8 0.7–1 and 8 0.8–2. In order to compare this prediction of 8 to observations one must be careful since most determinations of 8 have built in the assumption of Einstein gravity or CDM models. We use galaxy clustering, which in view of the results in Section 7.2 for higher-order moments, should provide a test of galaxy biasing independent of gravity being DGP or Einstein. Recent determinations of 8 from galaxy clustering in the SDSS survey [78] give ∗8 = 0.89 ± 0.02 for L∗ galaxies at an effective redshift of the survey zs = 0.1. We can convert this value to 8 for dark matter at z = 0 as follows. We evolve to z = 0 using a conservative growth factor, that of DGP for 0m = 0.2. In order to convert from L∗ galaxies to dark matter, we use the results of the bispectrum analysis of the 2dF survey [79] where b = 1.04 ± 0.11 for luminosity L 1.9L∗ . We then scale to L∗ galaxies using the empirical relative bias relation obtained in [80] that b/b∗ = 0.85 + 0.15(L/L∗ ), which is in very good agreement with SDSS (see Fig. 30 in Ref. [78]). This implies 8 = 1.00 ± 0.11. Even if we allow for another 10% systematic uncertainty in this procedure, the preferred value of
0m in DGP gravity that fits the supernovae data is about 2 away from that required by the growth of structure at z = 0. Nevertheless, the main difficulty for DGP gravity to simultaneously explain cosmic acceleration and the growth of structure is easy to understand: the expansion history is already significantly different from a cosmological constant, corresponding to an effective equation of state with weff = −(1 + m )−1 . This larger value of w suppresses the growth somewhat due to earlier epoch of the onset of acceleration. In addition, the new repulsive contribution to the force law suppresses the growth even more, driving 8 to a rather low value, in contrast with observations. If as error bars shrink
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the supernovae results continue to be consistent with weff = −1, this will drive the DGP fit to a yet lower value of
0m and thus a smaller value of 8 . For these reasons we expect the tension between explaining acceleration and the growth of structure to be robust to a more complete treatment of the comparison of DGP gravity against observations. Already more ideas for how to approach testing Newton’s constant on ultralarge scale and the self-consistency of the DGP paradigm for explaining cosmic accleration have been taken [81–84,61]. 8. Gravitational lensing 8.1. Localized sources One clear path to differentiating DGP gravity from conventional Einstein gravity is through an anomalous mismatch between the mass of a compact object as computed by lensing measurements, versus the mass of an object as computed using some measure of the Newtonian potential, such as using the orbit of some satellite object, or other means such as the source’s X-ray temperature or through the SZ-effect. The lensing of light by a compact matter source with metric Eqs. (5.9) and (5.10) may be computed in the usual way. The angle of deflection of a massless test particle is given by J A = dr 2 , (8.1) 2 2 r (E /N ) − (J 2 /r 2 ) where E = N 2 dt/d and J = r 2 d /d are constants of motion resulting from the isometries, and d is the differential affine parameter. Removing the effect of the background cosmology and just focussing on the deflection generated by passing close to a matter source, the angle of deflection is [55] rmax Rg (r) = + 2b dr √ , (8.2) 2 r r 2 − b2 b where b is the impact parameter. This result is equivalent to the Einstein result, implying light deflection is unaltered by DGP corrections, even when those corrections are large. This result jibes with the picture that DGP corrections come solely from a light scalar mode associated with brane fluctuations. Since scalars do not couple to photons, the trajectory of light in a DGP gravitational field should be identical to that in a Einstein gravitational field generated by the same mass distribution. Light deflection measurements thus probe the “true” mass of a given matter distribution. Contrast this with a Newtonian measurement of a matter distribution using the gravitational potential, Eq. (5.9) while incorrectly assuming general relativity holds. The mass discrepancy between the lensing mass (the actual mass) and that determined from the Newtonian force may be read directly from Eq. (5.9), M =
Mlens 1 −1= −1 . MNewt 1 + (r)
(8.3)
This ratio is depicted in Fig. 21 for both cosmological phases, with an arbitrary background cosmology chosen. When the mass is measured deep within the Einstein regime, the mass discrepancy simplifies to 1/3 r3 M = ∓ . (8.4) 2r02 Rg Solar system measurements are too coarse to be able to resolve the DGP discrepancy between lensing mass of the sun and its Newtonian mass. The discrepancy M for the sun at O(AU) scale distances is approximately 10−11 . Limits on this discrepancy for the solar system as characterized by the post-Newtonian parameter, − 1, are only constrained to be < 3 × 10−4 . In the solar system, this is much too small an effect to be a serious means of testing this theory, even with the most recent data from Cassini [68]. Indeed most state-of-the-art constraints on the post-Newtonian parameter will not do, because most of the tests of general relativity focus on the strong-gravity regime near r ∼ rg . However, in this theory, this is where the anomalous corrections to general relativity are the weakest.
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0.5
self-accelerating
0.4
0.3
δM
0.2
0.1
0 FLRW -0.1
-0.2 0
0.5
1
1.5
2
2 r/(r0 rg)1/3
Fig. 21. Mass discrepancy, M, for a static point source whose Schwarzschild radius is rg . The solid curve is for a self-accelerating background with H = r0−1 . The dashed curve is for a FLRW background with H = r0−1 . Figure from Ref. [55].
The best place to test this theory is as close to the weak-brane regime as we can get. A possibly more promising regime may be found in galaxy clusters. For 1014 → 1015 M clusters, the scale (r02 Rg )1/3 has the range 6 → 14 Mpc. For masses measured at the cluster virial radii of roughly 1 → 3 Mpc, this implies mass discrepancies of roughly 5 → 8%. X-ray or Sunyaev–Zeldovich (SZ) measurements are poised to map the Newtonian potential of the galaxy clusters, whereas weak lensing measurements can directly measure the cluster mass profile. Unfortunately, these measurements are far from achieving the desired precisions. If one can extend mass measurements to distances on the order of r0 , Fig. 21 suggests discrepancies can be as large as −10% for the FLRW phase or even 50% for the selfaccelerating phase; however, remember these asymptotic limits get smaller as (r0 H )−1 as a function of the redshift of the lensing mass. It is a nontrivial result that light deflection by a compact spherical source is identical to that in four-dimensional Einstein gravity (even with potentials, Eqs. (5.9)–(5.12), substantially differing from those of Einstein gravity) through the nonlinear transition between the Einstein phase and the weak-brane phase. As such, there remains the possibility that for aspherical lenses that this surprising null result does not persist through that transition and that DGP may manifest itself through some anomalous lensing feature. However, given the intuition that the entirety of the anomalous DGP effect comes from an extra gravitational scalar suggests that even in a more general situation, photons should still probe the “true” matter distribution. Were photon geodesics in DGP gravity to act just as in general relativity for any distribution of lenses, this would provide a powerful tool for proceeding with the analysis of weak lensing in DGP. Weak lensing would then provide a clean window into the true evolution of structure as an important method of differentiating it from general relativity. Tanaka’s analysis of gravitational fields in more general matter distributions in DGP gravity provides a first step to answering the nature of photon geodesics [85]. 8.2. The late-time ISW The late-time integrated Sachs–Wolfe (ISW) effect on the cosmic microwave background (CMB) may be viewed as another possible “lensing” signature. The Sachs–Wolfe effect is intrinsically connected to how photons move through and are altered by gravitational potentials (in this case, time-dependent ones). This effect is a direct probe of any possible alteration of the gravitational potentials over time; moreover, it is a probe of potential perturbations on the largest of distance scales, safely in the linear regime.
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Late-time ISW would then seem to be a promising candidate for modified-gravity theories of the sort that anticipate cosmic acceleration, as we see that the potentials are altered by substantial corrections. Indeed, for a particular set of modified-gravity theories, this assertion is indeed the case [86]. However, for DGP gravity, there is an unfortunate catch. Recall in Section 7, we argued that when considering linear potentials satisfied Eqs. (7.1) and (7.2), our results generalized beyond just spherical perturbations. In order to bring our linear potentials, Eqs. (7.1) and (7.2), into line with how the late-time ISW is normally treated, we need to identify the gravitational potentials as perturbations around a homogeneous cosmological background with the line element ds 2 = [1 + 2(, )] d2 − a 2 ()[1 + 2(, )][d2 + 2 d ] .
(8.5)
Here (, ) and (t, ) are the relevant gravitational potentials and is a comoving radial coordinate. In effect we want to determine and given n and a. Unlike the case of Einstein’s gravity, = −. One may perform a coordinate transformation to determine that relationship. We find that, assigning r = a(), and = n(, r) , dr =− a(, r) , r
(8.6) (8.7)
keeping only the important terms when rH >1. The quantity of interest for the ISW effect is the time derivative of − , the above the above analysis implies ∇ 2 ( − ) =
8 2 a , MP2
(8.8)
where ∇ is the gradient in comoving spatial coordinates, i . Just as we found for light deflection, this result is identical to the four-dimensional Einstein result, the contributions from the brane effects exactly cancelling. Again, the intuition of the anomalous DGP effects coming from a light gravitational scalar is correct in suggesting the microwave background photons probe the “true” matter fluctuations on the largest of scales. Thus, the late-time ISW effect for DGP gravity will be identical to that of a dark energy cosmology that mimics the DGP cosmic expansion history, Eq. (3.10), at least at scales small compared to the horizon. Our approximation does not allow us to address the ISW effect at the largest scales (relevant for the CMB at low multipoles), but it is applicable to the cross-correlation of the CMB with galaxy surveys [87,88]. At larger scales, one expects to encounter difficulties associated with leakage of gravity off the brane (for order-unity redshifts) and other bulk effects [28,62,89] that we were successfully able to ignore at subhorizon scales. 8.3. Leakage and depletion of anisotropic power There is an important effect we have completely ignored up until now. At scales comparable to r0 , when the Hubble expansion rate is comparable to r0−1 , i.e., O(1) redshifts, gravitational perturbations can substantially leak off the brane. This was the original effect discussed from the introduction of DGP braneworlds. At the same time, perturbations that exist in the bulk have an equal likelihood of impinging substantial perturbation amplitude onto the brane from the outside bulk world. This leads to a whole new arena of possibilities and headaches. There is little that can be done to control what exists outside in the free bulk. However, there are possible reasonable avenues one can take to simplify the situation. In Section 3.2 we saw how null worldlines through the bulk in the FLRW phase could connect different events on the brane. This observation was a consequence of the convexity of the brane worldsheet and the choice of bulk. Conversely, if one chooses the bulk corresponding to the self-accelerating phase, one may conclude that no null lightray through the bulk connects two different events on the brane. Gravity in the bulk is just empty space five-dimensional Einstein gravity, and thus, perturbations in the bulk must follow null geodesics. Consider again Fig. 5. If we live in the self-accelerating cosmological phase, then the bulk exists only exterior to the brane worldsheet in this picture. One can see that, unlike in the interior phase, null geodesics can only intersect our brane Universe once. That is, once a perturbation has left our brane, it can never get back. Therefore,
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if we assume that the bulk is completely empty and all perturbations originate from the brane Universe, that means that perturbations can only leak from the brane, the net result of which is a systematic loss of power with regard to the gravitational fluctuations. Let us attempt to quantify this depletion. As a crude estimate, we can take the propagator from the linear theory given by Eq. (2.3), and treat only fluctuations in the Newtonian potential, (x A ) where g00 = 1 + 2 . For modes inside the horizon, we may approximate evolution assuming a Minkowski background. For a mode whose spatial wavenumber is k, the equations of motion are j2 j2 − 2 + k2 = 0 , j2 jz subject to the boundary condition at the brane (z = 0) 2 j j 2 = −r − k
. 0 jz z=0 j2
(8.9)
(8.10)
Then for a mode initially localized near the brane, the amplitude on the brane obeys the following form [28]: | | = | |0 e
−1/2 /kr 20
,
(8.11)
when kr 20 . Imagine the late universe as each mode reenters the horizon. Modes, being frozen outside the horizon, are now free to evolve. For a given mode, k, the time spent inside the horizon is 1 3 = r0 1 − , (8.12) kr 0 in a late-time, matter-dominated universe and where we have approximated today’s cosmic time to be r0 . Then, the anomalous depletion resulting in DGP gravity is | | 1 1 1 1− −1 . (8.13) = exp − | | DGP 2 kr 0 (kr 0 )3 This depletion is concentrated at scales where kr 0 ∼ 1. It should appear as an enhancement of the late-time integrated Sachs–Wolfe effect at the largest of angular scales. A more complete analysis of the perturbations of the full metric is needed to get a better handle on leakage on sizes comparable to the crossover scale, r0 . Moreover, complicated global structure in the bulk renders the situation even more baffling. For example, the inclusion of an initial inflationary stage completely alters the bulk boundary conditions. Other subtleties of bulk initial and boundary condition also need to be taken into account for a proper treatment of leakage in a cosmological setting [28,62,89].
9. Prospects and complications We have presented a preliminary assessment of the observational viability of DGP gravity to simultaneously explain the acceleration of the universe and offer a promising set of observables for the near future. The theory is poised on the threshold of a more mature analysis that requires a new level of computational sophistication. In order to improve the comparison against observations a number of issues need to be resolved. The applicability of coarse graining is a pertinent issue that needs to be addressed and tackled. Understanding growth of structure into clustered objects is essential in order to apply promising observing techniques in the near future. To do a full comparison of the CMB power spectrum against data, it remains to properly treat the ISW effect at scales comparable to the horizon. Understanding how primordial fluctuations were born in this theory likewise requires a much more detailed treatment of technical issues.
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Fig. 22. It is the Birkhoff’s law property of general relativity that allows one take an arbitrary, though spherically symmetric matter distribution (left) but describe the evolution of observer B relatively to the origin of symmetry A by excising all matter outside a spherical ball of mass around the A and condensing the matter within the ball to a single point. DGP gravity, in a limited fashion, obeys the same property.
9.1. Spherical symmetry and Birkhoff’s law An important outstanding question in DGP cosmology is whether a universe driven by a uniform distribution of compact sources, such as galaxies or galaxy clusters, actually drives the same overall expansion history as a truly uniform distribution of pressureless matter. The problem is very close to the same problem as in general relativity (although in DGP, the system is more nonlinear) of how one recovers the expansion history of a uniform distribution of matter with a statistically homogeneous and isotropic distribution of discrete sources. What protects this coarse-graining in general relativity is that the theory possesses a Birkhoff’s law property (Fig. 22), even in the fullynonlinear case. It is clear from the force law generated from the metric components Eqs. (5.9)–(5.12) that Birkhoff’s law does not strictly apply in DGP gravity, even with a pressureless background energy-momentum component. The expressions in these equations make a clear distinction between the role of matter that is part of an overdensity and matter that is part of the background. However, there is a limited sense in which Birkhoff’s law does apply. We do see that the evolution of overdensities is only dependent on the quantity Rg (r). In this sense, because for spherically symmetric matter configurations, only the integrated mass of the overdensity matters (as opposed to more complicated details of that overdensity), Birkhoff’s law does apply in a limited fashion. So, if all matter perturbations were spherically symmetric, then coarse graining would apply in DGP gravity. Of course, matter perturbations are not spherically symmetric, not even when considering general relativity. We extrapolate that because we are concerned with perturbations that are at least statistical isotropic, that like in general relativity, coarse graining may be applied in this more general circumstance. It is widely believed that coarse graining may be applied with impunity in general relativity. Based on the arguments used here, we assume that the same holds true for DGP. However, it is worthwhile to confirm this in some more systematic and thorough way, as there are subtleties in DGP gravity that do not appear in general relativity. 9.2. Beyond isolated spherical perturbations Understanding the growth of structure in DGP gravity is an essential requirement if one is to treat it as a realistic or sophisticated counterpart to the standard cosmological model. We need to understand large-scale structure beyond a linear analysis and beyond the primitive spherical collapse picture. Without that understanding, analysis of anything involving clustered structures are suspect, particularly those that have formed during the epoch of O(1) redshift where the nonlinear effects are prominent and irreducible. This includes galaxy cluster counting, gravitational lens distributions, and several favored methods for determining 8 observationally. For analyses of DGP phenomenology such as Refs. [90–96,59,60] to carry weight, they need to include such effects. What is needed is the moral equivalent of the N -body simulation for DGP gravity. But the native nonlinearity coursing throughout this model essentially forbids a simple N -body approach in DGP gravity. The “rubber sheet”
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picture described at the end of Section 6 and in Fig. 17 must be developed in a more precise and formal manner. The truly daunting aspect of this approach is that it represents a full five-dimensional boundary value problem for a complex distribution of matter sources, even if one could use the quasistatic approximation. One possible simplification is the equivalent of employing the assumption, Eq. (5.7), but in a more general context of an arbitrary matter distribution. This allows one to reduce the five-dimensional boundary-value problem to a much simpler (though still terribly intractable) four-dimensional boundary-value problem. Tanaka has provided a framework where one can begin this program of extending our understanding of gravitational fields of interesting matter distributions beyond just spherically symmetric ones [85]. If this program can be carried out properly, whole new vistas of approaches can be taken for constraining DGP gravity and then this modified-gravity theory may take a place as a legitimate alternative to the standard cosmological model. 9.3. Exotic phenomenology There are a number of intriguing exotic possibilities for the future in DGP gravity. The model has many rich features that invite us to explore them. One important avenue that begs to be developed is the proper inclusion of inflation in DGP gravity. Key issues involve the proper treatment of the bulk geometry, how (and at what scale) inflationary perturbation in this theory develop and what role do brane boundary conditions play in the treatment of this system. An intriguing possibility in DGP gravity may come into play as the fundamental five-dimensional Planck scale is so low (corresponding to M ∼ 100 MeV, suggesting transplanckian effects may be important at quite low energies. Initial work investigating inflation in DGP gravity has begun [97–99]. Other work that has shown just a sample of the richness of DGP gravity and its observational possibilities include the possibility of nonperturbatively dressed or screened starlike solutions [100,101] and shock solutions [102,103]. Indeed, a possible glimpse of how strange DGP phenomenology may be is exhibited by the starlike solution posed in Refs. [100,101]. In these papers, the solution appears as four-dimensional Einstein when r>r∗ , just as the one described in Section 4. However, far from the matter source when r?r0 , rather than the brane having no influence on the metric, in this new solution, there is a strong backreaction from the brane curvature that screens the native mass, M, of the central object and the effective ADM mass appears to be 1/3 rg Meff ∼ M , (9.1) r0 where rg is the Schwarzschild radius corresponding to the mass M. Given the strongly nonlinear nature of this system, that two completely disconnected metric solutions exist for the same matter source is an unavoidable logical possibility.9 The consequences of such an exotic phenomenology are still to be fully revealed. 9.4. Ghosts and instability There has been a suggestion that the scalar, brane-fluctuation mode is a ghost in the self-accelerating phase [7,12]. Ghosts are troublesome because, having a negative kinetic term, high momentum modes are extremely unstable leading to the rapid devolution of the vacuum into increasingly short wavelength fluctuations. This would imply that the vacuum is unstable to unacceptable fluctuations, rendering moot all of the work done in DGP phenomenology. The status of this situation is still not resolved. Koyama suggests that, in fact, around the self-accelerating empty space solution itself, the brane fluctuation mode is not a ghost [104], implying that there may be a subtlety involving the self-consistent treatment of perturbations around the background. Nevertheless, the situation may be even more complicated, particularly in a system more complex than a quasistatic deSitter background. 9 Unfortunately, while there exists the possibility that multiple, but disconnected, Schwarzschild-like solutions exist, I do not believe that the
solution given in Refs. [100,101] has the necessary ingredients to be such an alternative solution. A coordinate transformation exists to bring the metric in the form given in these articles to that given by Eq. (4.19). From this point of view, it is clear that the form of the induced metric on the brane is constrained so that N (z = 0) = A−1 (z = 0). Like the solutions discussed in Section 5.4.2, the solution here is thus also completely constrained on the brane, leaving no degree of freedom there to ensure that the metric becomes asymptotically Minkowski (z → ∞) when integrating the bulk Einstein equations from the brane into the bulk. Even using their own coordinate system, one is lead to the same conclusion by counting degrees of freedom available on the brane.
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However, recall that the coupling of the scalar mode to matter is increasingly suppressed at sort scales (or correspondingly, high momentum). Even if the extra scalar mode were a ghost, it is not clear that the normal mechanism for the instability of the vacuum is as rapid here, possibly leaving a phenomenological loophole: with a normal ghost coupled to matter, the vacuum would almost instantaneously dissolve into high momentum matter and ghosts, here because of the suppressed coupling at high momentum, that devolution may proceed much more slowly. A more quantitative analysis is necessary to see if this is a viable scenario.
10. Gravity’s future What is gravity? Our understanding of this fundamental, universal interaction has been a pillar of modern science for centuries. Physics has primarily focused on gravity as a theory incomplete in the ultraviolet regime, at incomprehensibly large energies above the Planck scale. However, there still remains a tantalizing regime where we may challenge Einstein’s theory of gravitation on the largest of scales, a new infrared frontier. This new century’s golden age of cosmology offers an unprecedented opportunity to understand new infrared fundamental physics. And while particle cosmology is the celebration of the intimate connection between the very great and the very small, phenomena on immense scales in of themselves may be an indispensable route to understanding the true nature of our Universe. The braneworld theory of Dvali, Gabadadze and Porrati (DGP) has pioneered this new line of investigation, offering an novel explanation for today’s cosmic acceleration as resulting from the unveiling of an unseen extra dimension at distances comparable today’s Hubble radius. While the theory offers a specific prediction for the detailed expansion history of the universe, which may be tested observationally, it offers a paradigm for nature truly distinct from dark energy explanations of cosmic acceleration, even those that perfectly mimic the same expansion history. DGP braneworld theory alters the gravitational interaction itself, yielding unexpected phenomenological handles beyond just expansion history. Tests from the solar system, large-scale structure, lensing all offer a window into understanding the perplexing nature of the cosmic acceleration and, perhaps, of gravity itself. Understanding the complete nature of the gravitational interaction is the final frontier of theoretical physics and cosmology, and provides an indispensable and tantalizing opportunity to peel back the curtain of the unknown. Regardless of the ultimate explanation, revealing the structure of physics at the largest of scales allows us to peer into its most fundamental structure. What is gravity? It is not a new question, but it is a good time to ask.
Acknowledgements The author wishes to thank C. Deffayet, G. Dvali, G. Gabadadze, M. Porrati, A. Gruzinov, G. Starkman and R. Scoccimarro for their crucial interactions, for their deep insights and for their seminal and otherwise indispensable scientific contribution to this material. References [1] [Supernova Cosmology Project Collaboration], S. Perlmutter, et al., Measurements of Omega and Lambda from 42 High-Redshift Supernovae, Astrophys. J. 517 (1999) 565. [2] [Supernova Search Team Collaboration], A.G. Riess, et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 (1998) 1009. [3] [Supernova Search Team Collaboration], A.G. Riess, et al., Type Ia Supernova Discoveries at z > 1 from the Hubble Space Telescope: evidence for past deceleration and constraints on dark energy evolution, Astrophys. J. 607 (2004) 665. [4] G. Dvali, G. Gabadadze, M. Porrati, 4D gravity on a brane in 5D Minkowski space, Phys. Lett. B 485 (2000) 208. [5] G.R. Dvali, G. Gabadadze, M. Kolanovic, F. Nitti, The power of brane-induced gravity, Phys. Rev. D 64 (2001) 084004 [arXiv:hep-ph/0102216]. [6] G.R. Dvali, G. Gabadadze, M. Kolanovic, F. Nitti, Scales of gravity, Phys. Rev. D 65 (2002) 024031 [arXiv:hep-th/0106058]. [7] M.A. Luty, M. Porrati, R. Rattazzi, Strong interactions and stability in the DGP model, JHEP 0309 (2003) 029 [arXiv:hep-th/0303116]. [8] V.A. Rubakov, Strong coupling in brane-induced gravity in five dimensions, arXiv:hep-th/0303125. [9] M. Porrati, J.W. Rombouts, Strong coupling vs. 4-D locality in induced gravity, Phys. Rev. D 69 (2004) 122003 [arXiv:hep-th/0401211]. [10] G. Dvali, Infrared modification of gravity, arXiv:hep-th/0402130. [11] G. Gabadadze, Weakly-coupled metastable graviton, Phys. Rev. D 70 (2004) 064005 [arXiv:hep-th/0403161]. [12] A. Nicolis, R. Rattazzi, Classical and quantum consistency of the DGP model, JHEP 0406 (2004) 059 [arXiv:hep-th/0404159].
46
A. Lue / Physics Reports 423 (2006) 1 – 48
[13] A.O. Barvinsky, On covariant long-distance modifications of Einstein theory and strong coupling problem, Phys. Rev. D 71 (2005) 084007 [arXiv:hep-th/0501093]. [14] C. Deffayet, J.W. Rombouts, Ghosts, strong coupling and accidental symmetries in massive gravity, arXiv:gr-qc/0505134. [15] E. Kiritsis, N. Tetradis, T.N. Tomaras, Thick branes and 4D gravity, JHEP 0108 (2001) 012 [arXiv:hep-th/0106050]. [16] C. Middleton, G. Siopsis, Fat branes in infinite volume extra space, arXiv:hep-th/0210033. [17] M. Kolanovic, Gravity induced over a smooth soliton, Phys. Rev. D 67 (2003) 106002 [arXiv:hep-th/0301116]. [18] C. Middleton, G. Siopsis, The Schwarzschild solution in the DGP model, Mod. Phys. Lett. A 19 (2004) 2259 [arXiv:hep-th/0311070]. [19] M. Kolanovic, DGP brane as a gravity conductor, Phys. Rev. D 65 (2002) 124005 [arXiv:hep-th/0203136]. [20] H. Collins, B. Holdom, Brane cosmologies without orbifolds, Phys. Rev. D 62 (2000) 105009 [arXiv:hep-ph/0003173]. [21] Y.V. Shtanov, On brane-world cosmology, arXiv:hep-th/0005193. [22] S. Nojiri, S.D. Odintsov, Brane-world cosmology in higher derivative gravity or warped compactification in the next-to-leading order of AdS/CFT correspondence, JHEP 0007 (2000) 049 [arXiv:hep-th/0006232]. [23] C. Deffayet, Cosmology on a brane in Minkowski bulk, Phys. Lett. B 502 (2001) 199 [arXiv:hep-th/0010186]. [24] P. Binetruy, C. Deffayet, D. Langlois, Non-conventional cosmology from a brane-universe, Nucl. Phys. B 565 (2000) 269 [arXiv:hepth/9905012]. [25] T. Shiromizu, K.I. Maeda, M. Sasaki, The Einstein equations on the 3-brane world, Phys. Rev. D 62 (2000) 024012 [arXiv:gr-qc/9910076]. [26] P. Binetruy, C. Deffayet, U. Ellwanger, D. Langlois, Brane cosmological evolution in a bulk with cosmological constant, Phys. Lett. B 477 (2000) 285 [arXiv:hep-th/9910219]. [27] E.E. Flanagan, S.H.H. Tye, I. Wasserman, Cosmological expansion in the Randall–Sundrum brane world scenario, Phys. Rev. D 62 (2000) 044039 [arXiv:hep-ph/9910498]. [28] A. Lue, Global structure of Deffayet (Dvali–Gabadadze–Porrati) cosmologies, Phys. Rev. D 67 (2003) 064004 [arXiv:hep-th/0208169]. [29] N. Deruelle, T. Dolezel, Brane versus shell cosmologies in Einstein and Einstein–Gauss–Bonnet theories, Phys. Rev. D 62 (2000) 103502 [arXiv:gr-qc/0004021]. [30] C. Deffayet, G.R. Dvali, G. Gabadadze, Accelerated universe from gravity leaking to extra dimensions, Phys. Rev. D 65 (2002) 044023 [arXiv:astro-ph/0105068]. [31] C. Deffayet, S.J. Landau, J. Raux, M. Zaldarriaga, P. Astier, Supernovae, CMB, and gravitational leakage into extra dimensions, Phys. Rev. D 66 (2002) 024019 [arXiv:astro-ph/0201164]. [32] V. Sahni, Y. Shtanov, Braneworld models of dark energy, JCAP 0311 (2003) 014 [arXiv:astro-ph/0202346]. [33] U. Alam, V. Sahni, Supernova constraints on braneworld dark energy, arXiv:astro-ph/0209443. [34] A. Lue, G.D. Starkman, How a brane cosmological constant can trick us into thinking that W < − 1, Phys. Rev. D 70 (2004) 101501 [arXiv:astro-ph/0408246]. [35] P.P. Avelino, C.J.A. Martins, Astrophys. J. 565 (2002) 661 [arXiv:astro-ph/0106274]. [36] C. Deffayet, G.R. Dvali, G. Gabadadze, Comments on A Supernova Brane Scan, [arXiv:astro-ph/0106449]. [37] E.V. Linder, Exploring the expansion history of the universe, Phys. Rev. Lett. 90 (2003) 091301 [arXiv:astro-ph/0208512]. [38] O. Elgaroy, T. Multamaki, Cosmic acceleration and extra dimensions: constraints on modifications of the Friedmann equation, Mon. Not. R. Astron. Soc. 356 (2005) 475 [arXiv:astro-ph/0404402]. [39] V. Sahni, Y. Shtanov, A. Viznyuk, Cosmic mimicry: is LCDM a braneworld in disguise?, arXiv:astro-ph/0505004. [40] J.S. Alcaniz, N. Pires, Cosmic acceleration in brane cosmology, Phys. Rev. D 70 (2004) 047303 [arXiv:astro-ph/0404146]. [41] J.S. Alcaniz, D. Jain, A. Dev, Age constraints on brane models of dark energy, Phys. Rev. D 66 (2002) 067301 [arXiv:astro-ph/0206448]. [42] J.S. Alcaniz, Some observational consequences of brane world cosmologies, Phys. Rev. D 65 (2002) 123514 [arXiv:astro-ph/0202492]. [43] Y. Iwasaki, Consistency condition for propagators, Phys. Rev. D 2 (1970) 2255. [44] H. van Dam, M.J.G. Veltman, Massive and massless Yang–Mills and gravitational fields, Nucl. Phys. B 22 (1970) 397; V.I. Zakharov, JETP Lett. 12 (1970) 312 [45] V.I. Zakharov, JETP Lett. 12 (1970) 312. [46] C. Deffayet, G.R. Dvali, G. Gabadadze, A.I. Vainshtein, Nonperturbative continuity in graviton mass versus perturbative discontinuity, Phys. Rev. D 65 (2002) 044026 [arXiv:hep-th/0106001]. [47] A. Lue, Cosmic strings in a brane world theory with metastable gravitons, Phys. Rev. D 66 (2002) 043509 [arXiv:hep-th/0111168]. [48] A. Gruzinov, On the graviton mass, arXiv:astro-ph/0112246. [49] M. Porrati, Fully covariant van Dam–Veltman–Zakharov discontinuity, and absence thereof, Phys. Lett. B 534 (2002) 209 [arXiv:hepth/0203014]. [50] A.I. Vainshtein, To the problem of nonvanishing gravitation mass, Phys. Lett. B 39 (1972) 393. [51] D.G. Boulware, S. Deser, Can gravitation have a finite range?, Phys. Rev. D 6 (1972) 3368. [52] G. Gabadadze, A. Gruzinov, Graviton mass or cosmological constant?, arXiv:hep-th/0312074. [53] A. Vilenkin, Gravitational field of vacuum domain walls and strings, Phys. Rev. D 23 (1981) 852. [54] R. Gregory, Gravitational stability of local strings, Phys. Rev. Lett. 59 (1987) 740. [55] A. Lue, G. Starkman, Gravitational leakage into extra dimensions: probing dark energy using local gravity, Phys. Rev. D 67 (2003) 064002 [arXiv:astro-ph/0212083]. [56] A. Lue, R. Scoccimarro, G.D. Starkman, Probing Newton’s constant on vast scales: DGP gravity, cosmic acceleration and large scale structure, Phys. Rev. D 69 (2004) 124015 [arXiv:astro-ph/0401515]. [57] G. Kofinas, E. Papantonopoulos, I. Pappa, Spherically symmetric braneworld solutions with (4)R term in the bulk, Phys. Rev. D 66 (2002) 104014 [arXiv:hep-th/0112019]. [58] G. Kofinas, E. Papantonopoulos, V. Zamarias, Black hole solutions in braneworlds with induced gravity, Phys. Rev. D 66 (2002) 104028 [arXiv:hep-th/0208207].
A. Lue / Physics Reports 423 (2006) 1 – 48 [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99]
47
Y.S. Song, Looking for an extra dimension with tomographic cosmic shear, Phys. Rev. D 71 (2005) 024026 [arXiv:astro-ph/0407489]. L. Knox, Y.S. Song, J.A. Tyson, Two windows on acceleration and gravitation: dark energy or new gravity?, arXiv:astro-ph/0503644. M. Ishak, A. Upadhye, D.N. Spergel, Is cosmic acceleration a symptom of the breakdown of general relativity?, arXiv:astro-ph/0507184. C. Deffayet, On brane world cosmological perturbations, Phys. Rev. D 66 (2002) 103504 [arXiv:hep-th/0205084]. C. Deffayet, Cosmological perturbations of brane-induced gravity and the vDVZ discontinuity on FLRW space–times, Phys. Rev. D 71 (2005) 103501 [arXiv:gr-qc/0412114]. G. Dvali, A. Gruzinov, M. Zaldarriaga, The accelerated universe and the Moon, Phys. Rev. D 68 (2003) 024012 [arXiv:hep-ph/0212069]. K. Nordtvedt, Improving gravity theory tests with solar system [grand Fits], Phys. Rev. D 61 (2000) 122001. J.G. Williams, X.X. Newhall, J.O. Dickey, Relativity parameters determined from lunar laser ranging, Phys. Rev. D 53 (1996) 6730. A. Milani, D. Vokrouhlicky, D. Villani, C. Bonanno, A. Rossi, Testing general relativity with the BepiColombo radio science experiment, Phys. Rev. D 66 (2002) 082001. B. Bertotti, L. Iess, P. Tortora, A test of general relativity using radio links with the Cassini spacecraft, Nature 425 (2003) 374. E.G. Adelberger, B.R. Heckel, A.E. Nelson, Tests of the gravitational inverse-square law, Ann. Rev. Nucl. Part. Sci. 53 (2003) 77 [arXiv:hepph/0307284]. C.M. Will, N. Yunes, Testing alternative theories of gravity using LISA, Class. Quant. Grav. 21 (2004) 4367 [arXiv:gr-qc/0403100]. S.G. Turyshev, M. Shao, K.L. Nordtvedt, Optical design for the laser astrometric test of relativity, eConf C041213 (2004) 0306 [arXiv:grqc/0502113]. [LATOR Collaboration], S.G. Turyshev et al., Fundamental physics with the laser astrometric test of relativity, arXiv:gr-qc/0506104. L. Iorio, On the effects of the Dvali–Gabadadze–Porrati braneworld gravity on the orbital motion of a test particle, arXiv:gr-qc/0504053. J.D. Anderson, P.A. Laing, E.L. Lau, A.S. Liu, M.M. Nieto, S.G. Turyshev, Study of the anomalous acceleration of Pioneer 10 and 11, Phys. Rev. D 65 (2002) 082004 [arXiv:gr-qc/0104064]. C.M. Will, The confrontation between general relativity and experiment, Living Rev. Rel. 4 (2001) 4 [arXiv:gr-qc/0103036]. [WMAP Collaboration], D.N. Spergel, et al., First year Wilkinson microwave anisotropy probe observations: determination of cosmological parameters, Astrophys. J. Suppl. 148 (2003) 175 [arXiv:astro-ph/0302209]. G. Hinshaw, et al., First year Wilkinson microwave anisotropy probe (WMAP) observations: angular power spectrum, Astrophys. J. Suppl. 148 (2003) 135 [arXiv:astro-ph/0302217]. [SDSS Collaboration], M. Tegmark, et al., The 3D power spectrum of galaxies from the SDSS, Astrophys. J. 606 (2004) 702 [arXiv:astroph/0310725]. L. Verde, et al., The 2dF Galaxy Redshift Survey: the bias of galaxies and the density of the Universe, Mon. Not. R. Astron. Soc. 335 (2002) 432 [arXiv:astro-ph/0112161]. [The 2dFGRS Collaboration], P. Norberg, et al., The 2dF Galaxy Redshift Survey: luminosity dependence of galaxy clustering, Mon. Not. R. Astron. Soc. 328 (2001) 64 [arXiv:astro-ph/0105500]. E.V. Linder, Probing gravitation, dark energy, and acceleration, Phys. Rev. D 70 (2004) 023511 [arXiv:astro-ph/0402503]. C. Sealfon, L. Verde, R. Jimenez, Limits on deviations from the inverse-square law on megaparsec scales, Phys. Rev. D 71 (2005) 083004 [arXiv:astro-ph/0404111]. F. Bernardeau, Constraints on higher-dimensional gravity from the cosmic shear three-point correlation function, arXiv:astro-ph/0409224. E.V. Linder, Cosmic growth history and expansion history, arXiv:astro-ph/0507263. T. Tanaka, Weak gravity in DGP braneworld model, Phys. Rev. D 69 (2004) 024001 [arXiv:gr-qc/0305031]. A. Lue, R. Scoccimarro, G. Starkman, Differentiating between modified gravity and dark energy, Phys. Rev. D 69 (2004) 044005 [arXiv:astroph/0307034]. P. Fosalba, E. Gaztanaga, F. Castander, Detection of the ISW and SZ effects from the CMB-Galaxy correlation, Astrophys. J. 597 (2003) L89 [arXiv:astro-ph/0307249]. [SDSS Collaboration], R. Scranton et al., Physical evidence for dark energy, arXiv:astro-ph/0307335. C. Deffayet, A note on the wellposedness of scalar brane world cosmological perturbations, Phys. Rev. D 71 (2005) 023520 [arXiv:hepth/0409302]. D. Jain, A. Dev, J.S. Alcaniz, Brane world cosmologies and statistical properties of gravitational lenses, Phys. Rev. D 66 (2002) 083511 [arXiv:astro-ph/0206224]. J.A.S. Lima, J.V. Cunha, J.S. Alcaniz, Constraining the dark energy with galaxy clusters X-ray data, Phys. Rev. D 68 (2003) 023510 [arXiv:astroph/0303388]. H.J. Seo, D.J. Eisenstein, Probing dark energy with baryonic acoustic oscillations from future large galaxy redshift surveys, Astrophys. J. 598 (2003) 720 [arXiv:astro-ph/0307460]. Z.H. Zhu, Accelerating universe from gravitational leakage into extra dimensions: confrontation with SNeIa, Astrophys. J. 620 (2005) 7 [arXiv:astro-ph/0404201]. J.P. Uzan, N. Aghanim, Y. Mellier, The distance duality relation from X-ray and SZ observations of clusters, Phys. Rev. D 70 (2004) 083533 [arXiv:astro-ph/0405620]. S. Capozziello, V.F. Cardone, M. Funaro, S. Andreon, Constraining dark energy models using the lookback time to galaxy clusters and the age of the universe, Phys. Rev. D 70 (2004) 123501 [arXiv:astro-ph/0410268]. J.S. Alcaniz, Z.H. Zhu, Complementary constraints on brane cosmology, Phys. Rev. D 71 (2005) 083513 [arXiv:astro-ph/0411604]. E. Papantonopoulos, V. Zamarias, Chaotic inflation on the brane with induced gravity, JCAP 0410 (2004) 001 [arXiv:gr-qc/0403090]. H.S. Zhang, R.G. Cai, Inflation on Dvali–Gabadadze–Porrati brane, JCAP 0408 (2004) 017 [arXiv:hep-th/0403234]. M. Bouhmadi-Lopez, R. Maartens, D. Wands, Gravitational waves from brane-world inflation with induced gravity, Phys. Rev. D 70 (2004) 123519 [arXiv:hep-th/0407162].
48 [100] [101] [102] [103]
A. Lue / Physics Reports 423 (2006) 1 – 48
G. Gabadadze, A. Iglesias, Schwarzschild solution in brane induced gravity, arXiv:hep-th/0407049. G. Gabadadze, A. Iglesias, Short distance non-perturbative effects of large distance modified gravity, arXiv:hep-th/0508201. N. Kaloper, Brane-induced gravity’s shocks, Phys. Rev. Lett. 94 (2005) 181601 [arXiv:hep-th/0501028]. N. Kaloper, Gravitational shock waves and their scattering in brane-induced gravity, Phys. Rev. D 71 (2005) 086003 [Erratum-ibid. D 71, 129905 (2005)] [arXiv:hep-th/0502035]. [104] K. Koyama, Are there ghosts in the self-accelerating brane universe?, arXiv:hep-th/0503191.
Physics Reports 423 (2006) 49 – 89 www.elsevier.com/locate/physrep
Pentaquark +(1540) production in N → KKN reactions Yongseok Oha , K. Nakayamaa , T.-S.H. Leeb,∗ a Department of Physics and Astronomy, University of Georgia, Athens, GA 30602, USA b Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
Accepted 14 October 2005 Available online 28 November 2005 editor: G.E. Brown
Abstract Recent developments in the search of exotic pentaquark hadrons are briefly reviewed. We then focus on investigating how the exotic pentaquark (1540) baryon production can be identified in the N → KKN reactions, focusing on the influence of the background (non- production) mechanisms. By imposing the SU(3) symmetry and using various quark model predictions, we are able to fix the coupling constants for evaluating the kaon backgrounds, the KK production through the intermediate vector meson and tensor meson photoproduction, and the mechanisms involving intermediate (1116), (1405), (1520), (1193), (1385), and (1232) states. The vector meson photoproduction part is calculated from a phenomenological model which describes well the experimental data at low energies. We point out that the neutral tensor meson production cannot be due to 0 -exchange as done by Dzierba et al. [Phys. Rev. D 69 (2004) 051901] because of C parity. The neutral tensor meson production is estimated by considering the vector meson exchange and found to be too weak to generate any peak at the position near (1540). For (1540) production, 0 we assume that it is an isoscalar and hence can only be produced in n → K + K − n and p → K 0 K p reactions, but not in 0 p → K + K − p and n → K 0 K n. The total cross section data of p → K + K − p is thus used to fix the form factors which 0 regularize the background amplitudes so that the signal of (1540) in n → K + K − n and p → K 0 K p cross sections can be + − + + − predicted. We find that the predicted K K and K n invariant mass distributions of the n → K K n reaction can qualitatively reproduce the shapes of the JLab data. However, the predicted (1540) peak cannot be identified unambiguously with the data. High statistics experiments are needed to resolve the problem. We also find that an even-parity is more likely to be detected, while it will be difficult to identify an odd-parity , even if it exists, from the background continuum, if its coupling constants are small as in the present quark model predictions. © 2005 Elsevier B.V. All rights reserved. PACS: 13.60.Le; 13.60.Rj; 14.80.−j
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. KK pair photoproduction from the nucleon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. t-channel Drell-type diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Vector meson background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Tensor meson background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗ Corresponding author.
E-mail addresses:
[email protected] (Y. Oh),
[email protected] (K. Nakayama),
[email protected] (T.-S.H. Lee). 0370-1573/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2005.10.002
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2.3.1. Charged tensor meson photoproduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Neutral tensor meson photoproduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Hyperon background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Pentaquark contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Total cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Invariant mass distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Tensor meson photoproduction contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Double differential cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Tensor mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Interactions with pseudoscalar mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. Tensor meson radiative decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Effective Lagrangians for the hyperon backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1. Baryon octet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2. Baryon decuplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3. (1405) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4. (1520) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C. Form factors and current conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1. t-channel Drell-type diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2. Vector meson and tensor meson parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3. Intermediate hyperons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3.1. Intermediate spin-1/2 hyperons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3.2. Intermediate spin-3/2 hyperons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4. Intermediate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59 60 61 63 64 64 65 67 70 72 73 73 73 75 76 76 77 80 81 82 82 83 83 83 84 85 85
1. Introduction The recent interests in pentaquark baryons was initiated by the discovery of + (1540) by the LEPS Collaboration at SPring-8 [1] and the subsequent experiments [2–14]. Candidates for exotic pentaquark states (1862) and c (3099) were also observed [15,16]. However, the signals for those exotic states could not be found in several recent experiments [17–30]. In addition, more reports have been given on possible candidates for various crypto-exotic baryons [31–37]. The existence of pentaquark baryons is thus not conclusive at the present time.1 In spite of this situation, it seems worthwhile at this stage to collect together in one place the various references to the literature with a brief review of major experimental and theoretical works in this rapidly expanding area of investigation. The results from recent experiments are summarized in Tables 1 and 2. We see that the positive (Table 1) and negative (Table 2) reports on the existence of + (1540) are almost equally divided.2 In Fig. 1, the peak of the spectrum at the missing mass ∼ 1.54 GeV was identified by the LEPS group with the excitation of + (1540) in the n → K + K − n process taking place in the 12 C target. On the other hand, the spectrum obtained from the e+ e− → pK 0 X experiment at BaBar [21] does not show any resonance peak at the predicted position of + (1540) as seen in Fig. 2. It is useful to first briefly review all of the theoretical works on pentaquark baryons which could be pure exotic or crypto-exotic. The pure exotic states can easily be identified by their unique quantum numbers, but the crypto-exotic states are hard to be identified as their quantum numbers can also be generated by three-quark states. Therefore, it is crucial to have careful analyses for their decay channels and other properties. Historically, there have been many efforts to find pentaquark states with the development of quark models, which, however, failed to observe (1540).
1 The most recent experimental report from the STAR Collaboration [38] would suggest the existence of a narrow with isospin I = 0 at a mass 1528 ± 2 ± 5 MeV. 2 For the negative reports on the existence of (1860) and (3099), see, e.g., Refs. [39–41] and the references in Table 2. c
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
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Table 1 Positive reports on the evidence of + (1540) pentaquark state Expt.
Reaction
Reference
LEPS DIANA CLAS SAPHIR BBCN CLAS HERMES SVD COSY-TOF ZEUS JINR SVD
+ 12 C K + + Xe
[1] [2] [3] [4] [6] [7] [9] [10] [12] [13] [11] [14]
(2003) (2003) (2003) (2003) (2003) (2003) (2003) (2004) (2004) (2004) (2004) (2005)
+d +p ( ) + A +p
e+d p+A p+p e± + p p+A p+A
Table 2 Negative reports on the evidence of + (1540) pentaquark state Expt.
Reaction
Reference
BES PHENIX ALEPH SPHINX BaBar CDF HERA-B HyperCP Belle FOCUS CLAS WA89
e+ + e−
[17] [18] [19] [20] [21] [24] [25] [26] [27] [28] [29] [30]
(2004) (2004) (2004) (2004) (2004) (2004) (2004) (2004) (2004) (2004) (2005) (2005)
A+A e+ + e− p+A e+ + e− p + p¯ p+A p+A e+ + e− +A +p − + A
The efforts to search for pentaquark baryons until 1980s were summarized in Refs. [42,43]. (See also Ref. [44].)3 Early theoretical works on exotic baryons can be found, e.g., in Refs. [52–57]. Rigorous theoretical studies were then performed for heavy quark sector, i.e., pentaquark baryons with one anticharmed quark or anti-bottom quark. In the pioneering work of Lipkin [58] and Grenoble group [59], the anti-charmed pentaquark with one strange quark was shown to have the same binding energy as the H dibaryon in the heavy quark mass limit and in the SU(3) limit. Then it has been studied in more sophisticated quark models [60–63], which improved the simple prediction of Refs. [58,59] and some of them predicted no bound state. The heavy pentaquark systems are also investigated using the Skyrme model, which gives different results compared with the quark model. In this approach, the bound-state model of Callan and Klebanov [64] was applied to study the heavy pentaquark system after it was shown that the model can be successfully applied to the normal heavy-quark baryons [65,66]. In Ref. [67], Riska and Scoccola used the Skyrme Lagrangian with symmetry breaking terms to investigate the heavy pentaquark system and found that some of the nonstrange heavy pentaquarks can be deeply bound and therefore stable against the decays by strong interaction. This is quite a remarkable result compared with the quark model where a nonstrange pentaquark 3 In the literature we could find several resonances that were claimed to be crypto-exotic states. For example, X(1340), X(1450), and X(1640) were reported by Refs. [45,46] and X(3520) by Ref. [47]. X(1390), X(1480), and X(1620) that have isospin I 5/2 were observed by Ref. [48], and Ref. [49] reported (3170). Most of them were found to have narrow widths, but their existence was not confirmed and questioned by later experiments [50,51]. SPHINX Collaboration has reported the existence of X(2000), X(2050), and X(2400) that are expected to have the quark content of uuds s¯ [31,32], whose existence should be carefully re-examined by other experiments.
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Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
Fig. 1. Data of missing mass spectrum of 12 C(K + K − ) reaction from the experiment at LEPS [1].
22000 20000
Events /2 MeVc -2
18000 16000 14000
(1540)
12000 10000 8000 6000 4000 2000 0 1.4
1.45
1.5 0 S
1.55
1.6
2
(pK ) Mass [GeV/c ] Fig. 2. Data from e+ e− → pK 0 X experiment at BaBar [21].
baryon has no sufficient symmetry to be stable via the hyperfine interactions. However, this model does not satisfy the heavy quark symmetry at the infinite heavy quark mass limit [68–71] by integrating out the heavy vector meson field. It was shown that including the heavy vector meson fields explicitly is essential for satisfying heavy quark symmetry [72], and the model was successfully applied to heavy quark baryons [73–81]. This model was then used to study the heavy pentaquarks in Refs. [78,82], which gives stable nonstrange pentaquark baryons, although the binding energy and the mass formulas are quite different from those of Ref. [67]. The finite mass corrections and the soliton-recoil effects are discussed in Refs. [79,81]. The extension to strange heavy pentaquarks can be found in Ref. [83]. Following the first experimental search for heavy pentaquarks [84,85], the observation of + (1540) and c (3099) has brought new interests in the heavy pentaquarks [86–92]. In the light quark sector, pentaquark states were anticipated already in the Skyrme model [93–96]. The first detailed study on anti-decuplet was made by Diakonov et al. [97,98], which predicted a very narrow + with a mass around 1530 MeV by identifying N (1710) as the nucleon analogue of the anti-decuplet. After the discovery of + (1540) there
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
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have been lots of theoretical models and ideas to explain the structure of pentaquark baryons and to search for the other pentaquark states. The subsequent theoretical studies include the soliton models [99–107], QCD sum rules [108–111], large Nc QCD [112–115], and lattice calculation [116–119], etc. As the quark models have provided a cornerstone for hadron physics, it is legitimate to start with the quark models and study the structure of pentaquark baryons. In Ref. [120], Karliner and Lipkin suggested a triquark–diquark model, where, for example, + is a system of (ud)-(ud s¯ ). In Ref. [121], Jaffe and Wilczek advocated a diquark–diquark–antiquark model so that + is (ud)-(ud)-¯s . In this model, they also considered the mixing of the pentaquark anti-decuplet with the pentaquark octet, which makes it different from the SU(3) soliton models where the octet describes the normal (three-quark) baryon octet. Assuming that the nucleon and analogues are in the ideal mixing of the octet and anti-decuplet, the nucleon analogue is then identified as the Roper resonance N (1440). In Ref. [122], however, it was pointed out that the N (1710) should be excluded as a pure anti-decuplet state. This is because, within SU(3) symmetry, anti-decuplet does not couple to decuplet and meson octet, whereas N (1710) has a large branching ratio into channel. Therefore, mixing with other multiplets is required if one wants to identify N (1710) as a pentaquark crypto-exotic state. However, recent study for the ideal mixing between anti-decuplet and octet states shows that the ideally mixed state still has vanishing coupling with the channel [123,124], which excludes N (1440) as a pentaquark state. This shows the importance of reaction/decay studies in identifying especially crypto-exotic pentaquark states. More discussions on the quark model predictions based on the diquark picture can be found, e.g., in Refs. [122,125]. Predictions on the anti-decuplet spectrum in various quark models can be found, e.g., in Refs. [126–131] In quark model, pentaquark baryons form six multiplets, 1, 8, 10, 10, 27 and 35. The other type resonances are thus expected together with anti-decuplet, particularly the isovector belonging to 27-plet and isotensor as a member of 35-plet. The interest in this direction has been growing [126,132–135] and it is important to know the interactions and decay channels to search for the other pentaquark baryons if they exist. Furthermore, understanding the + properties such as spin-parity requires careful analyses of production ¯ [143–148], N → K¯ ∗ [149], N → K [150] N → K + K − N [151–156], mechanisms4 including N → K NN → Y [138,139,142,157], and KN → KN [158]. Most model predictions for those production processes, however, do not consider the intermediate pentaquark baryons in its production mechanisms as the unknown inputs like the electromagnetic and strong couplings of pentaquark baryons are required. Therefore, knowing the interaction Lagrangian of pentaquark baryons are necessary for understanding the production mechanisms. The physical pentaquark states would be mixtures of various multiplets as in the chiral soliton model [134]. Such a representation mixing is induced by SU(3) symmetry breaking and it can be studied in quark potential models. Therefore, it is desirable to obtain the full set of pentaquark wave functions in quark model for further investigation. There are several works in this direction and the flavor wave functions of anti-decuplet has been obtained in Refs. [123,128,129,159]. (See also Refs. [113–115] for the relation between the wave functions of pentaquark baryons in quark model and Skyrme model in the large Nc limit.) The SU(3) symmetric interactions for anti-decuplet have been studied in Refs. [122–124], which motivated the development of a chiral Lagrangian for anti-decuplet [160,161]. In a recent paper [162] the SU(3) quark model has been extended to obtain the flavor wave functions of all pentaquark states including singlet, octet, decuplet, anti-decuplet, 27-plet and 35-plet. Thus the SU(3) symmetric Lagrangian describing the pentaquark–three-quark and pentaquark–pentaquark interactions with meson octet has been obtained. From the above brief descriptions of theoretical works on pentaquark baryons, it is clear that the verification of the existence of + (1540) is a very important step in the development of hadron physics. To make progress, both experimental and theoretical efforts are needed. Experimentally, several high statistics experiments have been planned [163–167] and some data have been obtained and are under analyses. On the theoretical side, it is necessary to understand the reaction mechanisms of the considered reactions and to investigate how the pentaquark states production can be identified from the experimental observables. In particular, one must explore whether the resonance-like peaks near W ∼ 1540 MeV can be resulted from the background (non- production) mechanisms, as emphasized by Dzierba et al. [152]. In this paper, we report on the progress we have made in this direction concerning the N → KKN reactions. The N → KKN reaction has been used to observe the production of (1540) through its decay into KN [1,3,4,9,13,25].5 We will take the effective Lagrangian approach to assume that the amplitudes of this reaction can 4 See, however, Refs. [136–142], where efforts to determine these properties, in particular the parity of the + , in a model-independent way in photo- and hadro-production reactions have been reported. 5 Old experiments to find the pentaquark states using this reaction can be found, e.g., in Ref. [168].
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Table 3 Comparison of the models for N → KKN as a background for the (1540) production in the literature Models NT [151] DKSTS [152] Roberts [153] TEHN [155] This work
t-channel Drell diagrams √ √ √
Vector meson background √ √ √ √ √
Tensor meson background √ √ √
Hyperon background √ √ √
See the text for the details.
be computed from the tree-diagrams. In addition to the production amplitude, we compute all possible background (non- production) amplitudes whose parameters can be fixed by the SU(3) symmetry or taken from various quark model predictions. Undoubtedly, there are some tree-diagrams which are kinematically allowed, but cannot be evaluated because of the lack of information about the relevant coupling constants. So our effort represents just a step toward developing a model for more realistic production amplitudes. But it should be sufficient for exploring the question concerning how the pentaquark (1540) can be identified from the observables of N → KKN reaction. We classify the possible tree diagrams into four types: the (tree) t-channel Drell-type diagrams (Fig. 3), vector meson and tensor meson production background (Fig. 4), hyperon background (Fig. 7), and the (1540) production amplitude (Fig. 8).6 Our objective is to explore the interplay between these tree diagrams in determining the following reactions: p → K + K − p, 0
p → K 0 K p,
n → K + K − n , 0
n → K 0 K n .
(1)
The parameters as well as the form factors for evaluating these diagrams will be explained in detail in Section 2 and in Appendices. There exist several investigations of the reactions listed in Eq. (1) in conjunction with the observation of (1540) [151–153,155]. (See also Refs. [172,173].) The background tree-diagrams considered by these works are summarized in Table 3. In the very first investigation of n → K + K − n by Nakayama and Tsushima (NT) [151], the t-channel Drelltype diagrams, vector meson background, and hyperon background are considered. But the tensor meson background was not included. Their hyperon background includes the (1193) and (1660) states, while the baryon resonances do not come into play in the considered n → K + K − n reaction because of the isospin selection rule. The investigation of the n → K + K − n reaction by Dzierba et al. (DKSTS) [152] was motivated by an old experiment [174] on − p → K − X reaction which was aimed at searching for the baryon(s). With 8 GeV pion beams, this experiment found two peaks at W = 1590 and 1950 MeV in the KN channel. However, these peak positions moved to 1500 and 1800 MeV as the pion beam energy was lowered to 6 GeV. They hence concluded that the most natural explanation was to ascribe the peaks to the background production mechanisms, especially to the higher-spin meson production. Motivated by this analysis, Dzierba et al. claimed the possibility that the peak at 1540 MeV in n → K + K − n may come from the tensor meson photoproduction background. However, other production mechanisms such as t-channel Drell diagrams and hyperon backgrounds were not included in their investigation, as indicated in Table 3. We will discuss this work in more detail later. In Ref. [153], Roberts studied the contributions of to the invariant mass distributions of N → KKN reactions listed in Eq. (1). In addition to including many and baryons, he included vector meson production, but neglected the tensor meson production and the photo-transitions in the intermediate baryon states. He also explored how the ± ± N → KKN observables can be used to distinguish the spin-parity of (1540) by considering J P = 21 or 23 for (1540). However, a common form factor was used for simplicity and as a result the amplitudes are not constrained by experimental data, which should be improved for more realistic models. 6 The full amplitudes of the diagrams of Fig. 3 together with the hyperon and production diagrams constitute the so-called Drell diagrams ∗ [169], where the incoming photon is converted into a KK pair or a KK (KK ∗ ) pair and then a virtual K (or K ∗ ) diffractively scatters on the nucleon [170,171].
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
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Titov et al. (TEHN) [155] investigated the effects of the scalar meson, vector meson, and tensor meson production background on the N → KKN reactions. They found that the meson photoproduction due to the Pomeronexchange is the major background mechanism. This result was obtained from using a phenomenological model which includes the meson exchanges and Pomeron exchange and can describe well the photoproduction data. The main emphasis in Ref. [155] was to explore the sensitivity and/or insensitivity of various spin asymmetries to the parity of by improving on the work of Ref. [151]. However, the t-channel Drell diagrams and hyperon background were not included in this investigation. In addition, this work focused only on a special kinematic region, i.e., at the resonance point, so the invariant mass distribution, which is required to verify the existence of the state, could not be discussed. As indicated in the last row of Table 3, we consider in this work all four classes of the background amplitudes. Therefore, the present work represents an extension of the work initiated by Nakayama and Tsushima [151] toward developing a more realistic model of the N → KKN reactions in order to extract relevant information concerning the pentaquark . The main challenge here is to find the coupling constants which are needed to evaluate all possible tree diagrams. Undoubtedly, no progress can be made unless some truncations and approximations are taken. We impose the SU(3) symmetry and only keep the tree diagrams whose coupling constants can be either determined from using the information given by the Particle Data Group (PDG) or taken from various quark model predictions. Thus, we only consider the rather well-known hyperons, namely, (1116), (1405), (1520), (1193), and (1385) to evaluate the hyperon background (Fig. 7). The photo-transitions among those hyperons and the → N transition are also included. For the background amplitudes with vector meson and tensor meson production (Fig. 4), the N → (V , T )N amplitudes are generated from the available phenomenological models which are constrained by the total cross section data of photoproduction of vector mesons ( , , ) and tensor meson [a2+ (1320)]. We also improve the model for neutral tensor meson photoproduction of Ref. [155] by including all possible vector meson exchanges. Another feature of our approach is regularizing each vertex in the considered tree diagrams by a form factor which depends on the mass and squared momentum of the exchanged particles. Motivated by the methods of Refs. [177–180], the current conservation is recovered by introducing contact diagrams. The details can be found in Appendix B. The cutoff parameters of the form factors are determined by the available data of the p → K + K − p reaction, which are then used to compute various observables. Thus our procedure in introducing the form factors is different from all the models listed in Table 3, such as no form factor in the work of Ref. [151] and the use of the same form factor for all diagrams in the approach of Ref. [153]. We are also motivated by the question concerning the quantum numbers of (1540). In Ref. [4], SAPHIR Collaboration claims nonexistence of (1540) in K + p channel while they could confirm the peak of (1540) in K + n channel. This leads to the conclusion that the observed (1540) is isosinglet and it belongs to baryon anti-decuplet.7 However, the spin-parity quantum numbers of are still under debate. Theoretically, uncorrelated quark models [128], QCD sum rules [108,110], lattice QCD [116–118] favors odd-parity of (1540), while even-parity is predicted by correlated quark models [121,120,129] and soliton models [97]. In addition, there is a debate on the parity of in lattice calculation [119] and the parity-flip was claimed in QCD sum rules [92] with heavier anti-quark. (See also Refs. [181–184] for the status of QCD sum rules calculation.) However, most models identify the (1540) as a member of the anti-decuplet with spin-1/2. We therefore will use this assumption in this work. To be more specific, we follow the observations of Ref. [4] to assume that is of isosinglet and hence can only be produced via the mechanisms of Fig. 8 in 0 n → K + K − n and p → K 0 K p of Eq. (1). The other two processes in Eq. (1) will also be considered for providing information to constrain the background (non- production) amplitude by using the available data. We will also predict how the observables of n → K + K − n depend on the parity of (1540), as done in Refs. [137–139,143–149,157,158] for the other reactions. This paper is organized as follows. In Section 2, we present our model for KK pair photoproduction reactions listed in Eq. (1). The form of the employed effective Lagrangians is given explicitly and the determination of their coupling constants is also discussed in detail. We present our results on the total and differential cross sections in Section 3. A comment on the spin asymmetries is also made. A summary is given in Section 4 and the details for the tensor meson properties, effective Lagrangians with the couplings, and form factors are given in Appendixes.
7 Flavor SU(3) symmetry allows three kinds of baryons in quark models; isosinglet in anti-decuplet, isovector in 27-plet, and isotensor in 35-plet [162].
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2. KK pair photoproduction from the nucleon As mentioned in Section 1, we will investigate all four reactions listed in Eq. (1) by considering the tree-diagrams illustrated in Fig. 3 for the (tree) t-channel Drell mechanisms, Fig. 4 for the production through vector meson and tensor meson photoproduction, Fig. 7 for the hyperon background, and Fig. 8 for the production. The effective Lagrangians needed for calculating each class of these tree diagrams will be given explicitly in the following subsections and Appendixes. The determinations of the relevant coupling constants will be discussed in detail. 2.1. t-channel Drell-type diagrams ∗
For the t-channel Drell-type diagrams (Fig. 3), the incoming photon is converted into a KK pair or a KK (KK ∗ ) pair and then a virtual K (or K ∗ ) diffractively scatters on the nucleon [169–171], which gives a nonresonant background. The diagrams include the full t-channel scattering amplitudes for the KN → KN and K ∗ N → KN scattering, but here we consider the tree diagrams of one-meson exchange only. For the intermediate KK pair, we consider the vector meson exchanges. The effective Lagrangians for defining these amplitudes are LKK = −ieA (K − j K + − j K − K + ) ,
LV KK = −igV KK (j KV K − KV j K) ,
V LV NN = −gV NN N V − j V N , 2MN LV KK = −eg V KK K[(1 + 3 )/2, V ]+ KA ,
(2)
where [A, B]+ = AB + BA, V = ( , , ), A is the photon field, and the kaon isodoublets are defined as + 0 K K= , K = (K − , K ) . K0
(3)
We follow Ref. [185] to set8 g NN = 3.1, = 2.0 , g NN = 10.3, = 0.0 .
(4)
The exchange of meson is neglected by taking the simplest OZI rule prediction, g NN = 0. We next need to define the vertices connecting the kaons and vector mesons in Fig. 3. This is done by using the following SU(3) symmetric Lagrangian, igV P P LV P P = − √ Tr{V (P j P − j P P )} , 2 LV V P = gV V P ε Tr(j V j V P ) , γ
γ
K K, K*
(5)
*
K
K, K
M N (a)
K
γ
K
K
M N′
N (b)
K
M N′
N (c)
N′
Fig. 3. t-channel Drell-type diagrams for N → KKN . Here M stands for a vector meson (V = , ) or a pseudoscalar meson ( = , ).
8 The Bonn potential gives ≈ 6.0 [186], which is larger than the value in Eq. (4). We found that our results on the N → KKN reaction are not sensitive to this coupling constant.
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
where
P=
√1 + √1 · 6 2
K
,
− √2
K
V =
√1
2
6
+ K
√1 · 2 ∗
K∗ −
57
.
(6)
The K ∗ isodoublets are defined by the same way as in Eq. (3). The SU(3) symmetry relations lead to g KK = g KK = g /2 = 3.02 .
(7)
∗
For the intermediate KK or KK ∗ pair in Fig. 3, we consider the pseudoscalar meson exchanges. The photon coupling is defined by LK ∗ K = g0K ∗ K j A (j K∗0 K + j K∗0 K ∗0 ) 0
−
+ gcK ∗ K j A (j K∗− K + + j K∗+ K ) ,
(8)
with g0K ∗ K = −0.388 GeV−1 ,
gcK ∗ K = 0.254 GeV−1 ,
(9)
determined from the radiative decay widths of the neutral and charged K ∗ vector mesons. The couplings involving the pseudoscalar mesons and are defined by ∗
LK ∗ K = −igK ∗ K (Kj K ∗ − K j K) , gNN N 5 j N , LNN = 2MN
(10)
where = · , and N = (p, n)T . We choose the usual gNN = 13.4 and use the SU(3) relation to set gNN = 3.54. 2 2 3 By using the experimental value for (K ∗ → K) and (K ∗ → K) = (gK ∗ K /8MK ∗ )p , we use gK ∗ K = 6.56 .
(11)
This value is close to the SU(3) value, gK ∗ K = g = 6.04. For gK ∗ K , we use the SU(3) relation, √ gK ∗ K = 3gK ∗ K = 11.36 .
(12)
∗
In the diagrams of Fig. 3, the KK or KK ∗ intermediate states can also interact with the nucleon via vector meson exchanges. This can be calculated from the Lagrangian derived from LV V P of Eq. (5), LK ∗ KV = gK ∗ KV ε Kj V j K∗ + H.c. ,
(13)
where gK ∗ K = gK ∗ K = g /2,
√ gK ∗ K = g / 2 .
(14)
The above coupling constants can then be fixed by using the hidden gauge approach [187] to set g =
Nc g 2 82 f
= 14.9 GeV−1 ,
(15)
where Nc = 3, g = g , and f = 93 MeV. 2.2. Vector meson background As shown in Fig. 4, the vector meson background amplitude is determined by a vector meson photoproduction amplitude and the V → KK vertex function defined by LV KK of Eq. (2). The resulting N → KKN amplitude
58
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
K γ
V, T
K N′
N
Fig. 4. Vector and tensor meson photoproduction contributions to N → KKN . Here V stands for a vector meson (V = , , ) and T for a tensor meson (T = a2 , f2 ).
can be written as M = uN (p )M ε uN (p) ,
(16)
where uN (p) is the Dirac spinor of a nucleon with four-momentum p, ε is the photon polarization vector, p and p are the four-momenta of the initial and final nucleon, respectively. The dynamics of Eq. (16) is contained in the following invariant amplitude: M = M (N → V N)
gV KK (q1 + q2 )
2
− MV2
(i)
(q1 − q2 ) FV ,
(17)
(i)
where q1 and q2 are the momenta of the outgoing K and K, FV depends on the channel quantum number i as well as a form factor (28) which takes into account the off-shellness of the intermediate vector meson. The needed coupling constants gV KK with V = , have been given in Eq. (7). For decay, we here use g KK = −4.49 deduced from the experimental data of ( → K + K − ) [188], while the phase is taken from the SU(3) symmetry. The photoproduction amplitude M (N → V N) is generated from a phenomenological model within which the blob in Fig. 4 includes Pomeron exchange, , f2 , and other meson exchanges, and the direct and crossed nucleon terms. The details of this model can be found in Refs. [189–192] and will not be repeated here. But it should be mentioned that those models describe well the experimental data for vector meson photoproduction. The finite decay width of vector mesons is included by replacing MV by MV − iV /2 in Eq. (17). 2.3. Tensor meson background The tensor meson photoproduction contribution to N → KKN is particularly interesting since it was suggested in Ref. [152] that this mechanism can generate a peak near 1540 MeV in the KN invariant mass distribution of the n → K + K − n reaction and hence the discovery of (1540) pentaquark baryon is questionable. In this subsection, we explore this mechanism in more detail. As illustrated in Fig. 4, the tensor meson photoproduction contribution is very similar to the vector meson photoproduction contribution. Its contribution to the N → KKN amplitude is of the same structure of Eqs. (16)–(17) and can be written as M = uN (p )M ε uN (p) .
(18)
The main dynamics of Eq. (18) is contained in M = M, (N → T N)
P; 2GT KK (i) q q F , MT (q1 + q2 )2 − MT2 1 2 T
(19)
(i)
where q1 and q2 are the momenta of the outgoing K and K. FT includes the constant depending on the channel i and the form factor. Here the T → KK decay vertex is defined by the following tensor structure associated with a spin J = 2 particle, whose propagator contains P ; = 21 (g g + g g ) − 13 g g ,
(20)
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
59
with g = −g +
1 p p , MT2
(21)
where p is the momentum of the tensor meson. The coupling constant GT KK in Eq. (19) is defined by the Lagrangian LT KK = −
2GT KK j KT j K , MT
(22)
where the kaon isodoublets are defined in Eq. (3) and T = f or · a , with f and a denoting the isoscalar f2 (1275) and isovector a2 (1320) tensor meson field, respectively. Some details about LT KK can be found in Appendix A. Using the experimental data [188], (f2 → KK)expt. ≈ 8.6 MeV and (a2 → KK)expt. ≈ 5.24 MeV, Eq. (22) leads to Gf KK = 7.15,
GaKK = 4.89 .
(23)
In this work, we neglect the contribution from f2 (1525) meson since it can be produced favorably only at energies much higher than the region considered in this work. The finite decay width of tensor mesons is included by replacing MT by MT − iT /2 in its propagator. We now turn to discussing the calculations of the tensor meson photoproduction amplitude M, of Eq. (19). We note here that the tensor meson photoproduction mechanisms depend very much on the charge of the produced tensor mesons. In particular, the one-pion exchange is known [193–196] to be the dominant mechanism for charged tensor meson production, while it is not allowed in neutral tensor meson production because of C parity. Thus the claim made by Dzierba et al. [152] concerning the peak at W ∼ 1540 MeV generated by neutral tensor meson production must be re-examined. This will be our focus by exploring the contributions from the vector meson exchange mechanisms. 2.3.1. Charged tensor meson photoproduction We first calculate the charged tensor meson photoproduction, p → a2+ (1320)n, to explore the one-pion exchange model. This can be done by using the following interaction Lagrangian [197] which defines the a2 coupling, La2 =
ga2 ± ε j A a (j j ∓ ) . Ma2
(24)
The decay width (a2± → ± ) then reads (a2± → ± ) =
ga22 p5 40 Ma4
,
(25)
where p = (Ma2 − M2 )/2Ma . Using (a2 → )expt. ≈ 0.29 MeV, we get ga2 ≈ 0.96 .
(26)
Then the p → a2+ (1320)n amplitude due to one-pion-exchange is obtained as √ ,
M
=
2gNN ga2 1 ε k (q − k) (q − k) 5 F (M , (k − q)2 ) , 2 Ma (p − p )2 − M2
(27)
where k is the photon momentum and q is the tensor meson momentum. The form factor is introduced in the form of F (M, r) =
4 4 + (r − M 2 )2
2 .
(28)
60
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89 2.5
1.5
+
σ(γ p → a2 (1320)n) (µb)
2.0
1.0
0.5
0.0
2
3
4
5
6
7
8
9
10
E γ (GeV)
Fig. 5. Total cross section for charged tensor meson photoproduction, p → a2+ (1320)n. Experimental data are from Refs. [193] (◦) and [194] (•).
We adjust the cutoff of the form factor to fit the total cross section of p → a2+ (1320)n. With = 0.45 GeV, the result is shown in Fig. 5. Therefore, we confirm that the photoproduction of charged a2 meson can be reasonably described by the one-pion-exchange [193–196]. 2.3.2. Neutral tensor meson photoproduction In Ref. [152], the authors used one-pion-exchange for neutral tensor meson photoproduction by extending the model for charged tensor meson photoproduction. However, the one-pion-exchange is not allowed for neutral tensor meson photoproduction because of the C-parity. Instead, we expect that the vector-meson exchange is the dominant process at low energies since the lightest mesons with C = −1 are the neutral vector mesons, 0 and . At high energies, the Odderon exchange, a partner of the Pomeron with odd C parity, is suggested as the major production mechanism and in fact neutral tensor meson photoproduction process has been suggested to study the Odderon exchange [198]. Since the role of the Odderon exchange is not clearly known, especially at low energies, we will mainly consider the vector-meson exchange for the production mechanism of neutral tensor meson photoproduction in the energy region of our interest, i.e., E 3 GeV. To calculate the vector-meson exchange amplitude, we need to first define the T V coupling, where T = f2 , a2 and V = , , . This can be determined by considering the tensor meson decay amplitude which can be written in the most general form as [199]
(k)V (k )|T =
1 A (k, k ) , MT
(29)
where A (k, k ) =
fT V
[g (k · k ) − k k (k − k ) (k − k ) ] MT3 + gT V [g (k − k ) (k − k ) + g k (k − k ) + g k (k − k ) − g k (k − k ) − g k (k − k ) − 2k · k (g g + g g )] .
(30)
The above form is known to give better descriptions of the known tensor meson radiative decays [200]. For simplicity, we will take the assumption of tensor meson dominance which leads to fT V ≈ 0 [199]. Without the fT V term, Eq. (30) is equivalent to that of Ref. [155] except for the factor of 2 difference in the definition of the coupling constant gT V . In the present calculation, we take gT V from the covariant quark model predictions [201], which gives reasonable description of the known radiative decay widths of vector and tensor mesons. These values are listed in Table 4 along
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
61
Table 4 Decay widths and couplings for T → V decay in a covariant quark model [201] Decay
Decay width (keV)
gT V
→ → → → → →
254 27 1.3 4.8 ∼0 104
0.14 0.048 −0.022 0.0145 ∼0 0.10
a2 → a2 → a2 →
28 247 0.8
0.044 0.134 −0.015
f2 f2 f2 f2 f2 f2
with the predicted decay widths. The details on the calculation of the radiative decays of tensor mesons are given in Appendix A. With the T V coupling fixed and the VNN coupling defined in the previous subsection, we can write the p → T p amplitude as g 1 C 1 V NN V NN g − 2 (p − p ) (p − p ) A (k, q − k) M , = − MT MV (p − p )2 − MV2
V × − i (31) (p − p ) , 2MN with CV NN = −1 for nn, and CV NN = 1 for pp, pp, and nn. The form of A (k, q − k) is defined by Eq. (30). The form factor (28) is also used here to regularize each vertex. However, the experimental data for neutral tensor meson photoproduction is very limited and uncertain and cannot be used to fix the cutoff parameter .9 To estimate their contribution, we take a relatively large cutoff =0.9 GeV to calculate the total cross sections for f20 (1275) and a20 (1320) photoproduction. The results are shown in Fig. 6. We see that they are smaller than the charged tensor meson production cross sections shown in Fig. 5 when E 5 GeV. If a smaller cutoff such as =0.45 GeV employed in Fig. 5 is used, the predicted cross sections will be even smaller. We will use = 0.9 GeV in our calculation as an estimate of an upper bound of the neutral tensor meson photoproduction contribution. This will allow us to examine whether a peak near 1540 MeV in KN invariant mass can be generated by neutral tensor meson photoproduction process in the n → K + K − n reaction. From our results, we also found that in the case of a20 photoproduction, the meson exchange is dominant, while both the and exchanges are comparable in f20 photoproduction. This is because ga2 /ga2 ∼ 1/3 and gf2 /gf2 ∼ 3 as seen in Table 4, while g 0 NN /g NN ∼ 1/3. Thus the meson exchange amplitude in a20 photoproduction is suppressed by an order of magnitude than the meson exchange, while the and meson exchanges have similar magnitude in f2 photoproduction.10 2.4. Hyperon background The hyperon backgrounds for N → KKN were considered in Refs. [151,153]. The main difficulty in estimating the hyperon background arises from the uncertainties of the coupling parameters of the hyperon resonances (Y = , ). We, therefore, consider only the well-known hyperon resonances, i.e., (1116), (1405), (1520), (1193), and (1385). We take the pseudoscalar coupling of kaons with the spin J = 1/2 hyperons. For coupling with the spin 9 In this work, we do not use the data for the backward scattering cross sections for f meson photoproduction of Ref. [202], which was obtained 2 by hand-drawn curves and after corrections for unobserved decay modes. 10 Therefore, our results are different from those of Ref. [155], where only the exchange for a 0 photoproduction and only the exchange for 2 f2 photoproduction were considered. We also calculated neutral tensor meson photoproduction in Regge model by Reggeizing the vector meson exchange amplitudes. The obtained results show that the total cross section decreases with energy as expected, but we found that the maximum values obtained in Regge model without form factor are close to the ones given in Fig. 6 with the similar photon energy.
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89 0
0
γ p → a2 (1320) p 1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
σ (γ p → f2(1275)p) (µb)
γ p → f2 (1275) p 1.0
σ (γ p → a2 (1320)p) (µb)
62
0.0
2
4
8
6
(a)
2
4
6
(b)
E γ (GeV)
0.0 10
8
E γ (GeV)
Fig. 6. Total cross section for neutral tensor meson photoproduction, (a) p → f20 (1275)p and (b) p → a20 (1320)p. The dotted lines are from meson exchange and the dashed lines from meson exchange. The solid lines are their sums.
γ(k)
K(q2)
N′(p2)
Y
N(p1)
K(q1)
(a)
γ
N′
Y
N
(b) γ
N
K
K
Y′
Y
N′
(d) γ
K
N′′
N
N
Y
N′
γ
K
K
N′′
Y
N
N′
γ
N
K
N
K
N′
Y
(f)
γ
(h)
N′
Y
K
K
γ
K
K*
(g)
K
K
(c)
(e) K
γ
K
K
K
K* Y
N′
N
Y
N′
(i)
Fig. 7. S = −1 hyperon backgrounds to N → KKN .
J = 3/2 hyperons, we follow the Rarita–Schwinger formulation of Refs. [203–207]. The resulting hyperon background tree-diagrams are shown in Fig. 7. We note that Fig. 7(d) includes the photo-transitions among the hyperons, which were not included in the previous works. Furthermore, Figs. 7(c) and (e) contain strangeness S = 0 state N . We will only consider the possibilities that N is either the nucleon (N) or the Delta [(1232)], since the experimental information about the transitions from other higher mass nucleon resonances to hyperons is not well known. Since we are using the pseudoscalar coupling for spin-1/2 baryons, there is no KY N contact interaction and hence we have five
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89 γ(k)
N(p1)
K(q1)
Θ
γ
K(q2)
N′(p2)
N
(a)
K
K
γ
K
N′
Θ
K
Θ
N
γ
K
K
γ
K
N
N′
Θ
N′
K
K
(d)
K
(c)
(b) γ
63
N
Θ
N′
*
N
(e)
K
Θ
N′
(f) γ
K
K
K N
Θ
*
N′
(g) Fig. 8. Exotic S = +1 pentaquark contribution to N → KKN .
diagrams, Figs. 7(a)–(e) for the intermediate (1116), (1405), and (1193) when K ∗ intermediate state is neglected. But for spin-3/2 (1520) and (1385), the KY N contact interaction is induced and we have seven diagrams, namely, Figs. 7(a)–(g)in the absence of K ∗ intermediate state. Figs. 7(h)–(i) account for the effects due to K ∗ intermediate state. The Lagrangians defining the photon coupling with K and K ∗ in Figs. 7(a), (b), (h), and (i) have been specified in Eqs. (2) and (8). The other Lagrangians used in the calculation of the diagrams in Fig. 7 are given in Appendix B in detail. There, we also discuss how the other couplings are defined by using SU(3) symmetry, experimental information, and some hadron model predictions. 2.5. Pentaquark contribution The pentaquark (1540) contribution to N → KKN is depicted in Fig. 8. We first assume that it belongs to pentaquark baryon anti-decuplet, which means that the (1540) is an isosinglet state. In this case, (1540) can 0 contribute to, or can be observed in the reactions of p → K 0 K p and n → K + K − n. There is still no experimental clue on the existence of isovector or isotensor , which are members of pentaquark 27-plet and 35-plet, respectively. So we do not consider such cases and focus on the isosinglet + since the nonexistence of a peak at 1540 MeV in K + p channel strongly suggests the isosinglet nature of (1540), if exists. Next we assume that (1540) has spin-1/2 following most hadron model predictions. There may be its higher spin resonances, but it will not be investigated in this study. However, there are strong debates on the parity of (1540). Therefore, in order to study the sensitivity of the physical quantities in N → KKN on the parity of (1540), we allow the both parities. Thus the effective Lagrangians are LKN = −igKN ± K N + H.c. ,
TK ∗ N ∗c ∗c ± ∓ LK ∗ N = −gK ∗ N K − j K N + H.c. , MN + M
L = −e A − j A , 2MN c
(32)
64
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
where ± = and
Kc =
5 1
,
± =
0
−K K−
,
5
K ∗c =
,
(33)
∗0
−K K ∗−
.
(34)
+ Here, the upper components of ± and ± are for the even-parity and the lower components for the odd+ parity . For the couplings of (1540), we assume that () = 1 MeV [175,176], which is close to the value of Particle Data Group [188], 0.9 ± 0.3 MeV. We then have
gKN = 0.984 (0.137)
(35)
∗ for positive (negative) √ parity . There is no information about the √ K N couplings. As in Ref. [149], we use gK ∗ N /gKN = 3 for even parity and gK ∗ N /gKN = 1/ 3 for odd parity following the quark model predictions [123,129,159] while neglecting the tensor coupling terms. The dependence of our results on this ratio will be discussed later. As for the magnetic moment of + (1540), we use the prediction from Ref. [208], () ≈ 0.1 (0.4), which gives ≈ −0.9 (−0.6) for positive (negative) parity . However, we found that our results are not sensitive to the value of .
3. Results Like all of the effective Lagrangian approaches, the considered tree-diagrams shown in the previous section need to be regularized by introducing form factors. In this work, we use the form (28) which was already used in our investigation of tensor meson photoproduction in Section 2.1. Namely, a form factor of the form of Eq. (28) is introduced at each vertex for all tree-diagrams in Section 2, where M is the mass of the exchanged (off-shell) particle and r is its four-momentum squared. Therefore, when it is on its mass-shell (r = M 2 ), the form factor becomes 1. It is well known that introducing form factors breaks the charge conservation condition. In order to satisfy this constraint, we extend the methods of Refs. [177–180]. Namely, we introduce contact diagrams to restore the current conservation. The details on this procedure are given and discussed in Appendix C. Some of the cutoff parameters are already fixed by the available experimental data for meson photoproduction as discussed in the previous section. For simplicity, we set the remaining cutoff parameters the same for all tree-diagrams and adjust it to fit the available experimental data of N → KKN reactions. Obviously, it is not easy to interpret the resulting form factor theoretically. Rather it should be just considered as a part of our phenomenological approach. 3.1. Total cross sections Since we assume that the (1540) is a particle with isospin I = 0, strangeness S = +1 and charge Q = +|e|, the production mechanisms (Fig. 8) cannot take place in the p → K + K − p reaction. We thus determine the cutoff parameter of the form factors by fitting the available total cross section data for p → K + K − p. In this way the background (non- mechanism) amplitude can be fixed, such that the identification of the from the available data of n → K + K − n reaction can be assessed. With =0.9 GeV, our fit (solid curve) is shown in Fig. 9. Clearly, the data can be reproduced reasonably well. With the same cutoff parameter, we then predict the total cross sections for the other three 0 processes listed in Eq. (1). They are also shown in Fig. 9: p → K 0 K p (dashed line), n → K + K − n (dot–dashed 0 0 line), and n → K 0 K n (dotted line). We can see that the cross sections for neutral K 0 K pair photoproduction from p (dashed curve) or n (dotted curve) are almost indistinguishable, and the cross sections for charged K + K − pair 0 photoproduction (solid and dash–dot curves) are larger than those for neutralK 0 K pair production. The contributions from the (1540) are not visible in the calculated total cross sections. The can be identified only in the invariant mass distributions.
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
65
σ(γ N → KKN) (µb)
2
1
0 1
2
3 Eγ (GeV)
4
Fig. 9. Total cross sections for N → KKN reactions. The solid line is for p → K + K − p, dashed line for p → K 0 K p, dot–dashed line 0 for n → K + K − n, and dotted line for n → K 0 K n. The experimental data are for p → K + K − p reaction and from Ref. [209] (•) and Ref. [210] (◦). The dashed and dotted lines are close together and hard to be distinguished. 0
3.2. Invariant mass distributions With the cutoff = 0.9 GeV determined from the total cross section for the p → K + K − p reaction, we can now compare our predictions of the invariant mass distributions for the n → K + K − n reaction with the JLab data [3]. Unfortunately, the JLab data are not scaled properly for comparing with the absolute magnitudes of the predicted cross sections because the JLab data just show the invariant mass distributions in an arbitrary unit. We therefore simply scale the JLab data to see whether we can roughly reproduce the shape of the data within our model.11 This must be a very crude assumption but will be enough to study the shape, especially the peak, of the data. The results from assuming an even-parity (odd-parity) (1540) are presented in Fig. 10 (Fig. 11) at E = 2.3 GeV. In both cases, we can reproduce very well the peak of the K + K − mass distributions [Figs. 10(a) and 11(a)]. The predicted shape in other region of the K + K − invariant mass also qualitatively agrees with the data. The comparisons with the data of the K + n mass distributions are shown in Fig. 10(c) and Fig. 11(c). Note that the JLab data were obtained from removing the → K + K − decay contributions at the peak and hence should only be compared with the dashed curves which are obtained from turning off the and production contribution in our calculations.12 For completeness, our full predictions (solid curves) are also displayed in (c) of Figs. 10 and 11. We see that the peak in KN mass distribution arising from the production of the (1540) is not so much pronounced as in the case of the meson peak in KK invariant mass distribution. This is mainly due to the small coupling of the with KN and K ∗ N . We also observe in Fig. 11(c) that the (1540) peak is much smaller in the case of odd-parity . Shown in (b) of Figs. 10 and 11 are the K − n mass distribution. By comparing the dashed curves and the experimental data in Figs. 10(c) and 11(c), we can see that the shape of the data can be reproduced well by our model except in the region near the (1540) peak. The width of the peak cannot be simply judged since the broad width of the JLab data reflects the detector resolution. Besides, there are two possible interpretations of our results. First, the (1540) is produced and the discrepancy between the data and the obtained curves is due to the low statistics and limited resolution of the experiment. On the other hand, the discrepancy is perhaps due to the deficiency of our model in accounting for other possible non- mechanisms and the existence 11 It should also be mentioned that the JLab data were not obtained with a single photon energy, instead the data were taken by incident electrons of 2.474 and 3.115 GeV, and the nuclear effects such as final state interactions are not properly taken into account. 12 We turn off the contribution in the dashed curves in Figs. 10(c) and 11(c) in order to show the enhancement of the peak compared with the backgrounds.
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
1.4
10
(a)
1.5
mKN (GeV) 1.6
1.7
(b)
1.8 2
dσ/dmKN (µb/GeV)
66
8
6 0
(c)
4
2
2
dσ/dmKN (µb/GeV)
dσ/dmKK (µb/GeV)
1
1
0
1
1.2 mKK (GeV)
1.4
1.4
1.5
1.7 1.6 mKN (GeV)
0 1.8
Fig. 10. (a) KK, (b) KN , and (c) KN invariant mass distributions for n → K + K − n at E = 2.3 GeV. The experimental data are from Refs. [3]. The dashed line in (c) is obtained without the meson background and the contributions. Here we assume that the (1540) has even parity.
mKN (GeV) 10
(a)
1.5
1.6
1.7
(b)
1.8 2
8
6 0
(c)
4
2
2
0
1
1
1.2 mKK (GeV)
1.4
1.4
1.5
1.6 mKN (GeV)
1.7
dσ/dmKN (µb/GeV)
dσ/dmKK (µb/GeV)
1
dσ/dmKN (µb/GeV)
1.4
0 1.8
Fig. 11. (a) KK, (b) KN , and (c) KN invariant mass distributions for n → K + K − n at E = 2.3 GeV. The experimental data are from Refs. [3]. Notations are the same as in Fig. 10. Here we assume that the (1540) has odd parity.
of is questionable, which also includes the possibility of the contamination due to (1520). As discussed in the previous section, we are limited by the lack of information in calculating some allowed non- background mechanisms. Obviously, high statistics and high-resolution experiments are strongly required. The existence of (1540) can be unambiguously established if and only if a very sharp resonance peak, which is very unlikely due to the background amplitudes as predicted by our model, is observed. Furthermore, an even-parity is more likely to be detected, while it will be difficult to identify an odd-parity , even if it exists, from the background continuum. This is because the odd-parity has much smaller couplings with the KN channel than the even-parity when the same decay width is assigned. Therefore, if the observed (1540) has odd-parity, it must have larger decay width, at least, one order of magnitude larger than the current estimate in Ref. [188] or it should have a large coupling to the other channels [155].
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
15
(a)
1.5
m KN (GeV) 1.6 1.7
(b)
1.8 12 10 8
10
4 2 0
(c)
6 5
5
4 3 2 1 0
1
1.1 1.2 mKK (GeV)
1.3
1.4
1.5
1.6 mKN (GeV)
1.7
dσ/dm KN (µb/GeV)
dσ/dmKK (µb/GeV)
6
dσ /dm KN (µb/GeV)
1.4
67
0 1.8
Fig. 12. (a) KK, (b) KN , and (c) KN invariant mass distributions for p → K + K − p at E = 2.3 GeV. The dashed line in (c) is obtained without the meson background and the (1520) contribution.
To facilitate the future experimental searches, we now present our predictions for the other three processes listed in Eq. (1). In Fig. 12, we give the results for the p → K + K − p reaction at E = 2.3 GeV. Since we assume that (1540) is isoscalar and cannot contribute to this reaction, there is no peak in K + p mass distribution. The peak in Fig. 12(b) of the K − p mass distribution is due to the (1520) of hyperon background (Fig. 7). The dashed line in Fig. 12(c) is obtained when the contributions from the and (1520) are neglected in the calculation. Clearly, these two mechanisms are the major background production processes. Experimental test of our predictions presented in Fig. 12 will also be an important task to check our model of non- background mechanisms which must be understood before the predictions for n → K + K − n can be used to determine the existence of the (1540). 0 The results for p → K 0 K p are shown in Figs. 13 and 14 for the even and odd parity , respectively. We see that the (1540) peak in Fig. 13 is much smaller than that in the n → K + K − n reaction (Fig. 10). The peak from the 0 odd-parity (1540) is hardly to be seen in the KN invariant mass distribution in p → K 0 K p as shown in Fig. 14. 0 The results for n → K 0 K n are given in Fig. 15. Again, there is no contribution from the isoscalar to this reaction and therefore there is no peak in the KN invariant mass distribution [Fig. 15(c)]. Here we also find that the predicted cross sections [dashed curve in Fig. 15(c)] is greatly reduced if the contributions from the and (1520) productions are turned off. 3.3. Tensor meson photoproduction contributions In this subsection, we discuss in more detail the contribution of the tensor meson production to the n → K + K − n reaction, which is an important issue raised by Dzierba et al. [152], who indicated the possibility that the observed peak at 1540 MeV in the KN invariant mass distribution could be a false peak arising from tensor meson background. Their calculation of neutral tensor meson photoproduction is based on a model of pion trajectory exchange mechanism. However, as we discussed in Section 2, the pion exchange is not allowed for this process because of the C parity, and the lowest allowed exchanged particles are vector mesons. Although the production mechanism used by the authors of Ref. [152] is questionable, their claim should be checked by a calculation using the vector meson exchange mechanism, as formulated in Section 2.2. Our calculations of the vector meson and tensor meson contributions to the KK mass distributions of the four processes listed in Eq. (1) are displayed in Fig. 16. The contributions from tensor mesons (displayed in small windows
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Y. Oh et al. / Physics Reports 423 (2006) 49 – 89 mKN (GeV) 1.4
1.6
1.7
1.8 1.5
(b)
15
0.5 10 0
(c) 1 5 0.5
0 0.9
1
1.1 1.2 1.3 mKK (GeV)
1.4
1.4 1.5
1.5
1.6 mKN (GeV)
1.7
dσ/dmKN (µb/GeV)
dσ/dmKK (µb/GeV)
1
dσ/dmKN (µb/GeV)
(a)
1.5
0 1.8
0
Fig. 13. (a) KK, (b) KN , and (c) KN invariant mass distributions for p → K 0 K p at E = 2.3 GeV. The dashed line in (c) is obtained without the meson background and the (1540) contribution and magnified by a factor of 10. Here we assume that the (1540) has even parity.
mKN (GeV) 1.4
1.5
1.6
1.7
20
(b) 1
0.5
0
10
(c) 1 5 0.5
0 0.9
1
1.1 1.2 1.3 mKK (GeV)
1.4
1.4 1.5
1.5
1.6 mKN (GeV)
1.7
dσ/dmKN (µb/GeV)
dσ/dmKK (µb/GeV)
15
dσ/dmKN (µb/GeV)
(a)
1.8 1.5
0 1.8
0
Fig. 14. (a) KK, (b) KN , and (c) KN invariant mass distributions for p → K 0 K p at E = 2.3 GeV. The dashed line in (c) is obtained without the meson background and the (1540) contribution and magnified by a factor of 10. Here we assume that the (1540) has odd parity. Since the contribution from (1540) is suppressed, its peak is not seen in (c).
of Fig. 16) are clearly much smaller than the vector meson contributions. Furthermore, because of their large decay widths, (f2 (1275)) ≈ 185 MeV and (a2 (1320)) ≈ 107 MeV, the contributions from these two tensor mesons do not give distinguishable two peaks in KK mass distribution. We also note that the tensor meson peak in p → K + K − p reaction is much more pronounced than in the other reactions. This is due to the isospin factors associated with the coupling constants, which define the relative phases between different contributions in each process. As seen in
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
(a)
mKN (GeV) 1.6
1.7
1.8 1.2
(b)
0.9
dσ/dmKK (µb/GeV)
0.6 10 0.3 0 1.2
(c)
0.9
5
0.6 0.3 0 0.9
1
1.1 1.2 1.3 m KK (GeV)
1.4
dσ/dmKN (µb/GeV)
15
1.5
1.5
1.6 m KN (GeV)
dσ/dmKN (µb/GeV)
1.4
69
0 1.8
1.7
0
Fig. 15. (a) KK, (b) KN , and (c) KN invariant mass distributions for n → K 0 K n at E = 2.3 GeV. The dashed line in (c) is obtained without the
meson background and the (1520) contribution. mKK (GeV) 1.1
1.2
1.3
mKK (GeV) 0.9
(a)
1
1.1
1.2
1.3
1.4
15
(b)
10
5
0.02
0.02
0.015
0.015
0.01
0.01
0.005
0.005
0 1.2 1.25 1.3 1.35
10
0 1.2 1.25 1.3 1.35
0 15 dσ/dmKK (µb/GeV)
5
0 15
(c)
(d)
0.01
0.01
10
10 0.005
5
0 0.9
1.5
0.005
0 1.2 1.25 1.3 1.35
1
1.1 1.2 1.3 mKK (GeV)
0 1.2 1.25 1.3 1.35
0.9
1
1.1 1.2 1.3 mKK (GeV)
1.4
5
dσ/dmKK (µb/GeV)
dσ/dmKK (µb/GeV)
15
1
dσ/dmKK (µb/GeV)
0.9
0 1.5
Fig. 16. Vector meson and tensor meson contributions to KK invariant mass distributions for (a) p → K + K − p, (b) p → K 0 K p, 0 (c) n → K + K − n, (d) n → K 0 K n at E = 2.3 GeV. Shown in the small windows are the region of tensor meson peaks. 0
Table 5, the resulting relative phases lead to constructive interference in p → K + K − p reaction and destructive interference in the other reactions. As a result, the tensor meson peaks in the n → K + K − n reaction is smaller than those in the p → K + K − p reaction. Therefore, the claim raised by Ref. [152] can be checked by comparing the results from the above two reactions. Namely, if the peak at 1540 MeV in the n → K + K − n reaction is coming from the tensor meson contribution, one could expect a similar or even more apparent peak at around 1540 MeV in p → K + K − p reaction with the similar energy of the photon beam. The absence of such a peak in p → K + K − p
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Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
Table 5 Relative phases of tensor meson photoproduction contribution to the N → KKN reaction a20 photoproduction
exchange
exchange
exchange
a20 → KK
exchange
f20 → KK
+
+
+
+
+
+
+ +
+ −
− +
+ +
+ −
+ +
+
−
−
+
−
+
mKN (GeV) 1.4
dσ/dmKN (µb/GeV)
0.02
mKN (GeV) 1.4
1.6
1.8
2
(b)
(a)
0.02
0.01
0 0.02 dσ/dmKN (µb/GeV)
1.6
0.01
(c)
0 0.02
(d)
0.01
0 1.4
0.01
1.6 mKN (GeV)
1.4
dσ/dmKN (µb/GeV)
p → K + K − p 0 p → K 0 K p + n → K K − n 0 n → K 0 K n
f20 photoproduction
1.8 1.6 mKN (GeV)
2
dσ/dmKN (µb/GeV)
Reaction
0
Fig. 17. Tensor meson contribution to KN invariant mass distributions for (a) p → K + K − p, (b) p → K 0 K p , (c) n → K + K − n, (d) 0 n → K 0 K n at E = 2.3 GeV. The dot–dashed, solid, and dashed lines are at E = 2.0, 2.3, and 2.6 GeV, respectively. The dot–dashed lines in (b,c,d) are suppressed and hard to be seen within the given scale. 0
reported by the SAPHIR [4] and HERMES [9] Collaborations, therefore, seems to disfavor the possibility of ascribing the peak at 1540 MeV in the KN mass distribution of n → K + K − n to the tensor meson background. This, of course, should be further examined by other higher statistics experiments. In Fig. 17, we give our results for KN invariant mass distribution coming solely from the tensor meson photoproduction part. Although their maximal values locate at around 1.56 GeV at E = 2.3 GeV (solid curve), the shapes are very broad and most of the magnitudes are much smaller than the other backgrounds by about two orders of magnitude. Thus, the tensor meson contributions estimated within our model based on vector meson exchange are too weak to generate any narrow peak in the presence of other much larger background processes even if the meson background is removed. We therefore conclude that we could not verify the claim of Ref. [152]. Our results also show that, if the peak is from the tensor meson background, its position should change with different photon beam energies. 3.4. Double differential cross sections The double differential cross sections are calculated for the three cases; no pentaquark, even-parity (1540), and oddparity (1540). Since we are interested in the existence and parity of isoscalar , we now focus on the n → K + K − n 0 reaction, which has larger cross sections than p → K 0 K p. Shown in Fig. 18 is the double differential cross sections for the n → K + K − n reaction at E = 2.3 GeV and mKN = 1.54 GeV, i.e., at the resonance point, as a function of cos 1 , where 1 is the polar angle of K − in the N center
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
71
0.5
dσ/dmKN dΩ1 (µb/GeV⋅sr)
0.4
0.3
0.2
0.1
0 −1
−0.5
0 cos θ1
0.5
1
Fig. 18. Double differential cross section d/dmKN d1 as a function of cos 1 , where 1 is the polar angle of K − in the center of mass frame, for n → K + K − n with E = 2.3 GeV and mKN = 1.54 GeV. The dotted line is obtained without (1540), while the solid (dashed) line is with even (odd) parity of .
of mass frame of which z-axis is defined as the photon beam direction. In this figure, the solid line is for even-parity and the dashed line for odd-parity . The dotted line is for the background, i.e., without . Our result shows that the differential cross sections are enhanced by the presence of even-parity in the forward scattering region of K − meson direction. This suggests that the kinematic cut for the angle of K − would be useful to enhance the peak in KN mass distributions. But for the odd-parity , this enhancement is small. This difference is primarily due to the magnitude of the couplings. We have also calculated some spin asymmetries of this reaction. As discussed in Refs. [137–139], however, we cannot avoid model-dependence on the spin asymmetries in most reactions, which makes it very hard to study the parity of by the spin asymmetries except some special cases like the NN reaction near the threshold. Therefore, we do not present our results on the spin asymmetries and do not make a definite conclusion on the dependence of spin asymmetries on the parity of . Instead, we make a comment on the spin asymmetries of the model considered in this work. We have considered two spin asymmetries, the single photon beam asymmetry x and beam-target double asymmetry CBT for the n → K + K − n reaction at the same energies as in Fig. 18. (For their definitions, see Ref. [151].) The considered asymmetries, the photon beam asymmetry and beam-target double asymmetry, were found to be rather sensitive to the parity of the (1540). But in the case of beam-target double asymmetry, the difference between the odd-parity and the background processes is found to be not large. Although the model-dependence is unavoidable, our model calculation shows that these spin asymmetries may have different values depending on the parity of (1540), especially in the forward scattering region.13 13 Here we note that our result on the photon beam asymmetry agrees qualitatively with that of Ref. [151] which is based on a much simpler model. Although the numerical values and the structure at large K − angles are different, these two models are consistent at least qualitatively at forward angles of the K − momentum in the center of mass frame. In Ref. [155], it is claimed that the single and double spin asymmetries are not sensitive to the parity of the (1540). We found that this comes mainly from the role of the K ∗ exchanges, Fig. 8(f,g). In Ref. [155], the authors used the signal to background ratio to constrain the ratio ≡ gK ∗ N /gKN , which leads to = 1.875 and 8.625 for the even and odd parity . As a consequence, they have the K ∗ -exchange dominance in the production mechanism especially in√the case of √ odd-parity , and as a result the single and double asymmetries are not sensitive to the parity of . In our calculation we used = 3 and 1/ 3 for the even and odd parity following the quark model predictions. We also found that the spin asymmetries are sensitive to the parity of is < 1 for the odd parity . Therefore, this illustrates the model-dependence of the asymmetries and it is very important to estimate the correct order of magnitude of , which is, however, not possible at the present state.
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We have also calculated these asymmetries at a given angle 1 as functions of mKN . (1 = 30◦ for x and 1 = 0◦ for CBT ) We could find the resonance structure in these asymmetries when the has even parity. For odd-parity , as can be inferred from Fig. 18, its contribution is small and the resonance structure in the beam-target double asymmetry is not manifested. Thus, although the parity of (1540) may not be uniquely determined by these asymmetries because of model-dependence [137], our results suggest that the resonance structure in the asymmetries can be used to verify the existence of .
4. Summary In this work, we have investigated how the N → KKN reaction can be used to study the production and the properties of the pentaquark (1540) baryon. We study the case that (1540) is a spin J = 1/2 and isospin I = 0 particle, and have focused on the influence of the (non-) background amplitudes due to the (tree) t-channel Drell diagrams (Fig. 3), the KK production through the intermediate vector meson and tensor meson photoproduction (Fig. 4), and the mechanisms involving intermediate (1116), (1405), (1520), (1193), (1385), and (1232) states (Fig. 7). The vector meson photoproduction amplitude is calculated from a phenomenological model which describes well the experimental data at low energies. The charged tensor meson production amplitude is calculated from a one-pion-exchange model, which describes well the total cross section data of p → a2+ (1320)n. The neutral tensor meson production part is estimated by using the vector meson exchange mechanisms. The coupling constants needed for calculating all of the considered background mechanisms are deduced from the data of decay widths by imposing the SU(3) symmetry or making use of various quark model predictions. No attempt is made to calculate other background amplitudes which are kinematically allowed but cannot be computed because of the lack of experimental information. Thus the present work represents only a step toward a complete dynamical description of the N → KKN reaction. Nevertheless, some progress has been made in assessing the existing data concerning the existence of the (1540) pentaquark state. The (1540) production mechanism (Fig. 8) is calculated by taking 1 MeV as its decay width, which is close to the Particle Data Group value. With the background amplitude constrained by the total cross section data of p → K + K − p (the considered isoscalar (1540) is not allowed in the process), we find that the resulting K + K − and K + n invariant mass distributions of the n → K + K − n reaction can qualitatively reproduce the shapes of the JLab data although its magnitude cannot be compared. However, the predicted (1540) peak cannot be identified unambiguously with the data. There are two possible interpretations of our results. First, it is possible that the (1540) is produced and the discrepancy between the data and our results is due to the low statistics and limited resolution of the experiment. On the other hand, the discrepancy is perhaps due to the deficiency of our model in accounting for other possible non- mechanisms and the existence of (1540) is questionable. Obviously, high statistics and high-resolution experiments are needed. The existence of (1540) can be unambiguously established if and only if a very sharp resonance peak, which is very unlikely from the background amplitudes as predicted by our model, is observed. We also find that an even-parity is more likely to be detected, while it will be difficult to identify an odd-parity , even if it exists, from the background continuum, unless it has much larger coupling to the K ∗ N channel than the current quark model predictions and/or other relevant production mechanism(s) not considered in this work. We have analyzed in some detail the contributions of the tensor meson to n → K + K − n reaction, which is an important issue raised by Dzierba et al. [152]. (See also Ref. [211].) These authors indicated that the observed peak at 1540 MeV in the KN invariant mass distribution could be a false peak arising from tensor meson background. What we found are (1) the calculation of Ref. [152] is based on the 0 exchange mechanism which cannot take place in neutral tensor meson photoproduction if the C-parity is conserved. (2) Instead of 0 exchange, we estimate the neutral tensor meson contributions by using the vector meson exchange model, and have found that the neutral tensor meson contribution is too weak to generate any resonance peak which can be identified with the existing data of n → K + K − n reaction. Finally, we also present the double differential cross sections and make a comment on the spin asymmetries. Our results show that some kinematic cut would enhance the peak of , if exists. In the spin asymmetries, the major uncertainty comes from the ratio ≡ gK ∗ N /gKN , and our results suggest a resonance structure in some spin asymmetries, which can be measured at current experimental facilities.
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Acknowledgements We are grateful to K. Hicks, K. Joo, T. Nakano, and A.I. Titov for useful information and fruitful discussions. We also thank V. Burkert for valuable comments. Y.O. acknowledges fruitful discussions with Hungchong Kim. This work was supported by Forschungszentrum-Jülich, Contract No. 41445282 (COSY-058) and US DOE Nuclear Physics Division Contract No. W-31-109-ENG-38.
Appendix A. Tensor mesons In this Appendix, we discuss the couplings of tensor mesons to hadrons and their radiative decays. Among the spin-2 tensor mesons, we are interested in f2 (1270), f2 (1525), and a2 (1320) as they can decay into two kaons and close to the threshold energy region of the N → KKN reaction.14 The f2 (1270) and f2 (1525) have the quantum numbers I G (J P C ) = 0+ (2++ ). The f2 (1270) has mass Mf2 = 1275 MeV, width f2 = 185.1 MeV, and it mostly decays into two pions. Some of its branching ratios are BR(f2 → ) = (84.7 +2.4 −1.3 )%, BR(f2 → KK) = (4.6 ± 0.5)%, and BR(f2 → ) = (1.41 ± 0.13) × 10−5 . The f2 (1525) has Mf2 = 1525 MeV and f2 = 76 ± 10 MeV. It mostly decays into two kaons and some of its branching ratios are BR(f2 → KK) = (88.8 ± 3.1)% and BR(f2 → ) = (1.23 ± 0.17) × 10−6 . The a2 (1320) is an isovector tensor meson with Ma2 = 1318 MeV and a2 = 107 ± 5 MeV. It mostly decays into , but it decays also into two kaons, , and two photons with BR(a2 → KK) = (4.9 ± 0.8)%, BR(a2 → ± ) = (2.68 ± 0.31) × 10−3 , and BR(a2 → ) = (9.4 ± 0.7) × 10−6 . Note that a20 → 0 decay is not allowed because of C-parity. By the same reason, f2 → 0 is forbidden. In our calculation, we use the coupling constants determined from the experimental data and quark model predictions. Here, we discuss the way to determine the couplings and compare the values with those obtained by assuming SU(3) symmetry and vector meson dominance. A.1. Interactions with pseudoscalar mesons Since f2 and a2 are spin-2 tensor mesons, we are dealing with tensor meson nonet whose members are a2 (1320), K2 (1430), f2 (1275), and f2 (1525). This is analogous to vector meson nonet of , K ∗ , , and . The pseudoscalar octet is represented by an SU(3) matrix P which is defined in Eq. (6). Similarly, the tensor meson octet is represented by T8 as 1 √ f8 + √1 a2 K2 6 2 , (A.1) T8 = K2 − √2 f8 6
where a2 = a2 · . The Lorentz index is suppressed. Then the T8 P P interaction is obtained as
L = g Tr(T8 j P j P ) ,
(A.2)
which gives the SU(3) symmetry relations to the coupling constants. Since f2 (1275) and f2 (1525) are expected to be close to ideal mixing, we introduce the tensor meson singlet f0 , whose interaction with two pseudoscalar mesons is given by g Lf0 P P = √ Tr(f0 j P j P ) 3 g = √ f0 {j j + j · j + 2j Kj K} . 3
(A.3)
14 The f (1525), which mostly decays into KK, is expected to be in ss state. Therefore, its coupling to the nucleon would be suppressed. Because 2 of its higher mass and assuming the OZI rule as the first approximation, we do not include f2 photoproduction in our study of KK photoproduction processes.
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Now we consider the mixing of f8 and f0 . The physical states f2 and f2 are written as f = cos f8 − sin f0 , f = sin f8 + cos f0 ,
(A.4)
where f2 and f2 are denoted by f and f , respectively. Then we have g g 1 1 gf = √ √ cos − sin , gf = √ √ sin + cos , 3 2 3 2 g g 1 1 gf KK = − √ √ cos + 2 sin , gf KK = √ − √ sin + 2 cos . 3 2 3 2
(A.5)
The mixing angle can be estimated from the masses of tensor meson nonet; tan = 2
2 + M2 3Mf2 − 4MK a2 2 2 − M 2 − 3M 2 4MK a2 f 2
≈ 0.35 ,
(A.6)
where we have used the Gell-Mann–Okubo mass relation for squared masses of tensor mesons. This gives us ≈ 30.5◦ ,
(A.7)
where the ideal mixing angle is idealmixing ≈ 35.3◦ . Note that in the case of vector meson nonet, the mixing angle is V ≈ 40◦ . Therefore, we can see that the tensor meson nonet is as close to ideal mixing as the vector meson nonet.15 The 2+ 0− 0− interaction is obtained as L=−
2Gf j · j f , Mf
(A.8)
which gives the decay width of f2 → as 2
(f2 → ) = where pF = Mf we obtain
2 Gf 5 p , 5 Mf4 F
(A.9)
1/4 − M2 /Mf2 . Using (f2 → )expt. ≈ 156.9 MeV and (f2 → )expt. ≈ 0.623 MeV,
Gf = 5.76,
Gf = 0.33 = 0.06Gf .
(A.10)
Similarly, f → KK vertex can be obtained from L=−
2Gf KK j Kj Kf . Mf
(A.11)
Thus the decay width f2 → KK is estimated as 2
(f2 → KK) =
2 Gf KK 5 p , 15 Mf4 F
(A.12)
which gives Gf KK = 7.15 = 1.24Gf . 15 Note that we are assuming qq structure of the tensor meson, which may contain nonnegligible glueball component.
(A.13)
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
75
with (f2 → KK)expt. ≈ 8.6 ± 0.8 MeV. This expression can also be applied to f2 (1525) meson decay, and with (f2 → KK)expt. ≈ 65 ± 5 MeV we get Gf KK = −11.23 = −1.95Gf ,
(A.14)
where the interaction is given by L=−
2Gf KK j Kj Kf . Mf
(A.15)
Note that f2 and f2 has the same quantum number and they are anticipated to form ideal mixing as in the case for and mesons. This is because f2 mostly decays into two pions, while f2 mostly into two kaons. But the presence of f2 → KK decay implies the deviation from the ideal mixing. In the case of a2 (1320), because of its isovector nature, the interaction Lagrangian reads L=−
2GaKK j K · a j K . Ma
(A.16)
Using (a2 → KK) ≈ 5.24 MeV, we obtain GaKK = 4.89 = 0.85Gf .
(A.17)
The obtained results should be compared with the SU(3) symmetry relations, Gf = 0.10 Gf , Gf KK = 1.11 Gf , Gf KK = −1.60 Gf , and GaKK = 1.04 Gf . The deviation from the SU(3) symmetry relations implies the SU(3) symmetry breaking effects and possibly the nonnegligible glueball components in tensor mesons. Here, in the study of KK photoproduction, we use the coupling constants determined from the measured decay widths of tensor mesons. A.2. Tensor meson radiative decays First, we consider the 2+ 0− 1− interaction such as a2 → or a2 → decays. The interaction Lagrangian reads [197] ga ± La2 = 2 2 ε j A a (j j ∓ ) . (A.18) Ma The coupling constant ga2 is determined from the decay width of a2± → ± as ga2 ≈ 0.96 .
(A.19)
For the decays of a tensor meson into two photons, the most general form reads [199]
(k)(k )|T =
1 A (k, k ) , MT
where the form of A (k, k ) is given by Eq. (30). With the above interaction, we have Mf 1 2 2 (T → ) = + gT . f 20 24 T
(A.20)
(A.21)
Since (f2 → )expt. = 2.6 ± 0.24 keV , (a2 → )expt. = 1.0 ± 0.06 keV , (f2 → )expt. = 9.35 × 10−2 keV ,
(A.22)
we get gf = 0.011, assuming fT = 0.
ga = 6.9 × 10−3 ,
gf = −1.96 × 10−3 ,
(A.23)
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Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
The above interaction form can be used for the interactions of a tensor meson with a vector meson and a photon, which gives MT 2 x2 x (T → V ) = gT V (1 − x)3 1 + + , (A.24) 10 2 6 with x = MV2 /MT2 . There is no experimental data for this decay, so we use the predictions of Ref. [201] based on a covariant quark model, which gives a reasonable description of the known radiative decay widths of vector and tensor mesons. The predictions and the obtained coupling constants are given in Table 4. Appendix B. Effective Lagrangians for the hyperon backgrounds In this Appendix, we present the effective Lagrangians and coupling constants used in the calculation for the hyperon background diagrams, Fig. 7. B.1. Baryon octet The baryon octet included in our calculation of Figs. 7(a)–(e) are the nucleon (N), (1116), and (1193). Their interactions with the photon are defined by
1 1 + 3 N N LNN = −eN A − ( + v 3 ) j A N , 2 2MN s e L = j A , 2MN
1 L = −e A T3 − , (B.1) ( +
T ) j A v 3 2MN s where T 3 = diag(1, 0, −1) and 1
N s = 2 ( p + n ) = −0.06,
= −0.61,
1
N v = 2 ( p − n ) = 1.85 ,
1
s = 2 ( + + − ) = 0.65,
1
v = 2 ( + − − ) = 0.81 .
(B.2)
The numbers above are obtained by using the measured magnetic moments of the baryon octet [188], (p) = 2.79, (n) = −1.91, () = −0.61, (+ ) = 2.46, and (− ) = −1.16 in the nucleon magneton unit. To calculate the photo-transition of into in Fig. 7(d), we use e 0 (B.3) L = j A + H.c. , 2MN where = −1.61 ± 0.08 as given by the Particle Data Group [188]. For meson–baryon interactions, we use the pseudoscalar coupling to write LKN = −igNK N 5 K + H.c. , LKN = −igNK N5 · K + H.c.
(B.4)
The flavor SU(3) symmetry relations evaluated with d + f = 1 give gKN = − √1 (1 + 2f )gNN , 3
gKN = (1 − 2f )gNN .
(B.5)
By using the empirical value f/d = 0.575, which gives f = 0.365 and d = 0.635, and g2NN /4 = 14, we get gKN = −13.24,
gKN = 3.58 .
(B.6) K ∗,
we use the following Lagrangian: To calculate Figs. 7(h) and (i) with intermediate
K ∗ NY Y j K ∗ . LK ∗ NY = −gK ∗ NY N Y K ∗ + 2MN
(B.7)
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
77
We use the values from the new Nijmegen potential [212,213] to define the coupling constants in the above equation, gK ∗ N = −6.11 ∼ −4.26, gK ∗ N = −3.52 ∼ −2.46,
K ∗ N = 0.436 ∼ 0.474 ,
K ∗ N = −1.0 ∼ −0.412 .
(B.8)
For our numerical calculation, the values in the right boundary are used. B.2. Baryon decuplet We now consider the couplings involving the members of the baryon decuplet, (1232) and ∗ (1385), which are intermediate states in Figs. 7(a)–(g). The Lagrangians describing the photo-interaction of ∗ (1385) read ∗
,
L∗ ∗ = e A ∗ ∗ , LKN ∗ = −i where [204]16 A
,
√ ∗− √ + ∗− ef KN ∗ ∗0 A ( pK − + 2 nK − − K + p∗0 − 2K n ) , MK
∗ 1 1 ∗ = g − ( + ) A T3 − ( , s + v T3 ) j A g 6 2MN
(B.9)
(B.10)
and the coupling fKN ∗ will be explained later. Since there is no experimental information for the magnetic moments of (1385), we make use of the following quark model predictions [215]: (∗+ ) = 3.15,
(∗0 ) = 0.36,
(∗− ) = −2.43 ,
(B.11)
to obtain ∗
s = 0.36,
∗
s = 1.79 .
(B.12)
We next need to construct the photo-transition Lagrangians LN , L∗ , and L∗ . These can be fixed by considering the radiative decays of the decuplet baryons. The most well-studied is the → N transition which enters into Figs. 7(c) and (e). Here we follow Ref. [205] to write ieg 1N 1 3 3 1 O (Z) 5 I , N F + H.c. , LN = 2MN 2 2 3 1 eg L2N = − 2N2 O (Z)5 I 3 (B.13) , (j N )F + H.c. , 2 2 4MN where F is the field strength tensor of the photon, F = j A − j A , and we choose the off-shell parameters so that O (Z) = g . The isospin factor is calculated as 2 + 0 3 3 1 I (B.14) , N= ( p + n) . 2 2 3 The above Lagrangians lead to the following expression for calculating the radiative decay width of the , 2
2 p3 e M g1N (3M + MN ) − g2N (M − MN ) ( → N ) = 2MN 72M2 2MN
M 2 2 . + 3 g1N − g2N (M − MN ) 2MN
(B.15)
It is well known that the g2N term is sensitive to the E2/M1 ratio of → N decay. In this calculation, however, we assume that g2N = 0, which gives somewhat large value, ≈ −6%, of E2/M1. However, our results show that the 16 There is an ambiguity with defining the nonminimal anomalous magnetic moment term. Here we follow Ref. [204]. Other definitions are discussed in Refs. [203,214].
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Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
contributions from the N transition to N → KKN reaction are suppressed compared with the other contributions. Therefore the precise value of g2N is irrelevant to this calculation. Using ( → N )expt. ≈ 672 keV, we then get gN = g1N ≈ 4.9 ,
(B.16)
which is close to the fitted value, ≈ 5.0, of Ref. [205]. To calculate Y → Y transition in Fig. 7(d), we consider the following Lagrangian: LBD =
ieg BD D O (Z) 5 BF + H.c. , 2MN
(B.17)
where D is the ∗ (1385) and B denotes for (1116) or (1193). There is no experimental information on the radiative decay widths for decuplet ∗ (1385) except some upper bounds. We therefore make use of the following quark model predictions [216]: (∗0 → ) = 232 keV, (∗0 → 0 ) = 19 keV,
(∗+ → + ) = 104 keV , (∗− → − ) = 2.5 keV .
(B.18)
The above values then fix the coupling constants of LBD as g∗0 ≈ 3.01,
g∗+ + ≈ 3.38,
g∗0 0 ≈ 1.44,
g∗− − ≈ 0.52 .
(B.19)
We now will use the SU(3) symmetry to fix the couplings involving K mesons, baryon decuplet [∗ (1385), (1232)] and baryon octet [N, (1116), (1193)]. We start with the well-studied Lagrangian for the N interaction. We follow Refs. [206,207] to write f N 3 1 LN = O (Z)I (B.20) , · j N + H.c. , m 2 2 where the isospin transition matrix reads 3 1 (+1) (−1) (0) I , · = −I3/2,1/2 + + I3/2,1/2 − + I3/2,1/2 0 , 2 2 with
⎛√ 6 1 ⎜ 0 (+1) I3/2,1/2 = √ ⎝ 0 6 0
⎞ √0 2⎟ ⎠, 0 0
⎛
0 1 ⎜2 (0) I3/2,1/2 = √ ⎝ 6 0 0
⎞ 0 0⎟ ⎠, 2 0
(B.21)
⎛
0 1 ⎜ √0 (−1) I3/2,1/2 = √ ⎝ 2 6 0
⎞ 0 0 ⎟ ⎠ . 0 √ 6
(B.22)
With O (Z) = g [206,207] (see also Ref. [203]), the → N decay width can be written as ( → N) =
p3 fN 2 1 [(M + MN )2 − M2 ] . 24 M M2
(B.23)
Using M = 1232 MeV and ( → N ) = 120 MeV, we get17 fN = 2.23 .
(B.24)
With the N coupling fixed, we then use the SU(3) relations, fK f N =− , MK MN
1 fKN ∗ = √ fK , 6
17 If we use the pole mass M = 1211 MeV, we get f = 2.56. In Ref. [185], fN is estimated to be 2.05–2.12. N
(B.25)
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
79
to obtain
fK 3 1 I , · j K + H.c. , MK 2 2 ∗ f ∗ LKN ∗ = KN j K · N + H.c. , MK
LK =
(B.26)
where fK ≈ −7.88,
fKN ∗ ≈ −3.22 .
(B.27)
In the numerical calculation, we use fKN ∗ = −2.6, which is within the range of SU(3) symmetry breaking. The diagrams of Figs. 7(c) and (e) can have K interactions with the baryon decuplet (1385) and . To determine these couplings using SU(3) relations, we again start with the interaction, which reads f 3 3 L = O (Z) 5 I (B.28) , · j O (Z) . M 2 2 See Ref. [162] for the definition of I (3/2, 3/2). By using the quark model prediction [217] f /fNN = 4/5 (fNN = M ), we find f ≈ 0.8. The SU(3) relation gN N 2M N fK ∗ = −
3 MK f ≈ 3.46 , 2 M
then fixes the following Lagrangian for K∗ interaction: fK ∗ 3 1 LK ∗ = O (Z) 5 O (Z)I , · ∗ j K . MK 2 2
(B.29)
(B.30)
Finally, we consider the interactions of baryon decuplet [∗ (1358)] with the vector mesons (K ∗ ) in Figs. 7(h) and (i). In order to use the SU(3) symmetry relation, we start with the N interaction, f N 3 1 L N = i O (Z)5 I (B.31) , · (j − j )N , M 2 2 where the coupling constant can be fixed by the quark model relation [186], f N =
fN g NN M (1 + ) . fNN 2MN
(B.32)
By using g NN = 3.1 and = 1.0, we find f N ≈ 5.5. This should be compared with the range 3.5–7.8 of Ref. [218]. By using the SU(3) relation f N M K ∗ fK ∗ N ∗ = − √ ≈ 2.59 , 6 M
(B.33)
we then fix the K ∗ N ∗ coupling, LK ∗ N ∗ = i
fK ∗ N ∗ ∗ ∗ (j K − j K ∗ ) · O 5 N . MK ∗
(B.34)
With the above Lagrangians, we can evaluate all diagrams in Fig. 7 by specifying the propagator of spin-3/2 ∗ (1385) and (1232). Here we follow Refs. [203,204] and use the following Rarita–Schwinger form for the propagator of a spin-3/2 particle of a mass MY ∗ and momentum p, ˜ (Y ∗ , p) =
p2
−i (Y ∗ , p) , − MY2 ∗
(B.35)
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Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
where (Y ∗ , p) = (p/ + MY ∗ )S (Y ∗ , p) −
2(p 2 − MY2 ∗ ) 3MY2 ∗
[ p + p − (p/ − MY ∗ ) ] ,
(B.36)
with S (Y ∗ , p) = −g +
1 1 2 ( p − p ) + p p . + 3 3MY ∗ 3MY2 ∗
(B.37)
The decay width of the decuplet is included by replacing MY ∗ by MY ∗ − iY ∗ /2 in the propagator. Similar form of the propagator is also used for N = in evaluating Fig. 7(c) and (e). B.3. (1405) −
The (1405) with J P = 21 is also included in our calculation of Figs. 7(a)–(e). Its decay width is = 50 ± 2 MeV and it mostly decays into the channel. To get its coupling with K, we consider scalar coupling SU(3) Lagrangian, (B.38) L1 = −ig1 1 KN + K c + · + + H.c. , where 1 is the (1405) field. This leads to 2 3g 1 p ((1405) → ) = M2 + p2 + M , 4 M1 where M1 is the mass of (1405) and p = (M2 1 , M2 , M2 )/(2M1 ) with (x, y, z) = x 2 + y 2 + z2 − 2(xy + yz + zx) .
(B.39)
(B.40)
So using ((1405) → )expt. ≈ 50 MeV, we obtain g1 ≈ 0.13 and the needed KN(1405) coupling can be computed from Eq. (B.38). Fig. 7(d) can have 1 1 vertex. This is calculated from L1 1 =
e 1 1 j A 1 , 2MN
(B.41)
where 1 is the magnetic moment of (1405). There is no experimental information for the magnetic moment of (1405). So we have to reply on model predictions. It has been estimated to be 0.22–0.25 in Skyrme model [219] and 0.24–0.45 in a unitarized chiral perturbation theory [220]. Here we use 1 = 0.25 in the nucleon magneton unit. For the intermediate hyperon state in Fig. 7(d), we also include (1405) → (1116) and (1405) → (1193) couplings. They are defined by eg Y 1 1 5 Y F + H.c. , (B.42) LY 1 = 4(M1 + MY ) which leads to (1 → Y ) =
em g2Y 1 p3 (M1 + MY )2
,
(B.43)
where p = (M2 1 − MY2 )/(2M1 ). By using the quark model predictions [216] ((1405) → ) = 143 keV,
((1405) → 0 ) = 91 keV ,
(B.44)
the parameters for LY 1 are then fixed as g1 ≈ 2.67,
g1 ≈ 3.34 .
(B.45)
The predicted decay width ((1405) → ∗0 (1385)) = 0.3 keV is suppressed and not considered in this calculation.
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
81
B.4. (1520) −
We now consider the calculations of Fig. 7 with Y = (1520) which is a J P = 23 state. There is no information for the magnetic moment of (1520). We therefore neglect the tensor coupling of the electromagnetic interaction of (1520) by setting [(1520)] = 0. The coupling of (1520) with photon thus has the same form as the L∗ ∗ in Eq. (B.9) except that (1520) is a neutral particle. For the KN (1520) (denoted as ) and KN (1520) couplings, we write fKN 5 j KN + H.c. , MK f LKN = ie KN A 5 K − p + H.c. , MK LKN =
(B.46)
where is the (1520) field. This gives the decay width of (1520) → N K as [(1520) → NK] =
3 1 fKN 2 pK 2 −M . MN2 + pK N 6 MK M
(B.47)
Using [(1520) → N K]expt. ≈ 7 MeV, we have fKN ≈ 10.92 .
(B.48)
Fig. 7(d) also includes the photon transition between (1520) and other hyperons. This is calculated by using the following Lagrangian [207], LY =
ieg 1 eg 2 O Y F − O (j Y )F . 2MY 4MY2
(B.49)
Then we have
( → Y ) =
2 MR (M − M ) R Y 2MY 48MR2 MY2 MR 2 MR 2 2 . +3 g1 − g2 (MR − MY ) − 8g2 MR 2g1 − g2 2MY 2MY em p3
g1 (3MR + MY ) − g2
(B.50)
For L , we use the data [(1520) → ]expt. = 159 ± 33 ± 26 keV [221]18 and set g2 = 0 as in the case of → N to get g1 ≈ 1.46 .
(B.51)
For the photo-transitions of (1520) to other hyperons, we again are guided by the quark model predictions Ref. [216], [(1520) → ] = 74 keV, [(1520) → ∗ ] ∼ 0 .
[(1520) → (1405)] = 0.2 keV , (B.52)
As the decay widths of (1520) → ∗ and (1405) are negligible, we only consider the coupling with the channel. Using [(1520) → ] = 74 keV and also setting g2 = 0, we get L with g1 ≈ 1.39. With the above four subsections, we have constructed the Lagrangians for calculating all diagrams in Fig. 7. We now turn to discussing production mechanisms. +26
18 The new measurement of the CLAS collaboration [222], 167 ± 43 −12 keV, is consistent with the value of Ref. [221].
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Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
Appendix C. Form factors and current conservation Here, we discuss how we restore current conservation with the form factors in the form of Eq. (28). Introducing form factors breaks current conservation. So, motivated by the works of Refs. [177–180], we restore current conservation by introducing contact diagrams, which effectively changes the form factors of some diagrams into a universal form factor. For example, let us assume that we have two diagrams, a and b, and, when combined, these two diagrams satisfy current conservation without form factors. The corresponding form factors Fa and Fb as functions of Mandelstam variables break current conservation. Introducing contact diagrams effectively gives the universal form factor which is a constant [177] or in the form of [179], Fa , Fb → 21 (Fa + Fb )
(C.1)
so that the current conservation condition is satisfied. It was pointed out by Ref. [180] that such a choice would not satisfy crossing symmetry and another form was suggested, although not unique, Fa , Fb → 1 − (1 − Fa )(1 − Fb ) .
(C.2)
In this work, since we have three-body final state, we extend the above method to the diagrams which spoil current conservation due to form factors. But we do not make any modification to the purely transverse amplitudes, such as the terms with K ∗ exchanges and photo-transitions among hadrons, which are constructed to be gauge-invariant by themselves individually. For later use, we define the Mandelstam variables as s = (k + p1 )2 , s1 = (q1 + q2 )2 , s2 = (q1 + p2 )2 , s3 = (q2 + p2 )2 , t1 = (k − q1 )2 , t2 = (k − q2 )2 , t3 = (k − p2 )2 , t4 = (p1 − q1 )2 , t5 = (p1 − q2 )2 , t6 = (p1 − p2 )2 ,
(C.3)
where the momenta of the initial photon and the nucleon are k and p1 , respectively, while those of K, K, and the final nucleon are q1 , q2 , and p2 , respectively. C.1. t-channel Drell-type diagrams Among the possible (tree) t-channel Drell diagrams depicted in Fig. 3, those diagrams with intermediate K ∗ have transverse amplitude only, i.e., each diagram satisfied current conservation individually and introducing form factors does not spoil the current conservation condition. The current conservation problem occurs only when we have intermediate K and K mesons, namely, p → K + K − p and n → K + K − n, since they do not contribute to the reactions 0 0 of p → K 0 K p and n → K 0 K n. Then the amplitude takes a form of
M = Ma Fa + Mb Fb + Mc Fc ,
(C.4)
where 2 Fa = F(t2 , MK )F(t6 , MV2 ),
2 Fb = F(t1 , MK )F(t6 , MV2 ),
Fc = F(t6 , MV2 ) ,
(C.5)
F(t6 , MV2 ) = F (t6 , MV2 )2 ,
(C.6)
where 2 2 2 F(t2 , MK ) = F (t2 , MK ) ,
2 2 2 F(t1 , MK ) = F (t1 , MK ) ,
and the form of F (r, M 2 ) is defined in Eq. (28). One can verify that amplitude (C.4) satisfies current conservation condition when the form factors are set to be 1, k · M = k · (Ma + Mb + Mc ) = 0 .
(C.7)
The condition, k · M, is satisfied by introducing the contact diagrams, which effectively replaces Fa , Fb , and Fc by 1 − (1 − Fa )(1 − Fb )(1 − Fc ) .
(C.8)
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
83
C.2. Vector meson and tensor meson parts Since the form factors and their cutoff parameters are fixed by vector meson photoproduction and tensor meson photoproduction processes where the vector or tensor mesons are on mass-shell, we need to include the form factors in order to take into account the off-shellness of the intermediate vector/tensor mesons. Furthermore, the amplitudes M of Eq. (17) and M , of Eq. (19) are constructed to satisfy the current conservation condition. So the form factor which should be multiplied in addition to the form factors for vector/tensor meson photoproduction part reads F = F (s1 , M 2 )2 ,
(C.9)
where M is the (produced) vector meson or tensor meson mass. C.3. Intermediate hyperons C.3.1. Intermediate spin-1/2 hyperons In this case, since we adopt the pseudoscalar coupling, we have five diagrams as shown in Figs. 7(a)–(e). The possible intermediate states are (1116), (1405), and (1193). We first consider the case of (1193), which have the properties as k · Mnc = 0, +
k · M d = 0, 0
k · Mne = 0 ,
k · M d = k · Mb + k · Ma = −k · Me − k · Mc , p p k · Ma + k · Mc = 0, k · Mb + k · Me = 0 , p
p
(C.10)
where the superscripts p, n denote the proton and neutron, respectively, and the subscripts specify the diagram in Fig. 7. For p → K + K − p, we have
p,
0 ,
M = Ma Fa + Mb Fb + Mc Fc + Md
p,
Fd + M e Fe .
(C.11)
Because of the properties of Eq. (C.10), we have k · M = 0 without form factors. Since the amplitudes Ma and Mc correspond to a part of photoproduction, which should have nontrivial form factors, we replace the form factors as 2 2 ) )(1 − F (s, MN2 )2 )}F (s2 , M2 )2 . Fa , Fc → {1 − (1 − F (t2 , MK
(C.12)
Similarly, 2 2 ) )(1 − F (t3 , MN2 )2 )}F (t5 , M2 )2 , Fb , Fe → {1 − (1 − F (t1 , MK
(C.13)
so that current conservation is now satisfied with the form factors. 0 For p → K 0 K p, we have + ,
p,
M = 2(Mc Fc + Md
p,
Fd + M e Fe ) ,
(C.14)
where the form factors Fc , Fd , and Fe are replaced by 1 − (1 − Fc )(1 − Fd )(1 − Fe ) .
(C.15)
For n → K + K − n, we have
n,
− ,
M = 2(Ma Fa + Mb Fb + Mc Fc + Md
n,
Fd + M e Fe ) ,
(C.16)
where the form factors Fa , Fb , and Fd are replaced by 1 − (1 − Fa )(1 − Fb )(1 − Fd ) . Since
n, Mc
and
n, Me
are transverse, the form factors Fc and Fe are not changed.
(C.17)
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Y. Oh et al. / Physics Reports 423 (2006) 49 – 89 0
For n → K 0 K n, we have n,
0 ,
M = Mc Fc + Md
n,
Fd + M e Fe .
(C.18)
Since all the amplitudes are transverse, they are gauge invariant. The intermediate (1116) and (1405) states are the same as the case of 0 . As a consequence, these intermediate 0 states do not exist for the p → K 0 K p and n → K + K − n reactions. We replace the form factors as in Eq. (C.11) 0 for p → K + K − p. The amplitudes for n → K 0 K n are all transverse. C.3.2. Intermediate spin-3/2 hyperons The intermediate states (1385) and (1520) contain seven diagrams shown in Figs. 7(a)–(g), which satisfy ∗0
k · M d = 0,
k · Mnc = 0, ∗+
k · Mne = 0 ,
∗−
k · M = −k · M = −k · Mc − k · Me , d d p k · Ma + k · Mc + k · Mf = 0 , p k · Mb + k · M e + k · M g = 0 p
p
(C.19)
for the case of intermediate ∗ (1385). For p → K + K − p, we have
∗0 ,
p,
M = Ma Fa + Mb Fb + Mc Fc + Md
p,
F d + M e Fe + M f F f + M g Fg .
(C.20)
In this case, the direct application of the method of Ref. [180], Fa , Fc , Ff → 1 − (1 − Fa )(1 − Fc )(1 − Ff ),
Fb , Fe , Fg → 1 − (1 − Fb )(1 − Fe )(1 − Fg ) ,
(C.21)
cannot be used. This is because the above form factor becomes 1 if, for example, any of (Fa , Fc , Ff ) is 1. And this in fact happens when the intermediate ∗ becomes on mass-shell, s2 = M2 ∗ since Ff = F (s2 , M2 ∗ )2 . Since these diagrams are a part of photoproduction of K and ∗ , this means that those amplitudes have no form factor. Therefore, instead of Eq. (C.21), we replace the form factors as 2 2 ) )(1 − F (s, MN2 )2 )}F (s2 , M2 ∗ )2 , Fa , Fc , Ff → {1 − (1 − F (t2 , MK
2 2 Fb , Fe , Fg → {1 − (1 − F (t1 , MK ) )(1 − F (t3 , MN2 )2 )}F (t5 , M2 ∗ )2 .
(C.22)
0
For p → K 0 K p, we have ∗+ ,
p,
M = 2(Mc Fc + Md
p,
Fd + M e Fe ) .
(C.23)
The form factors Fc , Fd , and Fe are replaced by 1 − (1 − Fc )(1 − Fd )(1 − Fe ) .
(C.24)
For n → K + K − n, we have
n,
∗− ,
M = 2(Ma Fa + Mb Fb + Mc Fc + Md
n,
F d + M e Fe + M f F f + M g Fg ) ,
(C.25)
where the form factors Fa , Fb , Fd , Ff , and Fg are 1 − (1 − Fa )(1 − Fb )(1 − Fd )(1 − Ff )(1 − Fg ) ,
(C.26)
Ff = Fg = Ff Fg ,
(C.27)
and
which is chosen to avoid the problem mentioned above.
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
85
0
For n → K 0 K n, we have ∗0 ,
n,
M = Mc Fc + Md
n,
Fd + M e Fe .
(C.28)
In this case, all amplitudes are gauge-invariant, we do not introduce any contact diagrams. 0 The case of intermediate (1520) is the same as in the case of 0 (1385). So it does not contribute to the p → K 0 K p and n → K + K − n reactions. C.4. Intermediate This case is similar to the case of intermediate except that has positive charge. The K ∗ exchange diagrams, Figs. 8(f, g), are transverse, so current conservation is satisfied even with the presence of form factors. The amplitudes of Fig. 8 satisfy k · Mne = 0 , k · Mnc = 0, k · Ma + k · M b + k · M d = 0 , k · Mc + k · Md + k · Me = 0 .
(C.29)
0
For p → K 0 K p, we have p,
p,
M = Mc Fc + Md Fd + Me Fe + Mf Ff + Mg Fg ,
(C.30)
Fc , Fd , Fe → 1 − (1 − Fc )(1 − Fd )(1 − Fe ) .
(C.31)
where
For n → K + K − n, we have n,
n,
M = Ma Fa + Mb Fb Mc Fc + Md Fd + Me Fe + Mf Ff + Mg Fg ,
(C.32)
Fa , Fb , Fd → 1 − (1 − Fa )(1 − Fb )(1 − Fd ) .
(C.33)
where
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
(LEPS Collaboration), T. Nakano, et al., Phys. Rev. Lett. 91 (2003) 012002. (DIANA Collaboration), V.V. Barmin, et al., Yad. Fiz. 66 (2003) 1763 [Phys. At. Nucl. 66 (2003) 1715]. (CLAS Collaboration), S. Stepanyan, et al., Phys. Rev. Lett. 91 (2003) 252001. (SAPHIR Collaboration), J. Barth, et al., Phys. Lett. B 572 (2003) 127. (CLAS Collaboration), V. Kubarovsky, et al., AIP Conf. Proc. 698 (2004) 543. A.E. Asratyan, A.G. Dolgolenko, M.A. Kubantsev, Yad. Fiz. 67 (2004) 704 [Phys. Atom. Nucl. 67 (2004) 682]. (CLAS Collaboration), V. Kubarovsky, et al., Phys. Rev. Lett. 92 (2004) 032001. S. Armstrong, B. Mellado, S.L. Wu, J. Phys. G 30 (2004) 1801. (HERMES Collaboration), A. Airapetian, et al., Phys. Lett. B 585 (2004) 213. (SVD Collaboration), A. Aleev, et al., Yad. Fiz. 68 (2005) 1012 [Phys. Atom. Nucl. 68 (2005) 974]. P.Zh. Aslanyan, V.N. Emelyanenko, G.G. Rikhkvitzkaya, hep-ex/0403044; P.Zh. Aslanyan, Nucl. Phys. A 755 (2005) 375. (COSY-TOF Collaboration), M. Abdel-Bary, et al., Phys. Lett. B 595 (2004) 127. (ZEUS Collaboration), S. Chekanov, et al., Phys. Lett. B 591 (2004) 7. (SVD Collaboration), A. Aleev, et al., hep-ex/0509033. (NA49 Collaboration), C. Alt, et al., Phys. Rev. Lett. 92 (2004) 042003. (H1 Collaboration), A. Aktas, et al., Phys. Lett. B 588 (2004) 17. (BES Collaboration), M. Ablikim, et al., Phys. Rev. D 70 (2004) 012004. (for the PHENIX Collaboration), C. Pinkenburg, J. Phys. G 30 (2004) S1201. (ALEPH Collaboration), S. Schael, et al., Phys. Lett. B 599 (2004) 1. (SPHINX Collaboration), Yu.M. Antipov, et al., Eur. Phys. J. A 21 (2004) 455. (BABAR Collaboration), B. Aubert, et al., hep-ex/0408064. (for the BABAR Collaboration), T. Berger-Hryn’ova, Int. J. Mod. Phys. A 20 (2005) 3749.
86
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
[23] (for the CDF Collaboration), I.V. Gorelov, hep-ex/0408025. [24] (for the CDF Collaboration), D.O. Litvintsev, Nucl. Phys. B (Proc. Suppl.) 142 (2005) 374. [25] (HERA-B Collaboration), I. Abt, et al., Phys. Rev. Lett. 93 (2004) 212003. [26] (HyperCP Collaboration), M.J. Longo, et al., Phys. Rev. D 70 (2004) 111101. [27] (Belle Collaboration), K. Abe, et al., hep-ex/0411005. [28] (FOCUS Collaboration), K. Stenson, et al., Int. J. Mod. Phys. A 20 (2005) 3745. ¯ [29] (CLAS Collaboration), M. Battaglieri, et al., hep-ex/0510061. [30] (WA89 Collaboration), M.I. Adamovich, et al., hep-ex/0510013. [31] L.G. Landsberg, Phys. Rep. 320 (1999) 223. [32] (SPHINX Collaboration), Yu.M. Antipov, et al., Yad. Fiz. 65 (2002) 2131, [Phys. Atom. Nucl. 65 (2002) 2070]. [33] (BES Collaboration), M. Ablikim, et al., hep-ex/0405030. [34] V. Kuznetsov (for the GRAAL Collaboration), hep-ex/0409032. ˙ [35] P.A. Zołnierczuk, et al., Phys. Lett. B 597 (2004) 131. [36] B. Tatischeff, nucl-ex/0404042. [37] (for the STAR Collaboration), S. Kabana, hep-ex/0406032. [38] (for the STAR Collaboration), H.Z. Huang, hep-ex/0509037. [39] H.G. Fischer, S. Wenig, Eur. Phys. J. C 37 (2004) 133. [40] (WA89 Collaboration), M.I. Adamovich, et al., Phys. Rev. C 70 (2004) 022201. [41] J. Pochodzalla, hep-ex/0406077. [42] E. Golowich, Phys. Rev. D 4 (1971) 262. [43] (Particle Data Group), M. Aguilar-Benitez, et al., Phys. Lett. B 170 (1986) 1. [44] H. Gao, B.-Q. Ma, Mod. Phys. Lett. A 14 (1999) 2313. [45] T. Hirose, K. Kanai, S. Kitamura, T. Kobayashi, Nuovo Cim. 50A (1979) 120. [46] C. Fukunaga, R. Hamatsu, T. Hirose, W. Kitamura, T. Yamagata, Nuovo Cim. 58A (1980) 199. [47] V.M. Karnaukhov, V.I. Moroz, C. Coca, A. Mihul, Phys. Lett. B 281 (1992) 148. [48] A.V. Aref’ev, et al., Yad. Fiz. 51 (1990) 414, [Sov. J. Nucl. Phys. 51 (1990) 264]. [49] J. Amirzadeh, et al., Phys. Lett. 89B (1979) 125. [50] B.M. Abramov, et al., Yad. Fiz. 53 (1991) 179, [Sov. J. Nucl. Phys. 53 (1991) 114]. [51] D. Aston, et al., Phys. Rev. D 32 (1985) 2270. [52] R.L. Jaffe, Phys. Rev. D 15 (1977) 267. [53] D. Strottman, Phys. Rev. D 20 (1979) 748. [54] H. Hogaasen, P. Sorba, Nucl. Phys. B 145 (1978) 119. [55] M. De Crombrugghe, H. Hogaasen, P. Sorba, Nucl. Phys. B 156 (1979) 347. [56] K. Maltman, S. Godfrey, Nucl. Phys. A 452 (1986) 669. [57] A. Ferrer, V.F. Perepelitsa, A.A. Grigoryan, Z. Phys. C 56 (1992) 215. [58] H.J. Lipkin, Phys. Lett. B 195 (1987) 484. [59] C. Gignoux, B. Silvestre-Brac, J.M. Richard, Phys. Lett. B 193 (1987) 323. [60] S. Fleck, C. Gignoux, J.M. Richard, B. Silvestre-Brac, Phys. Lett. B 220 (1989) 616. [61] S. Zouzou, J.-M. Richard, Few-Body Syst. 16 (1994) 1. [62] Fl. Stancu, Phys. Rev. D 58 (1998) 111501. [63] M. Genovese, J.M. Richard, Fl. Stancu, S. Pepin, Phys. Lett. B 425 (1998) 171. [64] C.G. Callan, I. Klebanov, Nucl. Phys. B 262 (1985) 365. [65] M. Rho, D.O. Riska, N.N. Scoccola, Phys. Lett. B 251 (1990) 597. [66] Y. Oh, D.-P. Min, M. Rho, N.N. Scoccola, Nucl. Phys. A 534 (1991) 493. [67] D.O. Riska, N.N. Scoccola, Phys. Lett. B 299 (1993) 338. [68] M.B. Voloshin, M.A. Shifman, Yad. Fiz. 45 (1987) 463 [Sov. J. Nucl. Phys. 45 (1987) 292]. [69] M.B. Voloshin, M.A. Shifman, Yad. Fiz. 47 (1988) 801 [Sov. J. Nucl. Phys. 47 (1988) 511]. [70] N. Isgur, M.B. Wise, Phys. Lett. B 232 (1989) 113. [71] N. Isgur, M.B. Wise, Phys. Lett. B 237 (1990) 527. [72] E. Jenkins, A.V. Manohar, M.B. Wise, Nucl. Phys. B 396 (1993) 27. [73] Z. Guralnik, M. Luke, A.V. Manohar, Nucl. Phys. B 390 (1993) 474. [74] D.-P. Min, Y. Oh, B.-Y. Park, M. Rho, Int. J. Mod. Phys. E 4 (1995) 47. [75] Y. Oh, B.-Y. Park, D.-P. Min, Phys. Rev. D 49 (1994) 4649. [76] J. Schechter, A. Subbaraman, Phys. Rev. D 51 (1995) 2311. [77] M. Harada, F. Sannino, J. Schechter, H. Weigel, Phys. Rev. D 56 (1997) 4098. [78] Y. Oh, B.-Y. Park, D.-P. Min, Phys. Rev. D 50 (1994) 3350. [79] Y. Oh, B.-Y. Park, Phys. Rev. D 51 (1995) 5016. [80] Y. Oh, B.-Y. Park, Phys. Rev. D 53 (1996) 1605. [81] Y. Oh, B.-Y. Park, Z. Phys. A 359 (1997) 83. [82] Y. Oh, B.-Y. Park, D.-P. Min, Phys. Lett. B 331 (1994) 362. [83] C.-K. Chow, Phys. Rev. D 53 (1996) 5108. [84] (E791 Collaboration), E.M. Aitala, et al., Phys. Rev. Lett. 81 (1998) 44.
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89 [85] (E791 Collaboration), E.M. Aitala, et al., Phys. Lett. B 448 (1999) 303. [86] K. Cheung, Phys. Rev. D 69 (2004) 094029. [87] H.-Y. Cheng, C.-K. Chua, C.-W. Hwang, Phys. Rev. D 70 (2004) 034007. [88] X.-G. He, X.-Q. Li, Phys. Rev. D 70 (2004) 034030. [89] T.E. Browder, I.R. Klebanov, D.R. Marlow, Phys. Lett. B 587 (2004) 62. [90] I.W. Stewart, M.E. Wessling, M.B. Wise, Phys. Lett. B 590 (2004) 185. [91] B. Wu, B.-Q. Ma, Phys. Rev. D 70 (2004) 034025. [92] H. Kim, S.H. Lee, Y. Oh, Phys. Lett. B 595 (2004) 293. [93] M. Chemtob, Nucl. Phys. B 256 (1985) 600. [94] A.V. Manohar, Nucl. Phys. B 248 (1984) 19. [95] M. Praszałowicz, in: M. Jezabeck, M. Praszałowicz (Eds.), Skyrmions and Anomalies, World Scientific, Singapore, 1987, p. 112. [96] H. Weigel, Eur. Phys. J. A 2 (1998) 391. [97] D. Diakonov, V. Petrov, M. Polyakov, Z. Phys. A 359 (1997) 305. [98] M.V. Polyakov, A. Sibirtsev, K. Tsushima, W. Cassing, K. Goeke, Eur. Phys. J. A 9 (2000) 115. [99] H. Walliser, V.B. Kopeliovich, Zh. Eksp. Teor. Fiz. 124 (2003) 483, [J. Exp. Theor. Phys. 97 (2003) 433]. [100] M. Praszałowicz, Phys. Lett. B 575 (2003) 234. [101] N. Itzhaki, I.R. Klebanov, P. Ouyang, L. Rastelli, Nucl. Phys. B 684 (2004) 264. [102] B.K. Jennings, K. Maltman, Phys. Rev. D 69 (2004) 094020. [103] B.-Y. Park, M. Rho, D.-P. Min, Phys. Rev. D 70 (2004) 114026. [104] M. Praszalowicz, K. Goeke, Acta Phys. Pol. B 36 (2005) 2255. [105] T.D. Cohen, Phys. Lett. B 581 (2004) 175. [106] A. Cherman, T.D. Cohen, T.R. Dulaney, E.M. Lynch, hep-ph/0509129. [107] H. Walliser, H. Weigel, hep-ph/0510055. [108] S.L. Zhu, Phys. Rev. Lett. 91 (2003) 232002. [109] R.D. Matheus, F.S. Navarra, M. Nielsen, R. da Silva, S.H. Lee, Phys. Lett. B 578 (2003) 323. [110] J. Sugiyama, T. Doi, M. Oka, Phys. Lett. B 581 (2004) 167. [111] M. Eidemüller, Phys. Lett. B 597 (2004) 314. [112] T.D. Cohen, R.F. Lebed, Phys. Lett. B 578 (2004) 150. [113] E. Jenkins, A.V. Manohar, Phys. Rev. Lett. 93 (2004) 022001. [114] E. Jenkins, A.V. Manohar, JHEP 0406 (2004) 039. [115] A.V. Manohar, Phys. Rev. D 70 (2004) 014004. [116] F. Csikor, Z. Fodor, S.D. Katz, T.G. Kovács, JHEP 0311 (2003) 070. [117] S. Sasaki, Phys. Rev. Lett. 93 (2004) 152001. [118] N. Mathur, et al., Phys. Rev. D 70 (2004) 074508. [119] T.-W. Chiu, T.-H. Hsieh, Phys. Rev. D 72 (2005) 034505. [120] M. Karliner, H.J. Lipkin, hep-ph/0307243. [121] R. Jaffe, F. Wilczek, Phys. Rev. Lett. 91 (2003) 232003. [122] Y. Oh, H. Kim, S.H. Lee, Phys. Rev. D 69 (2004) 094009. [123] F.E. Close, J.J. Dudek, Phys. Lett. B 586 (2004) 75. [124] S.H. Lee, H. Kim, Y. Oh, J. Korean Phys. Soc. 46 (2005) 774. [125] D. Diakonov, V. Petrov, Phys. Rev. D 69 (2004) 094011. [126] R. Bijker, M.M. Giannini, E. Santopinto, Eur. Phys. J. A 22 (2004) 319. [127] Fl. Stancu, D.O. Riska, Phys. Lett. B 575 (2003) 242. [128] C.E. Carlson, C.D. Carone, H.J. Kwee, V. Nazaryan, Phys. Lett. B 573 (2003) 101. [129] C.E. Carlson, C.D. Carone, H.J. Kwee, V. Nazaryan, Phys. Lett. B 579 (2004) 52. [130] S.M. Gerasyuta, V.I. Kochkin, Phys. Rev. D 71 (2005) 076009. [131] N.I. Kochelev, H.J. Lee, V. Vento, Phys. Lett. B 594 (2004) 87. [132] M. Praszałowicz, Acta Phys. Pol. B 35 (2004) 1625. [133] B. Wu, B.-Q. Ma, Phys. Rev. D 69 (2004) 077501. [134] J. Ellis, M. Karliner, M. Praszałowicz, JHEP 0405 (2004) 002. [135] V. Dmitrašinovi´c, Fl. Stancu, hep-ph/0402190. [136] M.P. Rekalo, E. Tomasi-Gustafsson, J. Phys. G 30 (2004) 1459. [137] K. Nakayama, W.G. Love, Phys. Rev. C 70 (2004) 012201. [138] A.W. Thomas, K. Hicks, A. Hosaka, Prog. Theor. Phys. 111 (2004) 291. [139] C. Hanhart, et al., Phys. Lett. B 590 (2004) 39. [140] M.P. Rekalo, E. Tomasi-Gustafsson, Phys. Lett. B 591 (2004) 225; M.P. Rekalo, E. Tomasi-Gustafsson, Eur. Phys. J. A 22 (2004) 119. [141] Y.N. Uzikov, Phys. Lett. B 595 (2004) 277; Y.N. Uzikov, hep-ph/0402216. [142] C. Hanhart, J. Haidenbauer, K. Nakayama, U.-G. Meißner, Phys. Lett. B 606 (2005) 67. [143] W. Liu, C.M. Ko, Phys. Rev. C 68 (2003) 045203. [144] W. Liu, C.M. Ko, Nucl. Phys. A 741 (2004) 215.
87
88
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89
[145] S.I. Nam, A. Hosaka, H.-C. Kim, Phys. Lett. B 579 (2003) 43. [146] Y. Oh, H. Kim, S.H. Lee, Phys. Rev. D 69 (2004) 014009. [147] Q. Zhao, J.S. Al-Khalili, Phys. Lett. B 585 (2004) 91. [148] B.-G. Yu, T.-K. Choi, C.-R. Ji, Phys. Rev. C 70 (2004) 045205. [149] Y. Oh, H. Kim, S.H. Lee, Nucl. Phys. A 745 (2004) 129. [150] W. Liu, C.M. Ko, V. Kubarovsky, Phys. Rev. C 69 (2004) 025202. [151] K. Nakayama, K. Tsushima, Phys. Lett. B 583 (2004) 269. [152] A.R. Dzierba, D. Krop, M. Swat, S. Teige, A.P. Szczepaniak, Phys. Rev. D 69 (2004) 051901. [153] W. Roberts, Phys. Rev. C 70 (2004) 065201. [154] W. Roberts, nucl-th/0412041. [155] A.I. Titov, H. Ejiri, H. Haberzettl, K. Nakayama, Phys. Rev. C 71 (2005) 035203. [156] A. Sibirtsev, J. Haidenbauer, S. Krewald, U.-G. Meißner, A.W. Thomas, hep-ph/0509145. [157] Y. Oh, H. Kim, S.H. Lee, Phys. Rev. D 69 (2004) 074016. [158] T. Hyodo, A. Hosaka, E. Oset, Phys. Lett. B 579 (2004) 290. [159] C.E. Carlson, C.D. Carone, H.J. Kwee, V. Nazaryan, Phys. Rev. D 70 (2004) 037501. [160] P. Ko, J. Lee, T. Lee, J.-H. Park, Phys. Lett. B 611 (2005) 87. [161] Y.-R. Liu, A. Zhang, P.-Z. Huang, W.-Z. Deng, X.-L. Chen, S.-L. Zhu, Phys. Rev. D 70 (2004) 094045. [162] Y. Oh, H. Kim, Phys. Rev. D 70 (2004) 094022. [163] (for the CLAS Collaboration), H.G. Juengst, Nucl. Phys. A 754 (2005) 265. [164] K. Hicks, Int. J. Mod. Phys. A 20 (2005) 219. [165] (for the CLAS Collaboration), V.D. Burkert, R. de Vita, S. Niccolai, nucl-ex/0408019. [166] (for the CLAS Collaboration), V. Kubarovsky, P. Stoler, Nucl. Phys. B (Proc. Suppl.) 142 (2005) 356. [167] (for the LEPS Collaboration), Y. Ohashi, hep-ex/0402005. [168] J. Tyson, J.S. Greenberg, V.W. Hughes, D.C. Lu, R.C. Minehart, S. Mori, J.E. Rothberg, Phys. Rev. Lett. 19 (1967) 255. [169] P. Söding, Phys. Lett. 19 (1966) 702. [170] S.D. Drell, Phys. Rev. Lett. 5 (1960) 278. [171] S.D. Drell, Rev. Mod. Phys. 33 (1961) 458. [172] A. Szczurek, A.P. Szczepaniak, Phys. Rev. D 71 (2005) 054005. [173] A.R. Dzierba, C.A. Meyer, A.P. Szczepaniak, J. Phys. Conf. Ser. 9 (2005) 192. [174] E.W. Anderson, et al., Phys. Lett. 29B (1969) 136. [175] R.A. Arndt, I.I. Strakovsky, R.L. Workman, Phys. Rev. C 68 (2003) 042201, 69 (2004) 019901(E). [176] R.A. Arndt, I.I. Strakovsky, R.L. Workman, Nucl. Phys. A 754 (2005) 261. [177] K. Ohta, Phys. Rev. C 40 (1989) 1335. [178] H. Haberzettl, Phys. Rev. C 56 (1997) 2041. [179] H. Haberzettl, C. Bennhold, T. Mart, T. Feuster, Phys. Rev. C 58 (1998) 40. [180] R.M. Davidson, R. Workman, Phys. Rev. C 63 (2001) 025210. [181] M. Oka, hep-ph/0409295. [182] S.H. Lee, H. Kim, Y. Kwon, Phys. Lett. B 609 (2005) 252. [183] R.D. Matheus, S. Narison, hep-ph/0412063. [184] H.-J. Lee, N.I. Kochelev, V. Vento, Phys. Lett. B 610 (2005) 50. [185] T. Sato, T.-S.H. Lee, Phys. Rev. C 54 (1996) 2660. [186] R. Machleidt, K. Holinde, C. Elster, Phys. Rep. 149 (1987) 1. [187] M. Bando, T. Kugo, K. Yamawaki, Phys. Rep. 164 (1988) 217. [188] (Particle Data Group), S. Eidelman, et al., Phys. Lett. B 592 (2004) 1. [189] Y. Oh, A.I. Titov, T.-S.H. Lee, in: V.D. Burkert, L. Elouadrhiri, J.J. Kelly, R.C. Minehart (Eds.), NSTAR2000 Workshop: Excited Nucleons and Hadronic Structure, World Scientific, Singapore, 2000, pp. 255–262, nucl-th/0004055. [190] Y. Oh, A.I. Titov, T.-S.H. Lee, Phys. Rev. C 63 (2001) 025201. [191] Y. Oh, T.-S.H. Lee, Phys. Rev. C 66 (2002) 045201. [192] Y. Oh, T.-S.H. Lee, Phys. Rev. C 69 (2004) 025201. [193] Y. Eisenberg, et al., Phys. Rev. Lett. 23 (1969) 1322. [194] Y. Eisenberg, et al., Phys. Rev. D 5 (1972) 15. [195] G.T. Condo, T. Handler, W.M. Bugg, G.R. Blackett, M. Pisharody, K.A. Danyo, Phys. Rev. D 48 (1993) 3045. [196] A.V. Afanasev, A.P. Szczepaniak, Phys. Rev. D 61 (2000) 114008. [197] H. Högaasen, J. Högaasen, R. Keyser, B.E.Y. Svensson, Nouvo Cimento 42A (1966) 323. [198] E.R. Berger, A. Donnachie, H.G. Dosch, O. Nachtmann, Eur. Phys. J. C 14 (2000) 673. [199] B. Renner, Nucl. Phys. B 30 (1971) 634. [200] P. Singer, Phys. Rev. D 27 (1983) 2223. [201] S. Ishida, K. Yamada, M. Oda, Phys. Rev. D 40 (1989) 1497. [202] (Aachen-Hamburg-Heidelberg-München Collaboration), W. Struczinski, et al., Nucl. Phys. B 108 (1976) 45. [203] M. El Amiri, G. López Castro, J. Pestieau, Nucl. Phys. A 543 (1992) 673. [204] D. Lin, M.K. Liou, Z.M. Ding, Phys. Rev. C 44 (1991) 1819. [205] R.M. Davidson, N.C. Mukhopadhyay, R.S. Wittman, Phys. Rev. D 43 (1991) 71.
Y. Oh et al. / Physics Reports 423 (2006) 49 – 89 [206] [207] [208] [209] [210] [211] [212] [213] [214] [215] [216] [217] [218] [219] [220] [221] [222]
M. Benmerrouche, R.M. Davidson, N.C. Mukhopadhyay, Phys. Rev. C 39 (1989) 2339. M. Benmerrouche, N.C. Mukhopadhyay, J.F. Zhang, Phys. Rev. D 51 (1995) 3237. R. Bijker, M.M. Giannini, E. Santopinto, Phys. Lett. B 595 (2004) 260. (ABBHHM Collaboration), R. Erbe, et al., Phys. Rev. 188 (1969) 2060. J. Ballam, et al., Phys. Rev. D 5 (1972) 545. K. Hicks, V. Burkert, A.E. Kudryavtsev, I.I. Strakovsky, S. Stepanyan, Phys. Rev. D 71 (2005) 098501. Th.A. Rijken, V.G.J. Stoks, Y. Yamamoto, Phys. Rev. C 59 (1999) 21. V.G.J. Stoks, Th.A. Rijken, Phys. Rev. C 59 (1999) 3009. S. Deser, A. Waldron, V. Pascalutsa, Phys. Rev. D 62 (2000) 105031. D.B. Lichtenberg, Phys. Rev. D 15 (1977) 345. J.W. Darewych, M. Horbatsch, R. Koniuk, Phys. Rev. D 28 (1983) 1125. G.E. Brown, W. Weise, Phys. Rep. 22 (1975) 279. H. Kammano, M. Arima, Phys. Rev. C 69 (2004) 025206. C.L. Schat, N.N. Scoccola, C. Gobbi, Nucl. Phys. A 585 (1995) 627. D. Jido, A. Hosaka, J.C. Nacher, E. Oset, A. Ramos, Phys. Rev. C 66 (2002) 025203. (SPHINX Collaboration), Yu.M. Antipov, et al., Phys. Lett. B 604 (2004) 22. (CLAS Collboration), S. Taylor, et al., Phys. Rev. C 71 (2005) 054609.
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Physics Reports 423 (2006) 91 – 158 www.elsevier.com/locate/physrep
Flux compactifications in string theory: A comprehensive review Mariana Grañaa, b, c,∗ a Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France b Centre de Physique Théorique, Ecole Polytechnique, 91128 Palaiseau Cedex, France c Service de Physique Théorique, CEA/Saclay, 91191 Gif-sur-Yvette Cedex, France
Accepted 20 October 2005 Available online 9 December 2005 editor: A. Schwimmer
Abstract We present a pedagogical overview of flux compactifications in string theory, from the basic ideas to the most recent developments. We concentrate on closed-string fluxes in type-II theories. We start by reviewing the supersymmetric flux configurations with maximally symmetric four-dimensional spaces. We then discuss the no-go theorems (and their evasion) for compactifications with fluxes. We analyze the resulting four-dimensional effective theories, as well as some of its perturbative and non-perturbative corrections, focusing on moduli stabilization. Finally, we briefly review statistical studies of flux backgrounds. © 2005 Published by Elsevier B.V. PACS: 11.25.−w; 11.25.Mj
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2. Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3. Type-II supersymmetric backgrounds with flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.1. Supersymmetric solutions in the absence of flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.2. Supersymmetric backgrounds with fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.3. N = 1 Minkowski vacua (and beyond) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.4. Internal manifold and generalized complex geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.5. N = 1 flux vacua as generalized Calabi–Yau manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4. No-go theorems for compact solutions with fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.1. Four-dimensional Einstein equation and no-go . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2. Bianchi identities and equations of motion for flux: tadpole cancellation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.3. Bianchi identities and special type IIB solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5. Four-dimensional effective theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.1. Effective theory for compactifications of type-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2. Effective theory for Calabi–Yau orientifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.3. Flux-induced potential and gauged supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.4. Flux-induced superpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.5. Mirror symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 ∗ Corresponding author at: Service de Physique Théorique, CEA/Saclay, 91191 Gif-sur-Yvette Cedex, France.
E-mail address:
[email protected]. 0370-1573/$ - see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.physrep.2005.10.008
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6. Moduli stabilization by fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.1. Moduli stabilization in type IIB Calabi–Yau orientifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.2. Moduli stabilization in type IIB orientifolds of tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.3. Moduli stabilization in type IIA Calabi–Yau orientifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.4. Moduli stabilization in type-IIA orientifolds of tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7. Moduli stabilization including non-perturbative effects and De Sitter vacua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.1. Corrections to the low energy action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.2. Non-perturbative corrections to the superpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.3. Moduli stabilization including non-perturbative effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.4. De Sitter vacua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8. Distributions of flux vacua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9. Summary and future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Appendix A. Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
1. Introduction One of the central questions in string theory concerns the existence and viability of semi-realistic four-dimensional vacua. The current paradigm of particle phenomenology prefers an N = 1 matter sector with spontaneously broken supersymmetry at low energies. A huge amount of effort in string theory is devoted to finding such spontaneously broken N = 1 vacuum with a Standard Model sector. As soon as the E8 ×E8 and SO(32) heterotic theories were constructed, vacuum configurations with four-dimensional N = 1 supersymmetry were found by compactifying the heterotic string on Calabi–Yau manifolds [1]. Unbroken N = 1 supersymmetry at the compactification scale in the heterotic theory is a very stringent requirement. If the vacuum is a product of four-dimensional maximally symmetric space and some compact manifold, the former can only be Minkowski, and the latter is required to be Calabi–Yau. Furthermore, no vacuum expectation value for the NS field strength is allowed. The situation improves when a warped factor multiplying the space–time metric is taken into account [2]. The NS field can acquire a vacuum expectation value, but the price to pay was too high at the time: the internal manifold is no longer Kähler. Not much was known about non-Kähler manifolds, and, as a consequence, the resulting four-dimensional effective theory was largely unknown. The attraction was therefore concentrated on flux-less Calabi–Yau or toroidal orbifold compactifications of the heterotic theory (with, however, vevs for internal fluxes, which break the gauge group to the standard model or GUT groups) [3]. Supersymmetry is spontaneously broken in these models by four-dimensional non-perturbative effects, such as gaugino condensation [4]. Due to the lack of technologies to study non-perturbative phenomena at string level, the structure of non-perturbative effects, as well as the possibility to break supersymmetry spontaneously, are determined by field theoretic considerations. In spite of the enormous progress achieved over the years, the mechanism is not yet satisfactory, as it always leads to negative cosmological constants, and suffers from other cosmological problems [5]. However, heterotic or type-I string internal fluxes can, besides breaking the SO(32) group to the standard model one, trigger spontaneous supersymmetry breaking [6]. Small tadpoles for the metric and dilaton are not canceled at the classical level in this construction, but are hoped to be canceled by higher loop or non-perturbative corrections. Nevertheless, one gets by this mechanism a satisfactory theoretical control of supersymmetry breaking, and consequently a good description of the low-energy physics. The scene changed drastically after the discovery of D-branes as non-perturbative BPS objects in string theory [7]. D-branes can serve as ingredients in constructing four-dimensional standard-like models [8]. Additionally, they constitute the previously missing sources for RR fluxes. Very soon after their discovery, the possibility of finding new supersymmetric vacua for type-II string theories with non-vanishing vacuum expectation values for RR field strengths was envisaged [9,10]. Solutions with background fluxes became rapidly more interesting from the theoretical and phenomenological point of view. Non-vanishing vacuum expectation values for the field strengths were shown to serve as a way to partially break the N = 2 supersymmetry of Calabi–Yau (non) compactifications down to N = 1 by mass deformation [11]. In conformally flat six-dimensional spaces, fluxes can break the N = 4 supersymmetry to N = 3, 2, 1, 0 in a controlled and stable way [12–15]. Fluxes became an even more attractive mechanism of partially breaking supersymmetry after AdS/CFT correspondence was conjectured [16]: type-IIB solutions with 3-form fluxes could realize string theory duals
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of confining gauge theories [17–19]. While partially breaking supersymmetry, fluxes give vacuum expectation values to some of the typically large number of massless fields (“moduli”) arising in string theory compactifications [11,20,21]. In some IIA scenarios discussed recently [22–24], fluxes alone can stabilize all moduli classically in a regime where the supergravity approximation can be trusted. Fluxes generate at the same time warped metrics, which can realize large hierarchies [21] as in Randall–Sundrum-type models [25,26]. Fluxes cannot however be turned on at will in compact spaces, as they give a positive contribution to the energy momentum tensor [27,21]. As a consequence, negative tension sources (orientifold planes) should be added for consistent compactifications of type-II theories. The number of units of fluxes allowed has therefore always an upper bound, given by the geometry of the compactification manifold. This still leaves nevertheless a huge amount of freedom, making compactifications in background fluxes one of the most rich and attractive ingredients in the ultimate goal of realizing string-based models of particle physics and early universe cosmology. Looking at the story from the opposite perspective, flux compactifications are perhaps too rich. Despite the flurry of activity in the field, we still lack of an understanding of whether any of the large amount of available perturbative vacua (the dense “discreetuum” [28], or “landscape” [29]) is in any sense preferred over the rest (either dynamical, cosmological or antrhopically). In the absence of a vacuum selection principle, a statistical study of the landscape was advocated as possible guidance principle for the search of the right vacuum [30,31]. The purpose of this review is to provide a pedagogical exploration of the literature on flux compactifications, from the basic ideas to the recent developments. We do not plan (and cannot be) exhaustive, as the field has evolved enormously, and the amount of literature on the subject is huge. Although we give a large number of references, the citation list is clearly not exhaustive either. We decided to concentrate on type-II compactifications with N = 1 supersymmetric flux vacua. We discuss the effective four-dimensional theories, as well as some of its perturbative and non-perturbative corrections, focusing on moduli stabilization. We also give a brief overview of statistical studies of flux backgrounds. Inevitably, many recent and not so recent developments in flux compactifications will not be covered in this review. Among them, some of the main subjects not to be discussed (for practical reasons, not for lack of interest) are open string fluxes and open string moduli stabilization. Besides, not much will be said about M-theory flux vacua, and their potential to stabilize moduli. There has been a lot of very recent progress in understanding the open and M-theory landscapes [32], open moduli stabilization by open- and closed-string fluxes [33–35] and moduli stabilization in Mtheory [20,36,37], which is worth a review by itself. Neither do we discuss twisted moduli, and their stabilization mechanisms [23,38,39]. In the final summary we mention other topics not covered in this review. The paper is organized as follows. In Section 2, we give the basic definitions to be used all throughout the review. In Section 3, we discuss type-II N = 1 Minkowski backgrounds with flux. In Section 3.4, we review very briefly generalized complex geometry, with the purpose of describing the internal geometries of N = 1 vacua, which we do in Section 3.5. The reader interested in the main theme of compactifications on (conformally rescaled) Ricci-flat manifolds can skip these two sections, which are not needed to understand most of the rest of the review. In Section 4, we discuss the no-go theorems for compactifications with fluxes, and the way string theory avoids them. In Section 5, we review the four-dimensional effective theories in Calabi–Yau and Calabi–Yau orientifold compactifications of type-II theories. Flux-generated potentials and their superpotential origins are discussed in Sections 5.3 and 5.4. We end up the section with a brief discussion of mirror symmetry in flux backgrounds, done in Section 5.5. In Section 6, we review moduli stabilization by fluxes. We discuss the general mechanism of flux stabilization in IIB and IIA Calabi–Yau orientifolds in Sections 6.1 and 6.3, and illustrate with examples for the simpler cases of orientifolds of tori in Sections 6.2 and 6.4 for IIB and IIA, respectively. In Section 7.1, we discuss some corrections to the low-energy effective action, reviewing in Sections 7.3 and 7.4 moduli stabilization including these corrections, and de Sitter vacua. Finally, we give in Section 8 a very brief overview of the distributions of flux vacua. We finish by a summary, mentioning some topics not covered in the review.
2. Basic definitions In this section we give the basic definitions that will be used all along the review. The definitions of some parameters less frequently used, as well as conventions, are left to the Appendix. The massless bosonic fields of type-II superstring theory are the dilaton , the metric tensor gMN and the antisymmetric 2-tensor BMN in the NS–NS sector. The massless RR sector of type-IIA contains a 1-form and 3-form potentials
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CM , CMNP . That of type-IIB comprises the axion C0 , the 2-form potential CMN , and the four-form field CMNP Q with self-dual five-form field strength. In type-IIB, the two scalars C0 and can be combined into a complex field = C0 + ie− which parameterizes an SL(2, R)/U (1) coset space. 1 The fermionic superpartners are two Majorana–Weyl gravitinos A M , A=1, 2 of opposite chirality in IIA (11 IIA M = 1 2 2 A A IIA M ; 11 IIA M =−IIA M ) and the same chirality in type IIB (11 IIB M =IIB M ); and two Majorana–Weyl dilatinos A with opposite chirality than the gravitinos. Type-II theories have D = 10, N = 2 supersymmetry with two Majorana–Weyl supersymmetry parameters A of the same chirality as the corresponding gravitinos. The field strength for the NS flux is defined H = dB .
(2.1)
For the RR field strengths, we will use the democratic formulation of Ref. [40], who actually considers all RR potentials (C1 . . . C9 in IIA, and C0 , C2 . . . C10 in IIB), imposing a self-duality constraint on their field strengths to reduce the doubling of degrees of freedom. The RR field strengths are given by1 F (10) = dC − H ∧ C + meB = Fˆ − H ∧ C ,
(2.2)
(10) where F (10) is the formal sum of all even (odd) fluxes in IIA (IIB), Fˆ = dC + meB , and m ≡ F0 = Fˆ0 is the mass parameter of IIA. These RR fluxes are constrained by the Hodge-duality relation (10)
Fn(10) = (−1)Int[n/2] F10−n ,
(2.3)
where is a ten-dimensional Hodge star. The Bianchi identities for the NS flux and the democratic RR fluxes are dH = 0,
dF (10) − H ∧ F (10) = 0 .
(2.4)
When sources are present, there is no globally well-defined potential, and the integral of the field strength over a cycle does is not necessarily zero. When this is the case, there is a non-zero flux. Charges are quantized in string theory,2 and therefore the fluxes have to obey Dirac quantization conditions. Any flux with a standard Bianchi identity (NS or RR) should satisfy 1 Fˆp ∈ Z (2.5) √ p−1 p 2 for any p-cycle p . By Poincaré and Hodge duality, there are as many 2- as 4-cycles in homology, while 3-cycles come in pairs (A, B). We therefore define electric and magnetic fluxes for each field strength according to 1 1 h3 K H = m , H = e , K = 1, . . . , 3 3 K 2 (2)2 AK (2)2 B K 1 1 Fˆ3 = mK Fˆ3 = eRR K , RR , 2 (2) AK (2)2 B k 1 1 a ˆ F Fˆ4 = eRR a , a = 1, . . . , h2 . = m , (2.6) √ 2 √ 3 RR a 2 Aa B 2 We have not defined an integral flux for F1 and F5 because there are no non-trivial 1 and 5-cycles in Calabi–Yau 3-folds (which will be the manifold we will mostly deal with). The distinction between A- and B-cycles is conventional at this level. In non-compact Calabi–Yau’s, the A-cycles are compact, while the B-ones go off to infinity. 1 The notation F (10) is used to distinguish them from the purely internal fluxes F defined in (3.4) and used all throughout the review. It should not be confused with the supraindices (1), (3), (6), (8) in Tables 3 and 4 below, which denote a particular SU(3) representation. 2 From the pure supergravity point of view, the charges are continuous parameters. In the quantum theory, they are quantized, and the total number of quanta will play a particularly important role in Section 4.
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We define the Poincare duals to the cycles as K = [B K ], K = [AK ], wa = [B a ], w˜ a = [Aa ], or equivalently L K = K ∧ L = L , = L ∧ K = − L K K , AL
b = Aa
a ∧ ˜ b = ba ,
BK
Ba
˜b =
˜ b ∧ a = − ba ,
(2.7)
where an integral without a subindex indicates an integral over the whole six-dimensional manifold. These relations imply that the field strengths can be expanded in the following way: 1 (2)2 1 √
2
H3 = mK K − eK K ,
(2)2
K Fˆ3 = mK RR K − eRR K
1 ˆ ˜a √ 3 F4 = −eRR a 2
Fˆ2 = maRR a ,
√ 2 Fˆ0 = m0RR ,
1
1 ˆ √ 5 F6 = eRR 0 Vol6 . 2
(2.8)
In most of the text, we will take (2)2 = 1. Factors of (2)2 are written explicitly only in a few equations, when they are relevant. There is a symplectic Sp(2h(1,1) + 2, Z) and Sp(2h(2,1) + 2, Z) invariance, part of which correspond to electric– magnetic duality. We can define the symplectic vectors N = (eK , mK ),
IIB NRR = (eRR K , mK RR ),
IIA NRR = (eRR A , mA RR ) ,
(2.9)
0 A where (eA RR , mA RR ) = (eRR 0 , eRR a , mRR , mRR ).
3. Type-II supersymmetric backgrounds with flux In this section we review what are the possible configurations of fluxes and internal geometry that N = 1 supersymmetry allows. By analyzing integrability conditions, it was proved [41–43] that in the context of type-II supergravity, a background that is supersymmetric and whose fluxes satisfy Bianchi identities and equations of motion is a solution to the full equations of motion (whenever there are no mixed external–internal components of the Einstein tensor, which will be our case). In this section we concentrate on supersymmetry conditions, while Bianchi identities and the equations of motion for flux are discussed in Section 4. The analysis of supersymmetry conditions in (unwarped) compactifications of the heterotic string in the presence of NS flux has been carried out in the celebrated paper [1]. The absence of warp factor (in the Einstein frame) enforces the flux to vanish. Warped backgrounds with NS flux have been found to be consistent with supersymmetry for the heterotic theory in Ref. [2] (see also [44]), and have been taken up for type-II theories using the language of G-structures (to be reviewed in Section 3.2) in Refs. [42,45,46]. Supersymmetric M-theory compactifications on four-folds to three dimensions were first analyzed in Ref. [47]. M (and F-theory) compactifications with fluxes to three and four dimensions were discussed in Refs. [20,48], and analyzed using the language of G-structures in Refs. [49–51] (see Ref. [52] for a review and more references). Type-IIA (and M-theory) supersymmetric backgrounds on manifolds of G2 and SU(3) structure were first studied in Refs. [53–55] using G2 and SU(3) structure techniques. Supersymmetric type-IIB and F-theory backgrounds preserving a particular type of N = 1 supersymmetry with both NSNS and RR fluxes (such that the complex flux G3 is imaginary self-dual) where studied in Refs. [12,13] (for a review and more references see [56]). N = 1 type-IIB flux backgrounds preserving more general supersymmetries were studied in Ref. [57], while the most general N = 1 supersymmetric ansatz in manifolds with SU(3) structure (and some with SU(2) structure) has been studied in Refs. [58–60] (see also Ref. [61]). Twenty years after the seminal paper by Candelas, Horowitz, Strominger, Witten, and thanks to the work of many people, the most general type-II backgrounds compatible with N = 1 supersymmetry on manifolds of SU(3) structure
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are now known [60,58,43,64,61], and a lot is known about flux backgrounds on SU(2) structure manifolds [59,62,63]. In this section we will review type-II supersymmetric solutions in the absence of flux, and in the following sections we review their flux counterparts, following mostly Refs. [60,62]. Explicit examples of supersymmetric solutions will be given later, mostly in Section 6. Before starting to review the technical details, let us note that a classification of supersymmetric solutions from the Killing spinors (and G-structures) has been carried out for example in Ref. [41]. Besides, some supersymmetric solutions with holographic duals were constructed using the symmetries of the construction to make an ansatz for the Killing spinors and the bosonic fields [65]. This method is often referred to as “algebraic Killing spinor technique” [66]. In this review, we will discuss vacua whose four-dimensional space admits maximal space–time symmetry, i.e. Minkowski, anti-de Sitter space (AdS4 ) or de Sitter (dS4 ). These have respectively Poincare, SO(1, 4) and SO(2, 3) invariance. The most general ten-dimensional metric consistent with four-dimensional maximal symmetry is ds 2 = e2A(y) g˜ dx dx + gmn dy m dy n ,
= 0, 1, 2, 3, m = 1, . . . , 6 ,
(3.1)
where A is a function of the internal coordinates called warp factor, g˜ is a Minkowski, dS4 or AdS4 metric, and gmn is any six-dimensional metric. Demanding maximal symmetry requires the vacuum expectation value of the fermionic fields to vanish. The background should therefore be purely bosonic. As far as the fluxes, we are allowed to turn on only those who have either no leg or four legs along space–time. Therefore the NSNS flux H3 can only be internal, while from the RR fluxes, only F4 in IIA and F5 in IIB are allowed to have external components. A supersymmetric vacuum where only bosonic fields have non-vanishing vacuum expectation values should obey Q = = 0, where Q is the supersymmetry generator, is the supersymmetry parameter and is any fermionic A field. In type-II theories, the fermionic fields are two gravitinos A M , A = 1, 2 and two dilatinos . In the supergravity approximation, the bosonic parts of their supersymmetry transformations in the string frame3
M = ∇M +
1 1 (10) F/n M Pn , H / M P + e 4 16 n
(3.2)
1 1
= j/ + H (−1)n (5 − n)/ Fn(10) Pn . / P + e 2 8 n
(3.3) 1
In these equations M = 0, . . . , 10, M stands for the column vector M = ( M 2 ) containing the two Majorana–Weyl M spinors of the same chirality in type IIB, and of opposite chirality in IIA, and similarly for and . The 2 × 2 matrices (n/2) P and Pn are different in IIA and IIB: for IIA P = 11 and Pn = 11 1 , while for IIB P = −3 , Pn = 1 for 2 (n + 1)/2 even and Pn = i for (n + 1)/2 odd. A slash means a contraction with gamma matrices in the form F/n = (1/n!)FP1 ...PN P1 ...PN , and HM ≡ 21 HMNP NP . The NS and RR field strengths are defined in (2.1, 2.2). We are using the democratic formulation of Ref. [40] for the RR fields, as explained in Section 2. We want to study flux backgrounds that preserve maximal four-dimensional symmetry. We therefore require Fn(10) = Fn + Vol4 ∧ F˜n−4 .
(3.4)
Using the duality relation (2.3), the internal and external components are related by [60] F˜n−4 = (−1)Int[n/2] ∗ F10−n ,
(3.5)
where ∗ is a six-dimensional star. This allows to write the supersymmetry transformation in terms of internal fluxes (10) only Fn , n = 0, . . . , 6. For instance a non-zero F4 with only -type indices is traded for a “internal” F6 with m-type indices. In (3.2), (3.3) this gives twice the contribution for each flux but now n = 0, . . . , 6 only.
3 Throughout the paper we use mostly string frame. Whenever Einstein frame is used, it will be indicated explicitly.
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3.1. Supersymmetric solutions in the absence of flux When no fluxes are present, demanding zero VEV for the gravitino variation (3.2) requires the existence of a covariantly constant spinor on the ten-dimensional manifold, i.e. ∇M = 0. The space–time component of this equation reads ∇˜ +
1 2
(˜ 5 ⊗ ∇A) / =0 ,
(3.6)
where we have used the standard decomposition of the ten-dimensional gamma matrices (see Appendix A)] and ∇˜ and ˜ mean a covariant derivative and gamma matrix with respect to g˜ . This yields the following integrability condition: [∇˜ , ∇˜ ] = − 21 (∇m A)(∇ m A) .
(3.7)
On the other hand, [∇˜ , ∇˜ ] =
1 ˜ k R = , 4 2
(3.8)
where we have used that for a maximally symmetric space R = k(g g − g g ), with k negative for AdS, zero for Minkowski and positive for dS. Since is invertible, the integrability condition reads k + ∇m A∇ m A = 0 .
(3.9)
The only possible constant value of (∇A)2 on a compact manifold is zero, which implies that the warp factor is constant and the four-dimensional manifold can only be Minkowski space. To analyze the internal component of the supersymmetry variation, we need to split the supersymmetry spinors into four- and six-dimensional spinors. For reasons that will become clear shortly, we will use only one internal Weyl spinor (and its complex conjugate) to do the decomposition, which reads 1IIA = 1+ ⊗ + + 1− ⊗ − , 2IIA = 2+ ⊗ − + 2− ⊗ + ,
(3.10)
1,2 ∗ for type IIA, where 11 1IIA =1IIA and 11 2IIA =−2IIA , and the four- and six-dimensional spinors obey 1,2 − =(+ ) , and − = (+ )∗ . (By a slight abuse of notation we use plus and minus to indicate both four- and six-dimensional chiralities.) For type IIB both spinors have the same chirality, which we take to be positive, resulting in the decomposition A A A IIB = + ⊗ + + − ⊗ − ,
A = 1, 2 .
(3.11)
Inserting these decompositions in the internal component of the gravitino variation, Eq. (3.2), we get the following condition: ∇m ± = 0 .
(3.12)
The internal manifold should therefore have a covariantly constant spinor. This is a very strong requirement from the topological and differential geometrical point of view. It forces the manifold to have reduced holonomy. In the following section we will explain this in more detail (a more detailed pedagogical discussion of special holonomy relevant to the present context can be found for example in Ref. [67]). For the time being, we just state that for sixdimensional manifolds the holonomy group should be SU(3), or a subgroup of it. A six-dimensional manifold with SU(3) holonomy is a Calabi–Yau manifold [68,69]. Such manifolds admit one covariantly constant spinor. To have more than one covariantly constant spinor the holonomy group of the manifold should be smaller than SU(3), and this results in a larger number of supersymmetries preserved. For most of this review, we shall consider the case of manifolds having only one covariantly constant spinor (when turning on fluxes, the covariant constancy condition will be relaxed, but we will still work mostly with manifolds admitting only one non-vanishing spinor). This explains the use of only one internal spinor to decompose the ten-dimensional ones in Eqs. (3.10) and (3.11). When there is one covariantly constant internal spinor, the internal gravitino equation tells us that there are two four-dimensional supersymmetry parameters, 1 and 2 . This compactification preserves therefore eight supercharges,
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i.e. N = 2 in four dimensions. From the world-sheet point of view, a Calabi–Yau compactification yields a super conformal field theory with (2, 2) supersymmetry [69]. In summary, supersymmetric compactifications without fluxes are only possible to unwarped Minkowski fourdimensional space, with a Calabi–Yau manifold as internal space. These compactifications preserve N = 2 in four dimensions. Fluxes can break the N = 2 supersymmetry spontaneously down to N = 1 or even completely in a stable way. In the following sections we will review in detail how this works. As we mentioned before, in order to decompose the ten-dimensional supersymmetry parameters, the internal manifold should admit at least one nowhere vanishing internal spinor. This restricts the class of allowed manifolds to those having reduced structure. We will first review the concept of G-structures, which is central for the development of flux compactifications, and then look at flux backgrounds preserving N = 1 supersymmetry. 3.2. Supersymmetric backgrounds with fluxes In this section we review compactifications preserving the minimal amount of supersymmetry, i.e. N = 1 in four dimensions. In order to have some supercharges preserved, or even in the case when all of them are completely broken spontaneously by the fluxes, we need to have globally well-defined supercurrents. This requires to have globally welldefined spinors on the internal manifold, which is only possible when its structure group is reduced. Let us start by briefly reviewing the main facts about G-structures. For detailed explanations, we refer the reader to the mathematical references Refs. [70–73]. For a review of G-structures in the context of compactifications with fluxes, see Ref. [46]. In the absence of fluxes, supersymmetry requires a covariantly constant spinor on the internal manifold. This condition actually splits into two parts, first the existence of such a spinor (i.e., the existence of a non-vanishing globally welldefined spinor), and second the condition that it is covariantly constant. A generic spinor such as the supercurrent can be decomposed in the same way as the supersymmetry parameters, Eqs. (3.10) and (3.11). The first condition implies then the existence of two four-dimensional supersymmetry parameters and thus an effective N = 2 four-dimensional action, while the second implies that this action has an N = 2 Minkowski vacuum. As far as the internal manifold is concerned, the fist condition is a topological requirement on the manifold, while the second one is a differential condition on the metric, or rather, on its connection. Let us first review the impactions of the first condition. A globally well-defined non-vanishing spinor exists only on manifolds that have reduced structure [70,71]. The structure group of a manifold is the group of transformations required to patch the orthonormal frame bundle. A Riemannian manifold of dimension d has automatically structure group SO(d). All vector, tensor and spinor representations can therefore be decomposed in representations of SO(d). If the manifold has reduced structure group G, then every representation can be further decomposed in representations of G. Let us concentrate on six dimensions, which is the case we are interested in, and the group G being SU(3). On a manifold with SU(3) structure, the spinor representation in six dimensions, in the 4 of SO(6), can be further decomposed in representations of SU(3) as 4 → 3 + 1. There is therefore an SU(3) singlet in the decomposition, which means that there is a spinor that depends trivially on the tangent bundle of the manifold and is therefore well defined and non-vanishing. The converse is also true: a six dimensional manifold that has a globally well defined non-vanishing spinor has structure group SU(3). We can now go ahead and decompose other SO(6) representations, such as the vector 6, 2-form 15 and 3-form 20 ¯ 15 → 8 + 3 + 3¯ + 1, 20 → 6 + 6¯ + 3 + 3¯ + 1 + 1. We can see in representations of SU(3). This yields 6 → 3 + 3, that there are also singlets in the decomposition of 2-forms and 3-forms. This means that there is also a non-vanishing globally well-defined real 2-form, and complex 3-form. These are called, respectively J and . We can also see that there are no invariant vectors (or equivalently five forms), which means in particular that J ∧ = 0. A six-form is on the contrary a singlet (and there is only one of them, up to a constant), which means that J ∧ J ∧ J is proportional to ¯ We use the convention J ∧ J ∧ J = (3i/4) ∧ . ¯ J and determine a metric.4 ∧ . Raising one of the indices of J we get an almost complex structure, which is a map that squares to minus the identity, i.e. Jm p Jp n =− m n . A real matrix that squares to minus the identity has eigenvalues ±i, coming in pairs. The existence of an almost complex structure allows to introduce local holomorphic and antiholomorphic vectors jzi , jzi¯ , i = 1, 2, 3, which are the local eigenvectors with eigenvalues +i and −i. If their dual one-forms dzi are integrable, i.e., there exist local functions f such that dz = df , and if the transition functions between the different patches are holomorphic, then 4 says what are the holomorphic and antiholomorphic coordinates, and in expressed in these coordinates g = −iJ . i¯E i¯E
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the definition of complex coordinates is globally consistent. In that case, the complex structure is said to be integrable, or equivalently the manifold is complex. The condition for integrability of the almost complex structure can be recast in the vanishing of a tensor called Nijenhuis, defined as Nmn p = 2(Jm q ∇[q Jn] p − Jn q ∇[q Jm] p ) .
(3.13)
A complex structure is integrable if its associated Nijenhuis tensor vanishes. Due to the antisymmetrizations, the covariant derivatives in (3.13) can actually be replaced by an ordinary derivatives. The SU(3) structure is determined equivalently by the SU(3) invariant spinor , or by J and . The latter can actually be obtained from the spinor by Jmn = ∓2i†± mn ± ,
mnp = −2i†− mnp + .
(3.14)
Jmn is a (1, 1)-form with respect to the almost complex structure Jm p , while is a (3, 0)-form.5 We argued that supersymmetry imposes a topological plus a differential condition on the manifold. So far we have reviewed the topological condition, which amounts to the requirement that the manifold has SU(3) structure. Let us now see what the differential condition is. In the case of Calabi–Yau 3-folds (this means a Calabi–Yau manifold with three complex dimensions), which are manifolds of SU(3) structure, the SU(3) invariant spinor is also covariantly constant. The metric (or rather the Levi–Civita connection) is said to have SU(3) holonomy. The holonomy group of a connection is the subgroup of O(n) that includes all possible changes of direction that a vector suffers when being parallely transported around a closed loop. In the case of a manifold with SU(3) structure, one can show that there is always a metric compatible connection g = 0), possibly with torsion,6 which is also compatible with the structure and such (i.e., a connection satisfying ∇m np that ∇m = 0 [71]. This means that on a manifold with SU(3) structure there is always a connection with or without torsion that has SU(3) holonomy. In the case where this connection is torsionless, the manifold is a Calabi–Yau. The torsion tensor Tmn p ∈ 1 ⊗ (su(3) ⊕ su(3)⊥ ) ,
(3.15)
where 1 is the space of 1-forms, and comes from the upper index p, while the lower indices mn span the space of two forms, which is isomorphic to so(6), the Lie algebra of SO(6). We have also used so(6) = su(3) ⊕ su(3)⊥ . Acting on SU(3) invariant forms, the su(3) piece drops. The corresponding torsion is called the intrinsic torsion, and contains the following representations: 0 p ¯ ⊗ (1 ⊕ 3 ⊕ 3) ¯ Tmn ∈ 1 ⊗ su(3)⊥ = (3 ⊕ 3) ¯ ⊕ 2(3 ⊕ 3) ¯ = (1 ⊕ 1) ⊕ (8 ⊕ 8) ⊕ (6 ⊕ 6) W1 W2 W3 W4 , W5 .
(3.16)
W1 , . . . , W5 are the five torsion classes that appear in the covariant derivatives of the spinor, of J and of . W1 is a complex scalar, W2 is a complex primitive (1, 1) form (primitivity means (W2 )mn J mn = 0), W3 is a real primitive (2, 1) + (1, 2) form and W4 and W5 are real vectors (W5 is actually a complex (1, 0)-form, which has the same degrees of freedom). Antisymetrizing the covariant derivative of J and and decomposing into SU(3) representations, we can see that dJ should contain W1 , W3 and W4 , while W1 , W2 and W5 appear in d (see for example Ref. [72] for details). We can therefore write Im(W¯ 1 ) + W4 ∧ J + W3 , d = W1 J 2 + W2 ∧ J + W¯ 5 ∧ . dJ =
3 2
(3.17)
We give in Eq. (A.9) in the appendix the inverse relations, namely Wi in terms of dJ , d, J and . A manifold of SU(3) structure is complex if W1 = 0 = W2 . We can understand that this condition is necessary by noting that the pieces containing W1 and W2 in d are (2, 2)-forms, while itself is a (3, 0)-form. In a complex 5 This can be seen from (3.14) by the fact that is a Clifford vacuum annihilated by gamma matrices with holomorphic indices, i.e. i = 0. + + 6 The torsion is defined by [∇ , ∇ ]V = −R q q mnp Vq − 2Tmn ∇q Vp . m n p
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Table 1 Vanishing torsion classes in special SU(3) structure manifolds Manifold
Vanishing torsion class
Complex Symplectic Half-flat Special Hermitean Nearly Kähler Almost Kähler Kähler Calabi–Yau “Conformal” Calabi–Yau
W 1 = W2 = 0 W1 = W 3 = W 4 = 0 ImW1 = ImW2 = W4 = W5 = 0 W1 = W 2 = W 4 = W 5 = 0 W2 = W 3 = W 4 = W 5 = 0 W1 = W 3 = W4 = W 5 = 0 W1 = W2 = W3 = W4 = 0 W1 = W 2 = W 3 = W 4 = W 5 = 0 W1 = W2 = W3 = 3W4 − 2W5 = 0
manifold, the exterior derivative of a (p, q)-form should only have (p + 1, q) and (p, q + 1) pieces, which means that if the manifold was complex, d could only be a (3,1)-form. Therefore for the manifold to be complex, W1 and W2 must vanish. It can be shown that this condition is also sufficient, and is equivalent to requiring the Nijenhuis tensor defined in Eq. (3.13) to be zero. In a symplectic manifold, the fundamental 2-form, J, is closed. A symplectic manifold of SU(3) structure has therefore vanishing W1 , W3 and W4 . A Kähler manifold is complex and symplectic, which means that the only possible nonzero torsion is W5 . In that case, the Levi–Civita connection has U(3) holonomy. Finally, a Calabi–Yau has SU(3) holonomy, and all torsion classes vanishing. We summarize this and also give the vanishing torsion classes in other special manifolds in Table 1. 3.3. N = 1 Minkowski vacua (and beyond) We have discussed the topological condition on the internal manifold required in order to have some supersymmetry preserved. In this section, we will see what is the differential condition that N = 1 supersymmetry imposes. This differential condition will link the allowed torsion classes to the fluxes, i.e. given an SU(3) structure manifold with certain non-vanishing torsion classes, the allowed fluxes are completely determined by the torsion. We know that imposing SU(3) structure on the manifold allows us to decompose the two ten-dimensional spinors as in Eqs. (3.10), (3.11). We have to insert now these decompositions in the gravitino and dilatino variations, Eqs. (3.2)–(3.2). But before doing that, we should notice that if we leave the four-dimensional spinors 1,2 generic, then the supersymmetry preserved would be N = 2 instead of N = 1. We need therefore a relation between 1 and 2 . Demanding maximal four-dimensional symmetry only allows a trivial relation between 1 and 2 , namely they should be proportional. The (complex) constant of proportionality can actually be a function of internal space, which can be included in the definition of six-dimensional spinors. We will therefore use 1IIA = + ⊗ (a+ ) + − ⊗ (a ¯ − ), 2 ¯ IIA = + ⊗ (b− ) + − ⊗ (b+ ),
1IIB = + ⊗ (a+ ) + − ⊗ (a ¯ −) , 2 ¯ IIB = + ⊗ (b+ ) + − ⊗ (b− ) ,
(3.18)
where a and b are complex functions. N = 1 supersymmetry links a and b, and how they are related tells us how the N = 1 vacuum sits in the underlying N = 2 effective four-dimensional theory. When inserting these spinors in the supersymmetry variations, Eqs. (3.2), (3.3), the four-dimensional pieces can be factored out, and we get equations involving the six-dimensional parts of the spinors. It is useful to decompose the resulting spinors in a basis, given by + , m − , m + , − , where the first (last) two have positive (negative) chirality. We can write schematically the resulting equations for the positive chirality spinor as follows
: P + + Pm m − = 0 ,
m : Qm + + Qmn n − = 0 ,
: R+ + Rm m − = 0 .
(3.19)
P , Q and R contain contributions coming from the torsion, the NS and RR fluxes, warp factor e2A , cosmological constant and derivatives of the functions a and b used in the decomposition (3.18). In Table 2 we indicate how the different representations contribute to P .Q and R. We are using that + is a Clifford(6) vacuum, annihilated by i + =0.
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Table 2 Decomposition of supersymmetry equations representations Torsion
NSNS flux
RR flux
jm A
jm a
Qij ¯ Qi Qij Qij ¯
Qij ¯ ,R Q i , Ri Qij Qij ¯
P P i , Q i , Ri Qij Qij ¯
P -
Pi -
Qi -
1 3 6 8
Table 3 Possible N = 1 vacua in IIA a = bei (BC)
IIA
a = 0 or b = 0 (A)
1
W 1 = H3 = 0 (1) (1) (1) (1) F0 = ∓F2 = F4 = ∓F6
(1)
(1)
F2n = 0 Generic (8) W2+ = e F2 W2− = 0 (6) W 3 = H3 = 0 ¯ (3) ¯ , F2 = 2iW¯ 5 = −2ij¯ A = 23 ij W4 = 0
8 (8)
W2 = F 2
(8)
= F4
=0
(6)
W3 = ∓ ∗ 6 H3 ¯ (3) ¯ W¯ 5 = 2W4 = ∓2iH3 = j j¯ A = j¯ a = 0
6 3
=0
W2+ = e F2 W2− = 0
(8)
(8)
+ e F4
Table 4 Possible N = 1 vacua in IIB IIB
a = 0 or b = 0 (A)
1 8 6
W 1 = F 3 = H3 = 0 W2 = 0 (6) F3 = 0 (6) W3 = ± ∗ H3 ¯ (3) ¯ W¯ 5 =2W4 =∓2iH3 =2j ¯jA = j¯ a = 0
3
(1)
a = ±ib (B)
a = ±b (C)
W3 = 0 (6) (6) e F3 = ∓ ∗ H3 ¯ ( 3) e F5 = 23 iW¯ 5 =iW4 =−2ij¯ A=−4ij¯ log a ¯ =0 j
H3 = 0 (6) W3 = ±e ∗ F3 ¯ ( 3) ¯ ±e F =2iW¯ 5 =−2ij¯ A=−4ij¯ log a=−ij
(ABC)
(1)
¯ (3)
e F1
F
¯ (3)
= 2e F5
(6)
3
(3.20) (3.21)
¯ = iW¯ 5 = iW4 = ij
The explicit expressions (for the case = 0) for these tensors are given in Ref. [60]. Eqs. (3.19) give a relation between the torsion, fluxes, warp factor and cosmological constant in each representation. We will skip the details of the derivation done in [60] (see also Refs. [59,58,64,61,43]) and quote the results. Tables 3 and 4, taken from Ref. [60], give all the possible N = 1 Minkowski vacua with SU(3) structure for type-IIA and type-IIB theories7 (for AdS4 vacua, see for example Refs. [43,64,61]). The last column in Table 4 corresponds to intermediate (“ABC”) solutions, satisfying 2abW 3 = e (a 2 + b2 ) ∗ 6 F3
(6)
, (6)
(a − b )W3 = −(a + b ) ∗ 6 H3 2
2
(6) 2ab H3
2
= −e (a − b 2
2
2
(6) )F3
,
, (3.20)
7 + (−) in the first columns of Tables 3, 4 correspond to a = 0 (b = 0), W ± are the real and imaginary parts of W and all fluxes not written in 2 2
the tables are zero.
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¯ (3)
e F3 = j¯ A = − W¯ 5 = ¯ (3)
−4i ab(a 2 + b2 ) j¯ a, a 4 − 2ia 3 b + 2iab3 + b4
4(ab)2 j¯ a, a 4 − 2ia 3 b + 2iab3 + b4
2(a 4 − 4a 2 b2 + b4 ) j¯ a, a 4 − 2ia 3 b + 2iab3 + b4
H3 =
−2i(a 2 + b2 )(a 2 − b2 ) j¯ a . a 4 − 2ia 3 b + 2iab3 + b4
W4 = ¯ (3)
2(a 2 − b2 )2 j¯ a , a 4 − 2ia 3 b + 2iab3 + b4
e F5 = j¯ =
−4ab(a 2 − b2 ) j¯ a , a 4 − 2ia 3 b + 2iab3 + b4
2(a 2 + b2 )2 j¯ a , a 4 − 2ia 3 b + 2iab3 + b4 (3.21)
a and b are two complex functions, satisfying |a|2 + |b|2 = eA . There is also a gauge freedom in their phases: rescaling + → ei + (or equivalently (a, b) → ei (a, b)), then → e2i , leaving J invariant. As a consequence, (W1 , W2 ) → e2i (W1 , W2 ) and W5 → W5 + 2id. From the tables we can see that W1 is always zero in vacua, while only one of W2± is nonzero in some IIA solutions. Furthermore, the transformation of a, b and that of W5 cancel out in the supersymmetry transformations. This means that the overall phase of ab can be fixed by rotating W2 (Table 3 is given in a fixed gauge). As a consequence, from the four real parameters in a, b, one is fixed by the normalization condition and another one by the gauge choice and consequently only two are physical. All N = 1 vacua with SU(3) structure can therefore be parameterized by two angles, as argued in Ref. [58,60], in the form a = eA/2 cos ei( /2) , b = eA/2 sin e−i( /2) .
(3.22)
These two angles parameterize a U (1)R subspace in the SU(2)R symmetry of the N = 2 underlying effective theory [74]. Note that in IIA there are no intermediate solutions (the solutions on the second column of Table 3, for which the susy parameters are of “interpolating” type BC, do not depend on the interpolating parameter ). Type A corresponds to a solution with NS flux only (plus, in IIA, possible additional RR flux in singlet representations) which is common to IIA, IIB and the heterotic theory, found in Ref. [2] (see Ref. [75] for an extensive analysis). It involves a complex non-Kähler manifold (W1 = W2 = 0, but W3 = 0). In the second column, Type-BC, the solution has RR flux only, and corresponds to the dimensional reduction of an M-theory solution on a seven-dimensional manifold with G2 holonomy [54]. The fact that there are no intermediate solutions was explained in Ref. [60] by looking at the 11-dimensional origin of the solutions: M-theory compactifications on seven manifolds with G2 structure group where shown in Ref. [76] to forbid fluxes, thus leading to compactifications on manifolds of G2 holonomy. Their dimensional reduction gives the second column of Table 3. In order to allow non-trivial fluxes, the structure group on the seven-dimensional manifold should be further reduced to SU(3) or subgroups thereof. An SU(3) structure in seven dimensions involves a vector in addition to the fundamental 2-form and holomorphic 3-form of its six-dimensional counterpart. If the reduction to six dimensions involves a second vector, then the resulting structure group of the six-dimensional manifold is SU(2) rather than SU(3). In order to get SU(3) structure in six dimensions, the two vectors should coincide. In this case, the M-theory four-form flux reduces purely to NS three-form flux (plus possibly some additional RR flux in singlet representations, corresponding to M-theory flux along space–time) giving the first column in Table 3. In IIB, on the contrary, there are solutions with intermediate values of and . Types A, B and C are special because these angles are constant. Type-A solution in the first column is the same as the first column in IIA (setting the RR singlets in the latter case to zero), and corresponds to the solution with NS flux only [2]. Type-C, S-dual to type-A, has RR flux only, and the same non-vanishing torsion classes as type-A. Type B, on the other hand, have, besides RR 5-form flux, RR as well as NSNS 3-form fluxes. They are related by a Hodge duality [12,13], usually expressed in terms of the complex 3-form flux G3 = F3 − ie− H3 = Fˆ3 − H3
(3.23)
( = C0 + ie− the complex combination of axion–dilaton). In Type-B solutions, G3 is imaginary self-dual and has no singlet (0, 3) representation (no flux gets a vev in a singlet representation in IIB, as Table 4 shows) ∗G3 = iG3
and
G(0,3) = 0 .
(3.24)
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The complex 3-form flux G3 , being imaginary self-dual and having no singlet or vector representation is therefore (2, 1) and primitive with respect to the complex structure defined by . For the solution on the first row in the 3 representation (the one not labeled with an “F”), the six-dimensional manifold is a conformal Calabi–Yau (a manifold whose metric is related to that of a Calabi–Yau by a conformal factor). This can be seen from the torsion classes W1 = W2 = W3 = 0, 2W5 = 3W4 = −6j¯ A (cf. Table 1), which means that the six-dimensional metric is 6 (CY) . ds 26 = e−2A ds 2
(3.25)
The conformal factor is therefore the inverse of the warp factor, and it is related to the RR 5-form flux. This class of N = 1 solutions is the most “popular” one, as it involves a Calabi–Yau manifold, whose mathematical properties are very well known. For most of this review, we will concentrate on this very well explored class of solutions. They are dual to M-theory solutions on Calabi–Yau 4-folds found by Becker and Becker [47]. Finally, there is an F-theory-like type-B solution (labeled by an “F”), that involves imaginary self-dual 3-form flux, and additionally a non-constant holomorphic axion–dilaton = (z). The six-dimensional manifold is still complex, but no longer conformal Calabi–Yau, as the torsions W4 and W5 are equal, rather than having a ration of 2/3. Note that for the three types, A, B and C, there is always a complex flux–torsion combination that is (2, 1) and primitive: TypeA: TypeB: TypeC:
dJ ± iH3 F3 ∓ ie d(e
−
−
is (2,1) and primitive , H3 is (2,1) and primitive ,
J ) ± iF3
is (2,1) and primitive ,
(3.26)
where the ± correspond to the two possible relations between a and b in Table 4. Dp-branes and O-planes preserve supersymmetries such that 1 =⊥ 2 , where “⊥” stands for directions perpendicular to the D-brane or O-plane (see for example [77]). For D3-branes and O3-planes, ⊥ is the product of six gammas in Euclidean space, which has eigenvalues ±i. This means that D3-branes and O3-planes preserve supersymmetries of the type a = ±b, which is of type B (the plus (minus) corresponds to D3 (anti-D3)). (Note also that when a = ±ib, the complex spinor 1 +i2 has a definite positive (negative) four-dimensional chirality.) The anti-symmetric product of two gamma matrices has also ±i eigenvalues, and consequently D7-branes and O7-planes preserve type-B supersymmetries. The product of four gamma matrices have eigenvalues ±1, and therefore the supersymmetries preserved by D5/D9branes and O5/O9-planes is of type C. branes wrapped on collapsed two cycles have the supersymmetries of the lower dimensional brane, i.e. for example D5-branes wrapped on the collapsed 2-cycles of the conifold has type-B supersymmetries [21]. D3/D5 or D3/NS5 bound states have instead intermediate BC or AB supersymmetries [57]. In type-IIA, O6 planes wrapped on Special Lagrangian cycles preserve type BC supersymmetries, where the phase corresponds to the overall phase of , which is a gauge choice, as argued below Eq. (3.21). The explicit IIB solutions worked out so far are mostly in the classes A, B or C. Starting from non-compact cases, the prominent ones that have a holographic dual interpretation are Maldacena–Nuñez (MN) [19], Klebanov–Strassler (KS) [18] and Polchinski–Strassler (PS) solutions [17]. MN, corresponding to NS5-branes wrapped on 2-cycles, is a non-compact regular type-A N = 1 background. Its S-dual version, constructed also in Ref. [19], is a type-C solution. KS is a regular non-compact type-B solution, where the Calabi–Yau in question is the conifold. This solution can be “compactified” by adding orientifold 3-planes [21], in the sense that it can be used as a local IR throat geometry of a compact Calabi–Yau, as we will review in Sections 4, 6.1. PS solution, corresponding to D5-branes or NS5branes wrapped on finite 2-cycles with induced D3-charge does not have any of the above supersymmetries, neither an interpolating type. Despite the exact solution is not known yet (PS is constructed perturbatively), it is expected not to have SU(3) structure, but a more reduced one. So do the flow solutions of Ref. [65], which have SU(2) structure, and where obtained using the algebraic killing spinor technique [66]. In the following, we give the reference to some of the type-II solutions discussed in the literature. Type-B backgrounds of IIB involving Calabi–Yau hypersurfaces in weighted projective spaces were constructed for example in Refs. [78–80]. There are many compact type-B solutions involving manifolds with a smaller structure group than SU(3), but which still have supersymmetries obeying the type B condition, a = ib. In these cases, there is more than one well-defined spinor, out of which a subset can be preserved, leading to N = 1 up to N = 4 supersymmetries
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in four dimensions. The latter, N = 4, is a solution with 5-form but no 3-form flux, like AdS5 × S 5 .8 Turning on 3-form fluxes, type-B supersymmetries are preserved if the complex 3-form flux is (2, 1) and primitive with respect to any of the complex structures defined by the preserved spinors. A solution with N = 3 on a six-torus (which has trivial structure group) was constructed in this fashion in Ref. [14]. N = 2 (and N = 1) solutions on orientifolds of K 3 × T 2 (with structure group SU(2)), were studied for example in Refs. [20,81]. One of the first constructions of N = 1 solutions on orientifolds of tori, which will be discussed in detail in Section 6.2, is Ref. [82]. Solutions with various N on tori, and in particular the possibility of connecting them by spherical domain walls is discussed in Ref. [15]. N = 1 type-B flux solutions relevant to particle phenomenology, i.e.containing chiral matter arising at the intersections of magnetized D-branes in representations close to the Standard Model were constructed for example in Refs. [83,168] using tori (see also Ref. [84] for their construction from type I), and in Ref. [38] involving conifolds. Most of the remaining known compact solutions (type C in IIB, with O5-planes, and type BC in IIA, involving O6-planes) were obtained by T -dualizing type-B solutions of IIB, and involve manifolds of trivial structure. Refs. [85,86] for example, constructed IIB type-C solutions on twisted tori (see [87]), starting from type-B solutions on orientifolded (by O3’s) tori. The twisted tori in question are complex, non-Kähler (they have W3 = e ∗ F3 = 0). IIA type-BC solutions (supersymmetric and non-supersymmetric) on twisted tori were also constructed in Refs. [22,24] (the latter having some relevance to particle phenomenology) by minimizing the flux-induced potential. Ref. [24] showed that the supersymmetric Minkowski vacua have W2 = e F2 (implying that the internal manifold is not complex), while AdS vacua have additionally W1 = 0. More references for compact solutions will be seen in Section 6, when discussing moduli stabilization. As for (non-compact) solutions with intermediate supersymmetries, those corresponding to bound states of D3/D5 branes in flat space were obtained in Refs. [88,89] by T -dualizing D4-brane solutions of IIA. Ref. [90] found a oneparameter family of regular IIB solutions interpolating between Maldacena–Nuñez (type-C) and Klebanov–Strassler (type-B), using the interpolating ansatz for the metric and fluxes of Ref. [91]. Finally, let us comment that some of these flux solutions can be related by duality to flux-less solutions, as nicely shown in Refs. [92,93], and also to non-geometric backgrounds [94], as discussed in Ref. [95]. 3.4. Internal manifold and generalized complex geometry In this section we discuss a unifying mathematical description of all internal manifolds arising in supersymmetric flux backgrounds. This description involves generalized complex geometry [96,97]. Readers interested in flux compactifications on Ricci-flat manifolds like Calabi–Yau or tori can skip the following two subsections, and go directly to Section 4. Sections 4 and on involve mostly Ricci-flat manifolds, except some remarks made in Section 5, more precisely at the end of Sections 5.1, 5.3, 5.4 and in most of Section 5.5. Looking back at Table 4 we can see that for all IIB vacua with SU(3) structure, the internal manifold is complex (see Table 1). For IIA, on the other hand, there is solution involving a complex manifold, the one on the first column (type A), while all other solutions are symplectic. A single differential geometric description of the allowed internal manifolds once the back-reaction to the fluxes is taken to account should therefore unify complex and symplectic geometry. Amazingly enough, there is such a description: both are generalized complex manifolds in generalized complex geometry. Generalized complex geometry, proposed by Hitchin [96] and developed in detail by his student Gualtieri [97], was born out of the idea of adding the B-field to differential geometry. One of the first outcomes is that complex and symplectic manifolds are special cases of generalized complex manifolds, which means that generalized complex geometry not only contains complex and symplectic geometry, but it also extends it. In this section we review very briefly the basic ideas that are useful in the context of flux compactifications. For more detail, we refer the reader to the original references [96,97]. Generalized complex geometry has been used in the context of flux compactifications from the space–time point of view in Refs. [60,98–100,62,74]; from the world-sheet perspective in Ref. [101]; in topological strings, D-branes and mirror symmetry [102–104]; most of these papers contain some introduction to generalized complex geometry.
8 From the M × M point of view, AdS × S 5 solution corresponds to a conformally flat six-dimensional manifold, with conformal factor 4 w 6 5 e−2A and warp factor e2A = R 2 /r 2 , where r is a radial coordinate in the six-dimensional space, and R is proportional to the number of units of 5-form flux.
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Usual complex geometry deals with the tangent bundle of a manifold T, whose sections are vectors X, and separately, with the cotangent bundle T ∗ , whose sections are 1-forms . In generalized complex geometry the tangent and cotangent bundle are joined as a single bundle, T ⊕ T ∗ . Its sections are the sum of a vector field plus a one-form X + . The standard machinery of complex geometry can be generalized to this bundle. On this even-dimensional bundle, one can construct a generalized almost complex structure J, which is a map of T ⊕ T ∗ to itself that squares to −I2d (d is real the dimension of the manifold). This is analogous to an almost complex structure9 Im n which is a bundle map from T to itself that squares to −Id . As for an almost complex structure, J must also satisfy the hermiticity condition 0 1 t ∗ J GJ = G, with the respect to the natural metric on T ⊕ T , G = 1 0 . Usual complex structures I are naturally embedded into generalized ones J: take J to be I 0 J1 ≡ , (3.27) 0 −I t with Im n a regular almost complex structure (i.e. I 2 = −Id ). This J satisfies the desired properties, namely J2 = −I2d , Jt GJ = G. Another example of generalized almost complex structure can be built using a non-degenerate two-form Jmn , 0 −J −1 J2 ≡ . (3.28) J 0 Given an almost complex structure Im n , one can build holomorphic and antiholomorphic projectors ± = 21 (Id ± iI ). Correspondingly, projectors can be build out of a generalized almost complex structure, ± = 21 (I2d ± iJ). There is an integrability condition for generalized almost complex structures, analogous to the integrability condition for usual almost complex structures. For the usual complex structures, integrability, namely the vanishing of the Nijenhuis tensor, can be written as the condition ∓ [± X, ± Y ] = 0, i.e. the Lie bracket of two holomorphic vectors should again be holomorphic. For generalized almost complex structures, integrability condition reads exactly the same, with and X replaced respectively by and X + , and the Lie bracket replaced by the Courant bracket10 on T ⊕ T ∗ . The Courant bracket does not satisfy Jacobi identity in general, but it does on the i-eigenspaces of J. In case these conditions are fulfilled, we can drop the “almost” and speak of generalized complex structures. For the two examples of generalized almost complex structure given above, J1 and J2 , integrability condition turns into a condition on their building blocks, Imn and Jmn . Integrability of J1 enforces I to be an integrable almost complex structure on T, and hence I is a complex structure, or equivalently the manifold is complex. For J2 , which was built from a two-form Jmn , integrability imposes dJ = 0, thus making J into a symplectic form, and the manifold a symplectic one. These two examples are not exhaustive, and the most general generalized complex structure is partially complex, partially symplectic. Explicitly, a generalized complex manifold is locally equivalent to the product Ck × (Rd−2k , J ), where J = dx 2k+1 ∧ dx 2k+2 + · · · + dx d−1 ∧ dx d is the standard symplectic structure and k d/2 is called rank, which can be constant or even vary over the manifold (jump by two at certain special points or planes). There is an algebraic one to one correspondence between generalized almost complex structures and pure spinors of Clifford(6, 6). In string theory, the picture of generalized almost complex structures emerges naturally from the world-sheet point of view, while that of pure spinors arises on the space–time side. Spinors on T transform under Clifford(6), whose algebra is {m , n }=2g mn . There is a representation of this algebra in terms of forms. We can take m = dx m ∧ +g mn n .11 These satisfy the Clifford(d) algebra. The algebra of Clifford(d, d) is instead {m , n } = 0,
{m , n } = m n,
{m , n } = 0 .
9 In this subsection we denote by I the almost complex structure on T, to avoid confusion with the fundamental form J. In the rest of the paper we use J for both, but when referring to the almost complex structure, like for example in Eq. (3.13), we write the indices explicitly. 10 The Courant bracket is defined as follows: [X + , Y + ] = [X, Y ] + L − L − 1 d( − ). C X Y X Y 2 11 : p T ∗ → p−1 T ∗ , dx i1 ∧ · · · ∧ dx ip = p [i1 dx i2 ∧ · · · ∧ dx ip ] . n n n
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m and m are independent, they cannot be obtained from one another by raising or lowering indices with the metric. There is also a representation of this algebra in terms of forms, namely m = dx m ∧,
n = n .
(3.29)
The holomorphic 3-form is a good vacuum of Clifford(6, 6), as it is annihilated by the holomorphic i and the antiholomorphic ™¯ . These are half of the total gamma matrices, which implies that is a pure Clifford(6, 6) spinor. Acting with the other half, ™¯ and i we get forms of all possible degrees. Clifford(6, 6) spinors are therefore equivalent to (p, q)-forms. Using the Clifford map, a Clifford(6, 6) spinor can also be mapped to a bispinor [60,99]: C≡
1 (k) 1 (k) i ...i / ≡ i k . Ci1 ...ik dx ii ∧ · · · ∧ dx ik ←→ C C k! k! i1 ...ik k
(3.30)
k
On a space of SU(3) structure, there is a nowhere vanishing SU(3) invariant Clifford(6) spinor . Out of it, we can construct two nowhere vanishing SU(3, 3) invariant bispinors by tensoring with its dagger, namely [60,100] + = + ⊗ †+ ,
− = + ⊗ †−
(3.31)
(and its complex conjugates). Using Fierz identities, this tensor product can be written in terms of the bilinears in Eq. (3.14) by + ⊗ †±
6 1 1 † = ik ...i1 . 4 k! ± i1 ...ik +
(3.32)
k=0
Using the Clifford map (3.30) backwards, the tensor products in (3.31) are identified with regular forms. From now on, we will use ± to denote just the forms. The subindices plus and minus in ± denote the Spin(6, 6) chirality: positive corresponds to an even form, and negative to an odd form. Irreducible Spin(6, 6) representations are actually “Majorana—Weyl”, namely they are of definite parity—“Weyl”—and real—“Majorana”—. The explicit expression for the Clifford(6, 6) spinors in (3.31) in terms of the defining forms for the SU(3) structure is 1 + = + ⊗ †+ = e−iJ , 8
i − = + ⊗ †− = − . 8
(3.33)
The forms in (3.31), (3.33) are pure. This can be seen from writing the usual gamma matrices acting on the left of →
←
(denoted as m ) and on the right (denoted as m ) in terms of the Clifford(6, 6) gamma matrices (3.29) →
m = 21 (dx m ∧ +g mn n ),
←
m = 21 (dx m ∧ ±g mn n ) ,
(3.34) ←
where the ± sign depends on the parity of the spinor on which m acts. We can check now that forms (3.31) are indeed pure: the six gamma matrices that annihilate them are ( + iI )nm n + ⊗ †± = 0,
+ ⊗ †± n ( ∓ iI )nm = 0 ,
(3.35)
where I is the almost complex structure on the tangent bundle. On a space of SU(3) structure on T, there are therefore two SU(3, 3) invariant pure forms (and their complex conjugates), eiJ and . This implies that the structure group on T ⊕ T ∗ , which is generically SO(d, d), is reduced in this case to SU(3)×SU(3) [97,99].12 12 Two SU(3, 3) invariant pure spinors reduce the structure on T ⊕ T ∗ to SU(3)×SU(3) if they are compatible, namely if they have three annihilators in common. Spinors of form (3.31) are always compatible, as they have in common the three annihilators on the left of (3.35).
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There is a one to one correspondence between a pure spinor and a generalized almost complex structure J. It maps the +i-eigenspace of J to the annihilator space of the spinor . Under this correspondence i − = − ↔ J1 , 8 + = 18 e−iJ ↔ J2 ,
(3.36)
where J1 and J2 are defined in (3.27, 3.28). Integrability condition for the generalized complex structure corresponds on the pure spinor side to the condition J is integrable ⇔ ∃ vector v and 1-form such that d = (v + ∧) . A generalized Calabi–Yau [96] is a manifold on which a closed pure spinor exists: Generalized Calabi.Yau ⇔ ∃ pure such that d = 0 . From the previous property, a generalized Calabi–Yau has obviously an integrable complex structure. Examples of generalized Calabi–Yau manifolds are symplectic manifolds and complex manifolds with trivial torsion class W5 (i.e., if W1 = W2 = 0, and W¯ 5 = j¯ f —cf. (3.17)—then = e−f is closed). There is also the possibility of twisting by a closed three-form H. Using a three-form, the Courant bracket can be modified,13 and with it the integrability condition. In terms of “integrability” of the pure spinors , adding H amounts to twisting the differential conditions for integrability and for Generalized Calabi–Yau. More precisely, “ Twisted” Generalized Calabi.Yau ⇔ ∃ pure, and H closed such that (d − H ∧) = 0 . Decomposing in forms, = k k , the twisted Generalized Calabi–Yau condition implies dk − H3 ∧ k−2 = 0 for every k. Note that this twisted exterior derivative appeared already in the definition of the modified RR fields, Eq. (2.2) and in their Bianchi identities (d − H ∧)F (10) = 0. Before relating this discussion to N=1 flux vacua, let us say that for very little price, one can describe manifolds with SU(2) structure using the same formalism. SU(2) structures in six dimensions are defined by two nowhere vanishing spinors 1 , 2 that are never parallel. A bilinear of form (3.14) with one gamma matrix made out of them, defines a complex vector, namely 1+† m 2− = v m − iv
m
.
(3.37)
Therefore, differently from the SU(3) case, on manifolds with SU(2) structure there is a nowhere vanishing vector.14 It is possible to describe SU(3) and SU(2) structures on the same footing. For that, we define [100] 2+ = c1+ + (v + iv )m m 1− ,
(3.38)
where c is a function of the internal manifold, and we let the norm of the vector v + iv vary over the manifold. The SU(3) structure case corresponds to |v(p) + iv (p)| = 0 ∀p ∈ M (and therefore c = 1 to keep 2 normalized), while in the SU(2) case |v(p) + iv (p)| = 0 ∀p ∈ M. The intermediate cases, where the norm vanishes at some points on the manifold, is better described as an SU(3)×SU(3) structure on T ⊕ T ∗ [100], as we will explain shortly. Using 1 and 2 , we can build the pure spinors (3.31), where the one daggered is, say, 2 . Their explicit form is [100,62] + = 1+ ⊗ 2+† =
1 8
(ce ¯ −ij − i ) ∧ e−iv∧v ,
1 −ij + ic ) ∧ (v + iv ) . − = 1+ ⊗ 2† − = − 8 (e
(3.39)
These are given in terms of the local SU(2) structure defined by (1 , 2 ): j and are the (1, 1) and (2, 0)-forms on the local four-dimensional space orthogonal to v and v . The SU(3) structure defined by 1 is given by J = j + v ∧ v , = w ∧ (v + iv ). 13 [X + , Y + ] = [X + , Y + ] + H . H C X Y 14 This is possible only in manifolds of vanishing Euler characteristic, = 0.
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Very much like in (3.35) one can show that ± of (3.39) are pure: just replace I by I1 (I2 ) in the equation on the left (right) of (3.35), where I1 (I2 ) is the almost complex structures defined by 1 (2 ). One can similarly show that they are compatible (see footnote 12). Therefore, the existence of ± implies that the structure group on T ⊕ T ∗ is SU(3)×SU(3). SU(3) and SU(2) structures on T are just particular cases of SU(3)×SU(3) structures on T ⊕ T ∗ . 3.5. N = 1 flux vacua as generalized Calabi–Yau manifolds Following Refs. [96,62], we show in this section that the internal manifold for all the N = 1 vacua shown in Tables 3 and 4 (and also vacua with SU(2) structure, or more generally with SU(3)×SU(3) structure on T ⊕ T ∗ ) are generalized Calabi–Yau’s. As we reviewed in the previous sections, as a result of demanding m = = 0, supersymmetry imposes differential conditions on the internal spinor . These differential conditions turn into differential conditions for the pure Clifford(6, 6) spinors ± , defined in (3.33), (3.39). We quote the results of Ref. [62], skipping the technical details of the derivation. N = 1 supersymmetry requires ˜ +) = 0 , e−2A+ (d + H ∧)(e2A− ¯˜ − −2A+ 2A− ˜ − ) = dA ∧ (d + H ∧)(e e −
2 1 16 e [(|a|
− |b|2 )FIIA− − i(|a|2 + |b|2 ) ∗ FIIA+ ]
2 1 16 e [(|a|
− |b|2 )FIIB+ − i(|a|2 + |b|2 ) ∗ FIIB− ]
(3.40)
for type IIA, and ¯˜ + ˜ + ) = dA ∧ e−2A+ (d − H ∧)(e2A− + −2A+ 2A− ˜ (d − H ∧)(e e − ) = 0 ,
(3.41)
for type IIB. In these equations FIIA ± = F0 ± F2 + F4 ± F6 ,
FIIB ± = F1 ± F3 + F5 ,
(3.42)
˜ ± are unnormalized Clifford(6, 6) pure spinors. They are constructed as in (3.39), but out of unnormalized spinors and 1,2 ˜ defined by ˜ 1+ = a1+ ,
˜ 2+ = b2+ .
(3.43)
These are the internal spinors that build the N = 1 supersymmetry parameter (cf. Eq. (3.18) and Tables 3, 4, which ˜ ± are therefore related to ± in (3.33) or (3.39) by are specialized to the case 1 = 2 ). ˜ + = a b ¯ +,
˜ − = ab−
(3.44)
N = 1 supersymmetry imposes the following relation between these norms: d|a|2 = |b|2 dA,
d|b|2 = |a|2 dA ,
(3.45)
for both IIA and IIB. According to the definitions given in the previous section, Eqs. (3.40) and (3.41) tell us that all N = 1 vacua on manifolds with SU(3)×SU(3) structure on T ⊕ T ∗ (which includes the case of SU(3) and SU(2) structures on T) are twisted generalized Calabi–Yau’s. We can also see from (3.40), (3.41) that RR fluxes act a defect for integrability of the second pure spinor. Specializing to the pure SU(3) structure case, i.e. for ± given in Eq. (3.33), and looking at (3.36), we see that the generalized Calabi–Yau manifold is complex15 in IIB and (twisted) symplectic in IIA. For the general SU(3)×SU(3) case, N = 1 vacua can be realized in hybrid complex–symplectic manifolds, i.e. manifolds with k complex dimensions and 6 − 2k (real) symplectic ones. In particular, given the criralities of the preserved Clifford(6, 6) spinors, the rank k must be even in IIA and odd in IIB (equal respectively to 0 and 3 in the pure SU(3) case). One very important comment is in order: in the IIA and IIB type-A solutions of Tables 3, 4, either a or b vanishes. ˜ ± = 0. For this case, Eqs. (3.40), (3.41) just impose F = 0, which is indeed the case This implies, via Eq. (3.44), that 15 H in Eq. (3.41) does not “twist” the (usual) complex structure, as (d − H ∧) = 0 implies in particular d = 0.
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in type-A solutions, but they do not say anything about the integrability properties of the associated generalized almost complex structures. We know that for type-A the internal manifold is complex [2]. However, the relation W3 = ∗H3 , satisfied in type-A solutions, say that the manifold is not twisted symplectic (which needs W3 = H3 ). Therefore, we should be careful when saying that IIA solutions in manifolds of SU(3) structure are twisted symplectic: this is not true for type A, which is a valid IIA solution. The more general statement that all N = 1 vacua are twisted Generalized Calabi–Yau’s is nevertheless true, as type A involves a complex manifold, which is a particular case of Generalized Calabi–Yau. A final comment before we move on to discuss the problems related to compactification, is that N = 2 vacua with ˜ ± = ± , FIIA = FIIB = A = 0 [100]. NS fluxes only, where shown to satisfy Eqs. (3.40), (3.41) for 4. No-go theorems for compact solutions with fluxes The integrated equations of motion (or integrability conditions, of the type (3.7)) yield restrictive no-go theorems that under quite general conditions rule out warped compactifications to Minkowski or de Sitter spaces in the presence of fluxes [27,105,21,42,46]. We will review first the general argument given in Ref. [27], and then discuss the no-go’s from Bianchi identities, and the need to introduce localized sources. 4.1. Four-dimensional Einstein equation and no-go Ref. [27] showed that for any solution, supersymmetric or not, the flux contribution to the energy momentum tensor is always positive, ruling out compact internal spaces when these are turned on. This can be seen from the trace reversed Einstein’s equation in ten dimensions RMN = TMN − 18 gMN TLL .
(4.1)
For the metric (3.1), the four-dimensional components of Einstein’s equation imply R = R˜ − g˜ (∇ 2 A + 2(∇A)2 ) = T −
L 1 2A 8 e g˜ TL
.
(4.2)
Contracting with g˜ on both sides we find R˜ + e2A (−T + 21 TLL ) = 4(∇ 2 A + 2(∇A)2 ) = 2e−2A ∇ 2 e2A ,
(4.3)
where the contractions on the stress energy tensor are done with the full ten-dimensional metric. For Minkowski and de Sitter compactifications, R˜ 0. Defining Tˆ = −T +
1 2
TLL = 21 (−T + Tmm )
(4.4)
and using the expression for the energy momentum tensor for an n-form flux16 P ...Pn−1
TMN = FMP 1 ...Pn−1 FN 1
−
1 gMN F 2 , 2n
(4.5)
we arrive at n−1 2 Tˆ = −F P1 ...Pn−1 F P1 ...Pn−1 + F . 2n
(4.6)
In this equation, fluxes along space–time and internal fluxes make separate contributions,17 which means that we can consider them independently. Let us first consider internal components of the flux, for which the contribution to (4.6) is n−1 2 Tˆint = F 0 . 2n 16 The energy momentum tensor for some of the fluxes have powers of e , which do not affect the following argument. 17 We are referring here to the ten-dimensional fluxes F (10) (cf. Eq. (3.4)), but we suppressed the label (10) to lighten notation.
(4.7)
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All internal fluxes give therefore a strictly positive contribution to the trace of the energy momentum tensor,18 except for the one form flux, whose contribution vanishes. As for fluxes along space–time, the first term in 4.6 gives F P1 ...Pn−1 F P1 ...Pn−1 =
4 2 F . n
(4.8)
We find therefore (9 − n) 2 Tˆext = − F 0 , 2n
(4.9)
where we have used F 2 0 since we are considering temporal components of F. All fluxes with external components give also strictly positive contributions, expect for F9 , which gives a vanishing contribution (the same as its dual, purely internal, F1 ). We have therefore shown that all fluxes give a strictly positive contribution to the second term in (4.3), except for F1 , whose contribution vanishes, and F0 , whose consideration was shown in Ref. [27] to lead to similar no-go theorems. Multiplying (4.3) by e2A and integrating (4.3) over the internal manifold, we get the (in)famous no-go theorems: the right-hand side vanishes, while the left-hand side is non-negative for de Sitter or Minkowski spaces. We see therefore that without taking localized sources or higher derivative corrections to the equations of motion into account, a compactification to de Sitter space, for which the first term on the left-hand side of (4.3) is also positive, is completely ruled out. Compactifications to Minkowski space are allowed in the presence of one-form flux only, while in compactifications to anti de-Sitter spaces, the cosmological constant is related to the square of the fluxes. Let us now see, following Ref. [21], how the inclusion of localized sources modifies the argument. Localized sources give an extra contribution to (4.1) loc RMN = TMN − 18 gMN TLL + TMN − 18 gMN TLL loc .
(4.10)
Eq. (4.3) gets accordingly an additional term R˜ + e2A 21 (Tˆ flux + Tˆ loc ) = 2e−2A ∇ 2 e2A .
(4.11)
In order to avoid the no-go theorem, the sources should give a negative contribution to Tˆ , canceling that of the fluxes. For compactifications to Minkowski space, localized sources should obey the following identity: (Tˆiflux + Tˆiloc ) = 0 . (4.12) i
Ref. [21] showed that a p-brane extended along space–time and wrapped over a (p-3)-cycle has 7−p Tˆ loc = Tp () , 2
(4.13)
where √ −(p+1) e(p−3)/4 Tp = 2
(4.14)
is the positive Einstein-frame brane tension. This implies that for p < 7 the branes also give a positive contribution to Einstein’s equation. In order to compactify we need to include negative tension objects. String theory does have such negative tension objects: orientifold planes, and can therefore evade no-go theorems. Note that constructions involving only D7-branes and one form flux (F-theory) avoid no-go theorems, as none of these gives a contribution to the stress tensor. But F-theory does have its “no-go”, or rather, an upper bound for the number D7-branes, which arises due to their induced D3-charge. We will come back to this in Section 4.2. One final remark before moving on: the no-go theorems discussed in this section apply to any solution, regardless of its supersymmetry properties. 18 Eq. (4.7) does not apply to F , the mass parameter of massive IIA supergravity. This flux has nevertheless been shown independently to lead 0 to similar no-go theorems [27].
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4.2. Bianchi identities and equations of motion for flux: tadpole cancellation conditions We saw that the integrated Einstein’s equation gives a no-go theorem for compactifications with fluxes. In the case of supersymmetric flux backgrounds, on which we concentrate in this review, Einstein’s equation is automatically satisfied if in addition to supersymmetry, we demand Bianchi identity and the equation of motion for the fluxes. A proof of this for IIA is given for example in Ref. [43], and assumes that there is no crossed time-spatial component of the Einstein tensor, which is always the case in the compactifications we are interested in. The no-go theorems in supersymmetric solutions should therefore arise from integrated Bianchi identities or integrated equations of motion for the fluxes. This can be understood by the BPS nature of supersymmetric solutions, in which charges are equal to tensions. Einstein’s equation gives no-go theorems based on the effective tension of the fluxes, while Bianchi identities and the equations of motion for the fluxes restrict the magnetic and electric charges of the solution. In the case of IIB type-B solutions Bianchi identity (or the equation of motion) for the self-dual flux F5 is exactly the same as the trace of Einstein’s equation, Eq. (4.11). For other solutions like type-A or -C, Bianchi identities impose stronger conditions than (4.11), as the latter is just a single piece in the former. Besides imposing how much negative charge we need in order to allow a compact space, they tell us how this charge should be localized, or in other words, what supersymmetric cycles should the orientifolds wrap. The Bianchi identities for the NS flux and the “democratic” RR fluxes are given in (2.4). Due to the self-duality relation (2.3), Bianchi identity contains also the equation of motion for the fluxes, which is (10)
d(Fn(10) ) ± (−1)Int[n/2] H ∧ F8−n = 0,
+(−) : IIA(IIB) ,
(4.15)
or equivalently (10)
d(Fn(10) ) + H ∧ Fn+2 = 0
(4.16)
for both IIA and IIB. Inserting decomposition (3.4), using (3.5) and the warped metric (3.1) yields the following Bianchi identities and equation of motion for the internal RR fluxes: (d − H ∧), (d − H ∧),
F =0 , (e4A ∗ F ) = 0 .
(4.17)
Integrating these over the appropriate cycles leads to no-go type conditions: the integral of dF over a compact cycle is zero, while supersymmetry equations enforce relations of the type F ∼ ∗H , which yields a positive number after integrating over the same cycle. We will make this more precise in the next section. This is in general stronger than the no-go from Einstein’s equation applied to supersymmetric solutions. Bianchi identities’ no-go’s can be again avoided by including orientifolds, which are BPS sources of negative charge proportional to the tension. The cancellation conditions are often referred to as cancellation of NS or RR tadpoles: the net NS charge or RR charge of the solution has to be zero, where the charges correspond to localized, smeared or effective sources (fluxes) extending along space–time. Adding to (4.17) the contribution from the localized sources we get √ n−1 dFn = H3 ∧ Fn−2 + 2 loc 8−n ,
(4.18)
where loc 8−n is the dimensionless charge density of the 8 − n-dimensional (in space only) magnetic source for Fn , which x − xi ). contains a n+1 ( In type-IIA, tadpole cancellation conditions come from D4-, D6- and D8-brane sources extended along space–time. However, D4- and D8-branes would wrap 1- and 5-cycles, respectively, in the internal manifold. In Calabi–Yau, which is the case we will deal mostly in this review, there are no non-trivial 1- and 5-cycles, and therefore such tadpole cancellation conditions do not arise. The only tadpole cancellation condition in Calabi–Yau compactifications (10) (10) of IIA arise from D6-branes, which are electric sources for F8 , and magnetic sources for its dual field, F2 = F2 (cf. Eq. (3.4)). For localized sources consisting of D6-branes and O6-planes extended along space–time and wrapped
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˜ 3 , Bianchi identity (4.18) integrated over the dual cycle 3 (i.e. intersecting ˜ 3 once) yields the IIA on a 3-cycle tadpole cancellation condition19 ˜ 3 ) − 2 NO6 ( ˜ 3 ) + F√0 ND6 ( H3 = 0 , (4.19) 2 3 where F0 is the mass parameter of IIA, and ND6 , NO6 are the number of D6-branes and O6-planes wrapped on the ˜ 3 , dual to 3 . Explicitly, a D6-brane wrapping nK times the cycle AK and n K times the cycle B K (these are cycle defined in Eq. (2.6)), or in other words a D6-brane wrapping ˜ 3 = nK AK + nK B K , would contribute −nK units to (4.19) when 3 = B K , and nK for 3 = AK . In type-IIB, there are tadpole cancellation conditions coming from D3-, D5- and D7-branes. D7-branes, as we saw, (10) do not contribute to the energy momentum tensor, and neither does the flux F1 for which they are magnetic sources. They do contribute however to a tadpole for D3, as a wrapped D7-brane has induced D3-charge is we take into account the first correction to its action. This is best seen in the language of F-theory, as we will review shortly. ˜ 2 are electric sources for F (10) and D5-branes extended along space–time and wrapped on an internal 2-cycle 7 (10) magnetic sources for F3 = F3 . Bianchi identity (4.18) integrated over the dual 4-cycle 4 reads 1 ˜ ˜ ND5 (2 ) − NO5 (2 ) + H3 ∧ F 1 = 0 . (4.20) (2)2 4 D3-branes extended along space–time are electric sources for F˜1 , and magnetic sources for F5 . These are sixdimensional Hodge duals of each other, as can be seen from Eq. (3.5). D3-branes are point-like in six-dimensions, and therefore the tadpole cancellation condition involves an integral over the whole six-dimensional space. It reads 1 1 H3 ∧ F3 = 0 . ND3 − NO3 + (4.21) 4 (2)4 2 Using the integral fluxes of Eq. (2.6), the number of units of D3-charge induced by the 3-form fluxes is 1 K t Nflux = H3 ∧ F3 = (eK mK (4.22) RR − m eK RR ) = N NRR , (2)4 2 0 1 where we have used the symplectic vectors (2.9) and the symplectic matrix is = −1 0 . The models of Calabi–Yau orientifolds with D3 and/or D7-branes admit a description as F-theory [106] compactified on a Calabi–Yau four-fold X4 with an elliptic fibration structure over a three-fold base M. This corresponds to a type-IIB compactification on M with a dilaton–axion at a point p ∈ M equal to the complex structure modulus of the fiber −1 (p), and 7-branes at the singularities of the fibration . The tadpole condition for such a construction, which will be of use in Section 8, is ND3 + Nflux =
(X4 ) , 24
(4.23)
where Nflux is defined as in (4.22) (the integral being on the base M) and is the Euler number of the four-fold. The right-hand side arises from the induced D3 charge of the wrapped D7-branes. The orientifold limit [107] corresponds to the special case in which the singularities are D4 singularities, giving an O7-plane and four coincident D7-branes at each singularity. The tadpole cancellation condition for NS5 branes is common to IIA and IIB. These are magnetic sources for H3 , whose Bianchi identity in the presence of sources is dH3 = NS5 .
(4.24)
In type-I/heterotic theory the right-hand side of (4.24) gets the higher order correction (tr F ∧ F − tr R ∧ R) (see for example [77]). 19 The charge of an Op-plane is −2p−5 times the charge of a Dp-brane.
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4.3. Bianchi identities and special type IIB solutions Bianchi identities, when specialized to special type IIB A, B and C backgrounds in Table 4, give a particularly simple second order equation. In type-A, the relevant Bianchi identity is that for NS flux. This flux is related to the fundamental form J by [2] (we are taking the upper sign in Table 4) H3 = i(j − j¯ )J2 ,
(4.25)
where j is the holomorphic exterior derivative. Bianchi identity (4.24) then gives [2] dH3 = −2ijj¯ J2 = NS5 .
(4.26)
In type-C, which is the S-dual of type-A, the corresponding equation is [108] √ 2 dF3 = 2ijj¯ (e−2A J2 ) = H3 ∧ F1 + 2 loc 5 .
(4.27)
In type-B solutions, the relevant Bianchi identity is the one for F5 . F5 is related to the warp factor by (10)
F5
= (1 + ) [d(e4A ) ∧ dx 0 ∧ dx 1 ∧ dx 2 ∧ dx 3 ]
(4.28)
or equivalently F5 = e−4A ∗ d(e4A ) .
(4.29)
Let us rewrite the metric (3.1) in the form ds 2 = e2A dx dx + e−2A gˆ mn dx m dx n .
(4.30)
For the type-B solution with constant dilaton (i.e., the one not labelled by “F” in Table 4), the metric gˆ mn is Calabi–Yau (see Eq. (3.25)). Bianchi identity for F5 is then reduced to −∇ˆ 2 e−4A = ∗ˆ (H3 ∧ F3 + (2)4 2 loc 3 )
(4.31)
where a hat indicates an operation with respect to the metric gˆ mn . Using the ISD property of the complex 3-form flux in type-B solutions, Eq. (3.24), we obtain [12] e ¯ mnp , (4.32) Gmnp G 12 where 3 is here is a 0-form, equal to (x − x i ) for all D3 and O3 sources. Note that to arrive at (4.32) we have used only two of the specific features of type-B solutions, namely imaginary self-duality of G3 , and the relation between F5 and the warp factor (4.28). On the contrary, we have not used any of properties of g˜ mn . The two properties that we did use can however be derived directly from the equations of motion if we just demand a BPS-type inequality for the energy momentum tensor of all localized sources [21]: −∇ˆ 2 e−4A = (2)4 2 3 (x) +
1 2
Tˆ loc T3 3 (x) ,
(4.33)
where T3 is the tension of a D3-brane, given in (4.14). Giddings–Kachru–Polchinski (GKP) [21] (see also Ref. [109]) showed that this BPS-type condition determines the form of the solution completely. In order to show it, GKP start from the most general five form flux preserving fourdimensional Poincare symmetry, which is actually very similar to (4.28), but instead of e4A we should write a general (10) function f. The equation for f coming from the Bianchi identity for F5 would be very similar to (4.32), but Gmnp should be replaced by be i ∗ (G3 )mnp , since in going from (4.31) to (4.32) we have used ISD of the 3-form flux. GKP subtract this equation to the trace of the four-dimensional Einstein’s equation, Eq. (4.11) (with R˜ = 0). In (4.3), GKP
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insert in Tˆ flux the energy momentum tensor for the NS and RR 3-form fluxes. The substraction gives the following equation in the Einstein frame
4 ∇ˆ 2 (e4A − f ) = e2A+ 16 | ∗ G3 − iG3 |2 + e−6A |j(e4A − f )|2 + (2)8 e2A 21 Tˆ loc − T3 loc . (4.34) 3 The left-hand side of (4.34) integrates to zero on a compact manifold, while assuming (4.33), all the terms on the right-hand side are non-negative. This means that in the case that all the localized sources obey (4.33), a warped compactification to Minkowski space is allowed only if: (i) the warp factor and the four-form potential are related by e4A = f , i.e., the five-form flux is as in type-B, Eq. (4.28); (ii) the complex 3-form flux is imaginary self-dual; (iii) the inequality (4.33) is saturated. D3-branes, O3-planes and D7-branes and O7-planes wrapped on the four-cycles arising in F-theory saturate the inequality (4.33), while D5 and anti-D3-branes satisfy (but do not saturate) it. However, D5-branes wrapped on collapsed 2-cycles saturate the inequality. O5 and anti O3-planes, on the contrary, do not satisfy the inequality. We are obviously not allowed to add any number of the sources saturating the inequality as we want: although their contribution to (4.34) is zero, (4.21) or (4.23) should still be satisfied. Finally, the internal components of the Ricci tensor and the axion–dilaton must satisfy, in the Einstein frame, (2)8 4 2 1 7 D7 D7 ˆ Rmn = , e j{m jn} ¯ + (2) Tmn − gˆ mn T 4 8 ˆ ˆ 2 − 4(2)7 e−2 √1 SD7 . ∇ˆ 2 = −ie (∇) (4.35) −g ˆ In summary, if all localized sources satisfy (4.33), the necessary and sufficient conditions for a warped solution are (i) an internal manifold satisfying (4.35); (ii) five form flux given by (4.28); (iii) ISD complex 3-form flux; (iv) the inequality (4.33) saturated. Note that (i), (ii) and (iii) are the same as in type-B solutions. However, GKP have not at all imposed supersymmetry. Imaginary self-duality of the complex 3-form flux is less restrictive than the conditions for a supersymmetric type-B solution: it allows for a (1, 2) non-primitive component (i.e., a component of the form J2 ∧ w(0,1) , which is in the 3¯ representation of SU(3)), and a (0, 3)-singlet piece. Both pieces should be zero in a supersymmetric solution. Note that for an internal manifold with SU(3) structure, there are no nontrivial closed 1-forms, which means that 3¯ representations of the 3-form flux are not allowed. In the absence of singlet representations of the 3-form flux, the solution is exactly of the type B form. With no D7-branes, it corresponds to the solution without a label “F”, for which the internal space is conformal Calabi–Yau (2W5 = 3W4 ), and the dilaton is constant. When D7-branes are present, the internal space is no longer conformal Calabi–Yau, but obeys instead (4.35) with nontrivial ∇, and has W4 = W5 = j. From this, we conclude that we can obtain a non-supersymmetric solution starting from a type B solution of Table 4, and turning on a (0, 3) component of 3-form flux. From now on we will concentrate on flux compactifications on Ricci flat manifolds (mostly SU(3) holonomy, and therefore Calabi–Yau, except in Sections 6.2, 6.4, where we discuss moduli stabilization for flux compactifications on tori). We will come back briefly to the non Ricci-flat geometries at the end of Sections 5.3, 5.4 and in Section 5.5. 5. Four-dimensional effective theories To obtain the four-dimensional effective theory for a given compactification, we should perform a Kaluza–Klein (KK) reduction of the ten-dimensional type-II supergravities on a compact internal manifold, and keep only some finite set of massless modes. As is standard in KK reduction, massless modes for each supergravity field (metric, dilaton and B-field in the NS sector, and RR potentials in the RR sector) correspond to harmonic forms on the internal manifold. 5.1. Effective theory for compactifications of type-II When the internal manifold is Calabi–Yau, and no fluxes are turned on, the four-dimensional effective action is well known: it corresponds to an N = 2 ungauged supergravity, whose matter content depends on which type II theory we look at [110–113]. Let us very briefly review how this effective action.
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Table 5 Basis of harmonic forms in a Calabi–Yau manifold Cohomology group
Basis
H (1,1) H (0) ⊕ H (1,1) H (2,2) H (2,1) H (3)
wa wA = (1, wa ) w˜ a
a = 1, . . . , h(1,1) A = 0, . . . , h(1,1) a = 1, . . . , h(1,1) k = 1, . . . , h(2,1) K = 0, . . . , h(2,1)
k (K , K )
¯ and one (3, 3)-form A Calabi–Yau has one harmonic 0-form -a constant-, one (3, 0)-form --, one (0, 3)-form -, (1,1) (2,1) -the volume-. Additionally, it has h harmonic (1, 1) and (2, 2)-forms and h harmonic (2, 1) and (1, 2)-forms. The total number of harmonic 3-forms is therefore 2h(2,1) + 2. Finally, there are no harmonic 1 and 5-forms. Table 5 gives a basis of harmonic forms. The forms satisfy the normalizations given in (2.7). In the NS sector, the dilaton is “expanded” in the only scalar harmonic form. The B-field can have purely external or internal components. The former is expanded in the only internal scalar, while the latter is expanded in the basis wa . As for the metric, the 4D massless fields correspond to deformations that respect the Calabi–Yau condition. It was shown in Ref. [114] that the deformations gi¯E correspond to deformations of the fundamental form J2 , expanded in the basis of h(1,1) harmonic forms. gij correspond on the contrary to deformations of the complex structure, which are in one to one correspondence with the harmonic (2, 1)-forms. We have therefore the following expansions for the deformations of the fields in the NS sector: (x, y) = (x) , gi¯E (x, y) = iv a (x)( a )i¯E (y),
⎛ gij (x, y) = i¯zk (x) ⎝
¯¯
(¯k )i k¯ l¯kj l ||2
⎞ ⎠ (y) ,
B2 (x, y) = B2 (x) + ba (x) a (y) .
(5.1)
Here, all the x-dependent fields are the moduli of the 4D theory. In the NS sector we get a total of 2(h(1,1) + 1) + h(2,1) moduli. In the RR sector, we perform the following expansions C1 (x, y) = C10 (x) ,
C3 (x, y) = C1a (x) a (y) + K (x)K (y) − ˜ K (x) K (y)
(5.2)
for type-IIA, and C0 (x, y) = C0 (x) , C2 (x, y) = C2 (x) + ca (x) a (y) , C4 (x, y) = V1K (x)K (y) + a (x) ˜ a (y)
(5.3) (10)
for type-IIB. In the expansion of C4 we have used the self-duality of F5 , which connects the terms expanded in K to the ones that would be expanded in the forms K , and similarly for a , which are dual to h(1,1) tensors D2 coming from expanding in the basis a . These moduli arrange into the N = 2 multiplets shown in Tables 6 and 7, taken from Refs. [115,116]. Inserting the expansions (5.1), (5.2) in the ten-dimensional type-IIA action, and (5.1), (5.3) in the IIB one and integrating over the Calabi–Yau, one obtains a standard four-dimensional N = 2 ungauged supergravity action (for a review of N = 2 supergravity see for example Ref. [117]), whose forms is [10,112] (for details, see for example Refs. [115,116,118–122]) 1 1 1 (4) SIIA = − R ∗ 1 + Re NAB F A ∧ F B + Im NAB F A ∧ ∗F B 2 2 2 M4 a b u v − Gab dt ∧ ∗dt¯ − huv dq ∧ ∗dq , (5.4)
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Table 6 Type-IIA moduli arranged in N = 2 multiplets Gravity multiplet
1
(g , C10 )
Vector multiplets
h(1,1)
(C1a , v a , ba )
Hypermultiplets
h(2,1)
(zk , k , ˜ k )
Tensor multiplet
1
(B2 , , 0 , ˜ 0 )
Table 7 Type-IIB moduli arranged in N = 2 multiplets Gravity multiplet
1
(g , V10 )
Vector multiplets
h(2,1)
(V1k , zk )
Hypermultiplets
h(1,1)
(v a , ba , ca , a )
Tensor multiplet
1
(B2 , C2 , , C0 )
for type IIA, and 1 1 1 (4) − R ∗ 1 + Re MKL F K ∧ F L + Im MKL F K ∧ ∗F L SIIB = 2 2 2 M4 − Gkl dzk ∧ ∗d¯zl − hpq dq˜ p ∧ ∗dq˜ q .
(5.5)
Let us explain these expressions. In the gauge kinetic part, the field strengths are F A = dC1A = (dC10 , dC1a ) in the IIA action (5.4), and F K = dV1A = (dV10 , dV1k ) in the IIB action (5.5). The gauge kinetic coupling matrices N and M, given below in Eq. (5.16), depend on the scalars in the respective vector multiplets. In IIA, these are the complex combination of Kähler and B-field deformations t a , called complexified Kähler deformations, and defined B + iJ = (ba + iv a ) a ≡ t a a .
(5.6)
In IIB, the scalars in the vector multiplet moduli space are the complex structure deformations zk , or the periods, defined as [114] Z K = ∧ K = , FK = ∧ K = . (5.7) AK
BK
Using these, can be expanded as = Z K K − F K K .
(5.8)
It turns out that the Jacobian jl (Z k /Z 0 ) is invertible, and therefore Z K can actually be viewed as projective coordinates. One can introduce special coordinates zk = Z K /Z 0 , which are the h(2,1) complex structure deformations in (5.1). These are the scalars in the vector multiplets in type-IIB, Table 7, and also part of the scalars in the IIA hypermultiplets, Table 7. The scalars in the vector multiplets span a special Kähler manifold of complex dimension h1,1 and h2,1 in IIA and IIB, respectively, whose metric Gab and Gkl will be given shortly. The scalars in the hypermultiplets span a quaternionic manifold whose coordinates are q u , u = 0, . . . , h(2,1) and u = 0, . . . , h(1,1) for IIA and IIB, respectively. The explicit expression for the quaternionic metric huv was found in Ref. [111], and is given for example in Refs. [120,122]. The metric and Kähler potential in the vector multiplet moduli space will be important later, so let us give their explicit form. The Kähler potential for the vector multiplet moduli space in IIA, which is spanned by the complexified Kähler deformations t a , is given by [114,123] 4 i 4 a b c ¯ ¯ ¯ J ∧ J ∧ J = − ln Kabc (t − t ) (t − t ) (t − t ) = − ln K , (5.9) K = − ln 3 6 3
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where K is six times the volume of the Calabi–Yau manifold, and Kabc are the intersection numbers defined by Kabc = a ∧ b ∧ c , Kab = a ∧ b ∧ J = Kabc v c , Ka = a ∧ J ∧ J = Kabc v b v c , K = J ∧ J ∧ J = Kabc v a v b v c . (5.10) The metric Gab in the IIA vector multiplet moduli space obtained from the Kähler potential (5.9) is 3 3 Kab 3 Ka Kb Gab = jt a jt¯b K = − = − a ∧ ∗ b . 2 K 2 K2 2K
(5.11)
In type-IIB, the scalars in the vector multiplet moduli space are the complex structure deformations zk . They span again a Kähler manifold, with Kähler potential given by ¯ = − ln i[Z¯ K FK − Z K F ¯ K] . K = − ln i ∧ (5.12) The metric derived from this potential is given by [114] ∧ ¯ l Gkl = − k . ¯ ∧
(5.13)
Both Kähler potentials for the scalars in the vector multiplet moduli space (5.9) for IIA and (5.12) for IIB, can be derived from a holomorphic prepotential F, namely ¯ i jF i jF K(z) = i z¯ . (5.14) −z jzi j¯zi For type IIA, the prepotential is 1 t at bt c G = − Kabc , 6 t0
(5.15)
where t 0 is an extra coordinate set to 1 after differentiation, and introduced such that the prepotential is homogeneous of degree two. We are now ready to give the expression for the matrices M and N in (5.4), (5.5): 1 − 3 Kabc ba bb bc 21 Kabc bb bc Re N = , 1 b c −Kabc bc 2 Kabc b b K 1 + 4Gab ba bb −4Gab bb Im N = − , −4Gab bb 4Gab 6 MKL = FKL + 2i
(Im F)KM Z M (Im F)LN Z N , Z N (Im F)NM Z M
(5.16)
where the (0, 0), (0, 1), (1, 0) and (1, 1) elements in the matrix expression for N give its (0, 0), (0, a), (a, 0) and (a, b) components, FKL = jL FK . Before moving on to a discussion of Calabi–Yau orientifolds, let us make a short pause and discuss following Ref. [74] the effective four-dimensional theories arising from compactifications on manifolds of SU(3) structure (or more generally, SU(3)×SU(3) structure, as discussed in Sections 3.4, 3.5). Hitchin showed [96,124] that there is a Special Kähler structure on the space of generalized almost complex structures (for usual almost complex structures, this bundle is known as the twistor bundle [125]). The space of generalized complex structures is the space of stable,20 real, even or odd forms {Re + }, {Re − }. In the SU(3) structure case, these are 20 A real form is stable if any element in a neighborhood of is GL(6, R)-equivalent to . An equivalent statement is that lives in an open orbit under the action of GL(6, R).
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the spaces of symplectic structures J and complex structures = Re (see Eq. (3.33)). The spinor + can actually be truly complexified by adding the B-field. Explicitly, + =
i −(B+iJ ) , e 8
− =
1 , 8
(5.17)
where the relative factors of i with respect to (3.33) are introduced for later convenience. Re + , Re − belong to irreducible (“Majorana–Weyl”) Spin(6, 6) representations, as discussed just above (3.33). The imaginary part of the complex spinors ± is obtained from the real part by Im ± = ∗Re ± .21 The complex Clifford(6, 6) spinors ± = Re ± + i Im ± are pure. The Kähler metric for the space of generalized complex structures is obtained from the following Kähler potential ¯ ± , (5.18) K± = − ln i ± , where the “Mukai” pairing ·, · is defined + , + ≡ 6 ∧ 0 − 4 ∧ 2 + 2 ∧ 4 − 0 ∧ 6 , − , − ≡ 5 ∧ 1 − 3 ∧ 3 + 1 ∧ 5 ,
(5.19)
(the subscripts denote the degree of the component form). A straightforward but very important observation is that for the case of SU(3) structures, i.e. for ± in (5.17), the Kähler potentials in (5.18) have exactly the same expressions as their Calabi–Yau counterparts, given in Eqs. (5.9), (5.12). The (big) difference is that ± in (5.18) need not be closed, or in other words, need not correspond to an integrable structure, as it does in the case of Calabi–Yau structures. Furthermore, the Kähler potential (5.18) applies also to the general case of SU(3)×SU(3) structures (by using (3.39) for ± ), or in other words to hybrid complex–symplectic structures (again, not necessarily integrable). Let us now return to the more familiar case of Calabi–Yau manifolds, and orientifold them. 5.2. Effective theory for Calabi–Yau orientifolds The presence of orientifolds projects out part of the spectra shown in Tables 6 and 7. For type-IIA, the only orientifold consistent with supersymmetry in a Calabi–Yau is an O6, extended along space–time and wrapping a Special Lagrangian internal 3-cycle. In type-IIB, supersymmetry allows for O3, O5, O7 and O9 planes, the O5 and O7 wrapping holomorphic 2- and 4-cycles, respectively. O6 planes can be included when the Calabi–Yau has a symmetry which is involutive (2 = 1), isometric (leaves the metric invariant) and antiholomorphic (Jn m = −Jn m ). The antiholomorphic involution acts on the holomorphic ¯ where is some phase (and ∗ denotes the pull-back of ). One can eliminate by redefining 3-form as ∗ = e2i , . From now on we will take = 0. A consistent truncation of the spectrum is obtained when the theory is modded out by (−1)FL p , where FL is the space–time fermion number in the left moving sector, and p is the world-sheet parity, which exchanges left and right movers [126,127,115]. In type-IIB, the involution is also isometric but this time it is holomorphic. There are two possible actions on the holomorphic 3-form , namely ∗ = ±. The plus sign leads to O5/O9 planes, while the minus sign to O3/O7. The theory can be consistently modded out respectively by (−1)FL p and p [126,127,116]. Table 8 gives the transformations of NSNS and RR under the action of (−1)FL , and p . The massless states that survive the orientifold projection are those that are even under the combined action of (−1)FL p for O3/O7 as well as O6, and even under p for O5/O9. For example, the B-field is odd under (−1)FL p and is also odd under p only. This implies that in the presence of any O-plane, the only components of B that survive are those that are odd under . 21 Im can be obtained solely from Re (i.e. without making use of the metric, constructed out of both Re and Re ), by means of + + + − the Hithcin function [124]. See Refs. [124,74] for details.
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Table 8 Parity of the bosonic fields under the actions of (−1)FL , and p
(−1)FL
p
g
B2
C0
C1
C2
C3
C4
+ +
+ +
+ −
− −
− +
− +
− −
− −
Table 9 Type-IIA moduli arranged in N = 1 multiplets for O6 compactifications O6 Gravity multiplet
1
g
Vector multiplets
(1,1) h+ (1,1) h− h(2,1) + 1
C1
Chiral multiplets Chiral multiplets
(v a , ba ) (Re Z K , K )
Table 10 Type-IIB moduli arranged in N = 1 multiplets for O3/O7 and O5/O9 orientifold O3/O7 Gravity multiplet Vector multiplets
1 (2,1) h+
Chiral multiplets
h−
O5/O9 g V1
1 (2,1) h−
(2,1)
zk
h+
(1,1)
(v , )
h+
h−
(1,1)
(ba , ca )
1
(, C0 )
h+
g V1k
(2,1)
z
(1,1)
(v , c )
h−
(1,1)
(ba , a )
1
( , C 2 )
The space of harmonic p-forms, H p , splits into two eigenspaces under the action of with eigenvalues plus and minus one. In IIA, is antiholomorphic, which implies that the spaces H (p,q) are not in general eigenspaces of (i.e. (1,1) (2,2) sends H (p,q) into H (q,p) ). H (1,1) splits into H± and so does H (2,2) , which splits into H± . H (2,1) ⊕ H (1,2) (2,1) (3,0) splits into halves of dimension h each, with positive and negative eigenvalues, and H ⊕ H (0,3) splits into two one-dimensional spaces with positive and negative eigenvalues. As an example, from the massless modes of the B-field (1,1) given in Table 6, ba and B2 , only a subset of ba , namely those multiplying a 2-form in H− , survive. The same is true for v a , the Kähler deformations of the metric. Opposite to this, the vector C1a , which is in the same N = 2 multiplet than (1,1) the complexified Kähler deformations and comes from C3 , should be expanded in harmonic forms in H+ . We see that (1,1) (1,1) the vector and the scalars in the N = 2 vector multiplet split, building h− N = 1 vector multiplets, and h+ N = 1 chiral multiplets. The same is true for the rest: all the original N = 2 multiplets break into N = 1 multiplets. Table 9, taken from Ref. [122], shows the surviving IIA multiplets after the orientifold projections. The moduli K in Table 9 correspond to a combination of the N = 2 moduli (0 , k , ˜ k , ˜ 0 ), namely those in H+3 , of dimension h3+ = h(2,1) + 1. The modulus Re Z 0 in Table 9 corresponds to the dilaton (for details, see Refs. [115,122]). (p,q) In type-IIB, the involution is holomorphic. This implies that its eigenspaces are inside H (p,q) , i.e. H (p,q) =H+ ⊕ (p,q) H− . Table 10 shows the surviving IIB multiplets after the orientifold projections. The moduli spaces spanned by the scalars split again into that of the scalars in the vector multiplets, and that of the hypermultiplets. They are both Kähler, and are appropriate subspaces of the special Kähler and quaternionic spaces of the N = 2 moduli spaces of the previous section. The effective actions for the type-II Calabi–Yau orientifold
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compactifications have the standard N = 1 form [128], namely 1 1 1 ¯ (4) SN=1 = − R ∗ 1 + KI J¯ DM I ∧ ∗D M¯ J + Ref F ∧ ∗F + Im f F ∧ F + V ∗ 1 . 2 2 M4 2
(5.20)
Here M I denote the complex scalars in the chiral multiplets. The potential V is given in terms of the superpotential W and the D-terms D by ¯ V = eK (K I J DI W D J¯ W¯ − 3|W |2 ) +
1 2
(Ref )−1 D D ,
(5.21)
where we have used the Kähler covariant derivatives, defined as DI W = jI W + W jI K .
(5.22)
The expressions for the gauge kinetic coupling matrix f are truncated versions of their N = 2 counterparts NAB (1,1) (2,1) and MKL . These give the couplings of the h+ gauge fields C for type IIA, and for IIB the h+ gauge fields V (2,1) k and h− V for O3/O7 and O5/O9 projections, respectively. In order to write the Kähler potential for the scalars in the chiral multiplets, one needs to identify the good Kähler coordinates, i.e. the complex coordinates M I such that the action is of the form (5.20). It turns out that the subset of the scalars in the N = 2 vector multiplets that survive the projections, namely t a = ba + iv a in IIA and zk for IIB O3/O7 or z for O5/O9 are good Kähler coordinates. Their Kähler potential and metric also have the same formal expressions as (1,1) (2,1) in N = 2, namely (5.11) for type-IIA, but where a runs only in h− , and (5.13) in IIB, with k running up to h− for (2,1) O3/O7 and h+ for O5/O9. For the chiral multiplets, the story is more complicated. In type IIA, the Kähler coordinates (i.e., the coordinates such that the action takes the form 5.20) corresponding to the complex structure moduli and the dilaton are encoded in the expansion of the complex 3-form field c = C3 + 2i Re(C) = ( + 2i Re(CZ )) + (˜ + 2i Re(CF )) ≡ N + T ,
(5.23)
where C is a “compensator” field proportional to e− (see Ref. [115] for details), and ( , ) is a basis of even ˜ which determines how many of the ’s are even, is basis dependent, ˜ = h˜ + 1, . . . , h2,1 (h, 3-forms, = 0, . . . , h, but the total number of complex structure moduli is obviously basis independent, equal to h(2,1) + 1). Finally, the IIA Kähler potential is given by [115] 4 KO6 = − ln J ∧ J ∧ J − 2 ln Re(C) ∧ ∗Re(C) , (5.24) 3 where this should be written in terms of the right Kähler coordinates defined in Eqs. (5.6) and (5.23). In type-IIB, the Kähler coordinates depend on what kind of orientifold projection is performed. For O3/O7 projections, these are the complex structure moduli zk and [116] = C0 + ie− , Ga = ca − ba , 1 i ¯ c, Kbc Gb (G − G) T = K + i − 2 2( − ¯ )
(5.25)
where the intersection numbers Kbc and K are defined in the same way as the N = 2 counterparts (5.10), but taking the appropriate basis. The Kähler potential is 1 ¯ KO3/O7 = − ln −i (z) ∧ (¯z) − ln[−i( − ¯ )] − 2 ln K(, G, T ) , (5.26) 6 where K has the same formal expression as in (5.10) as a function of v , but v should be re-expressed in terms of the Kähler coordinates , Ga , T . This cannot be done explicitly, except for the case in which there is only one v (1,1) (1,1) (and therefore one T ≡ T ), i.e. when h+ = 1. If additionally h− = 0, we get a particularly simple and familiar expression for the Kähler potential [21], namely −2 ln K = −3 ln[T + T¯ ] .
(5.27)
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The Kähler potential for the chiral multiplets in type-IIB O3/O7 compactifications coming from N = 2 hypermultiplets, i.e. the last term in Eq. (5.26), satisfies a very important property, namely [116,130] ¯
jI KjJ¯ KK I J = 3 ,
(5.28)
for I = This is a no-scale type condition [129]. When there is a nontrivial superpotential (which is the case in the presence of fluxes), the condition (5.28) implies that the positive contribution to the potential (5.21) offsets the negative one −3|W |2 , and we therefore get V 0. This equality can be easily checked in the simple case of one Kähler modulus, Eq. (5.27). For O5/O9 orientifolds, the right Kähler coordinates are again the complex structure moduli z , and the combinations [115] (Ga , T
).22
t = e− v − ic ,
Aa = ab bb + ia ,
¯ b, S = 16 e− K + ih − 41 (Re −1 )ab Aa (A + A) where we have defined ab (t) = Kab t ,
1 C6 = h + a ba 2
(i.e. h is an appropriate dual to C2 ). The Kähler potential for O5/O9 is given by 1 1 −3 ¯ KO5/O9 = − ln −i ∧ − ln J ∧ J ∧ J − ln e− K(S, Aa , t ) , e 6 3
(5.29)
(5.30)
(5.31)
where the first term is a function of the complex structure moduli, the second a function of the moduli t only, and in the last one should solve for K in terms of (S, Aa , t ) using (5.30). 5.3. Flux-induced potential and gauged supergravity We showed in Section 3 that fluxes back react on the geometry, and a Calabi–Yau manifold is no longer a solution to the equations of motion. If however the typical energy scale of the fluxes is much lower than the KK scale, we can assume that the spectrum of Sections 5.1 and 5.2 is the same, except that some of the massless modes acquire a mass due to the fluxes. The energy scale of, for example, 3-form fluxes can be estimated using the quantization condition (2.6), and is given by Nflux /R 3 (where Nflux are the number of units √ of 3-form flux). The KK scale is 1/R. The former is much lower than the latter when the radius is much bigger than N flux times the string scale, which is in any case needed from the start in order to neglect -corrections to the action. This truncation of the spectrum to those modes that are massless in the absence of fluxes is standard in flux compactifications, and it is shown to yield a consistent N = 2 or 1 gauged supergravity action, depending whether one starts with a Calabi–Yau [9–11,131–133,118,119,135,78,14,136] or a Calabi–Yau orientifold [115,116]. A similar argument was used in Ref. [121] to show that is it possible to do a consistent truncation to a set of light modes in the case of compactifications on half flat manifolds with fluxes, and generalized in Ref. [74] to the case of any SU(3) structure manifold, such that the resulting action has the standard form of N = 2 gauged supergravity. In this section we will concentrate on the case of Calabi–Yau compactifications with fluxes, and only say a few comments about these more general constructions at the end. Turning on RR fluxes in type-IIA and keeping the same light spectrum of Calabi–Yau compactifications amounts to doing the replacements dC1 → dC1 + maRR a ,
dC3 → dC3 − eRR a ˜a .
(5.32)
Reducing the ten-dimensional action in the presence of these fluxes leads to the following new terms in the fourdimensional action [119] 1 1 SRR = −B2 ∧ J2 − M 2 B2 ∧ ∗B2 − MT2 B2 ∧ B2 − V , (5.33) 2 2 M4 22 Including the second term in (5.26) and summing over I = (, Ga , T ) gives 4 in the right-hand side of (5.28).
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Table 11 Scalars that get a potential in the presence of fluxes in CY compactifications IIA
RR flux NS flux
IIB
N=2
N = 1 (O6)
N=2
N = 1 (O3/O7)
(v a , ba )
(v a , ba )
zk
zk
(zk , 0 , )
(Re Z k , 0 )
z k , , C
z k , , C
F F
0
0
N = 1 (O5/O9) F F
zk zk
F D
Table 12 Effect of fluxes and torsion in the N = 2 four-dimensional action IIA
IIB
Electric RR-flux eRR Magnetic RR-flux mRR Electric NS-flux e0
Green–Schwarz coupling Massive tensor B2 Massive C10
Green–Schwarz coupling Massive tensor B2 1 Massive V1k
Magnetic NS-flux m0 Electric torsion eaK Magnetic torsion mK a
Massive C10 Massive C1a Massive C1a
Massive tensor C2 Massive V1k Massive tensors D2a
where B B J2 = −eRR A F A + mA RR (Im NAB ∗ F + Re NAB F ) , B MB2 2 = −mA RR Im NAB mRR , B A MT2 B2 = −mA RR Re NAB mRR + mRR eRR A
VIIA RR = −
e 4 AB (eRR B − N¯ BD mD (eRR A − N¯ AC mC RR ) . RR )(Im N) 2
(5.34)
We see from (5.33) that RR fluxes induce Green–Schwarz type couplings, regular and topological mass terms for the tensor B2 , and a potential that renders massive some of the scalars in the vector multiplets. When magnetic fluxes are present, the tensor B2 becomes massive by a Stückelberg- type mechanism, namely it “eats” one combination of the C1a gauge vectors, which becomes pure gauge once the magnetic fluxes are introduced. The potential in (5.34) depends on the scalars in the vector multiplets, namely the complexified Kähler deformations. The effect of RR fluxes is summarized in Tables 11 and 12. The extra terms in the action coming from turning on RR fluxes where shown to be consistent with a standard N = 2 gauged supergravity in Ref. [119] for the case mI = 0, when there are no massive tensors. The introduction of magnetic fluxes is also consistent with an N = 2 gauged supergravity with massive tensors [137], in which case the tensor is not dual to a scalar but rather to a massive vector. In IIB, RR fluxes are introduced by K dC2 → dC2 + mK RR K − eRR K .
(5.35)
Inserting this in the Lagrangian results in the same new terms as in IIA, Eq. (5.33), where the definitions as in IIA, Eq. (5.34), just replacing N by M, and the indices A, B, . . . by K, L, . . . (i.e. the sums are from 0 to h(2,1) ). Therefore, RR fluxes in IIB have the same effect as in IIA, namely induce Green–Schwarz type couplings, regular and topological mass terms for the tensor B2 , and a potential that renders massive some of the scalars in the vector multiplets. This is summarized in Tables 11 and 12. NS fluxes are introduced in IIA and IIB by modifying dB2 → dB2 + mK K − eK K .
(5.36)
In type-IIA, NS fluxes give gauge charges to the scalars a (dual to B2 ) and (K , ˜ K ) in the tensor and hypermultiplets respectively, whose ordinary derivatives in (5.4) are replaced by covariant derivatives. The only vector field participating
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in these gaugings is the graviphoton, who as a consequence acquires a mass. Fluxes also generate a potential for the scalars (, zk , K , ˜ K ) in the hyper and tensor multiplets. The potential is given by [119] VIIA NS = −
4 e 2 ¯ LN mN ) + e (mK ˜ K − eK K + e0 )2 . (eK + MKM mM )(Im M)KL (eL + M 4K 2K
(5.37)
Note that the matrix entering the NS flux potential is M, rather than N (cf. Eq. (5.34)), which means that the scalars that get a potential are all in hypermultiplets. Note also that among all the axions (˜ K , K ), only a single combination of them, namely mK ˜ K − eK K , gets a potential. We will come back to this in Section 6. In IIB, NS electric fluxes gauge the scalars in the tensor multiplet, namely the (appropriate) dual of B2 (see Ref. [113] for the redefinitions of the quaternionic coordinates) and the dual of C2 [10,118,121]. Opposite to the case in type-IIA, the vectors that gauge these scalars are those in the vector multiplets, and not the graviphoton. One combination of these vectors acquires therefore a mass. Electric fluxes also generate a potential for the scalars in the vector multiplet, and the axion–dilaton. Finally, magnetic fluxes give a mass to C2 . The flux generated potential is given by e 4 e−2 2 (5.38) C0 + eK (Im M)KL eL . VIIB NS = − 2K 2K The effect of fluxes is summarized in Tables 11 and 12. The scalars gauged by the fluxes are always “axions” in ˜ B2 in type IIA, and B2 , C2 in type IIB. The quaternionic metric in the hypermultiplet hyper (or tensor) multiplets: , , moduli space has translational isometries, corresponding to shifts of these scalars. Fluxes gauge these translational isometries. The partial derivatives in the action turn into covariant derivatives, namely u A C , j q u → D q u = j q u − kA
(5.39)
u are the Killing vectors, and C A are the vectors in the vector multiplet that participating in the gauging. In where kA type IIA, without magnetic RR flux, the gauging are [119]
kaB = 2eRR a ,
k0B = (mK ˜ K − eK K ),
K
k0 = mK u , K
K ˜
k0 = eK ,
(5.40)
where B is the scalar dual to B2 (massless in the absence of magnetic RR flux). In type IIB, for electric RR and NS flux the Killing vectors are B kK = 2eRR K + eK 0 ,
0 ˜
kK = eK .
(5.41)
Ref. [138] showed that the isometries corresponding to these Killing vectors are not lifted by quantum corrections (instantons) to the quaternionic metric. The flux generated potentials can be studied similarly in the case of Calabi–Yau orientifold compactifications [116,115]. In O6 compactifications of type IIA, the tensor B2 is projected out of the spectrum, so there are no massive tensors arising from the introduction of fluxes. The flux term that does survive the projection is the potential term in (1,1) (5.34), i.e. RR fluxes give potential terms to the scalars v a , ba ; a = 1, . . . , h− . NS fluxes can also be combined in a (2,1) potential term of the form (5.34) which depends on the scalars in the h + 1 chiral multiplets. Ref. [115] showed that in the language of N = 1 supergravity, this potential arises from a superpotential, to be reviewed in the next section, and no D-term. In type-IIB, O3/O7 orientifolds project out the tensors B2 , C2 and the graviphoton. This means that from the combined action of RR and NS fluxes, Eq. (5.34)—replacing N by M and A, B, . . . by K, L, . . .—and (5.38), only the potential survives. This potential is a truncated version of the sum of the potentials in (5.34) and (5.37), with indices running from (2,1) 0 to h− (i.e. there is a potential for the complex structure moduli zk and the axion–dilaton). Its explicit expression in terms of the coupling matrix M and the fluxes can be found in Ref. [116]. We will see in next sections that the potential can be derived from a superpotential, as in Eq. (5.21), and no D-term. In the case of O5/O9 planes, the tensor C2 is not projected out from the spectrum, and it acquires a mass when NS magnetic fluxes are present. The potential due to the fluxes is again a truncated version of (5.34) and (5.38). There are two distinguished pieces, one containing RR (2,1) fluxes, for which there is a sum from 0 to h+ (i.e., it corresponds again a potential for the complex structure moduli
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and the dilaton), and another one for NS fluxes, whose sum runs from 1 to h+ , which is also a potential for complex structure moduli. It was shown in Ref. [115] that the piece of the potential involving RR fluxes can be derived from a superpotential as we will review in the next section, while the NS fluxes give rise to a D-term. Table 11 shows which scalars get a nontrivial potential due to RR and NS fluxes, for Calabi–Yau, and Calabi–Yau orientifolds.23 Note that in type-IIB the 4 h(1,1) moduli in hypermultiplets (or 2 h(1,1) complexified Kähler moduli in the orientifolded theory) do not get a potential by fluxes. In type-IIA, the ones that do not get a potential are 2h(2,1) scalars in hypermultiplets (or h(2,1) scalars in chiral multiplets in the orientifolded theory). This means that fluxes could potentially be the only ingredient needed to stabilize all moduli in compactifications of type-IIA on rigid manifolds (h(2,1) = 0), but in type-IIB there is no way of stabilizing all moduli, as h(1,1) 1 (there is at least the volume modulus). We will come back to this issue at length in Section 6. Before discussing the flux-induced superpotentials, let us pose for a moment and discuss the effect of fluxes in the case of manifolds with SU(3) structure, not necessarily Calabi–Yau’s. As we mention at the beginning of this subsection, in a similar spirit than the one used to study the effect of fluxes on Calabi–Yau manifolds, Refs. [121,74] argued that in the case of manifolds with SU(3) structure, it is possible to do a consistent truncation to a finite set of light modes. The light modes are obtained by expanded in a set of p-forms, out of which some are not closed. The non closure is proportional to the torsion, which as shown in Refs. [121,74], plays a very similar role as the fluxes (see also Ref. [139]). We know in fact [140,121] that some of the torsion classes in Eqs. (3.16), (3.17) are mirror to NS flux, as we will briefly discuss in Section 5.5. By this procedure, Ref. [74] showed that the resulting action has the standard form of N = 2 gauged supergravity (we have shown already in Eq. (5.18) that the moduli space of complex structures and Kähler deformations is Kähler). To be more precise, torsion is encoded in the non-closure of the forms in the basis L d a = mK a K − eaL , d ˜a =0 , ˜a , dK = −eaK d K = mK ˜a . a
(5.42)
The forms used here are denoted in the same way as in the case of Calabi–Yau manifolds, but we should bear in mind that here they are clearly not harmonic (or at least some of them are not), and furthermore the indices a and K run from 1 to bJ and from 0 to b , where bJ and b are, respectively, the dimensions of the finite-dimensional set of light 2 and 3-forms, respectively. Furthermore, the “light” spectrum is shown to be the same as in the Calabi–Yau case, Tables 6 and 7, with h(1,1) , h(2,1) in IA (Table 6) replaced, respectively, by bJ , b , and the opposite in IIB (Table 7) (but differently from the CY case, these are not massless in the presence of RR and NS flux). We will not give the details of the derivation, but just quote that the resulting low energy action is consistent with N=2 gauged supergravity. Table 12, taken from Ref. [74], summarizes the effect of fluxes and torsion in the four-dimensional action.24 5.4. Flux-induced superpotential In the previous section we showed that fluxes induce a potential for certain moduli in type-II Calabi–Yau and typeII Calabi–Yau orientifolds (and we also briefly discussed the effect of fluxes on manifolds of SU(3) structure). We mentioned that in compactifications yielding N = 1 actions, this potential can be fully derived from a superpotential, except in the case of Calabi–Yau O5 compactifications, where the NS-flux generated potential comes from a D-term. A flux generated superpotential was proposed by Gukov–Vafa–Witten (GVW) [141] for M-theory compactifications to three dimensions on Calabi–Yau four-folds. GVW showed that the tension for BPS domain walls (five-branes wrapping a four cycle of the CY 4-fold) separating vacua coincided with the jump in the superpotential when going from one vacua to the other. Taylor and Vafa [142] showed a similar result for type-IIB Calabi–Yau 3-folds, where 23 The symbols F and D in Table 11 mean that the potential arises from an F or a D-term, respectively (see Eq. (5.21)). Additionally, 0 in the row corresponding to NS flux in IIA stands for the combination mK ˜ K − eK K . 24 The tensor D a comes from the expansion of C . In the text we have used a dual scalar instead of D a . 4 a 2 2
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the BPS domain walls correspond in that case to five branes wrapped around 3-cycles in the Calabi–Yau. They also proposed a type-IIA superpotential along the lines of Gukov’s for Calabi–Yau four-folds [143]. The type-IIB superpotential for compactifications of Calabi–Yau 3-folds, and Calabi–Yau O3/O7 generated by the fluxes is K WO3/O7 = G3 ∧ = (eK RR − ieK )Z K − (mK (5.43) RR − im )FK , where in the last equality we have used Eqs. (2.7), (2.8), (3.23) and (5.6). This superpotential depends on the complex structure moduli through , and on the dilaton–axion, by the definition of G3 . On the contrary, the Kähler moduli, v , , as well as the moduli coming from B2 and C2 , ba and ca do not appear in the superpotential. The potential is obtained from this superpotential by computing the Kähler covariant derivatives, Eq. (5.21). The Kähler potential for the chiral multiplets in O3/O7 compactifications is given in Eqs. (5.12) and (5.26). The Kähler covariant derivatives are given by i ¯ 3 ∧ + iGab ba bb W, DT W = −2 v W , D W = e G 2 K b DGa W = 2iGab b W, Dzk W = G3 ∧ k , (5.44) where we have used j = kk + k . jzk
(5.45) ¯
Inserting (5.44) in (5.21), we get the following potential for Calabi–Yau O3/O7 compactifications (the metric K I J is given explicitly for example in Ref. [144]) 18ie ¯ + Gkl G3 ∧ k G ¯ 3 ∧ G3 ∧ ¯ 3 ∧ ¯ l . VO3/O7 = 2 G (5.46) ¯ K ∧ As anticipated from the no-scale condition (5.28), this potential is positive semi-definite. It contains both RR and NS fluxes, and depends on the axion–dilaton and complex structure moduli, in agreement with the gauged supergravity result obtained by KK reduction, summarized in Table 11. Furthermore, inserting the expansions for all the forms given in the previous sections, it can be shown that it agrees with the sum of (5.34) (for type IIB, i.e. with N replaced by M) and (5.37). Another check of GVW superpotential comes from supersymmetry conditions [21]. A Minkowski vacuum should satisfy W = 0, DI W = 0. From (5.44) we see that these are satisfied provided ¯ W = 0, DI W = 0 ⇒ G3 ∧ = 0; G3 ∧ = 0; G3 ∧ k = 0 . (5.47) These equations imply that in a fixed complex structure, there are no (0, 3), (3, 0) or (1, 2) pieces in G3 , or in other words, G3 should be (2, 1). Since there are non-trivial 1-forms in a CY, G3 is automatically primitive. Then G3 satisfies the type B supersymmetric conditions discussed in Section 3.3 (see Table 4). It is easy to see that when G3 is (2, 1), the full potential (5.46) is zero. As we discuss below Table 4, the supersymmetries preserved by a type-B solution are the same as those for O3/O7 planes, so it is expected that supersymmetry conditions resulting from the GVW superpotential fall into type-B class. Note however that if one adds (2, 1) complex 3-form flux on a Calabi–Yau manifold, the backreacted geometry is no longer a product, but it is a warped product, and the internal manifold is no longer Calabi–Yau. It is nevertheless “as close as it gets”, namely the torsion classes W1 , W2 , W3 are zero, and the only non vanishing classes are in the 3 representation. Furthermore, for the case = const -no D7-branes- (corresponding in Table 4 to the row without an “F ”), the internal manifold is conformally Ricci-flat. As discussed below Table 4, the conformal factor is the inverse of the warp factor. The warp factor behaves at large radius like e2A ∼ 1 + O(gs N 2 r −4 ), where N is the number of D3-branes, or the units of 3-form flux. Therefore, it is a usually argued that in the large radius, weak coupling limit, where one trusts the supergravity approximation, the effect of the warping is negligible. An honest computation of supersymmetry conditions via a superpotential and Kähler potential should nevertheless include the warping.
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A first step toward this was put forward in Refs. [145–147], who considered KK-reductions in warped products with a conformal Calabi–Yau factor. In particular, it was claimed in Refs. [145,147]that the warp factor affects the N = 1 Kähler potential, but not the superpotential. In the case of O5/O9 planes, Ref. [116] showed that RR fluxes generate a superpotential, while NS fluxes generate a D-term. The superpotential for this case is the Gukov–Vafa–Witten one, setting H3 to zero, i.e. (5.48) WO5/O9 = Fˆ3 ∧ = eK RR Z K − mK RR FK . This superpotential depends on the complex structure moduli only, which implies that all its Kähler covariant derivatives except the one along the complex structure moduli are proportional to W itself. The derivative along the complex structure moduli, Dz W gives the same expression as in (5.44), with G3 replaced by F3 . This gives the F -term piece of the potential 18ie kl ¯ ˆ ˆ ˆ ˆ F3 ∧ F3 ∧ + G (5.49) F3 ∧ k F3 ∧ ¯ l , VO5/O9,F = 2 ¯ K ∧ which agrees with (5.34) after truncating appropriately the sum over moduli. Imposing the Minkowski supersymmetric vacuum conditions W = 0, DI W = 0, one gets that F3 should have no (0, 3) or (1, 2) piece. Since F3 is real, the F -term conditions would set all the components of F3 to zero. On the other hand, we argued that O5/O9 planes preserve type-C supersymmetries, and we therefore expect to get a type-C solution, which, according to Table 4, can have non-vanishing F3 . There is nevertheless no contradiction, because we see from Table 4 that a primitive component of F3 generates the torsion class W3 . The internal manifold is then no longer Calabi–Yau, in a much more drastic way than in type-B, namely, besides the torsion in 3 representations, there is torsion in the 6 representation. This means that in order to get supersymmetry conditions for type-C solutions, one should consider a more general superpotential for manifolds of SU(3) structure, not necessarily Calabi–Yau. We will come back to this issue shortly. In Type-IIA, the potential comes from a superpotential25 of the form [143,142,115,22] H3 ∧ + FˆA ∧ e(B+iJ ) WO6 = = T m − N e + e0 RR + ea RR t a +
1 2
Kabc maRR t b t c +
1 6
m0RR Kabc t a t b t c ,
(5.50)
where FA = F0 + F2 + F4 + F6 , t a is defined in Eq. (5.6) and N , T in Eq. (5.23). This superpotential depends on all the O6 moduli, and so does the corresponding potential, which agrees with the expression from previous section obtained by doing a KK reduction. This is a fundamental difference between IIA and IIB flux superpotential, and will become very important in next section when we discuss moduli stabilization. As in the case of Calabi–Yau O5/O9 compactifications, variations of this superpotential do not yield the N = 1 supersymmetry conditions for IIA shown in Table 3. The reason for this is the same as in O5/O9 case, namely N = 1 type-IIA vacua in manifolds of SU(3) structure have either NS flux and nonzero torsion W3 , or RR flux and nonzero W2 . In one case the manifold is non-Kähler, and in the other it is not even complex, so we do not expect to get the supersymmetry conditions in Table 3 from variations of superpotential for Calabi–Yau orientifolds 5.50. What should be varied instead is the general superpotential for manifolds of SU(3) structure, to which we turn. The superpotentials given so far, namely Eqs. (5.43), (5.48), (5.50), can be obtained from the supersymmetry variation of the gravitino [132], whose generic form is
= ∇ + ieK/2 W ∗ ,
(5.51)
where is the (four-dimensional) N = 1 supersymmetry parameter and K and W are the N = 1 Kähler potential and superpotential. Given the ten-dimensional supersymmetry transformation of the gravitino (3.2), inserting the decomposition of the ten-dimensional supersymmetry spinor (3.18), and the Kähler potentials (5.26), (5.31), (5.24), we obtain the superpotentials (5.43), (5.48), (5.50) [135]. The O3/O7, O5/O9 and O6 superpotentials (5.43), (5.48), 25 In massive IIA, there is an additional term in the potential proportional to F
conditions are satisfied.
0
Im ∧H3 [22,23], which vanishes when the tadpole cancellation
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(5.50) are actually obtained using respectively, a = ib, a = b and a = bei in (3.18), which are the supersymmetries preserved by O3, O5 and O6 planes (cf. Tables 3, 4). The reason why we mention this is that this procedure allows us to obtain the superpotential for SU(3) structure manifolds, not necessarily Calabi–Yau’s. In order to do this, we just need to insert the appropriate covariant derivative for the internal spinor. This was done in Ref. [148] for the heterotic theory, in Ref. [149] for M-theory, and in Ref. [74] for type-II theories (see also Ref. [150] for IIA). We quote the results of Ref. [74], and refer the reader to the original reference for details. The general N = 1 superpotential for unwarped compactifications26 manifolds of SU(3) structure is 2 − 2 − ¯ ¯ ¯ + , FˆIIA + , WIIA = a¯ e + , d− − b e + , d− + 2 a¯ b ¯ + − 2i a¯ b ¯ − , FˆIIB , WIIB = a¯ 2 e− − , d+ + b¯ 2 e− − , d (5.52) where ± are given in (5.17) and FˆIIA and FˆIIB are the sum of the RR fluxes, as in (3.42). This superpotential is similar to the one proposed in Ref. [151], which is expressed in terms of periods of pairs of Calabi–Yau mirror manifolds. Inserting a = ib in (5.52) we get the GVW superpotential for O3/O7 orientifolds of Calabi–Yau (5.43). With a = b, we recover the RR part of O5/O9 superpotential (5.48), which gets modified by a torsion piece, namely WO5/O9,nonCY = (e− dJ + Fˆ3 ) ∧ . (5.53) For a = 0 in IIA or IIB, we get the heterotic superpotential [75,148] Whet = e− (dJ + iH ) ∧ .
(5.54)
Finally, for a = ib in IIA we get, after integrating by parts the NS piece, the “torsional O6” superpotential − WO6,nonCY = e (dJ + iH ) ∧ Re + i FˆIIA ∧ eB+iJ ,
(5.55)
whose RR and NS pieces were proposed respectively in Refs. [143,121], and checked explicitly for twisted tori in Ref. [22]. These superpotentials have the right holomorphic dependence on the respective chiral multiplets. The superpotentials (5.52) were obtained for (unwarped) compactifications on manifolds of SU(3) structure. Nevertheless, Ref. [74] conjectures that (5.52) is also valid for manifolds of SU(3)×SU(3) structure on T ⊕ T ∗ (which comprises the cases of SU(3) and SU(2) structures on T), if we just replace ± by the appropriate Clifford(6, 6) spinors, given in (3.39) (where + would have to include a factor of e−B ). 5.5. Mirror symmetry In this section we review the state of the art about mirror symmetry for flux backgrounds. To start with, let us very briefly review the main ideas of mirror symmetry in Calabi–Yau compactifications. (See for example [152] for a review.) String theory compactified on a Calabi–Yau three-fold gives a four-dimensional N = 2 theory. From the world-sheet point of view, this compactification yields a two-dimensional (2, 2) superconformal field theory (SCFT) whose marginal operators belong to the (chiral,chiral) and (antichiral,chiral) rings, of respective dimensions h0,q (M, p T ) = hp,q (M) and h0,q (M, p T ∗ ) = h3−p,q (M). From the conformal field theory point of view, there is a trivial symmetry corresponding to the exchange of a relative sign between the two U(1) currents, by which (c, c) ↔ (a, c). However, on the geometrical level this symmetry is far from being trivial, as it amounts for example to the exchange h2,1 ↔ h1,1 on the Calabi–Yau cohomologies. It implies that for a given (2, 2) SCFT, there are two interpretations, as a string theory compactified on two topologically very different manifolds, M and M˜ such that ˜ . hp,q (M) = h3−p,q (M) 26 In deriving this result, the warp factor was set to zero.
(5.56)
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This very nontrivial symmetry is called “mirror symmetry”, and the manifolds M and M˜ are mirror pairs. IIA com˜ In particular, the complex structure moduli space of M is pactified on M is identical to a IIB compactification on M. ˜ as well as their respective prepotentials, i.e. [113] identified to the Kähler moduli space of M, Z K = (Z 0 , Z 0 zk ) ←→ t A = (1, t a ) , FK ←→ GA .
(5.57)
Furthermore, the whole multiplets are mapped, as the effective actions resulting from Calabi–Yau compactifications respect the exchange (see Tables 6, 7) (K , ˜ K ) ←→ (cA , A ) = (C2 , ca , C0 , a ) .
(5.58)
Mirror symmetry was also shown to hold at the level of effective actions in Calabi–Yau orientifolds in the large volume, large complex structure limit in Ref. [115]. The question we are interested in, is to what extent mirror symmetry survives when fluxes are present, or how should this symmetry be modified. We will try to answer this question from the effective supergravity point of view. There are several very important aspects of this question to take into account. The first one is that introducing fluxes generically breaks supersymmetry spontaneously. When fluxes are present, we are led to look for mirror symmetry of the effective actions rather than on vacua [121,74,153]. The second aspect is that from (5.56) we expect fluxes in even cohomologies to be mapped to fluxes in odd cohomologies. For RR fluxes this is fine, as IIA contains fluxes in even cohomologies, while those of IIB are in odd. NS flux however belongs to an odd cohomology, and its “mirror” is an even NS “flux”: torsion [140,121]. Therefore, the right setup to study mirror symmetry in the presence of fluxes is that of compactifications on manifolds with torsion, or more precisely on manifolds with SU(3) structure. This also relates to the first aspect: the effective action whose vacua are backgrounds with non zero flux should be those resulting from compactifications on SU(3) structure manifolds, rather than Calabi–Yau’s. Working on manifolds of SU(3) structure, Ref. [121] made precise the conjecture in Ref. [140], showing that the mirror of the NS flux H3 is the torsion of half-flat manifolds, namely Re W1 , Re W2 . Ref. [103] obtained the mirror symmetry map for general SU(3) structure manifolds, and Ref. [153] its topological version. These results are obtained by extending the Strominger–Yau–Zaslow (SYZ) procedure [154] for Calabi–Yau manifolds to manifolds of SU(3) structure. SYZ conjectured that every Calabi–Yau with a mirror is a T 3 Special Lagrangian fibration over a threedimensional base, and mirror symmetry is T -duality along the T 3 fiber. Assuming that the SU(3) structure manifolds in question have this T 3 fibration, Refs. [121,103,153] perform three T -dualities along the fiber. Let us first discuss mirror symmetry (or 3 T -dualities) in terms of the defining objects of the structure, J and (or equivalently the spinor ). Using the T -duality rules for the supersymmetry parameter [155], one can see that by 3 T -dualities there is an exchange of + with − . Using this in (3.31), it is natural to conjecture that mirror symmetry is an exchange of pure spinors, namely [103] + ←→ − eB+iJ ←→ ,
(5.59)
where in the last line we have specialized to manifolds of SU(3) structure (see (3.33), (5.17))). Ref. [103] checked this conjecture explicitly by performing 3 T -dualities on a T 3 -fibered metric and a B-field of fiber-base type. Mirror symmetry exchanges therefore the two Clifford(6, 6) pure spinors. We expect (5.59) to hold also for the general case of SU(3)×SU(3) structures on T ⊕ T ∗ . We want now to introduce fluxes on the SU(3) structure manifolds. The SYZ picture of mirror symmetry makes it clear that RR fluxes are mapped among themselves (even fluxes in IIA are mapped to odd fluxes in IIB), while NS fluxes are mixed with metric components via T -duality (see for example Refs. [156,157] for constructions relevant to this discussion). The explicit mirror symmetry map involving NS fluxes and torsion for manifolds of SU(3) structure found in Ref. [103] is ¯
¯
i(W3 + iH (6) )ij + ij k (W¯ 4 + iH (3) )k ←→ −2iW¯ i¯2E − 2gi¯E (W¯ 1 + 3iH (1) ) (W5 − W4 − iH (3) )i ←→ (W5 − W4 − iH (3) )™¯ ,
(5.60)
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or in a more compact version (∇J + H )ij k ←→ (∇J − H )i¯Ek¯ .
(5.61)
¯ ™¯ , This comes from the exchange of + and − under the T -dualities, which results in Qij ↔ Qi¯E and Qi ↔ −Q where the Q’s are defined in (3.19). Actually, all the matrices in (3.19) entering the full supersymmetry transformations
, M where shown to follow such an exchange [60], if in addition FIIA ←→ FIIB .
(5.62)
The N = 1 supersymmetry equations for the Clifford(6, 6) pure spinors (3.40), (3.41) respect the mirror symmetry maps (5.59), (5.62).27 Eqs. (5.60), (5.62) can be understood as the result of mirror symmetry exchange of SU(3) representations 6 + 3¯ ↔ 8 + 1 [103]. We turn now to the specific question of mirror symmetry of the N = 2 effective actions resulting from compactifications on SU(3) structure manifolds in the presence of fluxes. The Kähler potentials for the vector multiplet moduli space in SU(3) structure compactifications, spanned by the lines of pure spinors + and − for IIA and IIB, respectively, are indeed mapped to one another, as can be seen from (5.18). The N = 2 flux generated potential, which can be derived from the mass matrix for the gravitinos, and is an N = 2 version of the superpotential, was also shown in Ref. [74] to respect the mirror symmetry maps (5.59), (5.62). The N = 1 superpotential is comprised in its N = 2 version, namely it is obtained by projecting the N = 2 on a plane orthogonal to the N = 1 preserved spinor in the SU(2)R -symmetry space (the parallel projection gives the D-term). As a result, it also satisfies the mirror maps if we do appropriate mirror projections, as can be seen from (5.52). Performing the explicit expansion of ± and the RR potentials in the basis of “light modes” of Refs. [121,74], one can see that the kinetic terms in the effective actions are symmetric under the maps (5.57), (5.58) (remember nevertheless that these are not “moduli”, as some of them are massive due to the torsion and fluxes). In the presence of RR and only electric NS fluxes (the latter include torsion, see Eq. (5.42)), the IIA and IIB N = 2 superpotential are symmetric under (5.57), (5.58) if fluxes are mapped via [119,121,74] A (eRR K , mK RR ) ←→ (eRR A , mRR ) , eA K ←→ eK A ,
(5.63)
where NS flux and torsion have been combined in the flux eA K ≡ (eK , ea K ). For the magnetic NS fluxes, Ref. [74] shows that the N = 2 potentials are not mirror symmetric. In particular, as summarized in Table 12, magnetic fluxes give rise to a massive tensor in IIB, while in IIA this is not the case. This seems to contradict the conjecture (5.59), which has torsion and H -flux hidden in d± (see footnote 27). However, the expansion in light modes of Ref. [74] is specialized to the case of SU(3) structure, where − contains only a 3-form (see 5.17), and not all odd forms, as in the case of SU(3)×SU(3) structures (see 3.39). The seeming paradox is therefore expected to be resolved by considering the more general SU(3)×SU(3) structures. 6. Moduli stabilization by fluxes In the previous section we discussed the flux generated potentials and superpotentials in Calabi–Yau compactifications and in general SU(3) structure manifolds. From now on, we will concentrate on compactifications on Calabi–Yau orientifolds. The presence of the flux-induced potentials implies that some of the moduli of Calabi–Yau compactifications cease to be moduli, i.e. they acquire mass due to the fluxes. This is one of the main reasons, if not the main one, which makes compactifications with fluxes so attractive, being a very active domain of research over the past five years. Let us first explain why we need to understand the ultimate mechanism of moduli stabilization. Scalars which remain massless lead to long range interactions. The coupling of these scalars to matter (both in standard compactifications and in brane-world scenarios) is not universal. This implies that different types of matter get different accelerations from these long range forces, violating the principle of equivalence. High precision measurements of the principle 27 Note that by using e±B ˜ ± and H = dB, the twisted covariant derivative (d + H ∧) in (3.40), (3.41) can be replaced by an ordinary derivative. This has been used in writing (5.52).
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of equivalence have tested the ratio of inertial to gravitational mass up to 1 part in 1013 [158]. Such a “fifth force” should therefore be very weak, or sufficiently short ranged not to violate experiments. If moduli remain massless, we do not expect their couplings to all types of matter to be so much smaller than that of gravity, which implies that in any realistic theory, all “moduli” should be massive. Furthermore, if the vacuum expectation value of the moduli fields, most notably the volume modulus, can be anything, string theory looses any predictability. If the flux generated potential for moduli has local minima, moduli will be stabilized at the values where one of these minima lie. If the vacua is at a minimum of the potential which is not the absolute minimum, there is an instability against tunneling through a barrier to the absolute minimum (see Ref. [159] for a review of moduli and microphysics). In many cases, notably for the GVW superpotential, the resulting potential has flat or runaway directions. In such cases, fluxes fix only some of the moduli, and in order to obtain realistic theories one needs to invoke non-perturbative effects to stabilize the remaining moduli. Once moduli are stabilized, the overall consistency requires that the dilaton is fixed at small values, i.e. gs = e >1 and the overall volume, or average radius, at large values in string units, i.e. √ R ∼ K v v v 1/6 ? . We will start by reviewing moduli stabilization in orientifolds (O3) of Calabi–Yau manifolds, following Ref. [21], and specializing on a compact version of the conifold. Then, we will discuss moduli stabilization on tori, focusing on the case of O3 orientifolds of T 6 and following Ref. [82]. As mentioned in the introduction, we will not cover stabilization of moduli by open string fluxes, or stabilization of open string moduli. We refer the reader to some of the original references [35,160,33]. Besides, we will not cover moduli stabilization in M-theory [36,37], and in the heterotic theory [161,75,139]. 6.1. Moduli stabilization in type IIB Calabi–Yau orientifolds In this subsection we review the mechanism of moduli stabilization in type IIB Calabi–Yau orientifolds, following mainly Giddings–Kachru–Polchinski (GKP) [21], who focused on (a compact version of) the conifold. Later developments for other Calabi–Yau manifolds can be found for example in Refs. [78–80,162–164], who considered Calabi–Yau hypersurfaces in weighted projective spaces. The superpotential for Calabi–Yau O3 compactifications is given in Eq. (5.43). The conditions for a supersymmetric Minkowski vacuum W = 0,
DI W = 0 ,
(6.1)
(2,1) 2h−
result in + 2 real equations, since the covariant derivative of the superpotential along the Kähler moduli is proportional to W itself (see (5.44)). (Note that this implies that the condition W =0 comes automatically from demanding (2,1) supersymmetry, and therefore there are no supersymmetric AdS4 vacua in this construction). The 2h− + 2 equations a a are independent of the Kähler moduli (v , ), (b , c ), which remain unfixed. Turning on appropriate fluxes, it is (2,1) possible to fix 2h− + 2 real moduli, namely the complex structure zk and the dilaton–axion = C0 + ie− . The fact that the complex structure moduli and the dilaton are fixed in type-B solutions is easy to understand from the supersymmetry conditions: given a set of fluxes (eK , mK , eRR,K , mK RR ), there are only some fixed complex structures and axion–dilaton which make the complex 3-form flux G3 (2, 1) (the axion–dilaton enters in the definition of G3 ). If there is no complex structure and such that G3 is (2, 1), then either there is a solution but it is not supersymmetric (this would be the case if there is some complex structure and for which G3 is (2, 1) plus (0, 3)), or there is no solution at all for that set of fluxes. GKP discuss moduli stabilization in a compact version of Klebanov–Strasssler [18], i.e. on a manifold with a local region with a deformed conifold geometry, embedded in a compact manifold with an O3 identification. GKP assumed (1,1) (2,1) (1,1) h− = 0, i.e. no (ba , ca ) moduli, and h− = h+ = 1, i.e. one complex structure modulus and one Kähler modulus, (2,1) (1,1) although the results are easily generalized to any h− , h+ , as they explain. The deformed conifold is a cone over a space with topology S 2 × S 3 . It is described by complex coordinates (w1 , w2 , w3 , w4 ) subject to w12 + w22 + w32 + w42 = z ,
(6.2) (2,1)
where the complex parameter z is the complex structure modulus, which controls the size of the S 3 . Since h− = 1, there are four non-trivial 3-cycles. In the vicinity of the conifold, there are two relevant cycles: the S 3 , called A, which
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intersects once the dual cycle B.28 The Klebanov–Strassler solution has M units of F3 -flux through the A-cycle, and −K units of H3 on the B-cycle, i.e. e1 = −K, m1RR = M (cf. Eq. (2.6)). Using this in (5.43), we get W = −MF(z) + Kz ,
(6.3)
F(z) has an expansion of the form F(z) = (z/2i) ln z+ analytic terms. Inserting this in (6.3) we get from the Minkowski vacuum condition, W = 0, that the complex structure is fixed to an exponentially small value z ∼ e−2K/Mg s .
(6.4)
GKP show that this also satisfies Dz W = 0 in the regime K/gs ?1, K/Mg s ?1. In order to satisfy the other F -term constrain, D W = 0, GKP showed that in the compact case one needs to turn on additional fluxes on the two remaining 3-cycles A , B . Calling −K the number of units of H3 on the B cycle, and z (z) the period of along the A cycle (of order 1), all the susy conditions (6.1) stabilize the complex structure modulus and dilaton at ¯ =
MF(0) , K z (0)
z ∼ e(2K/K ) Im[F(0)/z (0)] .
(6.5)
This implies that by appropriate choices of the fluxes, the dilaton can be fixed at weak coupling, and z is small. The complex structure being stabilized at a small value has interesting phenomenological consequences, as the warp factor will be very small close to the end of the throat (located at the points where the S 3 shrinks to zero). This is because the warp factor, which solves (4.32), goes for D3-branes like e4A ∼ r 4 , where r is the conical coordinate. The resolution of the conifold cuts this off at r ∝ w 2/3 ∝ z1/3 , which means that there is a minimum, (but non zero) warp factor
e2Amin ∼ z2/3 ∼ e−4K/3K Im[F(0)/z (0)]
(6.6)
which generates a large hierarchy of scales. This a string realization of Randall–Sundrum type models [25] à la Verlinde [26], where the compact region plays the role of IR brane, and the size of hierarchy is a function of the fluxes. As in all type-B solutions, the Kähler moduli (in this case just the overall volume) is not stabilized by the fluxes. 6.2. Moduli stabilization in type IIB orientifolds of tori In this section we discuss moduli stabilization in compactifications of type IIB on tori. We will present one of the first examples discussed in the literature, that of Kachru–Schulz–Trivedi (KST) [82], consisting compactifications on T 6 with orientifold 3-planes. Other more complicated examples have been worked out in the literature after KST. In particular, moduli stabilization on orientifolds (O3) of T 6 /(Z2 ⊗ Z2 ) orbifolds, where there are twisted and untwisted sectors, is discussed in Refs. [165–168]. Stabilization in orientifolds of T 6 /ZN and in generic T 6 /(ZN ⊗ ZM ) is analyzed respectively in Refs. [169,39]. In these examples, the effect of D9—brane fluxes—which can stabilize some Kähler moduli and also lead to a chiral open string sector with potential interest for phenomenology- is additionally considered. Moduli stabilization in K3×T 2 /Z2 is discussed in Refs. [170,171,33], where in the latter D3 and D7-brane moduli and fluxes are also taken into account. KST study N = 1 compactifications on a T 6 /Z2 orientifold with NS flux and RR flux F3 . Since there are O3 planes, we expect the supersymmetries of the solution to be of type B. We should be careful however in applying the conditions from supersymmetry outlined Table 4, since a torus is a manifold of trivial structure, while the results of Table 4 concern manifolds of SU(3) structure. The fact that the structure group is more reduced than SU(3) means that there is more than one nowhere vanishing spinor that we can use in the decomposition (3.10). This can lead to solutions that are very different from those of Table 4, as we showed in Section 3.4 (for example, the internal manifold need not be complex). Nevertheless, KST showed that the supersymmetry parameter of their solution uses only one internal spinor, as in the case of SU(3) structure, Eq. (3.10). The supersymmetry conditions are therefore those of type-B in Table 4, except that when the structure group is more reduced than SU(3) there are harmonic 1-forms, and therefore the condition of G3 being primitive has to be further imposed (in the SU(3) structure case, the absence of harmonic 1-forms made this 28 The cycle A can be taken to be the S 3 on which all w’s are real, while the cycle B, which goes off to infinity in the non compact case, can be constructed by taking w1,2,3 imaginary and w4 real and positive.
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condition automatic, and we only needed to impose that G3 be (2, 1)). This will be very important, as it will allow to fix some of the Kähler moduli, which are otherwise unfixed in the Calabi–Yau case. The moduli for O3 compactifications in Calabi–Yau manifolds are given in Table 10. In the case of a torus, we have (2,1) (1,1) to take into account that the structure group is trivial. This implies that besides h− = 9, h+ = 9, we have to consider (1,0) (2,0) (3,1) (3,2) (0,1) also the cohomologies h− = h+ = h+ = h− = 3 and their conjugates (h− = 3, etc). Note that in T 6 /Z2 , (2,1) (1,1) all even (odd) forms are even (odd) with respect to the involution . Therefore, h+ = h− = 0. The additional cohomologies add 12 extra vectors coming from Bm and Cm , and six extra scalars from C(3,1) and C(1,3) . This gives a total of 12 vectors, 21 scalars from the metric (9 from the Kähler moduli v and 12 (real) from complex structure deformations,29 15 scalars from C4 (9 scalars plus 6 extra from C(3,1) and C(1,3) ) and 2 from the axion–dilaton. These arrange into multiplets of N = 4, as the existence of 4 nowhere vanishing vectors plus the orientifold projection implies that the effective four-dimensional action is N = 4. The graviton, 6 gauge bosons and the axion dilaton are in the N = 4 supergravity multiplet, and the others build, together with their fermionic partners, six vector multiplets with one vector and six scalars each. The scalars span the manifold MN=4 =
SU(1, 1) SO(6, 6) × , U(1) SO(6) × SO(6)
(6.7)
where the first factor corresponds to the dilaton–axion, and the second to the scalars in the vector multiplets. The explicit solution is constructed as follows. First, let x i , y i , i = 1, 2, 3 be six real coordinates on the torus, with periodicities x i ≡ x i + 1, y i ≡ y i + 1, and take the holomorphic 1-forms to be dzi = dx i + ij dy j .
(6.8)
The matrix ij specifies the complex structure. The holomorphic 3-form is = dz1 ∧ dz2 ∧ dz3 .
(6.9)
The basis (K , L ) from Eq. (2.7), where K = 0, . . . , 9 is taken to be 0 = dx 1 ∧ dx 2 ∧ dx 3 , ij = 21 ilm dx l ∧ dx m ∧ dy j , ij = − 21 j lm dy l ∧ dy m ∧ dx j 0 = dy 1 ∧ dy 2 ∧ dy 3 .
(6.10)
The holomorphic 3-form in (6.9) is given in this basis by = 0 + ij ij − ij (cof )ij + 0 (det ) ,
(6.11)
where (cof )ij ≡ (det ) −1,T =
kp mq 1 2 ikm jpq
.
(6.12)
The NS 3-form fluxes along these 3-cycles are denoted e0 , eij , m0 and mij (see Eq. (2.8)), and similarly for the RR fluxes, adding a subindex “RR”. The number of units of flux is constrained by the tadpole cancellation condition (4.21). In T 6 /Z2 , there are 26 O3-planes, giving a negative contribution of −16 to the tadpole (4.21), leading to the following condition: 1 1 1 K H3 ∧ F3 = ND3 + (eK mK (6.13) ND3 + RR − m eK RR ) = 16 , 2 (2)4 2 T 6 2 where the factor 1/2 comes from the volume of T 6 /Z2 , which is half the volume of T 6 . (2,1)
29 Since h = 9, one would expect nine complex structure deformations instead of six. The difference appears because in the torus (as well − as in any manifold with no-where vanishing vectors) not all complex structure deformations correspond to deformations of the metric. There are six real deformations of the complex structure that leave the metric invariant [82,74].
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The type B, KST solution has additionally 5-form flux, and non-trivial warp factor whose Laplacian is given by (4.32). The internal manifold is not a torus, but a conformally rescaled torus. However, as usually done in this class of examples [21,14], it is argued that at large radius and weak coupling one can neglect the warping (and five-form flux) and treat the moduli space as if it was that of a Calabi–Yau. One should nevertheless demand that the integrated Bianchi identity for F5 is satisfied, or in other words, satisfy Eq. (6.13). For the given setup, the GVW superpotential (5.43) is ij
W = (m0RR − m0 ) det − (mRR − mij )(cof )ij − (eij RR − eij ) ij − (e0 RR − e0 ) .
(6.14)
The supersymmetry conditions reduce to eleven equations, namely ij
W − j W = 0 ⇒ m0RR det − mRR (cof )ij − eij RR ij − e0 RR = 0
(6.15)
j W = 0 ⇒ m0 det − mij (cof )ij − eij ij − e0 = 0
(6.16)
ij
j
j ij W = 0 ⇒ (m0RR − m0 )(cof )kl − (mRR − mij )ikm j ln mn − (eij RR − eij ) ik l = 0 .
(6.17)
In these equations, the condition W = 0 is a consequence of demanding vanishing F -terms along the Kähler moduli, DT W = 0 (see (5.44)). When W = 0, the Kähler covariant derivatives reduce to ordinary derivatives. Eqs. (6.15) are eleven complex coupled non linear equations for ten complex variables, namely the axion–dilaton and the nine complex structure moduli ij . Generically, these cannot be solved, and supersymmetry is broken, or even more, for a given set of fluxes, there might be no solution at all to the equations of motion. Besides (6.15), we should additionally impose the 3-form flux to be primitive, i.e. J2 ∧ G3 = 0 .
(6.18)
These are six real equations (J ∧ G is a five-form, with six different components) for the nine Kähler moduli v , which means that generically, and differently from the Calabi–Yau case, only three of them remain unfixed. KST work out several N = 1 vacua. Let us briefly discuss one of them. Taking the flux matrices to be diagonal, namely ij
(eij , mij , eij RR , mRR ) = (e, m, eRR , mRR ) ij ,
(6.19)
the complex structure matrix that solves (6.15) should be proportional to the identity ij = ij .
(6.20)
This means that the torus factorizes as T 6 = T 2 × T 2 × T 2 with respect to the complex structure. turns out to be the root of a third degree polynomial equation where the coefficients are given by the fluxes. Additionally, is fixed by the third equation in (6.15). Let us take for example the set of fluxes ij
(e0 , eij , mij , m0 , e0 RR , eij RR , mRR , m0RR ) = (2, −2 ij , −2 ij , −4, 2, 0, 0, 2) .
(6.21)
These fluxes contribute Nflux = 12 to the tadpole cancellation condition (4.21), (4.22), and therefore for consistency we should add 4 D3-branes. KST showed that the complex structure modulus and the axion–dilaton moduli are fixed for these fluxes at = 21 = e2i/3 . Generically six of the Kähler moduli are fixed by the primitivity condition (6.18). However, in the example at hand, KST showed that only 3 are fixed, and we are left with 6 flat directions given by the 3 radii of the T 2 ’s (Ji¯™ ), plus the components J12¯ + J21¯ , and the same for 1, 3 and 2, 3. Note that the √ coupling constant gs in this example is fixed at a value where perturbative corrections are important, namely gs = 1/ 3. It is possible however to chose fluxes such that gs is fixed in the perturbative regime. 6.3. Moduli stabilization in type IIA Calabi–Yau orientifolds In this section we discuss stabilization of moduli in type IIA Calabi–Yau (O6) orientifolds. As already mentioned, there are two main differences between the mechanisms of moduli stabilization by fluxes in type-IIA and type-IIB.
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The first difference is that the potential in type-IIA depends on complex structure as well as Kähler moduli, while its IIB counterpart depends on the complex structure moduli, but not the Kähler moduli. This means that a priori fluxes alone could fix all the moduli in IIA. But most importantly, in IIA there are no vacua (neither Minkowski, nor AdS) that involve a Calabi–Yau manifold (or better a conformal Calabi–Yau, as is the case of type-B solutions in IIB). Even more, one can see from Table 3 that Minkowski vacua in IIA involving manifolds of SU(3) structure have either W3 or W2 non-zero, or in other words, are either non-Kähler, or non-complex. Besides, Ref. [64] showed that AdS vacua with internal manifolds of SU(3) structure are only possible with “type BC” supersymmetries (i.e., those in the second column of Table 3), with a = bei ), and have non-vanishing W1 (besides non-vanishing W2+ when there is 2-form flux in the 8 representation), which means that they are not complex. W1 , as well as the mass parameter of type IIA and the singlets in 4-form flux and NS flux, are all proportional to the cosmological constant. In summary, Minkowski or AdS vacua involving manifolds of SU(3) structure are far from being Calabi–Yau: they are either non-symplectic (W3 = 0) or non complex (W1 = 0 or W2 = 0 or both). This means that using the Calabi–Yau orientifold superpotential (5.50) to determine vacua is just not correct.30 One should consider instead the general superpotential for SU(3) structure manifolds, Eq. (5.52). Nevertheless, as Ref. [23] argues, one can use the Calabi–Yau superpotentials and Kähler potential to attempt to determine the vacua and dynamics in terms of the properly corrected superpotential and Kähler potential. In any case, the reason why we review moduli stabilization mechanism in type IIA Calabi–Yau orientifolds is to understand the next subsection, which deals with tori: when the SU(3) structure is broken to SU(2) or further31 a conformally Ricci-flat space is a possible vacuum of IIA [64]. Furthermore, when there are no non-trivial 1-forms on the manifold, i.e. when h1 = 0, their moduli spaces are the same as those of a Calabi–Yau (or Calabi–Yau orientifold), and moduli stabilization works in the same way as for Calabi–Yau’s (with the caveat that again for discussing moduli, we will have to neglect the effect of the warping—or conformal factor—). The example studied in Ref. [23], namely a T 6 /(Z3 ⊗ Z3 ), is precisely of this type. After this long discussion of the differences between the IIA and IIB flux-induced superpotentials and moduli stabilization mechanisms, let us review the technicalities of IIA moduli stabilization on Calabi–Yau orientifolds, following Ref. [23]. The superpotential for Calabi–Yau O6 is given in (5.50), and the proper Kähler coordinates are given in Eqs. (5.6) and (5.23). Supersymmetric vacua are given by the conditions DI W = 0, for I = (N , T , t a ). The conditions DN W = 0, DT W = 0 give e + 2ie2D W Im(CF ) = 0 , m + 2ie2D W Im(CZ ) = 0 ,
(6.22)
where e2D = 6e2 /( J ∧ J ∧ J ), is a real function of the dilaton and Kähler moduli t a . Given that C and D are real, vanishing of the imaginary part of (6.22) implies that the real part of the superpotential is zero: Re W = ˜ m − e + Re(e0 RR + ea RR t a +
1 2
Kabc maRR t b t c + 16 m0RR Kabc t a t b t c ) = 0 .
(6.23)
This single equation is the only condition involving the axions, which means that only one combination of them is fixed by the fluxes. Therefore, the remaining h(2,1) axions are not fixed by the fluxes, and have to be stabilized by other (non perturbative) mechanisms. These are nevertheless the only moduli that cannot be stabilized by fluxes in type IIA Calabi–Yau orientifolds. The real parts of (6.22) say that if any NS flux is non-zero, then Im W has to be nonzero. Given that Im W = 0, the real parts of (6.22) are h(2,1) real equations that generically fix the h(2,1) (real) complex structure moduli in terms of the dilaton. Note that there are h(2,1) + 1 real NS fluxes, so we did not expect to have the h(2,1) + 1 (real) complex structure at the same time as the h(2,1) + 1 axions fixed by the NS fluxes. The Kähler moduli t a appear in the RR piece of the superpotential. Its Kähler covariant derivative Dt a W also splits into a real and imaginary part. Since the Kähler potential for t a depends only on their real part, v a , the imaginary part
30 We remind the reader that in IIB, one of the possible Minkowski vacua (type-B) involves a conformal CY, and it was argued in Refs. [145,147] that the conformal factor does not enter the GVW superpotential, which remains that of Eq. (5.43). 31 We could also say that SU(3) structure is “enlarged” to SU(2), depending on the point of view, namely whether one looks at the number of generators of the group, or the number of invariant spinors.
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of the Kähler covariant derivative contains just the regular derivative. To be more explicit, we have to impose Im(Dt a W ) = Im(jt a W ) = 0 ⇒ ba = −
maRR m0RR
.
(6.24)
All the moduli ba are therefore fixed.32 The real part of the Kähler covariant derivative gives, after some algebraic (1,1) (1,1) manipulations, h− simple quadratic equations for the h− moduli v a , which are therefore generically fixed. Finally Ref. [23] showed that the dilaton also gets stabilized. In summary, in massive Type IIA Calabi–Yau O6 compactifications with fluxes, enforcing DI W = 0, for W given by (5.50), leads to an AdS4 supersymmetric vacuum with all Kähler moduli (v a , ba ) generically stabilized; all complex structure moduli and dilaton (Re(Z k ), Im(F )) stabilized, but only one combination of the axions ( , ˜ ) fixed, while the remaining h(2,1) stay massless. 6.4. Moduli stabilization in type-IIA orientifolds of tori In this section, we illustrate the mechanism of moduli stabilization in type IIA orientifolds of tori with a specific example: an orientifold (O6) of the orbifold T 6 /(Z3 ⊗ Z3 ), constructed in Ref. [23]. We will concentrate on supersymmetric vacua, but we note that Ref. [23] considers additionally stabilization of moduli in non supersymmetric cases, by inspecting the minima of the potential. At the end of the section, we mention briefly other constructions of O6 orientifolds of twisted tori (i.e., manifolds of trivial structure but not trivial holonomy). As stressed in the previous section, the back reaction of the fluxes in IIA allows for vacua involving conformally Ricci-flat manifolds (or orbifolds/orientifolds thereof) only when the structure group is more reduced than SU(3). This is the case of tori, whose structure group is trivial, and, as we will see, can support supersymmetric fluxes. Furthermore, as also stressed in previous sections, if h1 = 0, as is the case for the orbifold T 6 /(Z3 ⊗ Z3 ), the moduli spaces and moduli fixing mechanisms work as in Calabi–Yau manifolds. We will show, following Ref. [23], that differently from the case of type-B compactifications, fluxes fix all moduli (since h(2,1) = 0 for this orbifold), and they can do it at arbitrarily large volume and weak coupling. Let us review how this magic works. The orbifold T 6 /(Z3 ⊗ Z3 ) is constructed as follows: the torus is parameterized by 3 complex coordinates dzi = dx i + i dy i (the Z3 action does not leave any freedom in the choice of complex structure), with the periodicity condition zi ≡ zi + 1 ≡ zi +
(6.25)
with = ei/3 . The two Z3 actions are given by T : (z1 , z2 , z3 ) → 2 (z1 , z2 , z3 ) 1+ 4 2 1+ 3 1+ 1 2 3 2 1 Q : (z , z , z ) → z + , z + ,z + . 3 3 3
(6.26)
T has 27 fixed points, while Q, is freely acting. The result is a Ricci-flat manifold (with curvature concentrated at 9 Z3 singularities), and Euler number = 24. The Euler number being non zero implies that there are no nowhere vanishing vectors, i.e. h1 = 0, which in turn makes the whole Calabi–Yau moduli business work for this case. The other Hodge numbers are h(2,1) = 0, h(1,1) = 3u + 9t (meaning three untwisted moduli and nine twisted moduli). The twisted moduli are localized at the singularities, and correspond to blow-up modes. In this review article we will not discuss twisted moduli and twisted moduli stabilization. We refer the reader to Ref. [23] for the discussion of blow up mode stabilization (see also Refs. [38,39]). The O6 projection is given by (−1)FL p , for the involution : zi → −¯zi ⇒ x i → −x i ,
yi → yi .
(6.27)
32 The case m0 = 0 is not interesting as in that case all other RR fluxes must vanish as well, therefore leaving all Kähler moduli unfixed, or RR v a are driven to zero.
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The hu
= 3 untwisted normalized two- and four-forms and the h3 = 2h(2,1) + 2 = 2 normalized three-forms are
wi = 2(3)1/6 dx i ∧ dy i = i(3)1/6 dzi ∧ d¯zi (−) 0 = (12)1/4 (dy 1 ∧ dy 2 ∧ dy 3 − 21 ij k dx i ∧ dx j ∧ dy k ) 0
= (12) w˜ = i
(dx ∧ dx ∧ dx −
1 2 ij k
∧ dy ) ∧ (dx ∧ dy
) = − 13
1/4
j 4 3 (dx
1
2
j
3
k
k
dy ∧ dy ∧ dx ) i
j
k
(+) (−)
(dz ∧ d z¯ ) ∧ (dz ∧ d¯zk ) j
j
k
where in parenthesis we have indicated the parity under , and we have used dy 3 =
1 √ . 8 3
=
(+) , T 6 /(Z3 )2
(6.28) dx 1 ∧ dx 2 ∧ dx 3 ∧ dy 1 ∧ dy 2 ∧
The O6 is wrapped along A0 , the cycle dual to 0 . The holomorphic 3-form is given by
√1 2
(0 + i 0 ) = i(3)1/4 dzi ∧ dz2 ∧ dz3 .
(6.29)
For this orientifold, Table 9 tells us that the untwisted moduli are a total of 8 real scalars: 6 of them are in 3 chiral multiplets t i = bi + iv i , i = 1, 2, 3 (cf. Eq. (5.6)), and the remaining two are the dilaton (written in the table as Re Z 0 ) and an axion 0 coming from C3 along 0 (cf. Eq. (5.23)). Now we want to turn on fluxes on this orientifold. The NS flux H3 should be odd under , which implies that it should be along 0 . The RR fluxes F0 and F4 should be even, while F2 and F6 are odd. We can therefore turn on the following fluxes H3 = −e0 0 ,
F0 = m0RR ,
F2 = miRR wi ,
F4 = −eRR i w˜ i ,
F6 = eRR 0 .
(6.30)
The tadpole cancellation condition (4.19) enforces m0RR e0 = −2 .
(6.31)
This means that the NS flux and the mass parameter are basically fixed by the tadpole (up to four choices ±(1, −2) or ±(2, −1)), but we are free to add any number of F2 , F4 and F6 fluxes. We can use (6.31) to write the full solution in terms of RR fluxes only. From Im(Dt a W ) = 0, Eq. (6.24) we know that the Kähler moduli ba = Re(t a ) are stabilized at the ratio of the mass parameter and the two-form flux, namely bi = −
miRR m0RR
.
(6.32) (1,1)
We said that the condition Re(Dt a W ) = 0 gives h− quadratic equations for the moduli v a = Im(t a ). Ref. [23] showed (1,1) that the solution to the h− = 3 equations that one gets for the T 6 /Z23 are33 1/2 j mRR mkRR eˆRR 1 eˆRR 2 eˆRR 3 1 i 5 v = , e ˆ ≡ e − 81 . (6.33) RR i RR i 9|eˆRR i | m0RR m0RR We showed that the complex structure equation DN W = 0 splits into a real and an imaginary part. The imaginary part stabilizes the dilaton at34 1/2 √ √ 12 3 1 2 3 1/2 4 3 eˆRR 1 eˆRR 2 eˆRR 3 − e = . (6.34) (v v v ) = 5 5 241|m0RR | m0RR We argued that the real part of DN W = 0, Eq. (6.23) implies that only one axion is fixed by the fluxes. In the example at hand there is only one axion, bingo!. It is stabilized at eRR i miRR m1RR m2RR m2RR eRR 0 0 . (6.35) = −2 + − 162 m0RR (m0RR )2 (m0RR )3 33 The factor 81 comes from the triple intersection number K
123 = 81. 34 We are setting here and in the next equation the twisted fluxes to zero.
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In summary, the orientifold of the T 6 /(Z3 ⊗ Z3 ) orbifold worked out in Ref. [23] has all moduli stabilized. The Kähler moduli t i = bi + iv i are stabilized at the values given by Eqs. (6.32)–(6.33), and the fixed dilaton–axion is given in (6.34)–(6.35). Let us make a few very important comments about the solution. First, this example with trivial structure has the same moduli as a Calabi–Yau because h1 = 0. Second, all moduli can be fixed thanks to the property h(2,1) = 0, otherwise there would be h(2,1) axions unfixed. Third, all the vacuum expectation values of the scalars depend on the mass parameter of type-IIA m0RR , which by the tadpole cancellation condition (6.31) is fixed to be ±1 or ±2. Last but not least, the flux parameters eRR i can be anything, since they do not enter the tadpole cancellation condition (neither do the F2 fluxes miRR , which give the same qualitative behavior). If we take all eˆRR i ∼ N , the radii and the dilaton are stabilized at eˆRR i ∼ N ⇒ v i ∼ R ∼ N 1/4 , gs ∼ N −3/2 . (6.36) Therefore, the stabilization can be done at arbitrarily large radius of compactification and weak coupling. Furthermore, inserting the stabilized moduli in the superpotential (5.50), we get that the cosmological constant is parametrically small, namely = −3eK |W |2 ∼ N −9/2 , (6.37) where we have used exp(−2 ln[ Re(C) ∧ ∗Re(C)]) = e4 /(vol)2 (see Refs. [115,23] for details). Let us analyze these vacua from the point of view of supersymmetry conditions discussed in Section 3.3. First of all, the supersymmetric vacua are all AdS, since the superpotential is not zero. The NS flux has obviously only a singlet ¯ Then, setting the two-form RR fluxes mi = 0 for convenience, it is not hard component, proportional to i( − ). RR to show using (6.33) that the four-form flux F4 is proportional to J 2 ∝ ij k ei RR wj ∧ wk , namely it is also in the singlet representation. Therefore, this solution contains only singlets in the fluxes, which give a parametrically small cosmological constant, but not a torsion class W1 . This singlets are nevertheless not that innocent, as they allow us to fix all moduli, and in a region were supergravity approximation holds. Needless to say again how differently moduli stabilization works in type-IIA and type-IIB! Even if needless, let us stress again the two main differences: first, in IIB, the fluxes that stabilize moduli enter the tadpole cancellation, so we are not free to make them as large as we want. Secondly, in IIB Calabi–Yau orientifolds, no Kähler moduli are fixed. This is better in the case of tori, where some Kähler moduli are fixed by the primitivity condition J ∧ G = 0. However, there will always be at least one unfixed Kähler modulus, the overall volume, since as GKP showed, all the type-B conditions are invariant under rescalings of the metric. In type IIA the unfixed moduli are all but one axionic partners of the complex structure moduli and dilaton.35 In manifolds with a rigid complex structure, as the one discussed in this section, there is only one axion, and therefore all the moduli are fixed. All this agrees with the results from Section 5.3, summarized in Table 11. Other discussions of closed moduli stabilization in IIA tori are given in Refs. [22,24]. Ref. [22] studied moduli stabilization in O6 orientifolds of the orbifold T 6 /(Z2 ⊗ Z2 ) including torsion (usually called metric fluxes), and found by exploiting the underlying gauged supergravities, that all untwisted moduli can in principle be stabilized by fluxes and torsion. Ref. [24] studied moduli stabilization in general orientifolds of Calabi–Yau twisted tori (with torsion, or “metric fluxes”), by analyzing the superpotential (5.55). They found that some axions remain unfixed in Minkowski vacua, while all moduli can be stabilized in some AdS ones. 7. Moduli stabilization including non-perturbative effects and De Sitter vacua We saw in the previous sections that fluxes are usually not enough to stabilize all moduli. In particular, we reviewed in Section 6.1 that in type-IIB compactifications on Calabi–Yau orientifolds, fluxes stabilize the complex structure moduli and the dilaton, but leave Kähler moduli unfixed. There are nevertheless perturbative and non-perturbative corrections to the leading order Kähler potential and superpotential considered in previous sections that can help in stabilizing the remaining moduli. In this section we discuss these corrections, concentrating on their effect on moduli stabilization, and whether they lead to de Sitter vacua. We will mainly focus on type-IIB compactifications on Calabi–Yau orientifolds. 35 This fact has even been regarded as a “blessing” in Ref. [24], since the unfixed axions give masses to potentially anomalous U(1) brane fields in SM-like constructions.
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7.1. Corrections to the low energy action Corrections to the low energy effective supergravity action are governed by the Planck scale, which in string theory is given by MP8 = 1/gs2 ( )4 . In the low energy limit, the dimensionless parameter lP /R, where R is a characteristic length of the solution, controls the corrections. One then thinks of the corrections as a double series expansion in gs and . There are perturbative and non-perturbative corrections to the supergravity action. The non perturbative arise from world-sheet or brane instantons. A world-sheet or a p-brane wrapping a topologically non-trivial space-like 2-cycle or p-cycle on the internal manifold gives instanton corrections which are suppressed by e−Vol()/2 . This will be the main effect stabilizing the Kähler moduli. As we will briefly discuss, the number of fermion zero modes on the instanton world-volumes dictates whether these corrections are there or not. Let us discuss the perturbative corrections in the case of N=1 compactifications, concentrating on compactifications of IIB on Calabi–Yau orientifolds. The ten dimensional supergravity action is corrected by a series of terms, coming from higher derivative terms in the action: CS loc loc + S(0) + 2 S(2) , S = S(0) + S(3) + · · · + S(n) + · · · + S(0) 3
n
(7.1)
where S CS are the Chern–Simons terms, and S loc the localized p-brane actions. In addition to higher derivative corrections contributing to (7.1), there are string loop corrections to the action, both for the bulk and for the localized pieces, whose effect is less known. They are suppressed by powers of gs . Some of these corrections were computed for IIB orientifolds of various tori with D5/D9 and D3/D7 branes in Refs. [172,173] (see also Ref. [174] for the case of N = 2 compactifications). String loop corrections to the bulk effective action appear nevertheless at order 3 , so their effects are subsumed in expansion (7.1), as a further gs expansion of each term. The term S(3) contains R4 corrections to the action (where R is the Ricci scalar), as well as R − Fp terms mixing flux and curvature. These corrections break the no-scale structure of the flux generated potential, both at string tree level [175] and at one loop [172]. Considering the scalings of all possible 3 correction to the bulk type IIB action compactified on a Calabi–Yau orientifold in the presence of fluxes, Refs. [163,164] conclude that the leading term has a scaling O( 3 /R 6 ) relative to the zeroth order term, in agreement with the result of Ref. [175]. loc lead to a potential energy in the case of D7-branes, Higher derivative corrections to the localized sources, n S(n) but not D3-branes [164]. The 2 correction to the D7-brane action gives their effective D3-brane charge and tension [21]. In F-theory this effective charge is given in terms of the Euler number of the fourfold by QD7 3 = −/24. This adds the constant term in the F-theory version of the tadpole cancellation condition, Eq. (4.23). Higher corrections to the D7-brane action do not lead to potential energy. The corrections just outlined lead to corrections to the four dimensional Kähler potentials and superpotentials. The N = 1 Kähler potential receives corrections at every order in perturbation theory, while the superpotential receives non-perturbative corrections only. Considering leading corrections, the Kähler potential and superpotential can be written K = K0 + Kp + Knp , W = W0 + Wnp .
(7.2)
where Kp comes from the corrections discussed above, Knp comes mainly from fundamental string wordlsheet instantons, and Wnp comes from string non-perturbative effects such as D-brane instantons (or similarly, from gaugino condensation on D-branes). Refs. [176,163,164] analyze the effect of these corrections on the potential (5.21), namely V = V0 + VKp + VWnp + · · ·
(7.3)
where V0 ∼ W02 ,
VKp ∼ W02 Kp ,
2 VWnp ∼ Wnp + W0 Wnp .
(7.4)
As discussed above, in four dimensional supergravities arising from compactifications of type II theories on Calabi–Yau orientifolds with D-branes, we have more information about Wnp (coming from field theoretic considerations, though) than about the perturbative corrections to the Kähler potential, Kp . We wish therefore to see whether there is any
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regime in which the latter can be neglected, and moduli stabilization can be studied just including the non-perturbative corrections to the superpotential, as is the case in the KKLT scenario [184] to be discussed in next section. If the tree 2 , and we can safely level superpotential W0 vanishes, then the first contribution to the potential is proportional to Wnp ignore Kp . Similarly, if W0 >1 in suitable units, then the tree level superpotential can have similar magnitude than the non-perturbative superpotential, leading to 2 2 W0 ∼ Wnp ⇒ Wnp ∼ VWnp ?VKp ∼ Wnp Kp .
(7.5)
This is the relevant behavior for the KKLT scenario to be reviewed in Section 7.3. Finally, when Wnp /Kp < W0 >1, the perturbative effects dominate, and it is not consistent to neglect them. Considering non-perturbative corrections to the superpotential while at the same time neglecting perturbative corrections to the Kähler potential is therefore justified only when the flux generated superpotential W0 is zero, or of the same order of magnitude than the non-perturbative “correction”. 7.2. Non-perturbative corrections to the superpotential There are two classes of effects that lead to corrections to the superpotential depending on the Kähler moduli of IIB compactifications: gaugino condensation and D-brane instantons. Gaugino condensation arises in D7-branes: in the presence of 3-form flux, many or all of the world-volume matter fields acquire masses [33,177]. If at the same time fluxes stabilize the D7’s at coincident locus, gaugino condensation is expected to occur at energy scales much lower than the mass scale, where the low energy dynamics is that of pure N = 1 SYM. Euclidean D3 branes wrapping 4-cycles can also lead to non-perturbative potentials. Non-perturbative superpotentials will arise when the D3-branes in question lift to M5-branes wrapping a “vertical” divisor D (where vertical means that it wraps the fiber directions that shrink in the F-theory limit) and that supports two fermionic zero modes [178]. The behavior of the non-perturbative superpotential with the Kähler moduli is similar in both cases, gaugino condensation and D-brane instantons. Here we will discuss the brane instanton case. For euclidean D3-branes wrapping a four-cycle dual to n w , the non-perturbative superpotential is given by [179,180,151] T
Wnp = Bn e−2n
,
(7.6)
where Bn are one loop determinants that depend on the expectation values of the complex structure moduli, and T are the Kähler moduli defined in (5.25), whose real part gives the volume of the cycle [w ] (we are taking here (1,1) h− = 0). The non-perturbative superpotential depends therefore on Kähler moduli, which were absent in the flux induced GVW superpotential (5.43). We will see in next section that taking into account this non-perturbative correction to the superpotential can lead to Calabi–Yau O3/O7 compactifications with all moduli stabilized. The one loop determinants Bn are non-vanishing whenever the instantons support two fermionic zero modes. In the absence of background flux, this is translated into a condition on the M-theory divisor, which should have holomorphic Euler characteristic (called sometimes arithmetic genus) one, (D) = p (−1)p h(0,p) (D) = 1. It has been argued recently [181–183] that this requirement is relaxed in the presence of background flux, and branes wrapping divisors of holomorphic Euler characteristic (D)1 can contribute to the non-perturbative superpotential. For example, Ref. [183] showed that in type-IIB string theory compactified on a K3 × T 2 /Z2 orientifold, a D3-brane wrapped on K3 sitting at the O7 fixed point on the T 2 would have (D) = 2 in the absence of fluxes. The presence of complex 3-form flux with two legs along the holomorphic 2-form on K3 and the third leg along the antiholomorphic direction on T 2 (making it overall (2, 1) and primitive) lifts some zero modes, leaving (Dflux ) = 1, thus allowing for instanton contributions to the superpotential. 7.3. Moduli stabilization including non-perturbative effects Considering the non-perturbative corrections to the superpotential discussed in the previous section, all moduli can be stabilized in type-IIB compactifications on Calabi–Yau orientifolds. The idea was first put up in the seminal paper of KKLT [184], where it was shown that besides having all moduli stabilized in this type of compactifications, it is possible to obtain de Sitter vacua by adding a small number of anti-D3-branes. In this section we will first review
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how the inclusion of a non-perturbative superpotential can lead to moduli stabilization, following KKLT. We will then mention the explicit supersymmetric and non supersymmetric examples constructed in the literature, and finally, in the next section, we will review how de Sitter spaces arise in these construction. (1,1) (1,1) KKLT study IIB flux compactifications on Calabi–Yau O3 orientifolds with h(2,1) arbitrary, and h+ = 1, h− = 0, i.e. with any number of complex structure moduli but with a single Kähler modulus T, whose real part is Re T = 21 K2/3 ∼ R 4 (where K is the overall volume, R the radius of compactification), as in Eq. (5.27). 3-form fluxes generate a superpotential for the complex structure moduli and dilaton–axion. Using the quantization conditions (2.6) and the fact that the 3-cycles have volumes R 3 , the complex structure moduli and dilaton have masses of order m ∼ /R 3 . KKLT set the complex structure moduli and the dilaton axion equal to their VEVs, and concentrate then on the effective field theory for the volume modulus T. In the presence of D3-brane instantons there is a nonperturbative superpotential for the Kähler modulus T, Eq. (7.6). A single instanton will lead to the a total superpotential of the form (7.2), namely W = W0 + Be−2T ,
(7.7)
where W0 is the contribution coming from the fluxes. Setting the axion = 0, and defining ≡ Re T = supersymmetric minimum satisfying DT W = 0 is attained at 4 cr , W0 = −Be−2cr 1 + 3
1 2
K2/3 a
(7.8)
(to perform the Kähler covariant derivative we have used the Kähler potential (5.27)). The volume is therefore stabilized at Kcr = (2cr )3/2 , which can take reasonably large values for sufficiently small |W0 |. Inserting this in the potential (5.21), we get that the minimum leads to an AdS vacuum Vmin = −3(eK |W |2 )min = −
22 B 2 e−4cr . 3cr
(7.9)
A few comments are in order. First, note that we need W0 = 0 for the complete stabilization to work. In Section 6.1, when we discussed the supersymmetry conditions, we set DI W0 = 0, as well as W0 = 0. Looking at Eqs. (5.44), (1,1) a we see however that if we do not consider derivatives along the Kähler moduli T , and take h− = 0 (i.e., no G ¯ 3 ∧ = G3 ∧ k = 0, or in other words, G3 must be ISD—(2, 1) moduli), the supersymmetry conditions are just G primitive plus (0, 3). If we consider a (0, 3) piece of the complex 3-form flux, the superpotential does not vanish. If we had just the flux superpotential, this will break supersymmetry, as DT W0 = 0. However, KKLT have shown that taking into account the non perturbative corrections to the superpotential, (0, 3) pieces of the 3-form flux can lead to supersymmetric (AdS) vacua. This piece has to be fine tuned, though, to give eK |W0 |2 >1, otherwise there is no large radius minimum of the potential. Examples of flux vacua with eK |W0 |2 ∼ 10−3 and less were constructed in Refs. [78,179]. The statistical results, as we will review in Section 8, suggest that even smaller values are possible, and conclude that the fraction of vacua having eK0 |W0 |2 scales like [31]. There are several critiques to the KKLT procedure, to be discussed shortly. These critiques do not affect the main results, but they do affect the detailed physics, and therefore tell us that KKLT should be taken only as a toy model of complete moduli stabilization in compactifications of IIB Calabi–Yau orientifolds. The first critique is that the procedure of obtaining an effective potential for light moduli via non-perturbative corrections after integrating out moduli that are assumed to be heavy at the classical level is in general not correct (see for example Refs. [185,186,164]). In some cases, this two step procedure can fail, giving rise to tachyonic directions [185]. One should instead minimize the full potential, which has additional terms (mixing the light and heavy modes). This is a highly involved procedure, which has not been carried out in the explicit examples of full stabilization worked out in the literature [179,187,37]. Another critique, outlined in Section 7.1, is that the corrections to the Kähler potential, both perturbative and nonperturbative in nature, have not been taken into account. As reviewed in that section, Refs. [176,163,164] show that the corrections to the Kähler potential are subleading whenever W0 ∼ Wnp . Otherwise, the perturbative corrections to the Kähler potential dominate, and one should include them in order to analyze the details of the potential. At large
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volumes, the vev of the GVW superpotential, Eq. (7.8), is indeed larger than the non-perturbative one, and according to Refs. [176,163,164], perturbative corrections start to take over. Taking into account the known correction to the Kähler potential computed in Ref. [175], Refs. [176,163,164] show that there is a large volume minimum which for sufficiently small values of W0 coexists with the KKLT minimum. Finally, Ref. [179] argues that in order to stabilize the Kähler moduli at strictly positive radii, one needs a sufficient number of distinct divisors, which excludes the case of internal manifolds with h(1,1) = 1, as in the toy model of KKLT. In the past year, explicit examples of type IIB orientifold compactifications were constructed with all moduli (also twisted and open string moduli) stabilized [179,187,37]. Ref. [179] studied Calabi–Yau four-folds with Fano and P1 fibered base. Out of 92 models with fano base, they found 29 in which all Kähler moduli can be stabilized by arithmetic genus one divisors. The simplest one is F18 , which has 89 complex structure moduli. Ref. [187] discusses an orientifold of the Calabi–Yau resolved orbifold T 6 /Z22 , while Ref. [37] studies M-theory on K3 × K3, dual to type IIB on K3 × T 2 /Z2 , where the flux modification to the condition (D) = 1 plays a prominent role. We will not review these explicit constructions, but just remark that Ref. [187] argues that moduli can be stabilized supersymmetrically in the perturbative regime for reasonable values of W0 . Furthermore, the corrections to the Kähler potential Kp , Knp coming, respectively from the tree level 3 correction of Ref. [175] and string worlsheet instantons are estimated to be small. There is no quantitative analysis of string loop corrections, but Ref. [187] argues that there are no expected reasons to believe that these will destabilize the solution for the values of gs found. Refs. [163,164] find that if in addition to the non-perturbative superpotential one considers the 3 corrections to the Kähler potential of [175] in generic Calabi–Yau orientifolds with h(2,1) > h(1,1) > 1, all geometric moduli can be stabilized, and there are non-supersymmetric AdS minima at exponentially large volume. Taking the example of the orientifold of P4[1,1,1,6,9] , Refs. [163,164] show that as |W0 | increases, the perturbative corrections dominate the non-perturbative ones, and including these corrections there is a large volume minimum which for small values of W0 coexists with the KKLT minimum. Very recently, in Ref. [188] it was argued, along the lines of Refs. [173,189], that corrections to the Kähler potential (tree level and string loop) should be enough to stabilize all moduli in a IIB orientifold compactification, without the need of non-perturbative corrections. Explicit models with the volume stabilized at large radius are not yet constructed, though. 7.4. De Sitter vacua After having fixed all moduli, KKLT outline the construction of de Sitter vacua. In order to get de Sitter solutions from IIB flux compactifications, one should uplift the AdS vacuum found after having fixed all moduli, as discussed in the previous section. KKLT do this by adding a small number of anti-D3-branes at the bottom of the warped throat.36 Other uplifting mechanisms involve adding D7-brane fluxes [190]; starting with a local non-zero minimum of the no-scale potential (which does depend on the overall volume through the factor eK ) and expanding around it [191]; Let us review here the KKLT uplifting procedure. We start by cooking up together all the ingredients of the previous sections, namely ISD fluxes, D3-branes, orientifold planes and instantons (or gaugino condensation on D7-branes). Let us assume nevertheless that the tadpole cancellation condition (4.21) is not satisfied, and we need a small number of anti-D3-branes to satisfy it. Anti-D3-branes in the warped geometry created by ISD fluxes, break supersymmetry explicitly and do not have translational moduli (see for example Ref. [192]), and are driven to the end of the throat, where the warp factor is minimized. The potential energy of such D3-brane is proportional to e4A at the location of the brane, and inversely proportional to the square of the volume (see for example Refs. [192,144]). Adding a small number n of D3-branes, there is an extra contribution to the potential of the previous section, given by VD3 =
D D = , 3 (Re T )3
(7.10)
where the coefficient D is proportional to n and e4A at the position of the branes. Adding this to the potential from the previous section, obtained by inserting the superpotential (7.7) and Kähler potential (5.27) into the N = 1 36 The throat is the highly warped region around the D3-brane sources where the warp factor e2A is very small, or equivalently where the conformal factor multiplying the Calabi–Yau metric, e−2A becomes very large.
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1.2 1 V
0.8 0.6 0.4 0.2 100
150
200
250 σ
300
350
400
Fig. 1. Potential (7.11) multiplied by 1015 , taken from Ref. [184].
potential (5.21), we get 2 D Be−2 −2 Be 1 + + 3 . + W V = 0 2 3
(7.11)
This potential has few terms because the no scale structure of the Kähler potential, Eq. (5.28), gives the cancellation ¯ K T T jT KjT¯ K|W |2 − 3|W |2 = 0. There are two extrema of the potential, a local minimum at positive energy and a maximum separating the de Sitter minimum from the vanishing potential at infinity. By fine tuning D, it is easy to get very small positive energy, at large values of , i.e. at large volume. Fig. 1, taken from [184] shows the potential as a function of for W0 = −10−4 , B = 1, D = 3 × 10−9 , and KKLT take 2 = 0.1 in the non-perturbative superpotential (7.6). Actually, the value of that minimizes the potential, cr , is just slightly shifted from the one that gives the AdS minimum of the previous section, Eq. (7.9). The effect of the last term in (7.11), coming from the D3-branes, is therefore to lift the AdS minimum without changing too much the shape around the minimum. The different versions of the uplifting mechanisms (Refs. [190,191]) differ in the precise shape of the uplifting potential (7.10), but have the same overall behavior. The breaking of supersymmetry is spontaneous instead of explicit, as it is in KKLT. However, there are consistency requirements to combine D-term breaking with a non-perturbative superpotential (see Refs. [193,194] for details). Note that the de Sitter vacuum just obtained is metastable, as there is a runaway behavior to infinite volume. This is expected for many reasons. On one hand, it has become clear on entropy grounds that de Sitter space cannot be a stable state in any theory of quantum gravity [195]. On the other hand, the runaway behavior is a standard feature of all string theories [196]. Ref. [197] argues very generically that a positive vacuum energy in a space with extra dimensions implies an unstable universe toward decompactification. KKLT showed nevertheless that the lifetime of the dS vacuum is large in Planck times (it can be longer than the cosmological time scale ∼ 1010 years), and shorter than the recurrence time tr ∼ eS , where S is the dS entropy [195,197]. Ref. [198] explored the possible decay channels of the KKLT de Sitter vacuum, finding, in agreement with KKLT, that even the fastest decays have decay times much greater than the age of our universe. Let us note again that due to the critiques discussed in the previous section, KKLT is a toy model for getting de Sitter vacua in IIB compactifications. Differently from the case of AdS vacua, no explicit models with dS vacua were constructed so far (some partial constructions can be found for example in Ref. [199]). On top of the difficulties already discussed in stabilizing all moduli in a controlled way, there is an extra fine tuning needed in order to make the constant D sufficiently small, and get at the same a long lived vacuum. There are however no fundamental reasons to doubt the existence of such explicit dS vacua. Uplifts to dS of IIA rigid Calabi–Yau orientifold flux vacua of the type discussed in Section 6.4 were considered in Ref. [200] (see also Ref. [136] for moduli stabilization in IIA including corrections, and possible de Sitter vacua). DS are possible after taking into account non-perturbative corrections to the superpotential, and perturbative corrections to the Kähler potential. The latter give rise to a positive contribution analogous to the anti-D3-brane one of KKLT. Similarly,
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dS vacua can be obtained from heterotic M-theory [201], using background fluxes and membrane instantons/gaugino condensation on the hidden boundary. In a very different spirit, Ref. [202] finds dS vacua from flux compactifications on products of Riemann surfaces, while Ref. [203] finds dS vacua in supercritical string theories.
8. Distributions of flux vacua It is somewhat clear from all the previous sections that the number of possible string/M theory vacua is very large. Despite the fact that as of today none of them fully reproduces the Standard Model data (hierarchy of masses, CKM matrix, etc, which in string theory vacua depend on moduli vevs), the hope is that adding sufficient number of ingredients to the soup, many of them could. This raises the “vacuum selection problem”: among the very large number (not clear whether it is even finite) of possible vacua, we have no idea which one is relevant, and how to find it. If there is no vacuum selection principle, i.e. no a priori condition that points toward the right candidate vacuum, the only way to find our vacuum seems to be by just enumerating all possible vacua, and testing each one against present observations. Since all possible vacua are too many (may be even infinite) Douglas and collaborators [204,30,31,205] have advocated for a statistical study of the “landscape” of vacua, which could give some guidance principle for the search of the right vacuum. Let us review the basic ideas, and some of their results. Given a set of effective N = 1 supergravity theories, i.e. a set of Kähler potentials Ki and superpotentials Wi , with the same configuration space (the space in which the moduli take values), the first ingredient needed is the density of (susy or non susy) vacua. Integrating this density over a region R in configuration space gives the number of vacua which stabilize moduli in that region, i.e. Nvac,R =
d2n z(z) ,
(8.1)
R
where z are the n complex moduli fields. This density is given by (z) =
n (Vi (z))| det Vi (z)| ,
(8.2)
i
where Vi is the N = 1 potential (5.21) for Ki , Wi . If the vacua are supersymmetric, this can be written in terms of the Kähler covariant derivatives as
n (DW i (z)) n (DW i (¯z))| det D 2 Wi (z)| , (8.3) (z)susy = i
where the Jacobian D 2 W is a 2n × 2n matrix DI¯ DJ W (z) DI DJ W (z) D2 W = . DI¯ DJ¯ W¯ (¯z) DI DJ¯ W¯ (¯z)
(8.4)
One can also find a density of supersymmetric vacua with a given cosmological constant, by multiplying (8.3) by
( − (−3eKi |Wi (z)|2 )). The ensemble of vacua that Douglas and collaborators consider are flux compactifications of F-theory on Calabi–Yau four-folds, or their IIB orientifolds limit. In the case of IIB vacua with 3-form fluxes on a Calabi–Yau orientifold, the Kähler potentials Ki are all the same, Eq. (5.26) (or an equivalent expression with and z together for the four-fold), K K while the superpotentials Wi , given in (5.43) are labeled by the 4(h2,1 − + 1) fluxes (eK , m , eK RR , mRR ). The tadpole cancellation condition (4.23) gives an upper bound for the number of units of flux (assuming we do not want to introduce D3-branes), given by K Nflux = eK mK RR − m eK RR = N N N∗ ,
N∗ =
(X4 ) . 24
(8.5)
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3
2
1
-0.5
0.5
Fig. 2. Values of for rigid CY flux vacua with N∗ = 150, (Z 0 , F0 ) = (1, i) , taken from Ref. [31].
Let us illustrate this with the simplest example, considered in Ref. [30], of a rigid Calabi–Yau 3-fold (h2,1 = 0). The only modulus of the theory appearing the flux superpotential is , which is stabilized by the GVW superpotential (5.43) W = (m0 F0 − e0 Z 0 ) + (−m0RR F0 + e0 RR Z 0 ) ≡ C + D . The constants Z 0 = A ; F 0 = B are determined by the geometry of the Calabi–Yau. Supersymmetric vacua obey C ¯ + D D W D W = j W − =− = 0 ⇒ ¯ = − . − ¯ − ¯ C
(8.6)
(8.7)
One can now scan all the possible values of (e0 , m0 , e0, RR , m0RR ) satisfying the inequality (8.5), and get the corresponding value of the stabilized axion–dilaton. Taking Z 0 = 1, F0 = i, N∗ = 150 and doing if necessary an SL(2, Z) transformation to each resulting such that it is in the fundamental domain,37 Ref. [31] gets the distribution shown in Fig. 2. The simplest example of flux vacua already gives an intricate distribution, from which it is hard to obtain any number (although Douglas and collaborators succeed in doing so), like for example the total number of vacua. However, Refs. [30,31] show that for a sufficiently large region, the density of vacua per unit volume in moduli space can be well √ approximated by a constant, equal to 2N∗2 . This is a good approximation for disks of radii R in moduli space if R > 1/N∗ . The total number of vacua, Eq. (8.1), is therefore Nsusy vac (Nflux < N∗ ) = () d2 M
≈
M
2N∗2
d2 (2 Im )
= 2N∗2 2
M
g¯ d2 = 2N∗2
. 12
(8.8)
It is hard to believe looking at Fig. 2 that the density is constant. One sees on one hand an accumulation of vacua close to the boundary, at = 1, and with a sparser distribution for larger . Additionally, there are voids around the points = ni. The higher density for lower values of Im is just due to the modular invariant metric in moduli space, d 2 /(2Im )2 . Regarding the holes, these are interpreted in Ref. [31] as consequences of the special conical shape of 37 The fundamental domain is F = { ∈ C : Im > 0, || 1, |Re | < 1 }. 2
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the region containing vacua. In any case, there is a very large degeneracy of vacua at the points = ni (there are for example 240 vacua at = 2i) which offsets the empty spaces, making the constant density approximation good for sufficiently large radii. The rigid Calabi–Yau is a particularly simple case, we do not expect in general the density of vacua to be well approximated by a constant. But we do need a continuous approximation to the density in order to extract numbers out of Eq. (8.1). Such continuous approximation will replace the sum over the integer values of the fluxes by an integral, namely
→
2,1 h −
(e,m,eRR ,mRR )∈Z
deK dmk deK, RR dmK RR .
(8.9)
K=0
(For a detailed discussion about the limitations of this approximation, see Ref. [30].) This integral should be cut off at a value given by the upper bound (8.5), i.e. it should be supplemented by a step function (Nflux − N∗ ). Collecting all the pieces together, Ref. [30] arrives at (Nflux − N∗ ) Nsusy vac = susy vac
1 ≈ 2i
C
d N∗ e
2p
d z
d4p N e−(/2)N N p (DW (z))| det D 2 W (z)| ,
(8.10)
where we have used the Laplace transform in of the step function, and defined p = h2,1 − + 1. Skipping the details of the calculation, which the reader is welcome to follow from Refs. [30,31], we quote the result. A good approximation for the number of type-IIB O3/O7 supersymmetric vacua inside a region R in moduli space is given by (2N∗ )2p Nsusy vac (Nflux N∗ ) ≈ p det(R + JI), p = h2,1 (8.11) − +1 . (2p)! R In this equation, R is the curvature two-form in the 2p-dimensional moduli space expressed as a p × p matrix, namely k (R)ab = Rk lab zl (a, b are orthonormal frame indices, and k, l tangent space indices in moduli space); J is the ¯ dz ∧ d¯ Kähler two-form Jk l¯ dzk ∧ d¯zl and I is the p × p identity matrix ab . We see that the constant density for the case of a rigid Calabi–Yau is replaced by a “topological” density (z)susy =
(2N∗ )2p det(R + J I) . p (2p)!
(8.12)
This is actually an “index” density: it counts the number of vacua with signs, i.e. dropping the absolute value of the determinant in Eq. (8.3). It therefore gives a lower bound to the density of vacua. Ref. [30] argues that (8.12) is a good lower bound for the number of flux vacua for N∗ ?2p?1, and that the total number of vacua is probably (8.11) multiplied by c2p , with some c ∼ 1. The index density (8.12) agrees with the constant 2N∗2 per unit volume in M for the rigid (p = 1) case. This means that the vacuum expectation values of the dilaton, or the string coupling constant, is in good approximation uniformly distributed. Integrating the density over a fundamental domain in moduli space, gives the number of supersymmetric flux vacua for a given Calabi–Yau. Given the formula for the total number of vacua, the first question to ask is whether this number is finite. It has been conjectured that the volumes of these moduli spaces are finite [207], which restricts the question of finiteness to possible divergences coming from the curvature. A typical case in which the curvature diverges is the neighborhood of a conifold point. However, Ref. [30] shows that for the complex structure moduli space of the mirror quintic (expecting conifold points on other CYs to have the same behavior), in spite of R being singular, the integral is finite. With more moduli there might be more complicated degenerations, but there is reasonable hope that the number Nsusy vac (Nflux < N∗ ) is finite.38 38 For a recent and very nice discussion about finiteness of string vacua, see M. Douglas’ talk at Strings 2005 [208].
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Let us now illustrate with the next to simplest example: a moduli space of complex dimension p = 2, i.e. consisting of the dilaton–axion plus one complex structure modulus, M = M × MY . The metric R + J I is R0 + J0 + J1 0 R+J I= , (8.13) 0 R1 + J0 + J1 where we have defined R0 ≡ R¯ d ∧ d¯, R1 ≡ Rz¯z dz ∧ d¯z,
J0 ≡ J¯ d ∧ d¯ , J1 ≡ Jz¯z dz ∧ d¯z .
(8.14)
It is easy to check that for the metric on M , given by the Kähler potential K = − ln[−i( − ¯ )], R0 = −2J0 . The determinant of (8.14) is therefore given by det(R + J I) = (−J0 + J1 ) ∧ (R1 + J0 + J1 ) = −J0 ∧ R1 . Inserting this in (8.11), and using M J0 = /12 (cf. Eq. (8.8)), gives (2N∗ )4 (2N∗ )4 R dz ∧ d¯ z = − Nsusy vac (Nflux N∗ ) ≈ − (MY ) . z¯ z 2 4! 12 MY 4!6
(8.15)
(8.16)
In this next to simplest example, the number of vacua is therefore just proportional to the Euler characteristic of the 2p complex structure modulus space (and to the usual power N∗ = N∗4 ). Let us apply this to a manifold Y with a conifold degeneration [31]. As in the case of the deformed conifold of Section 6.1 (see below Eq. (6.3)), the periods are Z(z) = z, F(z) = 2zi + analytic terms. Inserting this in the Kähler potential (5.26), we get K ≈ − ln[ 21 |z|2 ln |z|2 ]. This gives the following metric near z = 0: gz¯z ≈ c ln
1 , |z|2
(8.17)
where c = eK0 /2, with K0 the Kähler potential at z = 0. The curvature is therefore Rz¯z ≈ −
1 |z|2 ln |z|2
.
(8.18)
This implies that the density (8.12) diverges at the conifold point, z = 0. The integral over a finite domain |z| < R is nevertheless finite, namely (2N∗ )4 (N∗ )4 Nsusy vac (Nflux N∗ ) ≈ − Rz¯z dz ∧ d¯z ≈ . (8.19) 2 4! 12 |z|
3
ln[−i(w p − w¯ p )] ,
(8.20)
p=0
where wp = (, i ). There are therefore four copies of the moduli space M . When building the 4 × 4 matrix (8.15), we can use Ri = −2 Ji , (see below Eq. (8.14)). We therefore have det(R + J I) = 8J0 ∧ J1 ∧ J2 ∧ J3 .
(8.21)
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This is a combinatoric factor times the volume form of the total moduli space. The fundamental region has a permutation symmetry, i.e. is preserved by SL(2, Z)p × Sp , so its volume is Vp =
1 p . p! 12
(8.22)
The number of vacua is therefore Nsusy vac (Nflux < N∗ ) = 8
(2N∗ )8 1 4 1 (2N∗ )8 . = 4 8! 4! 12 3 (12)4 8!
(8.23)
For N∗ = 16, which is the value for a T 6 with 64 O3-planes, we have about four million supersymmetric vacua. Typical Calabi–Yau’s have many more complex structure moduli, leading to many more possible vacua. For example, Ref. [78] finds several supersymmetric flux vacua for the Calabi–Yau 3-fold hypersurface in W P 1,1,1,1,4 . The 3-fold has h2,1 = 149, and its orientifold descends from the F-theory Calabi–Yau 4-fold in W P 1,1,1,1,8,12 , which has /24 = 972. We can roughly estimate the number of vacua to be of the order 10230 . After looking at these huge numbers of vacua, we can convince ourselves that searching for vacua one by one until we reach the right one does not seem to be a clever idea. It is true nevertheless that we are not imposing any observational constraint on these vacua (like cosmological constant, spectrum...). Imposing such constraints the numbers should reduce. Clearly, none of these vacua has a positive cosmological constant, but we could nevertheless demand that |W | is smaller than a given value such that the KKLT (or its variations thereof) uplifting mechanism could work. Let us see then how the constraint |W |2 ∗ reduces the numbers. |W |2 < is enforced by inserting a delta (|W |2 − ) in (8.10) and integrate over up to ∗ . This seems like an easy task to do. However, there is no nice topological formula such as (8.11) for the index density when one includes this restriction. Instead, Eq. (8.12) becomes (z, )susy = (N∗ − )
k p ck (z) (2p)! (N∗ − )2p−1 , (N∗ )p (2p − 1 − k)! N∗ −
(8.24)
k=0
where ck (z) are homogeneous polynomial functions. We see that the cosmological constant is smaller than N∗ , and the density for ∼ N∗ is suppressed by a factor (N∗ − )p−1 . Because of this suppression, for large p we expect the distribution to be peaked at small . Integrating √ (8.24) up to a small value ∗ , we see that the fraction of vacua with || ∗ behaves like ∗ /N∗ (in units of 1/(2 )4 ). This implies that the smallest cosmological constant is of order 2 /Nvac . Vacuum multiplicity would therefore help in solving the cosmological constant problem. Another important feature about the distribution of flux vacua, is that it is suppressed in the large complex structure ¯ region. With one complex structure modulus, and in the large complex structure limit, the Kähler potential ∧ has an expansion of the form − ln[i(z − z¯ )3 ], resembling the expression for the Kähler moduli t, Eq. (5.9). We saw that for this Kähler potential Rz¯z ∝ gz¯z , and therefore the density of vacua per unit volume in moduli space is constant. The index density (8.12) goes therefore like d(z) ∼ gz¯z d2 z ∼ d2 z/(z − z¯ )2 . This density behaves like that of Fig. 2. We see therefore that the large complex structure limit region is strongly suppressed. In the mirror IIA picture, Im z is mapped to Im t ∼ V 1/3 (where V is the volume). This gives a number of vacua that falls off with volume as V −1/3 . For n complex structure moduli (or many Kähler moduli in the mirror picture), there is a large volume falloff as V −n/3 . Ref. [206] studies the density of non supersymmetric (F -breaking) flux vacua, given by (8.2). We will not review the details here, but just quote one of their main results. The number of metastable vacua with F -term supersymmetry breaking scale Msu/sy < M∗ inside a region R is related to that of supersymmetric vacua by Nsu/sy,R (Msu/sy < M∗ ) = M∗2 Nsusy,R .
(8.25)
If one imposes additionally that the cosmological constant be smaller than a given small value ∗ , this ratio is Nsu/sy,R (Msu/sy < M∗ , < ∗ ) = M∗12 ∗ Nsusy,R ,
∗ < |F |, |W | .
Low scale supersymmetry breaking is therefore disfavored.
(8.26)
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As mentioned, there is no constraint imposed regarding the matter content of these flux vacua. In that respect, Ref. [204] gives a first estimate of the fraction of vacua that have Standard Model spectrum (see Ref. [209] for some explicit D-brane models statistics). The case analyzed corresponds to SM matter arising from D6-branes in IIA wrapping 3 cycles of Calabi–Yau’s. The fraction of vacua that have a U (3) × U (2) × U (1)2 gauge group is roughly 10−6 . While the estimates of Ref. [204] are very crude, the conjecture is that ignoring values for the couplings, the fraction of models which realize the standard model spectrum is closer to O(10−10 ) than to O(10−100 ). Let us summarize the prominent features of the distribution of supersymmetric vacua. • • • • •
The density of dilaton–axion vev’s is constant per unit volume in moduli space. Vacua accumulate near conifold points. The distribution of values of cosmological constant is uniform near zero. The distribution has a falloff at large complex structure, or in the mirror manifold, at large volume as N ∼ V −(p−1)/3 . High scale supersymmetry breaking is favored.
Statistics of M-theory vacua have been put forward in the same spirit in Ref. [218], while some “rudimentary statistics” in a simple IIA toy model are performed in Ref. [23]. Although still a bit premature to drive final conclusions from the distributions of flux vacua, Douglas and collaborators have made a big step toward developing a statistical approach to flux compactifications. The first conclusion seems to be that even after imposing metastability, acceptable supersymmetry breaking and a Standard Model spectrum, the number of vacua seems to be very large, probably too large to be explored one by one, making it useful to study their distribution statistically. This distribution shows many simple properties, such as large volume and low supersymmetry breaking suppression. To finish this section, let us point out that the distribution of supersymmetric IIB flux vacua (8.12), in particular its 2p behavior on the moduli space geometry as det(R + J I); its growth with N∗ as N∗ as well as its conifold attractor point have been tested using Monte Carlo experiments in Refs. [79,162] (see also Ref. [210]). Ref. [79] studies the mirror of an orientifold of the CY hypersurface in P1,1,1,1,4 , which has p = 2 (one complex structure modulus) and N∗ = 972. Refs. [162,211] analyze the mirror of the CY hypersurface P1,1,2,2,6 , which has two complex structure moduli.
9. Summary and future prospects It is clear that flux compactifications in string theory has been an extremely active and fruitful area of research, particularly in the past five years. The main reason for this flurry of activity is that fluxes are the only known perturbative mechanism that stabilizes moduli. In many setups, most notably in all IIB compactifications on Calabi–Yau’s or tori, closed string fluxes are not enough to stabilize all moduli. Therefore, other perturbative and non perturbative mechanisms, some of which we do not have enough theoretical control yet, have to be invoked. However, fluxes were shown very recently to be enough to stabilize all moduli in IIA compactifications on rigid tori and on twisted tori, and similarly all moduli can in principle be stabilized in IIB by considering additionally open string fluxes. While stabilizing moduli, fluxes break supersymmetry partially or completely in a stable way, and generate warp factors, giving stringy mechanisms of realizing large hierarchies. Fluxes have also served as an ingredient in the construction of semi-realistic four-dimensional vacua with standard model spectra. In all these models, the back-reaction of the geometry is basically neglected. The back-reacted internal geometry involves, in all flux backgrounds, manifolds with torsion. In spite of the recent progress in understanding their mathematical properties, and therefore the four-dimensional resulting physics, many of the very basic questions such as what are their moduli spaces are still widely unanswered. However, conformally Ricci-flat manifolds (like conformal Calabi–Yau’s or tori) allow for non trivial fluxes, and give us a way of studying the effective theories with the available mathematical tools (in most cases neglecting, however, the conformal factor). In the absence of fluxes, these leave N = 8, 4, 2 unbroken supersymmetries in four-dimensions, while turning on fluxes breaks these to N = 4, 3, 2, 1, 0. Most of the analysis of moduli stabilization has been carried out in this class of compactifications. There are explicit constructions of flux compactifications with all moduli stabilized. In type IIB, these use nonperturbative effects in sufficiently sophisticated internal manifolds that have the right cycles to support the corrections, and are non-supersymmetric. In type IIA, the explicit constructions involve rigid orientifolds of tori orbifolds, where
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all moduli can be stabilized in N = 1 supersymmetric AdS vacua, or orientifolds of twisted tori, where there are additional Minkowski vacua with all moduli fixed except for some axions. Besides, metastable dS vacua in IIB can also be attained using (supersymmetry breaking) fluxes and non-perturbative effects. Flux compactifications have many additional interconnected aspects/applications that we have not discussed in depth in this review, or have not discussed at all. First of all, there is much more to report about the gauged supergravity interpretation of certain flux compactifications [212,22] (for a short review, see [213]). Besides, we have not discussed the relation between flux compactifications on twisted tori and Scherk–Schwarz compactifications [214] (see Refs. [157,215]). Neither have we discussed non-geometric backgrounds [94], which are sometimes dual to geometric flux vacua. Another undiscussed area is that of topological strings [221], in particular the attractor mechanism [222] and the Ooguri–Strominger–Vafa conjecture [223]. On a more phenomenological level, fluxes generate supersymmetric and soft supersymmetry breaking terms on D-branes [144,216,177,217], which can stabilize open string moduli [33–35]. (In the presence of fluxes, the supersymmetric cycles wrapped by D-branes are also modified, in order to minimize the action [42,220].) Soft-supersymmetry breaking effects and moduli stabilization in KKLT type scenarios has been discussed extensively (see for example [164,185,219]). Some other phenomenological applications that we have not discussed much are, for example, the landscape of supersymmetry breaking vacua and supersymmetry breaking scales [224] (see Ref. [31] for a statistical analysis); and models of inflation embedded in flux compactification scenarios, started most notably by Ref. [225]. Some non-compact flux backgrounds play a very important role in AdS/CFT, as they can realize string duals of confining gauge theories (see [226] for a review). Latest developments along these lines involve the recent explicit construction of conical Calabi–Yau metrics over the Sasaki–Einstein manifolds Y p,q [227] and Lp,q,r [228]. Their dual conformal field theories were found in Ref. [229]. However, there are no complex structure deformations on these manifolds, and opposite to the case in Klebanov–Strassler’s solution, adding 3-form fluxes does not seem to lead to regular supersymmetric backgrounds [230]. By turning on appropriate fluxes (which requires sometimes some fine tuning), moduli are stabilized in the regime where we can trust the effective field theory approximation. This needs a (string frame) volume or radius much bigger than the KK and string scales, and a small coupling constant (or more the more restrictive condition gs Nflux >1). Nevertheless, specific (small) numbers like the cosmological constant in non-supersymmetric solutions are subject to corrections which can be of the same order of magnitude as the leading order value, and cannot therefore be fully trusted. Corrections are however expected to be small if the vacuum energy is itself much smaller than the string scale and KK scale to the fourth power. In any case, leaving aside the phenomenology of scales, it is conceivable that in some of these backgrounds a subsector of the theory develops an instability, and we are dropping in the effective Lagrangian a KK or stringy mode which is tachyonic. Backgrounds with NS fluxes can be studied in the other extreme regime, where the internal space is very small, but world-sheet techniques are powerful enough, like in orientifolds of Gepner models [231] (see Ref. [232] for a search of vacua with standard model spectra in Gepner models). The analysis of backgrounds with RR flux using world-sheet techniques is on the contrary still far from being available, although a lot of progress in this direction is expected (or hoped) to happen: a new formalism for a covariant quantization of the superstring which allows to study backgrounds with RR flux was introduced a few years ago [233]. As of today, however, loop amplitudes have only been obtained in highly symmetric backgrounds like AdS5 × S 5 or the pp-wave. A fair concluding remark would be that we have made huge progress in the past years, but there is still a lot of work ahead of us to see whether the available models of flux compactifications are as copious and rich as they appear today, and whether our four-dimensional world is one of them.
Acknowledgements We would like to thank Thomas Grimm, Jan Louis, Andrei Micu, Ruben Minasian, Michela Petrini, Alessandro Tomasiello, Stefan Theisen, Dan Waldram and Fabio Zwirner for useful discussions. This work has been supported by European Commission Marie Curie Postdoctoral Fellowship under contract number MEIF-CT-2003-501485. Partial support is provided additionally from INTAS grant, 03-51-6346, CNRS PICS 2530, RTN contracts MRTN-CT-2004005104 and MRTN-CT-2004-503369 and by a European Union Excellence Grant, MEXT-CT-2003-509661.
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Appendix A. Conventions • We use the following index notation: ◦ , , . . . = 0, . . . , 3 label external coordinates. ◦ m, n, . . . = 1, . . . , 6 label real internal coordinates. ◦ i, j = 1, 2, 3, ¯™, ¯E = 1, 2, 3 label internal complex coordinates. ◦ M, N, . . . = 0, . . . , 9 label all coordinates. • We switch back and forth between Einstein frame and string frame. Section 3 is entirely in string frame. Most of Section 4 is also in string frame, except Eqs. (4.14), (4.34) and (4.35) (all taken from Ref. [21]), which are in Einstein frame. Sections 5 and on are in Einstein frame. • The different RR field strengths used (standard, modified, internal, external) are ◦ Fˆn = dCn−1 is the standard RR field strength, and F (10) is the modified one. Explicitly, F (10) = dC − H ∧ C + m eB = Fˆ − H ∧ C .
(A.1)
◦ The modified flux is split into internal and external components according to Fn(10) = Fn + Vol4 ∧ F˜n−4 .
(A.2)
Hodge dualities among the components given in Eqs. (2.3), (3.5). • A symbol means ten-dimensional Hodge duality, while we use ∗ for internal (6D) Hodge duality. • Whenever factors of are not written explicitly, we are taking (2)2 = 1. • We use the standard decomposition of the ten-dimensional gamma matrices M = ( , m ) as = ⊗ 1,
= 0, 1, 2, 3,
m = 5 ⊗ m ,
m = 1, . . . , 6 ,
(A.3)
and 5 =
i , 4!
7 = −
i mnpqrs mnpqrs , 6!
11 = 5 7 .
(A.4)
The m are Hermitean, as are the , except 0 which is antihermitean. • A slash is defined in the following way: F/n =
1 FP ...P P1 ...PN , n! 1 N
P1 ...PN = [P1 . . . PN ] .
(A.5)
For the SU(3) structure, the norm of the normalized spinor is † = †+ + + †− − = 1 .
(A.6)
• The fundamental 2-form and holomorphic 3-form constructed from this spinor as in (3.14) obey J ∧J ∧J =
3i ¯ . ∧ 4
(A.7)
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• The decomposition of a 2-, 3-, 4-, 5- and 6-form in SU(3) representations used in Section 3.3 is (3)
¯ (3)
F 1 = F1 + F 3 (1)
(3)
(8)
F2 = 13 F2 J + Re(F2 ) + F2
,
¯ ¯ (1) ¯ (3) (3) (6) (6) + (F3 + F3 ) ∧ J + F3 + F3 , F3 = − 23 Im(F3 ) (1)
(3)
(8)
F4 = 16 F4 J ∧ J + Re(F4 ∧ ) + F4 (3)
,
¯ (3)
F5 = (F5 + F5 ) ∧ J ∧ J , (1)
F6 = 16 F6 J ∧ J ∧ J .
(A.8)
The inverse relations are (3)
(F1 )i = Fi F2 = 21 Fmn J mn = Fi¯E J i¯E (1) (1)
i F3 = − 36 F ij k ij k , (1)
F4 = 18 F mnpq Jmn Jpq , (3)
(F5 )i = (1)
F6 =
(3)
(F3 )i = 41 Fimn J mn , (3)
(F4 )k =
(6)
(F3 )ij = F kl (i j )kl
ij l 1 24 Fk ij l
1 mnpq Jmn Jpq 16 Fi
1 mnpqrs Jmn Jnp Jqr 48 F
.
(A.9)
These can be used to obtain the torsion classes in (3.17) in terms of dJ, d, J and . References [1] P. Candelas, G.T. Horowitz, A. Strominger, E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46. [2] C.M. Hull, Superstring Compactifications with Torsion and Space-Time Supersymmetry, Print-86-0251 (CAMBRIDGE); A. Strominger, Superstrings with torsion, Nucl. Phys. B 274 (1986) 253. [3] Some of the first constructions are E. Witten, New issues in manifolds of SU(3) holonomy, Nucl. Phys. B 268 (1986) 79; B.R. Greene, K.H. Kirklin, P.J. Miron, G.G. Ross, A three Generation Superstring model. 2. Symmetry breaking and the low-energy theory, Nucl. Phys. B 292 (1987) 606; I. Antoniadis, J.R. Ellis, J.S. Hagelin, D.V. Nanopoulos, An improved SU(5) X U(1) model from four-dimensional string, Phys. Lett. B 208 (1988) 209 [Addendum-ibid. B 213 (1988) 562]; For the most recent heterotic string Standard Model construction, see V. Braun, Y.H. He, B.A. Ovrut, T. Pantev, A heterotic standard model, Phys. Lett. B 618 (2005) 252 [arXiv:hep-th/0501070]. [4] J.P. Derendinger, L.E. Ibanez, H.P. Nilles, On the low-energy D = 4, N = 1, supergravity theory extracted from the D = 10, N = 1 superstring, Phys. Lett. B 155 (1985) 65; M. Dine, R. Rohm, N. Seiberg, E. Witten, Gluino condensation in superstring models, Phys. Lett. B 156 (1985) 55. [5] For a review, see for example F. Quevedo, Lectures on superstring phenomenology, arXiv:hep-th/9603074. [6] C. Bachas, A way to break supersymmetry, arXiv:hep-th/9503030. [7] J. Polchinski, Dirichlet–Branes and Ramond–Ramond charges, Phys. Rev. Lett. 75 (1995) 4724 [arXiv:hep-th/9510017]. [8] For a review, see for example A.M. Uranga, Chiral four-dimensional string compactifications with intersecting D-branes, Class. Quant. Grav. 20 (2003) S373 [arXiv:hep-th/0301032]; E. Kiritsis, D-branes in standard model building, gravity and cosmology, Fortsch. Phys. 52 (2004) 200 [arXiv:hep-th/0310001]; D. Lust, Intersecting brane worlds: a path to the standard model?, Class. Quant. Grav. 21 (2004) S1399 [arXiv:hep-th/0401156]. [9] J. Polchinski, A. Strominger, New vacua for type II string theory, Phys. Lett. B 388 (1996) 736 [arXiv:hep-th/9510227]. [10] J. Michelson, Compactifications of type IIB strings to four dimensions with non-trivial classical potential, Nucl. Phys. B 495 (1997) 127 [arXiv:hep-th/9610151]. [11] T.R. Taylor, C. Vafa, RR flux on Calabi–Yau and partial supersymmetry breaking, Phys. Lett. B 474 (2000) 130 [arXiv:hep-th/9912152]. [12] M. Graña, J. Polchinski, Supersymmetric three-form flux perturbations on AdS(5), Phys. Rev. D 63 (2001) 026001 [arXiv:hep-th/0009211]; Gauge/gravity duals with holomorphic dilaton, Phys. Rev. D 65 (2002) 126005 [arXiv:hep-th/0106014]. [13] S.S. Gubser, Supersymmetry and F-theory realization of the deformed conifold with three-form flux, arXiv:hep-th/0010010.
152
M. Graña / Physics Reports 423 (2006) 91 – 158
[14] A.R. Frey, J. Polchinski, N = 3 warped compactifications, Phys. Rev. D 65 (2002) 126009 [arXiv:hep-th/0201029]. [15] S. Kachru, X. Liu, M.B. Schulz, S.P. Trivedi, Supersymmetry changing bubbles in string theory, JHEP 0305 (2003) 014 [arXiv:hep-th/0205108]. [16] J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200]. [17] J. Polchinski, M.J. Strassler, The string dual of a confining four-dimensional gauge theory, arXiv:hep-th/0003136. [18] I.R. Klebanov, M.J. Strassler, Supergravity and a confining gauge theory: duality cascades and chiSB-resolution of naked singularities, J. High Energy Phys. 0008 (2000) 052 [arXiv:hep-th/0007191]. [19] J.M. Maldacena, C. Nuñez, Towards the large N limit of pure N = 1 super Yang Mills, Phys. Rev. Lett. 86 (2001) 588 [arXiv:hep-th/0008001]. [20] K. Dasgupta, G. Rajesh, S. Sethi, M theory, orientifolds and G-flux, JHEP 9908 (1999) 023 [arXiv:hep-th/9908088]. [21] S.B. Giddings, S. Kachru, J. Polchinski, Hierarchies from fluxes in string compactifications, Phys. Rev. D 66 (2002) 106006 [arXiv:hepth/0105097]. [22] J.P. Derendinger, C. Kounnas, P.M. Petropoulos, F. Zwirner, Superpotentials in IIA compactifications with general fluxes, Nucl. Phys. B 715 (2005) 211 [arXiv:hep-th/0411276]; G. Villadoro, F. Zwirner, N = 1 effective potential from dual type-IIA D6/O6 orientifolds with general fluxes, JHEP 0506 (2005) 047 [arXiv:hep-th/0503169]. [23] O. DeWolfe, A. Giryavets, S. Kachru, W. Taylor, Type IIA moduli stabilization, arXiv:hep-th/0505160. [24] P.G. Camara, A. Font, L.E. Ibanez, Fluxes, moduli fixing and MSSM-like vacua in a simple IIA orientifold, arXiv:hep-th/0506066. [25] L. Randall, R. Sundrum, A large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370 [arXiv:hep-ph/9905221]; L. Randall, R. Sundrum, An alternative to compactification, Phys. Rev. Lett. 83 (1999) 4690 [arXiv:hep-th/9906064]. [26] H.L. Verlinde, Holography and compactification, Nucl. Phys. B 580 (2000) 264 [arXiv:hep-th/9906182]. [27] J.M. Maldacena, C. Nuñez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [arXiv:hep-th/0007018]. [28] R. Bousso, J. Polchinski, Quantization of four-form fluxes and dynamical neutralization of the cosmological constant, JHEP 0006 (2000) 006 [arXiv:hep-th/0004134]. [29] L. Susskind, The anthropic landscape of string theory, arXiv:hep-th/0302219. [30] S. Ashok, M.R. Douglas, Counting flux vacua, JHEP 0401 (2004) 060 [arXiv:hep-th/0307049]. [31] F. Denef, M.R. Douglas, Distributions of flux vacua, JHEP 0405 (2004) 072 [arXiv:hep-th/0404116]; F. Denef, M.R. Douglas, Distributions of nonsupersymmetric flux vacua, JHEP 0503 (2005) 061 [arXiv:hep-th/0411183]. [32] B.S. Acharya, F. Denef, R. Valandro, Statistics of M theory vacua, JHEP 0506 (2005) 056 [arXiv:hep-th/0502060]. [33] D. Lust, P. Mayr, S. Reffert, S. Stieberger, F-theory flux, destabilization of orientifolds and soft terms on D7-branes, arXiv:hep-th/0501139. [34] R. Blumenhagen, M. Cvetic, F. Marchesano, G. Shiu, Chiral D-brane models with frozen open string moduli, JHEP 0503 (2005) 050 [arXiv:hep-th/0502095]. [35] I. Antoniadis, A. Kumar, T. Maillard, Moduli stabilization with open and closed string fluxes, arXiv:hep-th/0505260; J. Gomis, F. Marchesano, D. Mateos, An open string landscape, arXiv:hep-th/0506179; M.P. Garcia del Moral, A new mechanism of Kahler moduli stabilization in type IIB theory, arXiv:hep-th/0506116. [36] B.S. Acharya, A moduli fixing mechanism in M-theory, arXiv:hep-th/0212294. [37] P.S. Aspinwall, R. Kallosh, Fixing all moduli for M-theory on K3 × K3, arXiv:hep-th/0506014. [38] J.F.G. Cascales, M.P. Garcia del Moral, F. Quevedo, A.M. Uranga, Realistic D-brane models on warped throats: fluxes, hierarchies and moduli stabilization, JHEP 0402 (2004) 031 [arXiv:hep-th/0312051]. [39] D. Lust, S. Reffert, W. Schulgin, S. Stieberger, Moduli stabilization in type IIB orientifolds. I: orbifold limits, arXiv:hep-th/0506090. [40] E. Bergshoeff, R. Kallosh, T. Ortin, D. Roest, A. Van Proeyen, New formulations of D = 10 supersymmetry and D8–O8 domain walls, Class. Quant. Grav. 18 (2001) 3359 [arXiv:hep-th/0103233]. [41] J.P. Gauntlett, S. Pakis, The geometry of D = 11 Killing spinors (T ), JHEP 0304 (2003) 039 [arXiv:hep-th/0212008]; O.A.P. Mac Conamhna, Refining G-structure classifications, Phys. Rev. D 70 (2004) 105024 [arXiv:hep-th/0408203]; J. Gillard, U. Gran, G. Papadopoulos, The spinorial geometry of supersymmetric backgrounds, Class. Quant. Grav. 22 (2005) 1033 [arXiv:hepth/0410155]; M. Cariglia, O.A.P. Mac Conamhna, Null structure groups in eleven dimensions, arXiv:hep-th/0411079; U. Gran, G. Papadopoulos, D. Roest, Systematics of M-theory spinorial geometry, Class. Quant. Grav. 22 (2005) 2701 [arXiv:hep-th/0503046]; U. Gran, J. Gutowski, G. Papadopoulos, D. Roest, Systematics of IIB spinorial geometry, arXiv:hep-th/0507087. [42] J.P. Gauntlett, D. Martelli, S. Pakis, D. Waldram, G-structures and wrapped NS5-branes, Commun. Math. Phys. 247 (2004) 421 [arXiv:hepth/0205050]. [43] D. Lust, D. Tsimpis, Supersymmetric AdS(4) compactifications of IIA supergravity, JHEP 0502 (2005) 027 [arXiv:hep-th/0412250]. [44] B. de Wit, D.J. Smit, N.D. Hari Dass, Residual supersymmetry of compactified D = 10 supergravity, Nucl. Phys. B 283 (1987) 165. [45] G.L. Cardoso, G. Curio, G. Dall’Agata, D. Lust, P. Manousselis, G. Zoupanos, Non-Kaehler string backgrounds and their five torsion classes, Nucl. Phys. B 652 (2003) 5 [arXiv:hep-th/0211118]. [46] J.P. Gauntlett, D. Martelli, D. Waldram, Superstrings with intrinsic torsion, Phys. Rev. D 69 (2004) 086002 [arXiv:hep-th/0302158]. [47] K. Becker, M. Becker, M-theory on eight-manifolds, Nucl. Phys. B 477 (1996) 155 [arXiv:hep-th/9605053]. [48] G. Curio, A. Krause, Four-flux and warped heterotic M-theory compactifications, Nucl. Phys. B 602 (2001) 172 [arXiv:hep-th/0012152]. [49] K. Behrndt, C. Jeschek, Fluxes in M-theory on 7-manifolds and G structures, JHEP 0304 (2003) 002 [arXiv:hep-th/0302047]. [50] P. Kaste, R. Minasian, A. Tomasiello, Supersymmetric M-theory compactifications with fluxes on seven-manifolds and G-structures, JHEP 0307 (2003) 004 [arXiv:hep-th/0303127]. [51] D. Martelli, J. Sparks, G-structures, fluxes and calibrations in M-theory, Phys. Rev. D 68 (2003) 085014 [arXiv:hep-th/0306225].
M. Graña / Physics Reports 423 (2006) 91 – 158
153
[52] D. Constantin, Flux compactification of M-theory on compact manifolds with Spin(7) holonomy, arXiv:hep-th/0507104. [53] P. Kaste, R. Minasian, M. Petrini, A. Tomasiello, Kaluza–Klein bundles and manifolds of exceptional holonomy, JHEP 0209 (2002) 033 [arXiv:hep-th/0206213]. [54] P. Kaste, R. Minasian, M. Petrini, A. Tomasiello, Nontrivial RR two-form field strength and SU(3)-structure, Fortsch. Phys. 51 (2003) 764 [arXiv:hep-th/0301063]. [55] G. Dall’Agata, N. Prezas, N=1 geometries for M-theory and type IIA strings with fluxes, Phys. Rev. D 69 (2004) 066004 [arXiv:hep-th/0311146]. [56] A.R. Frey, Warped strings: self-dual flux and contemporary compactifications, arXiv:hep-th/0308156. [57] A.R. Frey, M. Graña, Type IIB solutions with interpolating supersymmetries, Phys. Rev. D 68 (2003) 106002 [arXiv:hep-th/0307142]. [58] A.R. Frey, Notes on SU(3) structures in type IIB supergravity, JHEP 0406 (2004) 027 [arXiv:hep-th/0404107]. [59] G. Dall’Agata, On supersymmetric solutions of type IIB supergravity with general fluxes, Nucl. Phys. B 695 (2004) 243 [arXiv:hep-th/0403220]. [60] M. Graña, R. Minasian, M. Petrini, A. Tomasiello, Supersymmetric backgrounds from generalized Calabi–Yau manifolds, JHEP 0408 (2004) 046 [arXiv:hep-th/0406137]. [61] K. Behrndt, M. Cvetic, P. Gao, General type IIB fluxes with SU(3) structures, Nucl. Phys. B 721 (2005) 287 [arXiv:hep-th/0502154]. [62] M. Grãna, R. Minasian, M. Petrini, A. Tomasiello, Generalized structures of N = 1 vacua, arXiv:hep-th/0505212. [63] J. Bovy, D. Lust, D. Tsimpis, N = 1, 2 supersymmetric vacua of IIA supergravity and SU(2) structures, arXiv:hep-th/0506160. [64] K. Behrndt, M. Cvetic, General N = 1 supersymmetric flux vacua of (massive) type IIA string theory, arXiv:hep-th/0403049; K. Behrndt, M. Cvetic, General N = 1 supersymmetric fluxes in massive type IIA string theory, Nucl. Phys. B 708 (2005) 45 [arXiv:hepth/0407263]. [65] K. Pilch, N.P. Warner, A new supersymmetric compactification of chiral IIB supergravity, Phys. Lett. B 487 (2000) 22 [arXiv:hep-th/0002192]; K. Pilch, N.P. Warner, N = 1 supersymmetric solutions of IIB supergravity from Killing spinors, arXiv:hep-th/0403005; N. Halmagyi, K. Pilch, C. Romelsberger, N.P. Warner, Holographic duals of a family of N = 1 fixed points, arXiv:hep-th/0506206. [66] C.N. Gowdigere, D. Nemeschansky, N.P. Warner, Supersymmetric solutions with fluxes from algebraic Killing spinors, Adv. Theor. Math. Phys. 7 (2004) 787 [arXiv:hep-th/0306097]; K. Pilch, N.P. Warner, Generalizing the N = 2 supersymmetric RG flow solution of IIB supergravity, Nucl. Phys. B 675 (2003) 99 [arXiv:hepth/0306098]. [67] S.S. Gubser, Special holonomy in string theory and M-theory, arXiv:hep-th/0201114. [68] For an introduction to Calabi–Yau manifolds see P. Candelas, Introduction to Complex Manifolds, Lectures at the 1987 Trieste Spring School. For a more detailed mathematical reference, see P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1978; K. Kodaira, Complex Manifolds and Deformations of Complex Structures, Springer, Berlin, 1986. [69] For a review of Calabi–Yau manifolds in string theory, see for example B. R. Greene, String theory on Calabi–Yau manifolds, arXiv:hepth/9702155. [70] S. Salamon, Riemannian Geometry and Holonomy Groups, Pitman Research Notes in Mathematics, vol. 201, Longman, Harlow, 1989. [71] D. Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, Oxford, 2000. [72] S. Chiossi, S. Salamon, The intrinsic torsion of SU(3) and G2 structures, in: Differential Geometry, Valencia, 2001, p. 115 [math.DG/0202282]. [73] T. Friedrich, S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math. 6 (2002) 303 [arXiv:mathdg/0102142]. [74] M. Graña, J. Louis, D. Waldram, Hitchin functionals in N = 2 supergravity, arXiv:hep-th/0505264. [75] K. Becker, M. Becker, K. Dasgupta, P.S. Green, Compactifications of heterotic theory on non-Kaehler complex manifolds. I, JHEP 0304 (2003) 007 [arXiv:hep-th/0301161]; K. Becker, M. Becker, P.S. Green, K. Dasgupta, E. Sharpe, Compactifications of heterotic strings on non-Kaehler complex manifolds. II, Nucl. Phys. B 678 (2004) 19 [arXiv:hep-th/0310058]. [76] P. Kaste, R. Minasian, A. Tomasiello, Supersymmetric M-theory compactifications with fluxes on seven-manifolds and G-structures, JHEP 0307 (2003) 004 [arXiv:hep-th/0303127]. [77] J. Polchinski, String Theory, vol. 2, Superstring Theory and Beyond, University Press, Cambridge, UK, 1998, 531p. [78] A. Giryavets, S. Kachru, P.K. Tripathy, S.P. Trivedi, Flux compactifications on Calabi–Yau threefolds, JHEP 0404 (2004) 003 [arXiv:hepth/0312104]. [79] A. Giryavets, S. Kachru, P.K. Tripathy, On the taxonomy of flux vacua, JHEP 0408 (2004) 002 [arXiv:hep-th/0404243]. [80] O. DeWlfe, Enhanced symmetries in multiparameter flux vacua, arXiv:hep-th/0506245. [81] P.K. Tripathy, S.P. Trivedi, Compactification with flux on K3 and tori, JHEP 0303 (2003) 028 [arXiv:hep-th/0301139]. [82] S. Kachru, M.B. Schulz, S. Trivedi, Moduli stabilization from fluxes in a simple IIB orientifold, JHEP 0310 (2003) 007 [arXiv:hep-th/0201028]. [83] J.F.G. Cascales, A.M. Uranga, Chiral 4d N = 1 string vacua with D-branes and NSNS and RR fluxes, JHEP 0305 (2003) 011 [arXiv:hepth/0303024]; F. Marchesano, G. Shiu, Building MSSM flux vacua, JHEP 0411 (2004) 041 [arXiv:hep-th/0409132]; F. Marchesano, G. Shiu, MSSM vacua from flux compactifications, Phys. Rev. D 71 (2005) 011701 [arXiv:hep-th/0408059]; J. Kumar, J.D. Wells, Surveying standard model flux vacua on T 6 /Z2 × Z2 , arXiv:hep-th/0506252. [84] M. Berg, M. Haack, B. Kors, An orientifold with fluxes and branes via T-duality, Nucl. Phys. B 669 (2003) 3 [arXiv:hep-th/0305183]. [85] S. Kachru, M.B. Schulz, P.K. Tripathy, S.P. Trivedi, New supersymmetric string compactifications, JHEP 0303 (2003) 061 [arXiv:hepth/0211182]. [86] M.B. Schulz, Superstring orientifolds with torsion: O5 orientifolds of torus fibrations and their massless spectra, Fortsch. Phys. 52 (2004) 963 [arXiv:hep-th/0406001]. [87] C.M. Hull, R.A. Reid-Edwards, Flux compactifications of string theory on twisted tori, arXiv:hep-th/0503114. [88] J.C. Breckenridge, G. Michaud, R.C. Myers, More D-brane bound states, Phys. Rev. D 55 (1997) 6438 [arXiv:hep-th/9611174].
154
M. Graña / Physics Reports 423 (2006) 91 – 158
[89] M.S. Costa, G. Papadopoulos, Superstring dualities and p-brane bound states, Nucl. Phys. B 510 (1998) 217 [arXiv:hep-th/9612204]. [90] A. Butti, M. Graña, R. Minasian, M. Petrini, A. Zaffaroni, The baryonic branch of Klebanov–Strassler solution: a supersymmetric family of SU(3) structure backgrounds, JHEP 0503 (2005) 069 [arXiv:hep-th/0412187]. [91] G. Papadopoulos, A.A. Tseytlin, Complex geometry of conifolds and 5-brane wrapped on 2-sphere, Class. Quant. Grav. 18 (2001) 1333 [arXiv:hep-th/0012034]. [92] M.B. Schulz, Calabi–Yau duals of torus orientifolds, arXiv:hep-th/0412270. [93] P.S. Aspinwall, An analysis of fluxes by duality, arXiv:hep-th/0504036. [94] S. Hellerman, J. McGreevy, B. Williams, Geometric constructions of nongeometric string theories, JHEP 0401 (2004) 024 [arXiv:hepth/0208174]; A. Flournoy, B. Wecht, B. Williams, Constructing nongeometric vacua in string theory, Nucl. Phys. B 706 (2005) 127 [arXiv:hep-th/0404217]; C.M. Hull, A geometry for non-geometric string backgrounds, arXiv:hep-th/0406102. [95] J. Shelton, W. Taylor, B. Wecht, Nongeometric flux compactifications, arXiv:hep-th/0508133. [96] N. Hitchin, Generalized Calabi–Yau manifolds, arXiv: math.DG/0209099. [97] M. Gualtieri, Generalized complex geometry, D.Ph. Thesis, Oxford University, 2004 [arXiv:math.DG/0401221]. [98] M. Graña, R. Minasian, M. Petrini, A. Tomasiello, Type II strings and generalized Calabi–Yau manifolds, Comptes Rendus Physique 5 (2004) 979 [arXiv:hep-th/0409176]. [99] F. Witt, Special metric structures and closed forms, D.Ph. Thesis, Oxford University, 2004 [arXiv:math.DG/0502443]. [100] C. Jeschek, F. Witt, Generalised G(2)-structures and type IIB superstrings, JHEP 0503 (2005) 053 [arXiv:hep-th/0412280]. [101] U. Lindstrom, R. Minasian, A. Tomasiello, M. Zabzine, Generalized complex manifolds and supersymmetry, Commun. Math. Phys. 257 (2005) 235 [arXiv:hep-th/0405085]; U. Lindstrom, M. Rocek, R. von Unge, M. Zabzine, Generalized Kaehler geometry and manifest N = (2, 2) supersymmetric nonlinear sigmamodels, arXiv:hep-th/0411186; M. Zabzine, Hamiltonian perspective on generalized complex structure, arXiv:hep-th/0502137. [102] A. Kapustin, Y. Li, Topological sigma-models with H-flux and twisted generalized complex manifolds, arXiv:hep-th/0407249; R. Zucchini, A sigma model field theoretic realization of Hitchin’s generalized complex geometry, JHEP 0411 (2004) 045 [arXiv:hepth/0409181]; Generalized complex geometry, generalized branes and the Hitchin sigma model, JHEP 0503 (2005) 022 [arXiv:hep-th/0501062]; V. Pestun, E. Witten, The Hitchin functionals and the topological B-model at one loop, arXiv:hep-th/0503083; P. Grange, R. Minasian, Modified pure spinors and mirror symmetry, arXiv:hep-th/0412086; O. Ben-Bassat, Mitya Boyarchenko, Submanifolds of generalized complex manifolds [arXiv:math.AG/0309013]; P. Koerber, Stable D-branes, calibrations and generalized Calabi–Yau geometry, arXiv:hep-th/0506154; L. Martucci, P. Smyth, Supersymmetric D-branes and calibrations on general N = 1 backgrounds, arXiv:hep-th/0507099. [103] S. Fidanza, R. Minasian, A. Tomasiello, Mirror symmetric SU(3)-structure manifolds with NS fluxes, Commun. Math. Phys. 254 (2005) 401 [arXiv:hep-th/0311122]. [104] O. Ben-Bassat, Mirror symmetry and generalized complex manifolds, [arXiv:math.AG/0405303]. [105] S. Ivanov, G. Papadopoulos, A no-go theorem for string warped compactifications, Phys. Lett. B 497 (2001) 309 [arXiv:hep-th/0008232]. [106] C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [arXiv:hep-th/9602022]. [107] A. Sen, Orientifold limit of F-theory vacua, Phys. Rev. D 55 (1997) 7345 [arXiv:hep-th/9702165]. [108] M. Graña, R. Minasian, M. Petrini, A. Tomasiello, work in progress. [109] A. Buchel, On effective action of string theory flux compactifications, Phys. Rev. D 69 (2004) 106004 [arXiv:hep-th/0312076]. [110] S. Cecotti, S. Ferrara, L. Girardello, Geometry of type II superstrings and the moduli of superconformal field theories, Int. J. Mod. Phys. A 4 (1989) 2475. [111] S. Ferrara, S. Sabharwal, Dimensional reduction of type II superstrings, Class. Quant. Grav. 6 (1989) L77; Quaternionic manifolds for type II superstring vacua of Calabi–Yau spaces, Nucl. Phys. B 332 (1990) 317. [112] M. Bodner, A.C. Cadavid, S. Ferrara, (2, 2) Vacuum configurations for type IIA superstrings: N = 2 supergravity Lagrangians and algebraic geometry, Class. Quant. Grav. 8 (1991) 789. [113] M. Bodner, A.C. Cadavid, Dimensional reduction of type IIB supergravity and exceptional quaternionic manifolds, Class. Quant. Grav. 7 (1990) 829. [114] P. Candelas, X. de la Ossa, Moduli space of Calabi–Yau manifolds, Nucl. Phys. B 355 (1991) 455. [115] T.W. Grimm, J. Louis, The effective action of type IIA Calabi–Yau orientifolds, Nucl. Phys. B 718 (2005) 153 [arXiv:hep-th/0412277]. [116] T.W. Grimm, J. Louis, The effective action of N = 1 Calabi–Yau orientifolds, Nucl. Phys. B 699 (2004) 387 [arXiv:hep-th/0403067]. [117] L. Andrianopoli, M. Bertolini, A. Ceresole, R. D’Auria, S. Ferrara, P. Fre, T. Magri, N = 2 supergravity and N = 2 super Yang–Mills theory on general scalar manifolds: symplectic covariance, gaugings and the momentum map, J. Geom. Phys. 23 (1997) 111 [arXiv:hep-th/9605032]; L. Andrianopoli, M. Bertolini, A. Ceresole, R. D’Auria, S. Ferrara, P. Fre’, General matter coupled N = 2 supergravity, Nucl. Phys. B 476 (1996) 397 [arXiv:hep-th/9603004]. [118] G. Dall’Agata, Type IIB supergravity compactified on a Calabi–Yau manifold with H -fluxes, JHEP 0111 (2001) 005 [arXiv:hep-th/0107264]. [119] J. Louis, A. Micu, Type II theories compactified on Calabi–Yau threefolds in the presence of background fluxes, Nucl. Phys. B 635 (2002) 395 [arXiv:hep-th/0202168]. [120] S. Gurrieri, N = 2 and N = 4 supergravities as compactifications from string theories in 10 dimensions, Ph.D. Thesis, arXiv:hep-th/0408044. [121] S. Gurrieri, J. Louis, A. Micu, D. Waldram, Mirror symmetry in generalized Calabi–Yau compactifications, Nucl. Phys. B 654 (2003) 61 [arXiv:hep-th/0211102]. [122] T.W. Grimm, The effective action of type II Calabi–Yau orientifolds, Ph.D. Thesis, arXiv:hep-th/0507153.
M. Graña / Physics Reports 423 (2006) 91 – 158
155
[123] A. Strominger, Yukawa couplings in superstring compactification, Phys. Rev. Lett. 55 (1985) 2547. [124] N.J. Hitchin, The geometry of three-forms in six and seven dimensions, arXiv:math.dg/0010054; N.J. Hitchin, Stable forms and special metrics, arXiv:math.dg/0107101. [125] S. Salomon, Harmonic and Holomorphic Maps, Lecture Notes in Mathematics, vol. 1164, Springer, Berlin, 1984. [126] I. Brunner, K. Hori, Orientifolds and mirror symmetry, JHEP 0411 (2004) 005 [arXiv:hep-th/0303135]. [127] B. Acharya, M. Aganagic, K. Hori, C. Vafa, Orientifolds, mirror symmetry and superpotentials, arXiv:hep-th/0202208. [128] J. Wess, J. Bagger, Supersymmetry and Supergravity, Princeton University Press, Princeton, 1992. [129] E. Cremmer, S. Ferrara, C. Kounnas, D.V. Nanopoulos, Naturally vanishing cosmological constant in N = 1 supergravity, Phys. Lett. B 133 (1983) 61; J.R. Ellis, A.B. Lahanas, D.V. Nanopoulos, K. Tamvakis, No-scale supersymmetric standard model, Phys. Lett. B 134 (1984) 429; R. Barbieri, E. Cremmer, S. Ferrara, Flat and positive potentials in N = 1 supergravity, Phys. Lett. B 163 (1985) 143. [130] R. D’Auria, S. Ferrara, M. Trigiante, c-map, very special quaternionic geometry and dual Kähler spaces, Phys. Lett. B 587 (2004) 138 [arXiv:hep-th/0401161]; R. D’Auria, S. Ferrara, M. Trigiante, Homogeneous special manifolds, orientifolds and solvable coordinates, Nucl. Phys. B 693 (2004) 261 [arXiv:hep-th/0403204]. [131] P. Mayr, On supersymmetry breaking in string theory and its realization in brane worlds, Nucl. Phys. B 593 (2001) 99 [arXiv:hep-th/0003198]. [132] G. Curio, A. Klemm, D. Lüst, S. Theisen, On the vacuum structure of type II string compactifications on Calabi–Yau spaces with H -fluxes, Nucl. Phys. B 609 (2001) 3 [arXiv:hep-th/0012213]; G. Curio, A. Klemm, B. Körs, D. Lüst, Fluxes in heterotic and type II string compactifications, Nucl. Phys. B 620 (2002) 237 [arXiv:hepth/0106155]. [133] M. Haack, J. Louis, M-theory compactified on Calabi–Yau fourfolds with background flux, Phys. Lett. B 507 (2001) 296 [arXiv:hepth/0103068]. [135] M. Becker, D. Constantin, A note on flux induced superpotentials in string theory, JHEP 0308 (2003) 015 [arXiv:hep-th/0210131]. [136] S. Kachru, A.K. Kashani-Poor, Moduli potentials in type IIA compactifications with RR and NS flux, JHEP 0503 (2005) 066 [arXiv:hepth/0411279]. [137] G. Dall’Agata, R. D’Auria, L. Sommovigo, S. Vaula, D = 4, N = 2 gauged supergravity in the presence of tensor multiplets, Nucl. Phys. B 682 (2004) 243 [arXiv:hep-th/0312210]; R. D’Auria, L. Sommovigo, S. Vaula, N = 2 supergravity Lagrangian coupled to tensor multiplets with electric and magnetic fluxes, JHEP 0411 (2004) 028 [arXiv:hep-th/0409097]. [138] A.K. Kashani-Poor, A. Tomasiello, A stringy test of flux-induced isometry gauging, arXiv:hep-th/0505208. [139] B. de Carlos, S. Gurrieri, A. Lukas, A. Micu, Moduli stabilisation in heterotic string compactifications, arXiv:hep-th/0507173. [140] C. Vafa, Superstrings and topological strings at large N , J. Math. Phys. 42 (2001) 2798 [arXiv:hep-th/0008142]. [141] S. Gukov, C. Vafa, E. Witten, CFT’s from Calabi–Yau four-folds, Nucl. Phys. B 584 (2000) 69 [Erratum-ibid. B 608 (2001) 477] [arXiv:hepth/9906070]. [142] T.R. Taylor, C. Vafa, RR flux on Calabi–Yau and partial supersymmetry breaking, Phys. Lett. B 474 (2000) 130 [arXiv:hep-th/9912152]. [143] S. Gukov, Solitons, superpotentials and calibrations, Nucl. Phys. B 574 (2000) 169 [arXiv:hep-th/9911011]. [144] M. Graña, T.W. Grimm, H. Jockers, J. Louis, Soft supersymmetry breaking in Calabi–Yau orientifolds with D-branes and fluxes, Nucl. Phys. B 690 (2004) 21 [arXiv:hep-th/0312232]. [145] O. DeWolfe, S.B. Giddings, Scales and hierarchies in warped compactifications and brane worlds, Phys. Rev. D 67 (2003) 066008 [arXiv:hepth/0208123]. [146] S.P. de Alwis, On potentials from fluxes, Phys. Rev. D 68 (2003) 126001 [arXiv:hep-th/0307084]. [147] S.B. Giddings, A. Maharana, Dynamics of warped compactifications and the shape of the warped landscape, arXiv:hep-th/0507158. [148] S. Gurrieri, A. Lukas, A. Micu, Heterotic on half-flat, Phys. Rev. D 70 (2004) 126009 [arXiv:hep-th/0408121]. [149] T. House, A. Micu, M-theory compactifications on manifolds with G(2) structure, Class. Quant. Grav. 22 (2005) 1709 [arXiv:hep-th/0412006]. [150] T. House, E. Palti, Effective action of (massive) IIA on manifolds with SU(3) structure, Phys. Rev. D 72 (2005) 026004 [arXiv:hep-th/0505177]. [151] P. Berglund, P. Mayr, Non-perturbative superpotentials in F -theory and string duality, arXiv:hep-th/0504058. [152] S. Hosono, A. Klemm, S. Theisen, Lectures on mirror symmetry, arXiv:hep-th/9403096. [153] A. Tomasiello, Topological mirror symmetry with fluxes, JHEP 0506 (2005) 067 [arXiv:hep-th/0502148]. [154] A. Strominger, S.T. Yau, E. Zaslow, Mirror symmetry is T -duality, Nucl. Phys. B 479 (1996) 243 [arXiv:hep-th/9606040]. [155] S.F. Hassan, T-duality space-time spinors and R-R fields in curved backgrounds, Nucl. Phys. B 568 (2000) 145 [arXiv:hep-th/9907152]. [156] P. Bouwknegt, J. Evslin, V. Mathai, T-duality: topology change from H -flux, Commun. Math. Phys. 249 (2004) 383 [arXiv:hep-th/0306062]. [157] A. Dabholkar, C. Hull, Duality twists, orbifolds, and fluxes, JHEP 0309 (2003) 054 [arXiv:hep-th/0210209]. [158] See for example C.M. Will, The confrontation between general relativity and experiment, Living Rev. Rel. 4 (2001) 4 [arXiv:gr-qc/0103036]. [159] E. Silverstein, TASI/PiTP/ISS lectures on moduli and microphysics, arXiv:hep-th/0405068. [160] P.G. Camara, L.E. Ibañez, A.M. Uranga, Flux-induced SUSY-breaking soft terms on D7-D3 brane systems, Nucl. Phys. B 708 (2005) 268 [arXiv:hep-th/0408036]. [161] G. Curio, A. Krause, G-fluxes and non-perturbative stabilisation of heterotic M-theory, Nucl. Phys. B 643 (2002) 131 [arXiv:hep-th/0108220]; G.L. Cardoso, G. Curio, G. Dall’Agata, D. Lust, BPS action and superpotential for heterotic string compactifications with fluxes, JHEP 0310 (2003) 004 [arXiv:hep-th/0306088]; E.I. Buchbinder, B.A. Ovrut, Vacuum stability in heterotic M-theory, Phys. Rev. D 69 (2004) 086010 [arXiv:hep-th/0310112]; S. Gukov, S. Kachru, X. Liu, L. McAllister, Heterotic moduli stabilization with fractional Chern–Simons invariants, Phys. Rev. D 69 (2004) 086008 [arXiv:hep-th/0310159]; G. Curio, A. Krause, D. Lust, Moduli stabilization in the heterotic/IIB discretuum, arXiv:hep-th/0502168.
156
M. Graña / Physics Reports 423 (2006) 91 – 158
[162] J.P. Conlon, F. Quevedo, On the explicit construction and statistics of Calabi–Yau flux vacua, JHEP 0410 (2004) 039 [arXiv:hep-th/0409215]. [163] V. Balasubramanian, P. Berglund, J.P. Conlon, F. Quevedo, Systematics of moduli stabilisation in Calabi–Yau flux compactifications, JHEP 0503 (2005) 007 [arXiv:hep-th/0502058]. [164] J.P. Conlon, F. Quevedo, K. Suruliz, large-volume flux compactifications: moduli spectrum and D3/D7 soft supersymmetry breaking, arXiv:hepth/0505076. [165] R. Blumenhagen, D. Lust, T.R. Taylor, Moduli stabilization in chiral type IIB orientifold models with fluxes, Nucl. Phys. B 663 (2003) 319 [arXiv:hep-th/0303016]. [166] J.F.G. Cascales, A.M. Uranga, Chiral 4d N = 1 string vacua with D-branes and NSNS and RR fluxes, JHEP 0305 (2003) 011 [arXiv:hepth/0303024]. [167] F. Marchesano, G. Shiu, MSSM vacua from flux compactifications, Phys. Rev. D 71 (2005) 011701 [arXiv:hep-th/0408059]; F. Marchesano, G. Shiu, Building MSSM flux vacua, JHEP 0411 (2004) 041 [arXiv:hep-th/0409132]. [168] M. Cvetic, T. Liu, Supersymmetric standard models, flux compactification and moduli stabilization, Phys. Lett. B 610 (2005) 122 [arXiv:hepth/0409032]; M. Cvetic, T. Li, T. Liu, Standard-like models as type IIB flux vacua, Phys. Rev. D 71 (2005) 106008 [arXiv:hep-th/0501041]. [169] A. Font, Z(N ) orientifolds with flux, JHEP 0411 (2004) 077 [arXiv:hep-th/0410206]. [170] P.K. Tripathy, S.P. Trivedi, Compactification with flux on K3 and tori, JHEP 0303 (2003) 028 [arXiv:hep-th/0301139]. [171] L. Andrianopoli, R. D’Auria, S. Ferrara, M.A. Lledo, 4-D gauged supergravity analysis of type IIB vacua on K3 × T ∗∗ 2/Z(2), JHEP 0303 (2003) 044 [arXiv:hep-th/0302174]. [172] M. Berg, M. Haack, B. Kors, Loop corrections to volume moduli and inflation in string theory, Phys. Rev. D 71 (2005) 026005 [arXiv:hepth/0404087]; M. Berg, M. Haack, B. Kors, String loop corrections to Kaehler potentials in orientifolds, arXiv:hep-th/0508043. [173] G. von Gersdorff, A. Hebecker, Kaehler corrections for the volume modulus of flux compactifications, arXiv:hep-th/0507131. [174] I. Antoniadis, R. Minasian, S. Theisen, P. Vanhove, String loop corrections to the universal hypermultiplet, Class. Quant. Grav. 20 (2003) 5079 [arXiv:hep-th/0307268]. [175] K. Becker, M. Becker, M. Haack, J. Louis, Supersymmetry breaking and alpha’-corrections to flux induced potentials, JHEP 0206 (2002) 060 [arXiv:hep-th/0204254]. [176] V. Balasubramanian, P. Berglund, Stringy corrections to Kaehler potentials, SUSY breaking, and the cosmological constant problem, JHEP 0411 (2004) 085 [arXiv:hep-th/0408054]. [177] P.G. Camara, L.E. Ibañez, A.M. Uranga, Flux-induced SUSY-breaking soft terms on D7-D3 brane systems, Nucl. Phys. B 708 (2005) 268 [arXiv:hep-th/0408036]; D. Lust, S. Reffert, S. Stieberger, Flux-induced soft supersymmetry breaking in chiral type IIb orientifolds with D3/D7-branes, Nucl. Phys. B 706 (2005) 3 [arXiv:hep-th/0406092]; D. Lust, S. Reffert, S. Stieberger, MSSM with soft SUSY breaking terms from D7-branes with fluxes, arXiv:hep-th/0410074. [178] E. Witten, Non-perturbative superpotentials in string theory, Nucl. Phys. B 474 (1996) 343 [arXiv:hep-th/9604030]. [179] F. Denef, M.R. Douglas, B. Florea, Building a better racetrack, JHEP 0406 (2004) 034 [arXiv:hep-th/0404257]. [180] L. Gorlich, S. Kachru, P.K. Tripathy, S.P. Trivedi, Gaugino condensation and nonperturbative superpotentials in flux compactifications, JHEP 0412 (2004) 074 [arXiv:hep-th/0407130]. [181] P.K. Tripathy, S.P. Trivedi, D3 brane action and fermion zero modes in presence of background flux, JHEP 0506 (2005) 066 [arXiv:hepth/0503072]. [182] N. Saulina, Topological constraints on stabilized flux vacua, Nucl. Phys. B 720 (2005) 203 [arXiv:hep-th/0503125]. [183] R. Kallosh, A.K. Kashani-Poor, A. Tomasiello, Counting fermionic zero modes on M5 with fluxes, JHEP 0506 (2005) 069 [arXiv:hepth/0503138]; E. Bergshoeff, R. Kallosh, A.K. Kashani-Poor, D. Sorokin, A. Tomasiello, An index for the Dirac operator on D3 branes with background fluxes, arXiv:hep-th/0507069. [184] S. Kachru, R. Kallosh, A. Linde, S.P. Trivedi, de Sitter vacua in string theory, Phys. Rev. D 68 (2003) 046005 [arXiv:hep-th/0301240]. [185] K. Choi, A. Falkowski, H.P. Nilles, M. Olechowski, S. Pokorski, Stability of flux compactifications and the pattern of supersymmetry breaking, JHEP 0411 (2004) 076 [arXiv:hep-th/0411066]. [186] S.P. de Alwis, Effective potentials for light moduli, arXiv:hep-th/0506266. [187] F. Denef, M.R. Douglas, B. Florea, A. Grassi, S. Kachru, Fixing all moduli in a simple F-theory compactification, arXiv:hep-th/0503124. [188] M. Berg, M. Haack, B. Kors, On volume stabilization by quantum corrections, arXiv:hep-th/0508171. [189] K. Bobkov, Volume stabilization via alpha’ corrections in type IIB theory with fluxes, JHEP 0505 (2005) 010 [arXiv:hep-th/0412239]. [190] C.P. Burgess, R. Kallosh, F. Quevedo, de Sitter string vacua from supersymmetric D-terms, JHEP 0310 (2003) 056 [arXiv:hep-th/0309187]. [191] A. Saltman, E. Silverstein, The scaling of the no-scale potential and de Sitter model building, JHEP 0411 (2004) 066 [arXiv:hep-th/0402135]. [192] S. Kachru, J. Pearson, H.L. Verlinde, Brane/flux annihilation and the string dual of a non-supersymmetric field theory, JHEP 0206 (2002) 021 [arXiv:hep-th/0112197]. [193] G. Villadoro, F. Zwirner, de Sitter vacua via consistent D-terms, arXiv:hep-th/0508167. [194] E. Dudas, S.K. Vempati, Large D-terms, hierarchical soft spectra and moduli stabilisation, arXiv:hep-th/0506172. [195] See for example N. Goheer, M. Kleban, L. Susskind, The trouble with de Sitter space, JHEP 0307 (2003) 056 [arXiv:hep-th/0212209]. [196] M. Dine, N. Seiberg, Is the superstring weakly coupled?, Phys. Lett. B 162 (1985) 299. [197] S.B. Giddings, The fate of four dimensions, Phys. Rev. D 68 (2003) 026006 [arXiv:hep-th/0303031]; S.B. Giddings, R.C. Myers, Spontaneous decompactification, Phys. Rev. D 70 (2004) 046005 [arXiv:hep-th/0404220]. [198] A.R. Frey, M. Lippert, B. Williams, The fall of stringy de Sitter, Phys. Rev. D 68 (2003) 046008 [arXiv:hep-th/0305018].
M. Graña / Physics Reports 423 (2006) 91 – 158
157
[199] R. Brustein, S.P. de Alwis, Moduli potentials in string compactifications with fluxes: mapping the discretuum, Phys. Rev. D 69 (2004) 126006 [arXiv:hep-th/0402088]. [200] F. Saueressig, U. Theis, S. Vandoren, On de Sitter vacua in type IIA orientifold compactifications, arXiv:hep-th/0506181. [201] M. Becker, G. Curio, A. Krause, de Sitter vacua from heterotic M-theory, Nucl. Phys. B 693 (2004) 223 [arXiv:hep-th/0403027]. [202] A. Saltman, E. Silverstein, A new handle on de Sitter compactifications, arXiv:hep-th/0411271. [203] E. Silverstein, (A)dS backgrounds from asymmetric orientifolds, arXiv:hep-th/0106209; A. Maloney, E. Silverstein, A. Strominger, de Sitter space in noncritical string theory, arXiv:hep-th/0205316. [204] M.R. Douglas, The statistics of string/M theory vacua, JHEP 0305 (2003) 046 [arXiv:hep-th/0303194]. [205] M.R. Douglas, Basic results in vacuum statistics, Comptes Rendus Physique 5 (2004) 965 [arXiv:hep-th/0409207]. [206] F. Denef, M.R. Douglas, Distributions of nonsupersymmetric flux vacua, JHEP 0503 (2005) 061 [arXiv:hep-th/0411183]. [207] J.H. Horne, G.W. Moore, Chaotic coupling constants, Nucl. Phys. B 432 (1994) 109 [arXiv:hep-th/9403058]. [208] M.R. Douglas, Talk given at Strings 2005, Toronto, July 2005, available online at http://www.fields.utoronto.ca/audio/05-06/strings/douglas/ [209] R. Blumenhagen, F. Gmeiner, G. Honecker, D. Lust, T. Weigand, The statistics of supersymmetric D-brane models, Nucl. Phys. B 713 (2005) 83 [arXiv:hep-th/0411173]. [210] O. DeWolfe, A. Giryavets, S. Kachru, W. Taylor, Enumerating flux vacua with enhanced symmetries, JHEP 0502 (2005) 037 [arXiv:hepth/0411061]. [211] A. Misra, A. Nanda, Flux vacua statistics for two-parameter Calabi–Yau’s, Fortsch. Phys. 53 (2005) 246 [arXiv:hep-th/0407252]. [212] Some references are L. Andrianopoli, R. D’Auria, S. Ferrara, M.A. Lledo, Gauging of flat groups in four dimensional supergravity, JHEP 0207 (2002) 010 [arXiv:hep-th/0203206]; S. Ferrara, M. Porrati, N = 1 no-scale supergravity from IIB orientifolds, Phys. Lett. B 545 (2002) 411 [arXiv:hep-th/0207135]; L. Andrianopoli, R. D’Auria, S. Ferrara, M.A. Lledo, 4-D gauged supergravity analysis of type IIB vacua on K3 × T ∗∗ 2/Z(2), JHEP 0303 (2003) 044 [arXiv:hep-th/0302174]; C. Angelantonj, S. Ferrara, M. Trigiante, New D = 4 gauged supergravities from N = 4 orientifolds with fluxes, JHEP 0310 (2003) 015 [arXiv:hep-th/0306185]; C. Angelantonj, R. D’Auria, S. Ferrara, M. Trigiante, K3 × T ∗∗ 2/Z(2) orientifolds with fluxes, open string moduli and critical points, Phys. Lett. B 583 (2004) 331 [arXiv:hep-th/0312019]; G. Dall’Agata, R. D’Auria, L. Sommovigo, S. Vaula, D = 4, N = 2 gauged supergravity in the presence of tensor multiplets, Nucl. Phys. B 682 (2004) 243 [arXiv:hep-th/0312210]; R. D’Auria, S. Ferrara, M. Trigiante, S. Vaula, Gauging the Heisenberg algebra of special quaternionic manifolds, Phys. Lett. B 610 (2005) 147 [arXiv:hep-th/0410290]. [213] L. Andrianopoli, S. Ferrara, M. Trigiante, Fluxes, supersymmetry breaking and gauged supergravity, arXiv:hep-th/0307139. [214] J. Scherk, J.H. Schwarz, How to get masses from extra dimensions, Nucl. Phys. B 153 (1979) 61; J. Scherk, J.H. Schwarz, Spontaneous breaking of supersymmetry through dimensional reduction, Phys. Lett. B 82 (1979) 60 [215] N. Kaloper, R.C. Myers, The O(dd) story of massive supergravity, JHEP 9905 (1999) 010 [arXiv:hep-th/9901045]; G. Dall’Agata, S. Ferrara, Gauged supergravity algebras from twisted tori compactifications with fluxes, Nucl. Phys. B 717 (2005) 223 [arXiv:hep-th/0502066]; L. Andrianopoli, M.A. Lledo, M. Trigiante, The Scherk–Schwarz mechanism as a flux compactification with internal torsion, JHEP 0505 (2005) 051 [arXiv:hep-th/0502083]. [216] M. Graña, MSSM parameters from supergravity backgrounds, Phys. Rev. D 67 (2003) 066006 [arXiv:hep-th/0209200]; P.G. Camara, L.E. Ibañez, A.M. Uranga, Flux-induced SUSY-breaking soft terms, Nucl. Phys. B 689 (2004) 195 [arXiv:hep-th/0311241]. [217] A. Lawrence, J. McGreevy, Local string models of soft supersymmetry breaking, JHEP 0406 (2004) 007 [arXiv:hep-th/0401034]. [218] B.S. Acharya, F. Denef, R. Valandro, Statistics of M theory vacua, JHEP 0506 (2005) 056 [arXiv:hep-th/0502060]. [219] O. Aharony, Y.E. Antebi, M. Berkooz, Open String Moduli in KKLT Compactifications, arXiv:hep-th/0508080. [220] J. Gutowski, G. Papadopoulos, P.K. Townsend, Supersymmetry and generalized calibrations, Phys. Rev. D 60 (1999) 106006 [arXiv:hepth/9905156]; J.F.G. Cascales, A.M. Uranga, Branes on generalized calibrated submanifolds, JHEP 0411 (2004) 083 [arXiv:hep-th/0407132]. [221] For reviews, see A. Neitzke, C. Vafa, Topological strings and their physical applications, arXiv:hep-th/0410178; M. Mariño, Chern–Simons theory and topological strings, arXiv:hep-th/0406005; M. Vonk, A mini-course on topological strings, arXiv:hep-th/0504147. [222] G.W. Moore, Arithmetic and attractors, arXiv:hep-th/9807087. [223] H. Ooguri, A. Strominger, C. Vafa, Black hole attractors and the topological string, Phys. Rev. D 70 (2004) 106007 [arXiv:hep-th/0405146]. [224] See for example M. Dine, E. Gorbatov, S. Thomas, Low energy supersymmetry from the landscape, arXiv:hep-th/0407043. [225] S. Kachru, R. Kallosh, A. Linde, J. Maldacena, L. McAllister, S.P. Trivedi, Towards inflation in string theory, JCAP 0310 (2003) 013 [arXiv:hep-th/0308055]. [226] F. Bigazzi, A.L. Cotrone, M. Petrini, A. Zaffaroni, Supergravity duals of supersymmetric four dimensional gauge theories, Riv. Nuovo Cim. 25N12 (2002) 1 [arXiv:hep-th/0303191]. [227] J.P. Gauntlett, D. Martelli, J. Sparks, D. Waldram, Sasaki–Einstein metrics on S(2) × S(3), Adv. Theor. Math. Phys. 8 (2004) 711 [arXiv:hepth/0403002]. [228] M. Cvetic, H. Lu, D.N. Page, C.N. Pope, New Einstein–Sasaki spaces in five and higher dimensions, arXiv:hep-th/0504225. [229] D. Martelli, J. Sparks, Toric geometry, Sasaki–Einstein manifolds and a new infinite class of AdS/CFT duals, arXiv:hep-th/0411238; S. Benvenuti, S. Franco, A. Hanany, D. Martelli, J. Sparks, An infinite family of superconformal quiver gauge theories with Sasaki–Einstein duals, JHEP 0506 (2005) 064 [arXiv:hep-th/0411264].
158
M. Graña / Physics Reports 423 (2006) 91 – 158
[230] D. Berenstein, C.P. Herzog, P. Ouyang, S. Pinansky, Supersymmetry breaking from a Calabi–Yau singularity, arXiv:hep-th/0505029; S. Franco, A. Hanany, F. Saad, A.M. Uranga, Fractional branes and dynamical supersymmetry breaking, arXiv:hep-th/0505040; M. Bertolini, F. Bigazzi, A.L. Cotrone, Supersymmetry breaking at the end of a cascade of Seiberg dualities, arXiv:hep-th/0505055. [231] See for example I. Brunner, K. Hori, K. Hosomichi, J. Walcher, Orientifolds of Gepner models, arXiv:hep-th/0401137. [232] T.P.T. Dijkstra, L.R. Huiszoon, A.N. Schellekens, Supersymmetric standard model spectra from RCFT orientifolds, Nucl. Phys. B 710 (2005) 3 [arXiv:hep-th/0411129]. [233] N. Berkovits, Super-Poincare covariant quantization of the superstring, JHEP 0004 (2000) 018 [arXiv:hep-th/0001035]; N. Berkovits, Covariant quantization of the superstring, Int. J. Mod. Phys. A 16 (2001) 801 [arXiv:hep-th/0008145].
Physics Reports 423 (2006) 159 – 294 www.elsevier.com/locate/physrep
A 3 + 1 perspective on null hypersurfaces and isolated horizons Eric Gourgoulhon∗ , José Luis Jaramillo Laboratoire de l’Univers et de ses Théories, UMR 8102 du CNRS, Observatoire de Paris, F-92195 Meudon Cedex, France Accepted 21 October 2005 editor: M.P. Kamionkowski
Abstract The isolated horizon formalism recently introduced by Ashtekar et al. aims at providing a quasi-local concept of a black hole in equilibrium in an otherwise possibly dynamical spacetime. In this formalism, a hierarchy of geometrical structures is constructed on a null hypersurface. On the other side, the 3 + 1 formulation of general relativity provides a powerful setting for studying the spacetime dynamics, in particular gravitational radiation from black hole systems. Here we revisit the kinematics and dynamics of null hypersurfaces by making use of some 3 + 1 slicing of spacetime. In particular, the additional structures induced on null hypersurfaces by the 3 + 1 slicing permit a natural extension to the full spacetime of geometrical quantities defined on the null hypersurface. This four-dimensional point of view facilitates the link between the null and spatial geometries. We proceed by reformulating the isolated horizon structure in this framework. We also reformulate previous works, such as Damour’s black hole mechanics, and make the link with a previous 3 + 1 approach of black hole horizon, namely the membrane paradigm. We explicit all geometrical objects in terms of 3 + 1 quantities, putting a special emphasis on the conformal 3 + 1 formulation. This is in particular relevant for the initial data problem of black hole spacetimes for numerical relativity. Illustrative examples are provided by considering various slicings of Schwarzschild and Kerr spacetimes. © 2005 Elsevier B.V. All rights reserved. PACS: 04.20.−q; 04.70.−s; 04.20.Ex; 04.25.Dm; 02.40.−k Keywords: General relativity; Black holes; Null hypersurfaces; 3 + 1 formalism; Numerical relativity
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 1.1. Scope of this article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 1.2. Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 1.2.1. Tensors: ‘index’ notation versus ‘intrinsic’ notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 1.2.2. Curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 1.2.3. Differential forms and exterior calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 2. Basic properties of null hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 2.1. Definition of a hypersurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 2.2. Definition of a null hypersurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 ∗ Corresponding author.
E-mail addresses:
[email protected] (E. Gourgoulhon),
[email protected] (J.L. Jaramillo). 0370-1573/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2005.10.005
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E. Gourgoulhon, J.L. Jaramillo / Physics Reports 423 (2006) 159 – 294 2.3. Auxiliary null foliation in the vicinity of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 2.4. Frobenius identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 2.5. Generators of H and non-affinity coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 2.6. Weingarten map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 2.7. Second fundamental form of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 3 + 1 formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 3.2. Spacetime foliation t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 3.3. Weingarten map and extrinsic curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 3.4. 3 + 1 coordinates and shift vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 3.5. 3 + 1 decomposition of the Riemann tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 3.6. 3 + 1 Einstein equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 3.7. Initial data problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 3 + 1-induced foliation of null hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.2. 3 + 1-induced foliation of H and normalization of l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.3. Unit spatial normal to St . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.4. Induced metric on St . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.5. Ingoing null vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 4.6. Newman–Penrose null tetrad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 4.6.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 4.6.2. Weyl scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 4.7. Projector onto H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 4.8. Coordinate systems stationary with respect to H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Null geometry in four-dimensional version. Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.1. Four-dimensional extensions of the Weingarten map and the second fundamental form of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5.2. Expression of ∇l: rotation 1-form and Há´"iˇcek 1-form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 5.3. Frobenius identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.4. Another expression of the rotation 1-form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.5. Deformation rate of the 2-surfaces St . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.6. Expansion scalar and shear tensor of the 2-surfaces St . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.7. Transversal deformation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 5.8. Behavior under rescaling of the null normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Dynamics of null hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.1. Null Codazzi equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.2. Null Raychaudhuri equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 6.3. Damour–Navier–Stokes equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.4. Tidal-force equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.5. Evolution of the transversal deformation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Non-expanding horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.1. Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 7.1.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 7.1.2. Link with trapped surfaces and apparent horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 7.1.3. Vanishing of the second fundamental form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 7.2. Induced affine connection on H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 7.3. Damour–Navier–Stokes equation in NEHs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 7.4. Evolution of the transversal deformation rate in NEHs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 7.5. Weingarten map and rotation 1-form on a NEH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 7.6. Rotation 2-form and Weyl tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.6.1. The rotation 2-form as an invariant on H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.6.2. Expression of the rotation 2-form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.6.3. Other components of the Weyl tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 7.7. NEH-constraints and free data on a NEH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.7.1. Constraints of the NEH structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.7.2. Reconstruction of H from data on St . Free data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 7.7.3. Evolution of ∇ˆ from an intrinsic null perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Isolated horizons I: weakly isolated horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 8.2. Basic properties of weakly isolated horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 8.2.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 8.2.2. Link with the 3 + 1 slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 8.2.3. WIH-symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
E. Gourgoulhon, J.L. Jaramillo / Physics Reports 423 (2006) 159 – 294 8.3. 8.4. 8.5. 8.6.
161
Initial (free) data of a WIH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Preferred WIH class [l] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Good slicings of a non-extremal WIH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Physical parameters of the horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 8.6.1. Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 8.6.2. Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 8.6.3. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 9. Isolated horizons II: (strongly) isolated horizons and further developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 9.1. Strongly isolated horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 9.1.1. General comments on the IH structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 9.1.2. Multipole moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 9.2. 3 + 1 slicing and the hierarchy of isolated horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 9.3. Departure from equilibrium: dynamical horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 10. Expressions in terms of the 3 + 1 fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 10.2. 3 + 1 decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 10.2.1. 3 + 1 expression of H’s fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 10.2.2. 3 + 1 expression of physical parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 10.3. 2+1 decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 10.3.1. Extrinsic curvature of the surfaces St . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 10.3.2. Expressions of and in terms of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 10.4. Conformal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 10.4.1. Conformal 3-metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 10.4.2. Conformal decomposition of K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 10.4.3. Conformal geometry of the 2-surfaces St . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 10.4.4. Conformal 2 + 1 decomposition of the shift vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 10.4.5. Conformal 2 + 1 decomposition of H’s fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 10.4.6. Conformal 2 + 1 expressions for , and viewed as deformation rates of St ’s metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 11. Applications to the initial data and slow evolution problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 11.1. Conformal decomposition of the constraint equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 11.1.1. Lichnerowicz–York equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 11.1.2. Conformal thin sandwich equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 11.2. Boundary conditions on a NEH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 11.2.1. Vanishing of the expansion: = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 11.2.2. Vanishing of the shear: ab = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 11.3. Boundary conditions on a WIH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 11.3.1. WIH-compatible slicing: = const. Evolution equation for the lapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 11.3.2. Preferred WIH class: Ll (k) = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 11.3.3. Fixing the slicing: 2 D · = h. Dirichlet boundary condition for the lapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 11.3.4. General remarks on the WIH boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 11.4. Other possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 12. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Appendix A. Flow of time: various Lie derivatives along l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 A.1. Lie derivative along l within H: HLl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 A.2. Lie derivative along l within St : S Ll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 Appendix B. Cartan’s structure equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 B.1. Tetrad and connection 1-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 B.2. Cartan’s first structure equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 B.3. Cartan’s second structure equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 B.4. Ricci tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Appendix C. Physical parameters and Hamiltonian techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 C.1. Well-posedness of the variational problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 C.1.1. Phase space and canonical transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 C.2. Applications of examples (C.1) and (C.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 C.2.1. Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 C.2.2. Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Appendix D. Illustration with the event horizon of a Kerr black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 D.1. Kerr coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 D.2. 3 + 1 quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 D.3. Unit normal to St and null normal to H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 D.4. 3 + 1 evaluation of the surface gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
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D.5. 3 + 1 evaluation of the Há´"iˇcek 1-form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 D.6. 3 + 1 evaluation of and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Appendix E. Symbol summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
1. Introduction 1.1. Scope of this article Black holes are currently the subject of intense research, both from the observational and theoretical points of view. Numerous observations of black holes in X-ray binaries and in the center of most galaxies, including ours, have firmly established black holes as ‘standard’ objects in the astronomical field [121,142,126]. Moreover, black holes are one of the main targets of gravitational wave observatories which are currently starting to acquire data: LIGO [75], VIRGO [2], GEO600 [153], and TAMA [6], or are scheduled for the next decade: advanced ground-based interferometers and the space antenna LISA [115]. These vigorous observational activities constitute one of the main motivations for new theoretical developments on black holes, ranging from new quasi-local formalisms [18,31] to numerical relativity [3,25], going through perturbative techniques [140] and post-Newtonian ones [28]. In particular, special emphasis is devoted to the computation of the merger of inspiralling binary black holes, a not yet solved cornerstone which is par excellence the 2-body problem of general relativity, and constitutes one of the most promising sources for the interferometric gravitational wave detectors [25]. In this article, we concentrate on the geometrical description of the black hole horizon as a null hypersurface embedded in spacetime, mainly aiming at numerical relativity applications. Let us recall that a null hypersurface is a 3-dimensional surface ruled by null geodesics (i.e. light rays), like the light cone in Minkowski spacetime, and that it is always a “one-way membrane”: it divides locally spacetime in two regions, A and B let’s say, such that any future directed causal (i.e. null or timelike) curve can move from region A to region B, but not in the reverse way. Regarding black holes, null hypersurfaces are relevant in two contexts. Firstly, wherever it is smooth, the black hole event horizon is a null hypersurface of spacetime1 [39,40,48]. Let us stress that the event horizon constitutes an intrinsically global concept, in the sense that its definition requires the knowledge of the whole spacetime (to determine whether null geodesics can reach null infinity). Secondly, a systematic attempt to provide a quasi-local description2 of black holes has been initiated in the recent years by Hayward [93,94] (concept of future trapping horizons) and Ashtekar and collaborators [10–17] — see Refs. [18,31] for a review — (concepts of isolated and dynamical horizons). Restricted to the quasi-equilibrium case (isolated horizons), the quasi-local description amounts to model the black hole horizon by a null hypersurface. This line of research finds its motivations and subsequent applications in a variety of fields of gravitational physics such as black hole mechanics, mathematical relativity, quantum gravity and, due to its quasi-local character, numerical relativity [65,108,99,58,19]. The geometry of a null hypersurface H is usually described in terms of objects that are intrinsic to H. At least some of them admit no natural extension outside the hypersurface H. In the case of the isolated horizon formalism this leads, in a natural way, to a discussion which is eminently intrinsic to H in a twofold manner. On one hand, the derived expressions are generically valid only on H without canonical extensions to a neighborhood of the surrounding spacetime. On the other hand, since in this setting the hypersurface H can be seen as representing the history of a spacelike 2-sphere (a world-tube in spacetime), the study of H’s geometry from a strictly intrinsic point of view leads to a strategy in which one firstly discuss evolution concepts, and then one considers the initial conditions on the 2-sphere which are compatible with such an evolution. We may call this an up-down strategy. On the contrary, the dynamics of black hole spacetimes is mostly studied within the 3+1 formalism (see e.g. [25,171] for a review), which amounts in the foliation of spacetime by a family of spacelike hypersurfaces. In this case, one deals with a Cauchy problem, starting from some initial spacelike hypersurface 0 and evolving it in order to construct the proper spacetime objects. In particular, this applies to the construction of the horizon H as a worldtube. We may call this a down-up strategy. 1 More generally the event horizon is an achronal set [90]. 2 By quasi-local we mean an analysis restricted to a submanifold of spacetime (typically a three-dimensional hypersurface with compact sections, but also a single compact two-dimensional surface).
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In this article, we analyze the dynamics of null hypersurfaces from the 3 + 1 point of view. As a methodological strategy, we adopt a complete four-dimensional description, even when considering objects which are actually intrinsic to a given (hyper)surface. This facilitates the link between the null horizon hypersurface and the spatial hypersurfaces of the 3 + 1 slicing. Moreover, the 3 + 1 foliation of spacetime induces an additional structure on H which allows to normalize unambiguously the null normal to H and to define a projector onto H. Let us recall that a distinctive feature of null hypersurfaces is the lack of such canonical constructions, contrary to the spacelike or timelike case where one can unambiguously define the unit normal vector and the orthogonal projector onto the hypersurface. Null hypersurfaces have been extensively studied in the literature in connection with black hole horizons, from many different points of view. In the seventies, Há´"iˇcek conducted geometrical studies of non-expanding null hypersurfaces to model stationary black hole horizons [82–84]. By studying the response of the (null) event horizon to external perturbations, Hawking and Hartle [91,85,86] introduced the concept of black hole viscosity. This hydrodynamical analogy was extended by Damour [59–61]. Electromagnetic aspects were studied by Damour [59–61] and Znajek [175,176]. These studies led to the famous membrane paradigm for the description of black holes ([141,165] and references therein). In particular, this paradigm represents the first systematic 3 + 1 approach to black hole physics. Whereas these studies all dealt with the event horizon (the global aspect of a black hole), a quasi-local approach, based on the notion of trapped surface [132], has been initiated in the nineties by Hayward [93,94], in the framework of the 2 + 2 formalism. Closely related to these ideas, a systematic quasi-local treatment has been developed these last years by Ashtekar and collaborators [10–17] (see Refs. [18,31] for a review), giving rise to the notion of isolated horizons and more recently to that of dynamical horizons, the latter not being constructed on a null hypersurface, but on a spacelike one. One purpose of this article is to fill the gap existing between the mathematical techniques used in null geometry and the standard expertise in the numerical relativity community. Consequently, an important effort will be devoted to the derivation of explicit expressions of null-geometry quantities in terms of 3 + 1 objects. More generally, the article is relatively self-contained, and requires only an elementary knowledge of differential geometry, at the level of introductory textbooks in general relativity [90,123,167]. We have tried to be quite pedagogical, by providing concrete examples and detailed derivations of the main results. In fact, these explicit developments permit to access directly to intermediate steps, which might be useful in actual numerical implementations. We rederive the basic properties of null hypersurfaces, taking advantage of our 3 + 1 perspective, namely the unambiguous definition of the null normal and transverse projector provided by the 3 + 1 spacelike slicing. Therefore the present article should not be considered as a substitute for comprehensive formal presentations of the intrinsic geometry of null surfaces, as Refs. [73,72,109,101,103]. Likewise, it is not the aim here to review the isolated horizon formalism and its applications, something already carried out in a full extent in Ref. [18]. Despite the length of the article, some important topics are not treated here, namely electromagnetic properties of black holes or black hole thermodynamics. In particular, we will not develop the Hamiltonian description of black hole mechanics in the isolated horizon scheme, except for the minimum required to discuss the physical parameters associated with the black hole. We do not comment either on the application of the isolated horizon framework beyond Einstein–Maxwell theory to include, e.g. Yang–Mills fields. Even though these fields are not expected to be relevant in an astrophysical setting, their inclusion involves a major conceptual and structural interest; we refer the reader to Chapter 6 in Ref. [18] for a review on the achievements in this line of research, namely on the mass of solitonic solutions. The plan of the article is as follows. After setting the notations in the next subsection, we start by reviewing the basic properties of null hypersurfaces in Section 2. Then the spacelike slicing of the 3 + 1 formalism widely used in numerical relativity is introduced in Section 3. The additional structures induced by this slicing on a given null hypersurface H are discussed in Section 4; in particular, this involves a privileged null normal, a null transverse vector and the associated projector onto H. Equipped with these tools, we proceed in Section 5 to describe the kinematics of null hypersurfaces, namely relations involving the first “time” derivative of their degenerate metric. The next logical step corresponds to dynamics, namely the second order derivatives of the metric, which is explored in Section 6. The Einstein equation naturally enters the scene at this level. In particular, we recover in Section 6 previous results from the membrane paradigm, like Damour’s Navier–Stokes equation or the tidal-force equation. Then in Section 7 we move to the quasi-local approach of black holes by restricting to null hypersurfaces with vanishing expansion, which are the “perfect horizons” of Há´"iˇcek and constitute the first step in Ashtekar et al. hierarchy leading to isolated horizons. The next levels in the hierarchy are studied in Sections 8 and 9, where we discuss the weakly and strongly isolated horizon structures. Due to the extension of the material, these two sections rely more explicitly on the existing literature and, as
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a consequence, the intrinsic point of view of the geometry of H (the up-down strategy referred above) acquires there a more important role than in the rest of the article. In Section 10, we express basic objects of null geometry in terms of the 3 + 1 quantities, including the standard conformal decompositions of 3 + 1 objects. This allows to translate in Section 11 the isolated horizon prescriptions into boundary conditions for the relevant 3 + 1 fields on some excised sphere, making the link with numerical relativity. Some technical details are treated in appendices: the relationship between different derivatives along the null normal is given Appendix A; Appendix B is devoted to the complete computation of the spacetime Riemann tensor. In contrast with some works on null hypersurfaces, we do not make use of the Newman–Penrose formalism but rely instead on Cartan’s structure equations. Appendix C briefly presents, with the aid of examples, the basics of the Hamiltonian description. Appendix D provides the concrete example of the horizon of a Kerr black hole, while simpler examples, based on Minkowski or Schwarzschild spacetimes, are provided throughout the main text. Finally Appendix E gathers the different symbols used throughout the article. 1.2. Notations and conventions For the benefit of the reader, we give here a somewhat detailed exposure of the notations used throughout the article. This is also the occasion to recall some concepts from elementary differential geometry employed here. We consider a spacetime (M, g) where M is a real smooth (i.e. C∞ ) manifold of dimension 4 and g a Lorentzian metric on M, of signature (−, +, +, +). We denote by ∇ the affine connection associated with g, and call it the spacetime connection to distinguish it from other connections introduced in the text. At a given point p ∈ M, we denote by Tp (M) the tangent space, i.e. the (4-dimensional) space of vectors at p. Its dual space (also called cotangent space) is denoted by Tp∗ (M) and is constituted by all linear forms at p. We denote by T(M) (resp. T∗ (M)) the space of smooth vector fields (resp. 1-forms) on M. The experienced reader is warned that T(M) does not stand for the tangent bundle of M (it rather corresponds to the space of smooth cross-sections of that bundle). No confusion may arise since we shall not use the notion of bundle in this article. 1.2.1. Tensors: ‘index’ notation versus ‘intrinsic’ notation Since we will manipulate geometrical quantities which are not well suited to the index notation (like Lie derivatives or exterior derivatives), we will use quite often an index-free notation. When dealing with indices, we adopt the following conventions: all Greek indices run in {0, 1, 2, 3}. We will use letters from the beginning of the alphabet (, , , . . .) for free indices, and letters starting from ( , , , . . .) as dumb indices for contraction (in this way the tensorial degree (valence) of any equation is immediately apparent). All capital Latin indices (A, B, C, . . .) run in {0, 2, 3} and lower case Latin indices starting from the letter i (i, j, k, . . .) run in {1, 2, 3}, while those starting from the beginning of the alphabet (a, b, c, . . .) run in {2, 3} only. For the sake of clarity, let us recall that if (e ) is a vector basis of the tangent space Tp (M) and (e ) is the associate dual basis, i.e. the basis of Tp∗ (M) such that e (e ) = , the components T 1 ...p ... of a tensor T of type 1 q with respect to the bases (e ) and (e ) are given by the expansion T = T 1 ...p ... e1 ⊗ · · · ⊗ ep ⊗ e1 ⊗ · · · ⊗ eq . 1 q
p q
(1.1)
The components ∇ T 1 ...p ... of the covariant derivative ∇T are defined by the expansion 1 q ∇T = ∇ T 1 ...p ... e1 ⊗ · · · ⊗ ep ⊗ e1 ⊗ · · · ⊗ eq ⊗ e . 1 q
(1.2)
Note the position of the “derivative index” : e is the last 1-form of the tensorial product on the right-hand side. In this respect, the notation T 1 ...p ... ; instead of ∇ T 1 ...p ... would have been more appropriate. This index 1 q 1 q convention agrees with that of MTW [123] [cf. their Eq. (10.17)]. As a result, the covariant derivative of the tensor T along any vector field u is related to ∇T by ∇u T = ∇T( ., . . . , . , u) . p+q slots
The components of ∇u T are then u ∇ T 1 ...p ... . 1 q
(1.3)
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Given a vector field v on M, the infinitesimal change of any tensor field T along the flow of v, is given by the Lie derivative of T with respect to v, denoted by Lv T and whose components are 1 ...p
(Lv T )
1 ...q =v
1 ...p
∇ (Lv T )
1 ...q −
p
T
1 ... ...p
1 ...q ∇ v
i
i=1
+
q
T 1 ...p ... ... ∇i v , 1 q
(1.4)
i=1
where the connection ∇ can be substituted by any other torsion-free connection. Actually let us recall that the Lie derivative depends only upon the differentiable structure of the manifold M and not upon the metric g nor a particular affine connection. In this article, extensive use will be made of expression (1.4), as well as of its straightforward analogues on submanifolds of M (see also Appendix A). We denote the scalar product of two vectors with respect to the metric g by a dot: ∀(u, v) ∈ Tp (M) × Tp (M),
u · v := g(u, v) .
(1.5)
We also use a dot for the contraction of two tensors A and B on the last index of A and the first index of B (provided of course that these indices are of opposite types). For instance if A is a bilinear form and B a vector, A · B is the linear form which components are (A · B) = A B .
(1.6)
However, to denote the action of linear forms on vectors, we will use brackets instead of a dot: ∀(, v) ∈ Tp∗ (M) × Tp (M),
, v = · v = v .
(1.7)
Given a 1-form and a vector field u, the directional covariant derivative ∇u is a 1-form and we have [combining the notations (1.7) and (1.3)] ∀(, u, v) ∈ T∗ (M) × T(M) × T(M),
∇u , v = ∇(v, u) .
(1.8)
Again, notice the ordering in the arguments of the bilinear form ∇. Taking the risk of insisting outrageously, let us stress that this is equivalent to say that the components (∇) of ∇ with respect to a given basis (e ⊗ e ) of T∗ (M) ⊗ T∗ (M) are ∇ : ∇ = ∇ e ⊗ e ,
(1.9)
this relation constituting a particular case of Eq. (1.2). The metric g induces an isomorphism between Tp (M) (vectors) and Tp∗ (M) (linear forms) which, in the index notation, corresponds to the lowering or raising of the index by contraction with g or g . In the present article, an index-free symbol will always denote a tensor with a fixed covariance type (e.g. a vector, a 1-form, a bilinear form, etc. . .). We will therefore use a different symbol to denote its image under the metric isomorphism. In particular, we denote by an underbar the isomorphism Tp (M) → Tp∗ (M) and by an arrow the reverse isomorphism Tp∗ (M) → Tp (M): (1) for any vector u in Tp (M), u stands for the unique linear form such that ∀v ∈ Tp (M),
u, v = g(u, v) .
(1.10)
However, we will omit the underlining on the components of u, since the position of the index allows to distinguish between vectors and linear forms, following the standard usage: if the components of u in a given basis (e ) are denoted by u , the components of u in the dual basis (e ) are then denoted by u [in agreement with Eq. (1.1)].
stands for the unique vector of Tp (M) such that (2) for any linear form in Tp∗ (M), ∀v ∈ Tp (M),
g(,
v) = , v .
As for the underbar, we will omit the arrow over the components of
by denoting them .
(1.11)
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(3) we extend the arrow notation to bilinear forms on Tp (M): for any bilinear form T : Tp (M) × Tp (M) → R, we
the (unique) endomorphism Tp (M) → Tp (M) which satisfies denote by T ∀(u, v) ∈ Tp (M) × Tp (M),
T(u, v) = u · T(v) .
(1.12)
with If T are the components of the bilinear form T in some basis e ⊗ e , the matrix of the endomorphism T respect to the vector basis e (dual to e ) is T . 1.2.2. Curvature tensor We follow the MTW convention [123] and define the Riemann curvature tensor of the spacetime connection ∇ by Riem : T∗ (M) × T(M)3 −→ C∞ (M, R) (, w, u, v) −→ , ∇u ∇v w − ∇v ∇u w − ∇[u,v] w ,
(1.13)
where C∞ (M, R) denotes the space of smooth scalar fields on M. As it is well known, the above formula does define a tensor field on M, i.e. the value of Riem(, w, u, v) at a given point p ∈ M depends only upon the values of the fields , w, u and v at p and not upon their behaviors away from p, as the gradients in Eq. (1.13) might suggest. We denote the components of this tensor in a given basis (e ), not by Riem , but by R . The definition (1.13) leads then to the following writing (called Ricci identity): ∀w ∈ T(M),
(∇ ∇ − ∇ ∇ )w = R w .
(1.14)
From the definition (1.13), the Riemann tensor is clearly antisymmetric with respect to its last two arguments (u, v). The fact that the connection ∇ is associated with a metric (i.e. g) implies the additional well-known antisymmetry: ∀(, w) ∈ T∗ (M) × T(M), Riem(, w, ·, ·) = −Riem(w, ,
·, ·) .
(1.15)
In addition, the Riemann tensor satisfies the cyclic property ∀(u, v, w) ∈ T(M)3 , Riem(·, u, v, w) + Riem(·, w, u, v) + Riem(·, v, w, u) = 0 .
(1.16)
The Ricci tensor of the spacetime connection ∇ is the bilinear form R defined by R : T(M) × T(M) −→ C∞ (M, R) (u, v) −→ Riem(e , u, e , v) .
(1.17)
This definition is independent of the choice of the basis (e ) and its dual counterpart (e ). Moreover the bilinear form R is symmetric. In terms of components: R = R .
(1.18)
Note that, following the standard usage, we are denoting the components of both the Riemann and Ricci tensors by the same letter R, the number of indices allowing to distinguish between the two tensors. On the contrary we are using different symbols, Riem and R, when dealing with the ‘intrinsic’ notation. Finally, the Riemann tensor can be split into (i) a “trace-trace” part, represented by the Ricci scalar R := g R , (ii) a “trace” part, represented by the Ricci tensor R [cf. Eq. (1.18)], and (iii) a “traceless” part, which is constituted by the Weyl conformal curvature tensor, C: R = C + 21 (R g − R g + R − R ) + 16 R(g − g ) .
(1.19)
The above relation can be taken as the definition of C. It implies that C is traceless: C = 0 .
(1.20)
The other possible traces are zero thanks to the symmetry properties of the Riemann tensor. It is well known that the 20 independent components of the Riemann tensor distribute in the 10 components in the Ricci tensor, that are fixed by Einstein equation, and 10 independent components in the Weyl tensor.
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1.2.3. Differential forms and exterior calculus In this article, we will make use of p-forms, mostly 1-forms and 2-forms. Let us recall that a p-form is a type p0 tensor field which is antisymmetric with respect to all its p arguments. In other words, it is a multilinear form field T(M) × · · · × T(M) −→ C∞ (M, R) which is fully antisymmetric. We follow the convention of MTW [123], Wald [167], and Straumann [157] textbooks for the exterior product (wedge product) between p-forms: if and are two 1-forms [i.e. two elements of T∗ (M)], ∧ is the 2-form defined by ∧ := ⊗ − ⊗ .
(1.21)
Note that this definition disagrees with that of Hawking and Ellis [90], which would require a factor 1/2 in front of the r.h.s. of (1.21) [cf. the equation on p. 21 of Ref. [90], and Ref. [38] for a discussion]. The exterior derivative of a differential form is defined by induction starting from df being the 1-form gradient of f for any scalar field (0-form) f. For any (p + q)-form that can be written as the exterior product of a p-form by a q-form , the exterior derivative is the (p + q + 1)-form defined by d( ∧ ) = d ∧ + (−1)p ∧ d .
(1.22)
This equation agrees with that in Box 4.1 of MTW [123]. It constitutes a version of Leibnitz rule altered by the factor (−1)p ; for this reason the exterior derivative is sometimes called an antiderivation (e.g. Definition 4.2 of Ref. [157]). The components of the exterior derivative of a 1-form with respect to some coordinate system (x ) on M are (d) = j − j ,
(1.23)
where the partial derivative j can be replaced by any covariant derivative operator without torsion on M (for instance the spacetime derivative ∇ ). Taking into account Eqs. (1.9) and (1.8), we can then write ∀ (u, v) ∈ T(M) × T(M),
d(u, v) = ∇(v, u) − ∇(u, v) = ∇u , v − ∇v , u .
(1.24) (1.25)
A very useful relation that we shall employ throughout the article is Cartan identity, which relates the Lie derivative of a p-form along a vector field v to the exterior derivative of : Lv = v · d + d(v · ) .
(1.26)
˜ on M (not necessarily the spacetime connection ∇ associated Given a 1-form ∈ T∗ (M) and a connection operator ∇ with the metric g), the exterior derivative d can be viewed as (minus two times) the antisymmetric part of the gradient ˜ The symmetric part is given by (half of) the Killing operator Kil(∇, ˜ .), such that Kil(∇, ˜ ) is the symmetric ∇. ∞ bilinear form T(M) × T(M) → C (M, R) defined by ˜ )(u, v) = ∇(u, ˜ ˜ Kil(∇, v) + ∇(v, u) ,
(1.27)
for any (u, v) ∈ T(M) × T(M). Combining Eqs. (1.27) and (1.24), we have the decomposition ∀ ∈ T∗ (M),
˜ = 1 [ Kil(∇, ˜ ) − d] . ∇ 2
(1.28)
˜ As stated before, the antisymmetric part, d, is independent of the choice of the connection ∇.
2. Basic properties of null hypersurfaces There is no doubt about the central role of null hypersurfaces in general relativity, and they have been extensively studied in the literature. We review here some of their elementary properties, referring the reader to Refs. [72,73,109] and [21,101–103,128] for further details or alternative approaches. Let us mention that the properties described here, as well as in the subsequent Sections 3–6, are valid for any kind of null hypersurface and do not require any link with a black hole horizon. For instance they are perfectly valid for a light cone in Minkowski spacetime.
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Fig. 1. Embedding of the 3-dimensional manifold H0 into the 4-dimensional manifold M, defining the hypersurface H = (H0 ). The push-forward ∗ v of a vector v tangent to some curve C in H0 is a vector tangent to (C) in M.
2.1. Definition of a hypersurface A hypersurface H of M is the image of a 3-dimensional manifold H0 by an embedding : H0 → M (Fig. 1): H = (H0 ) .
(2.1)
Let us recall that embedding means that : H0 → H is a homeomorphism, i.e. a one-to-one mapping such that both and −1 are continuous. The one-to-one character guarantees that H does not “intersect itself”. A hypersurface can be defined locally as the set of points for which a scalar field on M, u let us say, is constant: ∀p ∈ M,
p ∈ H ⇐⇒ u(p) = 1 .
(2.2)
For instance, let us assume that H is a connected submanifold of M with topology3 R × S2 . Then we may introduce locally a coordinate system of M, x = (t, u, , ), such that t spans R and (, ) are spherical coordinates spanning S2 . H is then defined by the coordinate condition u = 1 [Eq. (2.2)] and an explicit form of the mapping can be obtained by considering x A = (t, , ) as coordinates on the 3-manifold H0 : : H0 −→ M (t, , ) −→ (t, 1, , ) .
(2.3)
In what follows, we identify H0 and H = (H0 ) (consequently, can be seen as the inclusion map : H −→ M). The embedding “carries along” curves in H to curves in M. Consequently it also “carries along” vectors on H to vectors on M (cf. Fig. 1). In other words, it defines a push-forward mapping ∗ between Tp (H) and Tp (M). Thanks to the adapted coordinate systems x = (t, u, , ), the push-forward mapping can be explicited as follows: ∗ : Tp (H) −→ Tp (M) v = (v t , v , v ) −→ ∗ v = (v t , 0, v , v ) ,
(2.4)
where v A = (v t , v , v ) denotes the components of the vector v with respect to the natural basis j/jx A of Tp (H) associated with the coordinates (x A ). Conversely, the embedding induces a pull-back mapping ∗ between the linear forms on Tp (M) and those on Tp (H) as follows ∗ : Tp∗ (M) −→ Tp∗ (H) −→ ∗ : Tp (H) → R v → , ∗ v .
(2.5)
Taking into account (2.4), the pull-back mapping can be explicited: ∗ : Tp∗ (M) −→ Tp∗ (H) = (t , u , , ) −→ ∗ = (t , , ) ,
(2.6)
3 This is the case we will consider in Section 7 and in the subsequent ones, whereas all results up to Section 7 are independent of the topology of H.
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where denotes the components of the 1-form with respect to the basis dx associated with the coordinates (x ). The pull-back operation can be extended to the multi-linear forms on Tp (M) in an obvious way: if T is a n-linear form on Tp (M), ∗ T is the n-linear form on Tp (H) defined by ∀(v1 , . . . , vn ) ∈ Tp (H)n ,
∗ T(v1 , . . . , vn ) = T(∗ v1 , . . . , ∗ vn ) .
(2.7)
Remark 2.1. By itself, the embedding induces a mapping from vectors on H to vectors on M (push-forward mapping ∗ ) and a mapping from 1-forms on M to 1-forms on H (pull-back mapping ∗ ), but not in the reverse way. For instance, one may define “naively” a reverse mapping F : Tp (M) −→ Tp (H) by v = (v t , v u , v , v ) −→ F v = (v t , v , v ), but it would then depend on the choice of coordinates (t, u, , ), which is not the case of the pushforward mapping defined by Eq. (2.4). For spacelike or timelike hypersurfaces, the reverse mapping is unambiguously provided by the orthogonal projector (with respect to the ambient metric g) onto the hypersurface. In the case of a null hypersurface, there is no such a thing as an orthogonal projector, as we shall see below (Remark 2.3). A very important case of pull-back operation is that of the bilinear form g (i.e. the spacetime metric), which defines the induced metric on H: q := ∗ g
,
(2.8)
q is also called the first fundamental form of H. In terms of the coordinate system4 x A =(t, , ) of H, the components of q are deduced from (2.6): qAB = gAB .
(2.9)
2.2. Definition of a null hypersurface The hypersurface H is said to be null (or lightlike, or characteristic or to be a wavefront) if, and only if, the induced metric q is degenerate. This means if, and only if, there exists a non-vanishing vector field l in T(H) which is orthogonal (with respect to q) to all vector fields in T(H): ∀v ∈ T(H),
q(l, v) = 0 .
(2.10)
The signature of q is then necessarily (0, +, +). An equivalent definition of a null hypersurface demands any vector field l in T(M) which is normal to H [i.e. orthogonal to all vectors in T(H)] to be a null vector with respect to the metric g: g(l, l) = l · l = 0
.
(2.11)
We adopt the same notation l than in the previous definition, since this l is nothing but the pushed-forward by ∗ of the l in T(H). Indeed, by saying that l is orthogonal to itself, Eq. (2.11) states that l is tangent to H. A distinctive property of null hypersurfaces is that their normal vectors are both orthogonal and tangent to them. Since the hypersurface H is defined by a constant value of the scalar field u [Eq. (2.2)], the gradient 1-form du is normal to H, i.e. ∀v ∈ T(M),
v ∈ T(H) ⇐⇒ du, v = 0 .
(2.12)
As a side remark notice that, in terms of the components v of v with respect to the natural basis associated with the coordinates (x ), du, v = v u and the above property is equivalent to ∀v ∈ T(M),
v ∈ T(H) ⇐⇒ v u = 0 ,
4 Let us recall that by convention capital Latin indices run in {0, 2, 3}.
(2.13)
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which agrees with (2.4). From (2.12), it is obvious that the 1-form l associated with the normal vector l by the standard metric duality [cf. notation (1.10)] must be collinear to du: l = e du
,
(2.14)
where is some scalar field on H. We have chosen the coefficient relating l and du to be strictly positive, i.e. under the form of an exponential. This is always possible by a suitable choice of the scalar field u. The characterization of Tp (H) as a hyperplane of the vector space Tp (M) can then be expressed as follows: ∀v ∈ Tp (M),
v ∈ Tp (H) ⇐⇒ l, v = l · v = 0
.
(2.15)
Remark 2.2. Since the scalar square of l is zero [Eq. (2.11)], there is no natural normalization of l, contrary to the case of spacelike hypersurfaces, where one can always choose the normal to be a unit vector (scalar square equal to −1). Equivalently, there is no natural choice of the factor in relation (2.14). In Section 4, we will use the extra-structure introduced in M by the spacelike foliation of the 3 + 1 formalism to set unambiguously the normalization of l. Remark 2.3. Another distinctive feature of null hypersurfaces, with respect to spacelike or timelike ones, is the absence of orthogonal projector onto them. This is a direct consequence of the fact that the normal l is tangent to H. Indeed, suppose we define “naively” := 1+all, . (or in index notation : := +a ) as the “orthogonal projector” with some coefficient a to be determined (a = 1 for a spacelike hypersurface and a = −1 for a timelike hypersurface, if l is the unit normal). Then it is true that for any v ∈ Tp (H), (v) = v, but if v ∈ / Tp (H), l · (v) = l · v = 0, which shows that (v) ∈ / Tp (H), hence the endomorphism is not a projector on Tp (H), whatever the value of a. This lack of orthogonal projector implies that there is no canonical way, from the null structure alone, to define a mapping Tp (M) −→ Tp (H) (cf. Remark 2.1). 2.3. Auxiliary null foliation in the vicinity of H The null normal vector field l is a priori defined only on H and not at points p ∈ / H. However within the fourdimensional point of view adopted in this article, we would like to consider l as a vector field not confined to H but defined in some open subset of M around H. In particular this would permit to define the spacetime covariant derivative ∇l, which is not possible if the support of l is restricted to H. Following Carter [43], a simple way to achieve this is to consider not only a single null hypersurface H, but a foliation of M (in the vicinity of H) by a family of null hypersurfaces, such that H is an element of this family. Without any loss of generality, we may select the scalar field u to label these hypersurfaces and denote the family by (Hu ). The null hypersurface H is then nothing but the element H = Hu=1 of this family [Eq. (2.2)]. The vector field l can then be viewed as defined in the part of M foliated by (Hu ), such that at each point in this region, l is null and normal to Hu for some value of u. The identity (2.14) is then valid for this “extended” l, and is now a scalar field defined not only on H but in the open region of M around H which is foliated by (Hu ). Obviously the family (Hu ) is non-unique but all geometrical quantities that we shall introduce hereafter do not depend upon the choice of the foliation Hu once they are evaluated at H. 2.4. Frobenius identity The identity (2.14) which expresses that the 1-form l is normal to a hypersurface u = const, leads to a particular form for the exterior derivative of l. Indeed, taking the exterior derivative of (2.14) (considering l defined in a open neighborhood of H in M, cf. Section 2.3) and applying rule (1.22) (with e = 0-form) leads to dl = e d ∧ du + e ddu .
(2.16)
Since dd = 0 is a basic property of the exterior derivative, the last term on the right-hand side of (2.16) vanishes [this is also obvious by applying Eq. (1.23) to the 1-form du]. Hence, after replacing du by e− l, one is left with dl = d ∧ l
.
(2.17)
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This reflects the Frobenius theorem in its dual formulation (see e.g. Theorem B.3.2 in Wald’s textbook [167]): the exterior derivative of the 1-form l is the exterior product of l itself with some 1-form (d in the present case) if, and only if, l defines hyperplanes of T(M) [by Eq. (2.15)] which are integrable in some hypersurface (H in the present case). 2.5. Generators of H and non-affinity coefficient Let us establish a fundamental property of null hypersurfaces: they are ruled by null geodesics. Contracting Eq. (2.17) with l and using the fact that l is null, gives l · dl = d , ll − l, l d = d , ll .
(2.18)
=0
On the other side if we express the exterior derivative dl in terms of the covariant derivative ∇ associated with the spacetime metric g, the left-hand side of the above equation becomes (l · dl) = ∇ − ∇ = ∇ = (∇l l) , where we have used
∇
= 1/2∇
(
) = 0.
(2.19)
Hence Eq. (2.18) leads to
∇l l = d , ll ,
(2.20)
or by the metric duality between 1-forms and vectors: ∇l l = l
,
(2.21)
where is the scalar field defined on H by := ∇l = d , l
.
(2.22)
In the case where H is the horizon of a Kerr black hole, l can be normalized to become a Killing vector of (M, g), of the form l = 0 + H 1 , where H = const and 0 and 1 are the Killing vectors associated with respectively the stationarity and axisymmetry of Kerr spacetime and normalized so that the parameter length of 1 ’s orbits is 2 and 0 asymptotically coincides with the 4-velocity of an inertial observer. is then called the surface gravity of the black hole (see Appendix D for further details). Since Eq. (2.21) involves only the derivative of l along l, i.e. within H, the definition of is intrinsic to (H, l) and does not depend upon the choice of the auxiliary null foliation (Hu ). Eq. (2.21) means that l remains collinear to itself when it is parallely transported along its field lines. This implies that these field lines are spacetime geodesics. Indeed, by a suitable choice of the renormalization factor such that l = l, Eq. (2.21) can be brought to the classical “equation of geodesics” form: ∇l l = 0 .
(2.23)
This is immediate since ∇l l = [∇l l + (∇l )l] = 2 ( + ∇l ln )l
(2.24)
and one can choose to get Eq. (2.23) by requiring it to be a solution of the following first order differential equation along the field lines of l, ∇l ln = − .
(2.25)
If = 0, Eq. (2.21) means that the parameter associated with l by = dx /d is not an affine parameter of the geodesics. For this reason, we may call the non-affinity coefficient. Note that (2.24) gives the following scaling law for : l → l = l
⇒
→ = ( + ∇l ln ) .
(2.26)
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Fig. 2. Null hypersurface H with some null normal l and the null generators (thin lines).
Fig. 3. Outgoing light cone in Minkowski spacetime. The null hypersurface H under consideration is the member u = 1 of the family (Hu ) of light cones emitted from the origin (x, y, z) = (0, 0, 0) at successive times t = 1 − u.
Having established that the field lines of l are geodesics, it is obvious that they are null geodesics (for l is null). They are called the null generators of H. Note that whereas l is not uniquely defined, being subject to the rescaling law l → l = l, the null generators, considered as one-dimensional curves in M, are unique (see Fig. 2). In other words, they depend only upon H. Example 2.4 (Outgoing light cone in Minkowski spacetime). The simplest example of a null hypersurface one may think of is the light cone in Minkowski spacetime (Fig. 3). More precisely, let us consider for H the outgoing light cone from a given point O, excluding O itself to keep H smooth. If (x ) = (t, x, y, z) denote standard Minkowskian coordinates with origin O, the scalar field u defining H as the level set u = 1 is then u(t, x, y, z) := r − t + 1
with r :=
x 2 + y 2 + z2 .
(2.27)
Note that u generates not only H, but a full null foliation (Hu ) as the level sets of u (cf. Section 2.3). The member Hu of this foliation is then nothing but the light cone emanating from the point (−u + 1, 0, 0, 0) (cf. Fig. 3). In terms of components with respect to the coordinates (x ), the gradient 1-form du is ∇ u = (−1, x/r, y/r, z/r). Hence, from
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Eq. (2.14), the null normal to H is = e (1, x/r, y/r, z/r). For simplicity, let us select = 0. Then x y z x y z = 1, , , and = −1, , , . r r r r r r
173
(2.28)
The gradient bilinear form ∇l is easily computed since, for the coordinates (t, x, y, z), ∇ = j /jx : ⎞ 0 0 0 2 +z xy xz ⎜0 − 3 − 3 ⎟ ⎟ ⎜ r3 r r ⎟ ( = row index ; ⎜ 2 2 ⎜ x +z xy yz ⎟ ∇ = ⎜ − 3 − 3 ⎟ ⎟ = column index) . ⎜0 3 r r r ⎠ ⎝ xz yz x2 + y2 0 − 3 − 3 r r r3 We may check immediately on this formula that ∇ = 0, which leads to ⎛0
=0 ,
y2
(2.29)
(2.30)
in accordance with = ∇l [Eq. (2.22)] and our choice = 0. Actually it is easy to check that the coordinate t is an affine parameter of the null geodesics generating H and that l is the associated tangent vector, hence the vanishing of the non-affinity coefficient . Example 2.5 (Schwarzschild horizon in Eddington–Finkelstein coordinates). The next example of null surface one might think of is the (future) event horizon of a Schwarzschild black hole. The corresponding spacetime is often (partially) described by two sets of coordinates: (i) the Schwarzschild coordinates (tS , r, , ), in which the metric components are given by 2m 2m −1 2 dr + r 2 (d2 + sin2 d2 ) , (2.31) dtS2 + 1 − g dx dx = − 1 − r r where m is the mass of the black hole, and (ii) the isotropic coordinates (tS , r˜ , , ), resulting in the metric components ⎛ m ⎞2 1− ⎜ 2˜r ⎟ dt 2 + 1 + m 4 dr˜ 2 + r˜ 2 (d2 + sin2 d2 ) . (2.32) g dx dx = −⎝ ⎠ S m 2˜r 1+ 2˜r m 2 The relation between the two sets of coordinates is given by r = r˜ (1+ 2˜ r ) . As it is well known, the above two coordinate systems are singular at the event horizon H, which corresponds to r = 2m, r˜ = m/2 and tS → +∞. In particular the hypersurfaces of constant time tS , which constitute a well known example of maximal slicing (cf. Section 3), do not intersect H, except at a 2-sphere (named the bifurcation sphere), where they also cross each other (this is illustrated by the Kruskal diagram in Fig. 4). A coordinate system, well known for being regular at H, is constructed with the ingoing Eddington–Finkelstein coordinates (V , r, , ), where the coordinate V is constant on each ingoing radial null geodesic and is related to the r Schwarzschild coordinate time tS by V = tS + r + 2m ln | 2m − 1|. The coordinate V is null, but if we introduce r t := V − r = tS + 2m ln − 1 , (2.33) 2m
we get a timelike coordinate. The system (t, r, , ) is called the 3 + 1 Eddington–Finkelstein coordinates. These coordinates are well behaved in the vicinity of H, as shown in Fig. 4, and yields the following metric components: 2m 4m 2m g dx dx = − 1 − (2.34) dt 2 + dt dr + 1 + dr 2 + r 2 (d2 + sin2 d2 ) . r r r It is clear on this expression that the 3 + 1 Eddington–Finkelstein coordinates are regular at the event horizon H, which is located at r = 2m (cf. Fig. 5). However, we cannot use u = r − 2m + 1 for the scalar field defining H, because the
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Fig. 4. Kruskal diagram representing the Schwarzschild spacetime; the hypersurfaces of constant Schwarzschild time tS (dashed lines) do not intersect the future event horizon H, except at the bifurcation 2-sphere B (reduced to a point in the figure), whereas the hypersurfaces of constant Eddington–Finkelstein time t (solid lines) intersect it in such a way that t can be used as a regular coordinate on H (figure adapted from Fig. 3.1 of Ref. [161]).
Fig. 5. Event horizon H of a Schwarzschild black hole in 3 + 1 Eddington–Finkelstein coordinates. The dashed lines represents the hypersurfaces of constant Schwarzschild time tS shown in Fig. 4.
hypersurfaces r = const are not null, except for r = 2m, whereas we have required in Section 2.3 all the hypersurfaces u = const to be null. Actually, a family of null hypersurfaces encompassing H is given by the constant values of the outgoing Eddington–Finkelstein coordinate r r U = tS − r − 2m ln − 1 = t − r − 4m ln − 1 . 2m 2m
(2.35)
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The event horizon corresponds to U =+∞; to get finite values, let us replace U by the null Kruskal–Szekeres coordinate (shifted by 1) U u := ± exp − +1 , (2.36) 4m where the + sign (resp. − sign) is for r 2m (resp. r < 2m). Then r r −t u= − 1 exp +1 2m 4m
(2.37)
and all the hypersurfaces Hu defined by u=const are null (they are drawn in Fig. 5), the event horizon H corresponding to u = 1. The null normals to Hu are deduced from the gradient of u by = e ∇ u [Eq. (2.14)]. We get the following components with respect to the 3 + 1 Eddington–Finkelstein coordinates (x ) = (t, r, , ): r −t r r 1 = exp + 1+ , − 1, 0, 0 . (2.38) 4m 4m 2m 2m Let us choose such that t = 1. Then t −r r = − ln 1 + + ln(4m) , 4m 2m r − 2m 2m − r = 1, , 0, 0 and = , 1, 0, 0 . r + 2m r + 2m
(2.39) (2.40)
H
Note that on the horizon, =(1, 0, 0, 0), i.e. H
l=t ,
(2.41)
where t = j/jt is a Killing vector associated with the stationarity of Schwarzschild solution. The gradient of l is obtained by a straightforward computation, after having evaluated the connection coefficients from the metric components given by Eq. (2.34): ⎛ m 2m − r m(3m2 + 4mr − 3r 2 ) ⎞ 0 0 2 2 ⎜ r r + 2m ⎟ r 2 (r + 2m) ⎜ ⎟ ⎜ ⎟ m(3r + 2m) m ⎜ ⎟ 0 0 ⎜ ⎟ 2 2 r r (r + 2m) ∇ = ⎜ ⎟ ⎜ ⎟ r(r − 2m) ⎜ ⎟ 0 0 0 ⎜ ⎟ r + 2m ⎝ ⎠ (r − 2m)r sin2 0 0 0 r + 2m ( = row index; = column index) . (2.42) We deduce from these values that 4m 4m(r − 2m) ∇ = , , 0, 0 . (2.43) (r + 2m)2 (r + 2m)3 Comparing with the expression (2.40) for , we deduce the value of the non-affinity coefficient [cf. Eq. (2.21)]: =
4m (r + 2m)2
.
(2.44)
As a check, we can recover by means of formula (2.22), evaluating ∇l from expression (2.39) for . Note that on the horizon, 1 , 4m which is the standard value for the surface gravity of a Schwarzschild black hole. H
=
(2.45)
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2.6. Weingarten map As for any hypersurface, the “bending” of H in M (also called extrinsic curvature of H) is described by the Weingarten map (sometimes called the shape operator), which is the endomorphism of Tp (H) which associates with each vector tangent to H the variation of the normal along that vector, with respect to the spacetime connection ∇: : Tp (H) −→ Tp (H) v −→ ∇v l
(2.46)
This application is well defined (i.e. its image is in Tp (H)) since l · (v) = l · ∇v l = 21 ∇v (l · l) = 0 ,
(2.47)
which shows that (v) ∈ Tp (H) [cf. Eq. (2.15)]. Moreover, since it involves only the derivative of l along vectors tangent to H, the definition of is clearly independent of the choice of the auxiliary null foliation (Hu ) introduced in Section 2.3. Remark 2.6. The Weingarten map depends on the specific choice of the normal l, in contrast with the timelike or spacelike case, where the unit length of the normal fixes it unambiguously. Indeed a rescaling of l acts as follows on : l → l = l ⇒ → = + d, ·l ,
(2.48)
where the notation d, ·l stands for the endomorphism Tp (H) −→ Tp (H), v −→ d, vl. The fundamental property of the Weingarten map is to be self-adjoint with respect to the metric q [i.e. the pull-back of g on T(H), cf. Eq. (2.8)]: ∀(u, v) ∈ T(H) × T(H),
u · (v) = (u) · v
,
(2.49)
where the dot means the scalar product with respect to q [considering u and v as vectors of T(H)] or g [considering u and v as vectors of T(M)]. Indeed, one obtains from the definition of u · (v) = u · ∇v l = ∇v (u · l) − l · ∇v u = 0 − l · (∇u v − [u, v]) = − ∇u (l · v) + v · ∇u l + l · [u, v] = 0 + v · (u) + l · [u, v] ,
(2.50)
where use has been made of l · u = 0 and l · v = 0. Now the Frobenius theorem states that the commutator [u, v] of two vectors of the hyperplane T(H) belongs to T(H) since T(H) is surface-forming (see e.g. Theorem B.3.1 in Wald’s textbook [167]). It is straightforward to establish it: l · [u, v] = l, [u, v] = u ∇ v − v ∇ u = −u v ∇ + v u ∇
= (∇ − ∇ )u v = (d) u v = (∇ − ∇ )u v
l · [u, v] = 0 ,
(2.51)
where use has been made of expression (2.17) for the exterior derivative of l and the last equality results from u = 0 and v = 0. Inserting (2.51) into (2.50) establishes the self-adjointness of the Weingarten map. Let us note that the non-affinity coefficient is an eigenvalue of the Weingarten map, corresponding to the eigenvector l, since Eq. (2.21) can be written (l) = l
.
(2.52)
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2.7. Second fundamental form of H The self-adjointness of implies that the bilinear form defined on H’s tangent space by : Tp (H) × Tp (H) −→ R (u, v) −→ u · (v)
(2.53)
is symmetric. It is called the second fundamental form of H with respect to l. Note that could have been defined for any vector field l, but it is symmetric only because l is normal to some hypersurface (since the self-adjointness of originates from this last property). If we make explicit the value of in the definition (2.53), we get [see Eq. (1.8)] ∀ (u, v) ∈ Tp (H) × Tp (H),
(u, v) = u · (v) = u · ∇v l = ∇v l, u = ∇l(u, v) ,
(2.54)
from which we conclude that is nothing but the pull-back of the bilinear form ∇l onto Tp (H), pull-back induced by the embedding of H in M [cf. Eq. (2.7)]: = ∗ ∇l
.
(2.55)
It is worth to note that although the bilinear form ∇l is a priori not symmetric on Tp (M), its pull-back on Tp (H) is symmetric, as a consequence of the hypersurface-orthogonality of l (which yields the self-adjointness of ). The bilinear form is degenerate, with a degeneracy direction along l (as the first fundamental form q), since ∀v ∈ Tp (H),
(l, v) = v · (l) = v · l = 0 .
(2.56)
Remark 2.7. As for , depends on the choice of the normal l. However its transformation under a rescaling of l is simpler than that of : from Eq. (2.48) and the orthogonality of l with respect to Tp (H), we get l → l = l ⇒ → = .
(2.57)
Remark 2.8. To get rid of the dependence upon the normalization of l in the definitions of and , some authors [109,72,73,101] introduce the following equivalence class R on Tp (H): u ∼ v iff u and v differ only by a vector collinear to l. Then the Weingarten map and the second fundamental form can be defined as unique geometric objects in the quotient space Tp (H)/R. However we do not adopt such an approach here because we plan to use some spacetime slicing by spacelike hypersurfaces (the so-called 3 + 1 formalism) to fix in a natural way the normalization of l, as we shall see in Section 4. 3. 3 + 1 formalism 3.1. Introduction The 3+1 formalism of general relativity is aimed at reducing the resolution of Einstein equation to a Cauchy problem, namely (coordinate) time evolution from initial data specified on a given spacelike hypersurface. This formalism originates in the works of Lichnerowicz (1944) [113], Choquet-Bruhat (1952) [70], Arnowitt et al. (1962) [9] and has many applications, in particular in numerical relativity. We refer the reader to York’s seminal article [171] for an introduction to the 3 + 1 formalism and to Baumgarte and Shapiro [25] for a recent review of applications in numerical relativity. Here we simply recall the most relevant features of the 3 + 1 formalism which are necessary for our purpose. 3.2. Spacetime foliation t The spacetime (or at least the part of it under study, in the vicinity of the null hypersurface H) is supposed to be foliated by a continuous family of spacelike hypersurfaces (t ), labeled by the time coordinate t (Fig. 6). The t ’s
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Fig. 6. Spacetime foliation by a family of spacelike hypersurfaces t . The t ’s can be considered as the level sets of some smooth scalar field t, such that the gradient dt is timelike.
can be considered as the level sets of some smooth scalar field t, such that the gradient dt is timelike. We denote by n the future directed timelike unit vector normal to t . It can be identified with the 4-velocity of the class of observers whose worldlines are orthogonal to t (Eulerian observers). By definition the 1-form n dual to n [cf. notation (1.10)] is parallel to the gradient of the scalar field t: n = −Ndt
.
(3.1)
The proportionality factor N is called the lapse function. It ensures that n satisfies the normalization relation n · n = n, n = −1 .
(3.2)
The metric induced by g on each hypersurface t (first fundamental form of t ) is given by =g+n⊗n
.
(3.3)
Since t is assumed to be spacelike, is a positive definite (i.e. Riemannian) metric. Let us stress that the writing (3.3) is fully four-dimensional and does not restrict the definition of to Tp (t ): it is a bilinear form on Tp (M). The endomorphism Tp (M) → Tp (M) canonically associated with the bilinear form by the metric g [cf. notation (1.12)] is the orthogonal projector onto t :
= 1 + n, .n
(3.4)
(in index notation: = + n n , whereas (3.3) writes = g + n n ). The existence of the orthogonal projector makes a great difference with the case of null hypersurfaces, for which such an object does not exist (cf. Remark 2.3). In particular we can use it to map any multilinear form on Tp (t ) into a multilinear form on Tp (M), which is in the direction inverse of that of the pull-back mapping induced by the embedding of t in M. We denote this mapping Tp∗ (t ) → Tp∗ (M) by ∗ and make it explicit as follows: given a n-linear form A on Tp (t ), ∗ A is the n-linear form acting on Tp (M)n defined by
∗ A : Tp (M)n −→ R (v1 , . . . , vn ) −→ A( (v1 ), . . . , (vn ))
.
(3.5)
Actually we extend the above definition to all multilinear forms A on Tp (M) and not only those restricted to Tp (t ). The index version of this definition is ( ∗ A)1 ...n = A 1 ... n 1 1 · · · n n .
(3.6)
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In particular, we have
∗ g = and ∗ n = 0 .
(3.7)
There exists a unique (torsion-free) connection on t associated with the metric , which we denote by D: D = 0. If we consider a generic tensor field T of type pq lying on t (i.e. such that its contraction with the normal n on any of its indices vanishes), then, from a four-dimensional point of view, the covariant derivative DT can be expressed as the full orthogonal projection of the spacetime covariant derivative ∇T on t [see Eq. (1.2)]: D T 1 ...p ... = 1 1 · · · p p 1 · · · q ∇ T 1 ... p 1 ... q 1 q 1 q
.
(3.8)
In the following,we shall make extensive use of this formula, without making explicit mention. In the special case of a tensor of type q0 , i.e. a multilinear form, the definition of DT amounts to, thanks to Eq. (3.6), DT = ∗ ∇T .
(3.9)
3.3. Weingarten map and extrinsic curvature As for the hypersurface H, the “bending” of each hypersurface t in M is described by the Weingarten map which associates with each vector tangent to t the covariant derivative (with respect to the ambient connection ∇) of the unit normal n along this vector [compare with Eq. (2.46)]: K : Tp (t ) −→ Tp (t ) v −→ ∇v n .
(3.10)
The computations presented in Section 2.6 for the Weingarten map of H can be repeated here,5 by simply replacing the normal l by the normal n, the field u by the field t and the coefficient e by −N [compare Eqs. (2.14) and (3.1)]. They then show that K is well defined [i.e. its image is in Tp (t )] and that it is self-adjoint with respect to the metric . A difference with the Weingarten map of H is that the Weingarten map K can be naturally extended to Tp (M) thanks to the orthogonal projector [Eq. (3.4)], which did not exist for H (cf. Remark 2.3), by setting K : Tp (M) −→ Tp (t ) v −→ ∇ (v) n ,
(3.11)
or in index notation: K = ∇ n .
(3.12)
We then define the extrinsic curvature tensor K of the hypersurface t as minus the second fundamental form [compare with Eq. (2.53)]: K : Tp (M) × Tp (M) −→ R (u, v) −→ −u · K(v) ,
(3.13)
or in index notation K = −∇ n .
(3.14)
Since the image of K is in Tp (t ), we can write K(u, v) = − (u) · K( (v)). It follows then immediately from the self-adjointness of K that K is symmetric and that the following relation holds: K = −∇ n , 5 Indeed the computations in Section 2.6 did not make use of the fact that H is null, i.e. that l is tangent to H.
(3.15)
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which we can write, thanks to Eq. (3.6) and the symmetry of K, K = − ∗ ∇n
.
(3.16)
Replacing in Eq. (3.14) n by its expression (3.1) in terms of the gradient of t leads to K = ∇ (N∇ t) = (∇ N∇ t + N∇ ∇ t)
= (∇ N ∇ t + N ∇ ∇ t) = D N∇ t + N ∇ (−N −1 n ) = − n N −1 D N − N ∇ (N −1 ) n −∇ n =0
= − n D ln N − ∇ n − ∇ n n n .
(3.17)
=0
Hence K = −∇ n − n D ln N
.
(3.18)
or, taking into account the symmetry of K and Eq. (1.9) K = −∇n − D ln N ⊗ n
.
(3.19)
In the following, we will make extensive use of this formula, without explicitly mentioning it. Inserting n as the second argument in the bilinear form (3.19) and using K(., n) = 0 as well as n, n = −1 results in the important formula giving the 4-acceleration of the Eulerian observers: ∇n n = D ln N
.
(3.20)
Another useful formula relates K to the Lie derivative of the spatial metric along the normal n: K = − 21 Ln
.
(3.21)
This formula follows from Eq. (3.19) and the symmetry of K by a direct computation, provided that the Lie derivative along n is expressed in terms of the connection ∇ via Eq. (1.4): (Ln ) = n ∇ + ∇ n + ∇ n . 3.4. 3 + 1 coordinates and shift vector We may introduce on M a coordinate system adapted to the (t ) foliation by considering on each hypersurface t a coordinate system (x i ), such that (x i ) varies smoothly from one hypersurface to the next one. Then, (x ) = (x 0 = t, x i ) constitutes a well behaved coordinate of M. The coordinate time vector of this system is t :=
j jt
(3.22)
and is such that each spatial coordinate x i is constant along its field lines. t can be seen as a vector “dual” to the gradient 1-form dt, in the sense that dt, t = 1 .
(3.23)
Then, from Eq. (3.1), n · t = −N and we have the orthogonal 3 + 1 decomposition t = Nn +
with n · = 0 .
(3.24)
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Fig. 7. Constant spatial coordinate lines x i = const cutting across the foliation (t ) and defining the coordinate time vector t and the shift vector . Also represented are the unit normal to each hypersurface t , n, and the lapse function N as giving the metric distance d between two neighboring hypersurfaces t and t+dt via d = N dt.
The vector := (t) is called the shift vector of the coordinate system (x ). The vectors t and are represented in Fig. 7. Given a choice of the coordinates (x i ) in an initial slice 0 , fixing the lapse function N and shift vector on every t fully determines the coordinates (x ) in the portion of M covered by these coordinates. We refer the reader to Ref. [152] for an extended discussion of the choice of coordinates based on the lapse and the shift. The components g of the metric tensor g with respect to the coordinates (t, x i ) are expressible in terms of the lapse N, the components i of the shift vector and the components ij of the spatial metric, according to g dx dx = −N 2 dt 2 + ij (dx i + i dt)(dx j + j dt) .
(3.25)
Example 3.1 (Lapse and shift of Eddington–Finkelstein coordinates). Returning to Example 2.5 (Schwarzschild spacetime in Eddington–Finkelstein coordinates), the lapse function and shift vector of the 3 + 1 Eddington–Finkelstein coordinates are obtained by comparing Eqs. (3.25) and (2.34): N=√
1 , 1 + 2m/r
= 0,
1 , 0, 0 1 + r/(2m)
(3.26)
and =
2m 4m2 , , 0, 0 r(r + 2m) r
.
(3.27)
H √ H Note that, on H (r = 2m), N = 1/ 2 and r = 1/2. The expression for the unit timelike normal to the hypersurfaces t is deduced from N and : ⎛ ⎛ ⎞ ⎞ ⎜ ⎜ ⎟ ⎟ 2m 1 2m n = ⎜ − , 0, 0⎟ , n = ⎜ , 0, 0, 0⎟ (3.28) ⎝ 1 + r ,− ⎝ ⎠ ⎠ . 2m 2m r 1+ 1+ r r
3.5. 3 + 1 decomposition of the Riemann tensor We present here the expression of the spacetime Riemann tensor Riem (cf. Section 1.2.2) in terms of 3 + 1 objects, in particular the Riemann tensor 3 Riem of the connection D associated with the spatial metric . This is a step required to get a 3 + 1 decomposition of the Einstein equation in next section. Moreover, this allows to gain intuition on the analogous (but null) decomposition that will be introduced in Section 6, when studying the dynamics of a null hypersurface.
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As a general strategy, calculations start from the three-dimensional objects and then use is made of Eqs. (3.8) and (3.3), together with the Ricci identity (1.14). Since these techniques will be explicitly exposed in Section 6, we present the following results without proof (see, for instance, Ref. [171]). The 3 + 1 writing of the spacetime Riemann tensor thus obtained can be viewed as various orthogonal projections of Riem onto the hypersurface t and along the normal n: R = 3 R + K K − K K ,
(3.29)
n R = D K − D K ,
(3.30)
n n R =
1 1 L(N n) K + K K + D D N . N N
(3.31)
From the symmetries of the Riemann tensor, all the other contractions involving n either are equivalent to one of the Eqs. (3.30)–(3.31), or vanish. For instance, a contraction with three times n would be zero. Eq. (3.29) is known as the Gauss equation, and Eq. (3.30) as the Codazzi equation. The third equation, (3.31), is sometimes called the Ricci equation [not to be confused with the Ricci identity (1.14)]. The Gauss and Codazzi equations do not involve any second order derivative of the metric tensor g in a timelike direction. They constitute the necessary and sufficient conditions for the hypersurface t , endowed with a 3-metric and an extrinsic curvature K, to be a submanifold of (M, g). Contracted versions of the Gauss and Codazzi equations turn out to be very useful, especially in the 3 + 1 writing of the Einstein equation. Contracting the Gauss equation (3.29) on the indices and leads to an expression that makes appear the Ricci tensors R and 3 R associated with g and , respectively [cf. Eq. (1.17)] R + n n R = 3 R + KK − K K ,
(3.32)
where K is the trace of K, K . Taking the trace of this equation with respect to , leads to an expression that involves the Ricci scalars R := g R and 3 R := 3 R , again respectively associated with g and : R + 2R n n = 3 R + K 2 − K K .
(3.33)
This formula, which relates the intrinsic curvature 3 R and the extrinsic curvature K of t , can be seen as a generalization to the four-dimensional case of Gauss’ famous Theorema egregium (see e.g. Ref. [26]). On the other side, contracting the Codazzi equation on the indices and leads to n R = D K − D K .
(3.34)
3.6. 3 + 1 Einstein equation We are now in position of presenting the 3 + 1 splitting of Einstein equation: R − 21 Rg = 8T
,
(3.35)
where T is the total (matter + electromagnetic field) energy–momentum tensor. The 3 + 1 decomposition of the latter is T = En ⊗ n + n ⊗ J + J ⊗ n + S ,
(3.36)
where the energy density E, the momentum density J and the strain tensor S, all of them as measured by the Eulerian
observer of 4-velocity n, are given by the following projections E := T n n , J := − T n , S := T . Einstein equation (3.35) splits into three equations by using, respectively, (i) the twice contracted Gauss equation (3.33), (ii) the contracted Codazzi equation (3.34), (iii) the combination of the Ricci equation (3.31) with the contracted
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Gauss equation (3.32): 3R
+ K 2 − K K = 16E
D K − D K = 8J
,
(3.37)
,
(3.38)
L(N n) K = −D D N + N {3 R − 2K K + KK +4[(S − E) − 2S ]}
.
(3.39)
These equations are known as the Hamiltonian constraint, the momentum constraint and the dynamical 3+1 equations, respectively. The Hamiltonian and momentum constraints do not contain any second order derivative of the metric in a timelike direction, contrary to Eq. (3.39) [remember that K is already a first order derivative of the metric in the timelike direction n, according to Eq. (3.21)]. Therefore they are not associated with the dynamical evolution of the gravitational field and represent constraints to be satisfied by and K on each hypersurface t . The dynamical equation (3.39) can be written explicitly as a time evolution equation, once a 3 + 1 coordinate system (t, x i ) is introduced, as in Section 3.4. Then N n is expressible in terms of the coordinate time vector t and the shift vector associated with these coordinates: N n = t − [cf. Eq. (3.24)], so that the Lie derivative in the left-hand side of Eq. (3.39) can be written as L(N n) = Lt − L .
(3.40)
Now, if one uses tensor components with respect to the coordinates (x i ), Lt Kij = jKij /jt, Eq. (3.39) becomes j jt Kij
− L Kij = −Di Dj N + N {3 Rij − 2Kik K k j + KK ij +4[(S − E) ij − 2Sij ]}
.
(3.41)
Similarly, relation (3.21) between K and Ln becomes j jt ij
− L ij = −2N K ij
,
(3.42)
where one may use the following identity [cf. Eq. (1.4)]: L ij = Di j + Dj i . 3.7. Initial data problem In view of the above equations, the standard procedure of numerical relativity consists in firstly specifying the values of and K on some initial spatial hypersurface 0 (Cauchy surface), and then evolving them according to Eqs. (3.41) and (3.42). For this scheme to be valid, the initial data must satisfy the constraint Eqs. (3.37)–(3.38). The problem of finding pairs (, K) on 0 satisfying these constraints constitutes the initial data problem of 3 + 1 general relativity. The existence of a well-posed initial value formulation for Einstein equation, first established by Choquet-Bruhat more than 50 years ago [70], provides fundamental insight for a number of issues in general relativity (see e.g. Refs. [68,22] for a mathematical account). In this article we aim at underlining those aspects related with the numerical construction of astrophysically relevant spacetimes containing black holes. In this sense, the 3+1 formalism constitutes a particularly convenient and widely extended approach to the problem (for other numerical approaches, see for instance [96,97,169]). Consequently, the first step in this numerical approach consists in generating appropriate initial data which correspond to astrophysically realistic situations. For a review on the numerical aspects of this initial data problem see [51,135]. If one chooses to excise a sphere in the spatial surface t for it to represent the horizon of a black hole, appropriate boundary conditions in this inner boundary must be imposed when solving the constraint Eqs. (3.37)–(3.38). This particular aspect of the initial data problem constitutes one of the main applications of the subject studied here, and
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will be developed in Section 11. In order to carry out such a discussion, the conformal decomposition of the initial data introduced by Lichnerowicz [113], particularly successful in the generation of initial data, will be presented in Section 10.
4. 3 + 1-induced foliation of null hypersurfaces 4.1. Introduction In Sections 2.6 and 2.7, we have introduced two geometrical objects on the 3-manifold H: the Weingarten map and the second fundamental form . These objects are unique up to some rescaling of the null normal l to H. Following Carter [41–43], we would like to consider these objects as four-dimensional quantities, i.e. to extend their definitions from the 3-manifold H to the four-manifold M (or at least to the vicinity of H, as discussed in Section 2.3). The benefit of such an extension is an easier manipulation of these objects, as ordinary tensors on M, which will facilitate the connection with the geometrical objects of the 3 + 1 slicing. In particular this avoids the introduction of special coordinate systems and complicated notations. For instance, one would like to define easily something like the type 0 3 tensor ∇, where ∇ is the spacetime covariant derivative. At the present stage, this is not possible even when restricting the definition of ∇ to H, because there is no unique covariant derivation associated with the induced metric q, since the latter is degenerate. We have already noticed that, from the null structure of H alone, there is no canonical mapping from vectors of M to vectors of H, and in particular no orthogonal projector (Remarks 2.1 and 2.3). Such a mapping would have provided natural four-dimensional extensions of the forms defined on H. Actually in order to define a projector onto Tp (H), we need some direction transverse to H, i.e. some vector of Tp (M) not belonging to Tp (H). We may then define a projector along this transverse direction. The problem with null hypersurfaces is that there is no canonical transverse direction since the normal direction is not transverse but tangent. However if we take into account the foliation provided by some family of spacelike hypersurfaces (t ) in the standard 3 + 1 formalism introduced in Section 3, we have some extra-structure on M. We may then use it to define unambiguously a transverse direction to H and an associated projector . Moreover this transverse direction will be, by construction, well suited to the 3 + 1 decomposition. 4.2. 3 + 1-induced foliation of H and normalization of l In the general case, each spacelike hypersurface t of the 3 + 1 slicing discussed in Section 3 intersects6 the null hypersurface H on some two-dimensional surface St (cf. Fig. 8): St := H ∩ t
.
(4.1)
More generally, considering some null foliation (Hu ) in the vicinity of H (cf. Section 2.3), we define the 2-surface family (St,u ) by St,u := Hu ∩ t .
(4.2)
St is then nothing but the element St,u=1 of this family. (St,u ) constitutes a foliation of M (in the vicinity of H) by 2-surfaces. This foliation is of type null-timelike in the terminology of the 2 + 2 formalism [96,98]. A local characterization of St follows from Eq. (2.2) and the definition of t as the level set of some scalar field t: ∀p ∈ M,
p ∈ St ⇐⇒ u(p) = 1 and t (p) = t .
(4.3)
6 Note however the existence of slicings of the “exterior” region of H which actually do not intersect H, such as the standard maximal slicing of Schwarzschild spacetime defined by the Schwarzschild time tS and illustrated in Fig. 4.
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Fig. 8. Foliation (St ) of the null hypersurface H induced by the spacetime foliation (t ) of the 3 + 1 formalism.
The subspace Tp (St ) of vectors tangent to St at some point p ∈ St is then characterized in terms of the gradients of the scalar fields u and t by ∀v ∈ Tp (M),
v ∈ Tp (St ) ⇐⇒ du, v = 0 and dt, v = 0 .
(4.4)
As a submanifold of t , each St is necessarily a spacelike surface. Until Section 7, we make no assumption on the topology of St , although we picture it as a closed (i.e. compact without boundary) manifold (Fig. 8). In the absence of global assumptions on St or H, we define the exterior (resp. interior) of St , as the region of t for which u > 1 (resp. u < 1). In the case another criterion is available to define the exterior of St (e.g. if St has the topology of S2 , t is asymptotically flat and the exterior of St is defined as the connected component of t \St which contains the asymptotically flat region), we can always change the definition of u to make coincide the two definitions of exterior. The St ’s constitute a foliation of H. The coordinate t can then be used as a parameter, in general non-affine, along each null geodesic generating H (cf. Section 2.5). Thanks to it, we can normalize the null normal l of H by demanding that l is the tangent vector associated with this parametrization of the null generators: =
dx . dt
(4.5)
An equivalent phrasing of this is demanding that l is a vector field “dual” to the 1-form dt (equivalently, the function t can be regarded as a coordinate compatible with l): dt, l = ∇l t = 1
.
(4.6)
A geometrical consequence of this choice is that the 2-surface St+ t is obtained from the 2-surface St by a displacement
tl at each point of St , as depicted in Fig. 9. Indeed consider a point p in St and displace it by a infinitesimal quantity
tl to the point p = p + tl (cf. Fig. 9). From the very definition of the gradient 1-form dt, the value of the scalar field t at p is t (p ) = t (p + tl) = t (p) + dt, tl = t (p) + t dt, l =t (p) + t .
(4.7)
=1
p
This last equality shows that ∈ St+ t . Hence the vector tl carries the surface St into the neighboring one St+ t . One says equivalently that the 2-surfaces (St ) are Lie dragged by the null normal l. Let s be the unit vector of t , normal to St and directed toward the exterior of St (cf. Fig. 10); s obeys to the following properties: s·s=1 ,
(4.8)
n·s=0 ,
(4.9)
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Fig. 9. Lie transport of the surfaces St by the vector l.
Fig. 10. Null vector l normal to H, unit timelike vector n normal to t , unit spacelike vector s normal to St and ingoing null vector k normal to St .
du, s > 0 , ∀v ∈ Tp (t ),
(4.10) v ∈ Tp (St ) ⇐⇒ s · v = 0 .
(4.11)
Let us establish a simple expression of the null normal l in terms of the unit vectors n and s. Let b ∈ Tp (t ) be the orthogonal projection of l onto t : b := (l) [cf. Eq. (3.4)]. Then l = an + b, with a coefficient a to be determined. By means of Eq. (3.1), dt, l = a/N, so that the normalization condition (4.6) leads to a = N , hence l = Nn + b .
(4.12)
For any vector v ∈ Tp (St ), l · v = 0. Replacing l by the above expression and using n · v = 0 results in b · v = 0. Since this equality is valid for any v ∈ Tp (St ), we deduce that b is a vector of t which is normal to St . It is then necessarily collinear to s: b = s, with = s · b = s · l = l, s = e du, s > 0, thanks to Eq. (4.10). The condition l · l = 0 then leads to = N, so that finally l = N (n + s)
.
(4.13)
In particular, the three vectors l, n and s are coplanar (see Fig. 10). Moreover, since (l) = N s with N > 0, (l) is directed toward the exterior of St . We say that l is an outgoing null vector with respect to St . Remark 4.1. Since n is a unit timelike vector, s a unit spacelike vector and they are orthogonal, it is immediate that the vector n+s is null. The relation (4.13) implies that this vector is tangent to H. Therefore, another natural normalization of the null normal to H would have been to consider l = n + s, instead of dt, l = 1 [Eq. (4.6)]. Both normalizations
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are induced by the foliation (t ). Only the normalization (4.6) has the property of Lie dragging the surfaces (St ). On the other side, at a given point p ∈ H, the normalization l = n + s can be defined by a single spacelike hypersurface intersecting H, whereas the normalization (4.6) requires the existence of a family (t ) in the neighborhood of p. In what follows, we will denote by lˆ the null vector 1 lˆ := √ (n + s) , (4.14) 2 √ where the factor 1/ 2 is introduced for later convenience. 4.3. Unit spatial normal to St Eq. (4.13) can be inverted to express the unit spatial normal to the surface St ,7 s, in terms of l and n: 1 (4.15) s= l−n . N When combined with l = e du [Eq. (2.14)] and n = −N dt [Eq. (3.1)] this leads to the following expression of the 1-form s associated with the normal s: s = N dt + Mdu
,
(4.16)
where we have introduced the factor e = ln(MN ) M := , so that N
.
(4.17)
Eq. (4.16) implies [cf. definition (3.5) of the operator ∗ ]
∗ s = M ∗ du , ∗
because dt
= −N −1
(4.18) ∗
∗
∗
n = 0. Now, since s ∈ Tp (t ), s = s. Moreover, from Eq. (3.9), du = Du, so that we get
s = M ∗ du = MDu
.
(4.19)
4.4. Induced metric on St The metric h induced by t ’s metric on the 2-surfaces St is given by a formula analogous to Eq. (3.3), except for the change of the + sign into a − one, to take into account the spacelike character of the normal s (whereas the normal n was timelike): h := − s ⊗ s = g + n ⊗ n − s ⊗ s .
(4.20)
Let us consider a pair of vectors (u, v) in Tp (H). Denoting by u0 and v0 their respective projections along l on the vector plane Tp (St ), we have the unique decompositions u = u0 + l
and v = v0 + l ,
(4.21)
where and are two real numbers. Since n · u0 = n · v0 = s · u0 = s · v0 = 0, one has h(u, v) = g(u, v) + n, un, v − s, us, v = g(u, v) + [n · (u0 + l)] × [n · (v0 + l)] − [s · (u0 + l)][s · (v0 + l)] = g(u, v) + (n · l )2 − (s · l )2 , =−N
=N
7 Note that the definition of the vector s can be extended to the 2-surfaces S t,u in the vicinity of H. This permits to extend the objects constructed by using s to a neighborhood of H, in the spirit of Section 2.3. We will refer in the following to St , keeping in mind that the results can be extended to the whole foliation (St,u ).
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h(u, v) = g(u, v) .
(4.22)
This last equality shows that h and g coincide on Tp (H). In other words, the pull-back of h on H equals the pull-back of g, that we have denoted q in Section 2 [see Eq. (2.8)]: ∗ h = ∗ g = q. We may then take h as the four-dimensional extension of q and write Eq. (4.20) as q=−s⊗s
(4.23)
q=g+n⊗n−s⊗s
.
(4.24)
Consequently we abandon from now on the notation h in profit of q. To summarize, on Tp (M), q is the symmetric bilinear form given by Eq. (4.24), on Tp (t ) it is the symmetric bilinear form given by Eq. (4.23), on Tp (H) it is the degenerate metric induced by g, and on Tp (St ) it is the positive definite (i.e. Riemannian) metric induced by g. The endomorphism Tp (M) → Tp (M) canonically associated with the bilinear form q by the metric g [cf. notation (1.12)] is the orthogonal projector onto the 2-surface St : q = 1 + nn, . − ss, .
,
(4.25)
in the very same manner in which defined by Eq. (3.4) was the orthogonal projector onto t . 4.5. Ingoing null vector As mentioned in Section 4.1, we need some direction transverse to H to define a projector Tp (M) → Tp (H). The (t ) slicing has already provided us with two different transverse directions: the timelike direction n and the spacelike direction s, both normal to the 2-surfaces St (cf. Fig. 10). These two directions are indeed transverse to H since n ∈ / Tp (H) (otherwise H would be a timelike hypersurface) and s ∈ / Tp (H) (otherwise H would be a spacelike hypersurface, coinciding locally with t ). However n and s are not the only natural choices linked with the (t ) foliation: we may also think about the null directions normal to St , i.e. the trajectories of the light rays emitted in the radial directions from points on St . The light rays emitted in the outgoing radial direction (as defined in Section 4.2) define the null vector l tangent to H already introduced. But those emitted in the ingoing radial direction define (up to some normalization factor) another null vector: 1 kˆ := √ (n − s) 2
(4.26)
√ [compare with Eq. (4.14)]. kˆ is transverse to H, since l · kˆ = − 2N = 0. In fact we will favor this transverse direction, rather than those arising from n or s, because its null character leads to simpler formulæ for the description of the null hypersurface H. √ Let us renormalize the vector kˆ by dividing it by 2N to get the null vector k=
1 (n − s) 2N
.
(4.27)
The normalization has been chosen so that k satisfies the relation l · k = −1
,
(4.28)
which will simplify some of the subsequent formulæ. Eqs. (4.13) and (4.27) can be inverted to express n and s in terms of the null vectors l and k: n=
1 l + Nk , 2N
(4.29)
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Fig. 11. View of the plane orthogonal to the 2-surface St : the timelike vector n, the spacelike vector s and the null vectors l and k all belong to this plane. The dyad (l, k) defines the intersection of this plane with the light cone emerging from St ’s points. Intersections of the hypersurfaces H and t with this plane are also shown.
s=
1 l − Nk . 2N
(4.30)
Each pair (n, s) or (l, k) forms a basis of the vectorial plane orthogonal to St : Tp (St )⊥ = Span(n, s) = Span(l, k) .
(4.31)
This plane is shown in Fig. 11. Remark 4.2. All the null vectors at a given point p ∈ H, except those collinear to l are transverse to H (see Fig. 10 where all these vectors form the light cone emerging from p). It is the slicing (St ) of H which has enabled us to select a preferred transverse null direction k, as the unique null direction normal to St and different from l. Let us consider the 1-form k canonically associated with the vector k by the metric g. By combining Eqs. (4.27), (3.1) and (4.16), one gets k = −dt −
M du 2N
.
(4.32)
Remark 4.3. Eqs. (2.14), (3.1), (4.16) and (4.32) show that the 1-forms l, n, s and k are all linear combinations of the exact 1-forms dt and du. This simply reflects the fact that the vectors l, n, s and k are all orthogonal to St [Eq. (4.31) above] and that (dt, du) form a basis of the two-dimensional space of 1-forms normal to St [see Eq. (4.4)]. An immediate consequence of (4.32) is that the action of k on vectors tangent to H is identical (up to some sign) to the action of the gradient 1-form dt: ∀v ∈ Tp (H),
k, v = −dt, v .
(4.33)
An equivalent phrasing of this is: the pull-back of the 1-form k on H and that of −dt coincide: ∗ k = −∗ dt
.
(4.34)
Remark 4.4. The null vector k can be seen as “dual” to the null vector l in the following sense: (i) l belongs to Tp (H), while k does not, and (ii) ∗ k is a non-trivial exact 1-form in T∗ (H), while ∗ l is zero. Example 4.5 (Slicing of Minkowski light cone). In continuation of Example 2.4 (H = light cone in Minkowski spacetime), the simplest 3 + 1 slicing we may imagine is that constituted by hypersurfaces t = const, where t is a standard Minkowskian time coordinate. The lapse function N is then identically one and the unit normal to t has trivial components with respect to the Minkowskian coordinates (t, x, y, z): n = (1, 0, 0, 0) and n = (−1, 0, 0, 0).
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t In Example 2.4, we have already normalized l so that = dt, l = 1 [Eq. (4.6)], [cf. Eq. (2.28)]. The 2-surface St is the sphere r := x 2 + y 2 + z2 = t in the hyperplane t and its outward unit normal has the following components with respect to (t, x, y, z): x y z x y z , s = 0, , , . (4.35) s = 0, , , r r r r r r
We then deduce the components of the ingoing null vector k from Eq. (4.27): 1 1 x y z x y z k = ,− ,− ,− , k = − , − , − , − , 2 2r 2r 2r 2 2r 2r 2r
(4.36)
and the components of q from Eq. (4.24): ⎞ ⎛0 0 0 0 2 2 xy xz ⎜0 y + z − 2 − 2 ⎟ ⎟ ⎜ 2 r r r ⎟ ⎜ 2 2 ⎜ x +z xy yz ⎟ q = ⎜ ⎟ . 0 − − ⎜ 2 2 2 ⎟ r r r ⎠ ⎝ x2 + y2 xz yz 0 − 2 − 2 r r r2
(4.37)
Example 4.6 (Eddington–Finkelstein slicing of Schwarzschild horizon). As a next example, let us consider the 3 + 1 slicing of Schwarzschild spacetime by the hypersurfaces t =const, where t is the Eddington–Finkelstein time coordinate considered in Example 2.5. This slicing has been already represented in Fig. 4. The corresponding lapse function has been exhibited in Example 3.1. In Example 2.5, we have already normalized the null vector l to ensure t =dt, l=1, so Eq. (2.40) provides the correct expression for the null normal induced by the 3 + 1 slicing. From the metric components given by Eq. (2.34), we obtain immediately the expression of the unit normal to St lying in t : ⎛ ⎛ ⎞ ⎞ ⎜ ⎜ ⎟ ⎟ 1 2m 2m s = ⎜ , 0, 0⎟ (4.38) , 0, 0⎟ , s = ⎜ , 1+ ⎝0, ⎝ ⎠ . ⎠ r 2m 2m 1+ r 1+ r r Inserting this value into formula (4.27) and making use of expression (3.26) for N and (3.28) for n, we get the ingoing null vector k: 1 m 1 m 1 m 1 m k = + , − − , 0, 0 , k = − − , − − , 0, 0 . (4.39) 2 r 2 r 2 r 2 r H
H
Note that on H, k =(−1, −1, 0, 0), so that we verify property (4.34), which is equivalent to (kt , k , k ) =(−1, 0, 0). The vectors n, s, l and k are represented in Fig. 12. The 2-surface St is spanned by the coordinates (, ) and the expression of the induced metric on St is obtained readily from the line element (2.34): q = diag(0, 0, r 2 , r 2 sin2 ) .
(4.40)
4.6. Newman–Penrose null tetrad 4.6.1. Definition The two null vectors l and k are the first two pieces of the so-called Newman–Penrose null tetrad, which we briefly present here. We complete the null pair (l, k) by two orthonormal vectors in Tp (St ), (ea ) = (e2 , e3 ) let’s say, to get a basis of Tp (M), such that l · l = 0, l · k = −1, k · k = 0, k · ea = 0 , ea · eb = ab .
l · ea = 0 , (4.41)
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Fig. 12. Null vector l normal to H, unit timelike vector n normal to t , unit spacelike vector s normal to St and ingoing null vector k normal to St for the 3 + 1 Eddington–Finkelstein slicing of Schwarzschild horizon.
The basis (l, k, e2 , e3 ) is formed by two null vectors and two spacelike vectors. At the price of introducing complex vectors, we can modify it into a basis of four null vectors. Indeed let us introduce the following combination of e2 and e3 with complex coefficients: 1 m := √ (e2 + ie3 ) . 2
(4.42)
Then the complex conjugate defines another vector, which is linearly independent from m: 1 ¯ = √ (e2 − ie3 ) . m 2
(4.43)
¯ are null vectors (with respect to the metric g). Both m and m ¯ constitutes a basis of Tp (M) made of null vectors only: any vector of Tp (M) admits a The tetrad (l, k, m, m) ¯ is unique expression as a linear combination (possibly with complex coefficients) of these four vectors. (l, k, m, m) called a Newman–Penrose null tetrad [127] (see also p. 343 of Ref. [90] or p. 72 of [156]). This tetrad obeys to l · l = 0, l · k = −1, l · m = 0, ¯ =0 , k · k = 0, k · m = 0, k · m ¯ =1 , m · m = 0, m · m ¯ ·m ¯ =0 . m
¯ =0 , l·m (4.44)
Since (e2 , e3 ) is an orthonormal basis of Tp (St ), the metric induced in St can be written as ¯ +m ¯ ⊗m . q = e2 ⊗ e2 + e3 ⊗ e3 = m ⊗ m
(4.45)
Moreover (n, s, e2 , e3 ) is an orthonormal tetrad of Tp (M). The spacetime metric can then be written as g = −n ⊗ n + s ⊗ s + e2 ⊗ e2 + e3 ⊗ e3 .
(4.46)
It can also be expressed in terms of the Newman–Penrose null tetrad: ¯ +m ¯ ⊗m . g = −l ⊗ k − k ⊗ l + m ⊗ m
(4.47)
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Comparing Eqs. (4.47) and (4.45), we get an expression of q in terms of g and the null dyad (l, k): q=g+l⊗k+k⊗l
.
(4.48)
This expression is alternative to Eq. (4.24). It can be obtained directly by inserting Eqs. (4.29) and (4.30) in Eq. (4.24). The related expression for the orthogonal projector q onto the 2-surface St is q = 1 + lk, . + k l, .
,
(4.49)
which constitutes an alternative to Eq. (4.25). 4.6.2. Weyl scalars In Section 1.2.2 we have introduced the Weyl tensor C and have indicated that it encodes 10 of the 20 independent components of the Riemann tensor. The null tetrad previously introduced permits to write these free components as five independent complex scalars n (n ∈ {0, 1, 2, 3, 4}), known as Weyl scalars. They are defined as 0 := C(l, m, l, m) = C m m , 1 := C(l, m, l, k) = C m k ,
¯ k) = C m m 2 := C(l, m, m, ¯ k , ¯ k) = C k m ¯ k , 3 := C(l, k, m,
¯ k, m, ¯ k) = C m 4 := C(m, ¯ k m ¯ k .
(4.50)
As we will see in the following sections, some relevant geometrical quantities are naturally expressed in terms of (some of) these scalars. For an account of the Newman–Penrose formalism in which they are naturally defined, see [154–156,45] and references therein. 4.7. Projector onto H Having introduced the transverse null direction k, we can now define the projector onto H along k by : Tp (M) −→ Tp (H) v −→ v + (l · v)k
.
(4.51)
This application is well defined, i.e. its image is in Tp (H), since ∀v ∈ Tp (M),
l · (v) = l · v + (l · v) (l · k) =0 .
(4.52)
=−1
Moreover, leaves invariant any vector in Tp (H): ∀v ∈ Tp (H),
(v) = v
(4.53)
and (k) = 0 .
(4.54)
These last two properties show that the operator is the projector onto H along k. The projector can be written as = 1 + kl, . . It can be considered as a type = + k .
(4.55) 1 1
tensor, whose components are (4.56)
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Comparing Eqs. (4.55) and (4.49) leads to the following relation: q = + lk, . .
(4.57)
Remark 4.7. The definition of the projector does not depend on the normalization of l and k as long as they satisfy the relation l · k = −1 [Eq. (4.28)]. Indeed a rescaling l → l = l would imply a rescaling k → k = −1 k, leaving invariant. In other words, is determined only by the foliation St of H and not by the scale of H’s null normal. Note that such a foliation-induced tranverse projector onto a null hypersurface has been already used in the literature (see e.g. Ref. [30]). Since is a well defined application Tp (M) → Tp (H), we may use it to map any linear form in Tp∗ (H) to a linear form in Tp∗ (M), in the very same way that in Section 2.1 we used the application ∗ : Tp (H) → Tp (M) to map linear forms in the opposite way, i.e. from Tp∗ (M) to Tp∗ (H). Indeed, and more generally, if T is a n-linear form on Tp (H)n , we define ∗ T as the n-linear form ∗ T : Tp (M)n −→ R (v1 , . . . , vn ) −→ T((v1 ), . . . , (vn ))
.
(4.58)
Note that since any multilinear form on Tp (M)n can also be regarded as a multilinear form on Tp (H)n , thanks to the pull-back mapping ∗ if we identify ∗ T with T abusing of the notation [cf. Eq. (2.7)], we may extend the definition (4.58) to any multilinear form T on Tp (M). In index notation, we have then (∗ T )1 ...n = T 1 ... n 1 1 · · · n n .
(4.59)
Note that we are again abusing of the notation, since ∗ T here should be properly denoted as (∗ ◦ ∗ )T. In particular, for a 1-form, the expression (4.55) for yields: ∀ ∈ Tp∗ (M),
∗ = + , kl .
(4.60)
For = l, we get immediately ∗ l = 0 ,
(4.61)
which reflects the fact that l restricted to Tp (H) vanishes. On the contrary, for the 1-form k we have ∗ k = k .
(4.62)
Collecting together Eqs. (4.53) (for v = l), (4.54), (4.61) and (4.62), we recover the duality between l and k mentioned in Remark 4.4: (l) = l ∗ l = 0
and and
(k) = 0 ∗ k = k
, .
(4.63) (4.64)
In index notation, the above relations write respectively = = 0
k = 0 ,
(4.65)
and k = k .
(4.66)
and
The various mappings introduced so far between the vectorial spaces Tp (H) and Tp (M) and their duals are represented in Fig. 13. Remark 4.8. The vector k is a special case of what is called more generally a rigging vector [118], i.e. a vector transverse to H everywhere, which allows to define a projector onto H whatever the character of H (i.e. spacelike, timelike, null or changing from point to point).
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Fig. 13. Mappings between the space Tp (H) (resp. Tp (M)) of vectors tangent to H (resp. M) and the space Tp∗ (H) (resp. Tp∗ (M)) of 1-forms on H (resp. M): ∗ and ∗ are respectively the push-forward and the pull-back mapping canonically induced by the embedding of H in M; is the projector onto H along the null transverse direction k and ∗ the induced mapping of 1-forms; g and g−1 denote the standard duality between vectors and 1-forms induced by the spacetime metric g. Note that since the metric q on H is degenerate, it provides a mapping Tp (H) → Tp∗ (H), but not in the reverse way. is the Weingarten map, defined in Section 2.6, which is an endomorphism of Tp (H).
4.8. Coordinate systems stationary with respect to H Let us consider a 3 + 1 coordinate system (x ) = (t, x i ), with the associated coordinate time vector t and shift vector , as defined in Section 3.4. It is useful to perform an orthogonal 2+1 decomposition of the shift vector with respect to the surface St , according to = bs − V
with s · V = 0 .
(4.67)
In other words, b = s · and V = − q() ∈ Tp (St ) (the minus sign is chosen for later convenience). Combining the two 3 + 1 decompositions l = N (n + s) [Eq. (4.13)] and t = N n + [Eq. (3.24)], we get l = t + V + (N − b)s
.
(4.68)
We say that (x ) is a coordinate system stationary with respect to the null hypersurface H iff the equation of H in this coordinate system involves only the spatial coordinates (x i ) and does not depend upon t, i.e. iff there exists a scalar function f (x 1 , x 2 , x 3 ) such that ∀p = (t, x 1 , x 2 , x 3 ) ∈ M,
p ∈ H ⇐⇒ f (x 1 , x 2 , x 3 ) = 1 .
(4.69)
This means that the location of the 2-surface St is fixed with respect to the coordinate system (x i ) on t , as t varies. The gradient of f is normal to H and thus parallel to du: H
du = df ,
(4.70)
H
where = means that this identity is valid only at points on H and is some scalar field on H. Eq. (4.70) and the independence of f from t imply ju H =0 . jt
(4.71)
This has an immediate consequence on the coordinate time vector t: du, t =
ju ju H ju
t = t = =0 , jx
jx jt
(4.72)
which implies that t is tangent to H [cf. Eq. (2.12)]. Consequently, for a coordinate system stationary with respect to H, lˆ · t = 0 .
(4.73)
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Fig. 14. Same as Fig. 10 but with the addition of the coordinate time vector t, the shift vector and H’s surface velocity vector V with respect to a coordinate system (x ) stationary with respect to H.
Replacing8 lˆ and t by their respective 3 + 1 decompositions (4.14) and (3.24) and using b = s · leads to b=N .
(4.74)
Thus, for a coordinate system stationary with respect to H, the decomposition (4.68) simplifies to l=t+V .
(4.75)
In the case where H is the event horizon of some black hole and (x ) is stationary with respect to H, V ∈ Tp (St ) is called the surface velocity of the black hole by Damour [59,60]. More generally, we will call V the surface velocity of H with respect to the coordinate system (x ) stationary with respect to H. To summarize, we have the following: ju H =0 jt ⇐⇒ t ∈ Tp (H)
((x ) stationary w.r.t. H) ⇐⇒
H
⇐⇒ lˆ · t = 0 H
⇐⇒ b = N H
⇐⇒ l = t + V
(4.76) (4.77) (4.78) (4.79) (4.80)
Notice that for a coordinate system stationary with respect to H, the scalar field u defining H is not necessarily such that u = u(x 1 , x 2 , x 3 ) everywhere in M, but only on H [Eq. (4.76)]. The vectors t, and V of a coordinate system stationary with respect to H are shown in Fig. 14. A special case of a coordinate system stationary with respect to H is a coordinate system (x ) for which the function f in Eq. (4.69) is simply one of the coordinates, x 1 let’s say: f (x 1 , x 2 , x 3 ) = x 1 . We call such a system a coordinate system adapted to H. For instance, if the topology of H is R × S2 , an adapted coordinate system can be of spherical type (x i ) = (r, ϑ, ), where r is such that H corresponds to r = 1. Another special case of coordinate system stationary with respect to H is a coordinate system (x ) for which V = 0 (in addition to the stationarity condition t ∈ Tp (H)). We call such a system a coordinate system comoving with H. From Eq. (4.80) this implies H
t=l , which shows that the null generators of H are some lines x i = const. 8 Expression (4.73) is equivalent to l · t = 0 whenever N = 0 on H.
(4.81)
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Example 4.9. The Minkowskian coordinates (t, x, y, z) introduced in Examples 2.4 and 4.5 are not stationary with respect to the light cone H. In particular, ju/jt = −1 = 0 and N = 1 = b = 0 for these coordinates. On the contrary, the Eddington–Finkelstein coordinates (t, r, , ) introduced in Examples 2.5, 3.1 and 4.6 are stationary with respect to the event horizon H of a Schwarzschild black hole. In particular, from the expression (2.37) for u, we notice that the H H √ requirement (4.76) is fulfilled, and from Eqs. (3.27) and (4.38), we get b=2m/r (1+2m/r)−1/2 so that b = N = 1/ 2, in agreement with (4.79). Moreover, the Eddington–Finkelstein coordinates are both adapted to H and comoving with H. Indeed, the equation for H can be defined by r = 2m (instead of u = 1), which shows the adaptation, and we have H
already noticed that t = l [Eq. (2.41)], which shows the comobility (this can also be seen from the shift vector which is collinear to s, according to Eqs. (3.27) and (4.38), implying V = 0). If (x ) is a coordinate system adapted to H, then9 (x A ) = (t, x a ) = (t, x 2 , x 3 ) is a coordinate system for H. In terms of it, the induced metric element on H is ds 2 |H = qAB dxA dx B = gtt dt 2 + 2gta dt dx a + gab dx a dx b .
(4.82)
Now, from Eqs. (4.68) and (4.80) gtt = t · t = (l − V) · (l − V) = V · V = Va V a
(4.83)
and, from Eqs. (4.67) and (4.16) gta = a = bs a − Va = −Va .
(4.84)
Besides, gab = qab [cf. Eq. (2.9)]. Thus the above line element can be written ds 2 |H = qAB dx A dx B = qab (dx a − V a dt)(dx b − V b dt)
.
(4.85)
This equation agrees with Eq. (I.50c) of Damour [60] (or Eq. (6) of Appendix of Ref. [61]).10 Example 4.10. For the Eddington–Finkelstein coordinates (t, r, , ) considered in Examples 2.5, 3.1, 4.6 and 4.9, (x A ) = (t, , ) constitutes a coordinate system for the event horizon H. Taking into account that r = 2m on H, we read from the line element (2.34) that ds 2 |H = r 2 (d2 + sin2 d2 ) ,
(4.86)
in agreement with Eq. (4.85), with, in addition V a = 0, since the Eddington–Finkelstein coordinates are comoving with H.
5. Null geometry in four-dimensional version. Kinematics In this section we consider the spacetime first derivatives of the null vectors l and k and of the associated 1-forms l and k, as well as the Lie derivatives of the induced metric q along l and k. This is what we mean by “kinematics”. The first derivative of l has been represented by the Weingarten map of H in Section 2.6. We start by extending the definition of this map to the whole four-dimensional vector space Tp (M), whereas its original definition was restricted to the three-dimensional subspace Tp (H).
9 Remember the index convention given in Section 1.2. 10 Note that Damour’s convention for indices A, B, . . . is the same than ours for indices a, b, . . ., namely they run in {2, 3} (whereas our convention ¯ B, ¯ . . .). for A, B, . . . is that they run in {0, 2, 3}, same as Damour’s A,
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5.1. Four-dimensional extensions of the Weingarten map and the second fundamental form of H Having introduced in Section 4.7 the projector onto H, we can extend the definition of the Weingarten map of H (with respect to the null normal l) to all vectors of Tp (M) at any point of H, by setting : Tp (M) −→ Tp (H) v −→ H ((v))
,
(5.1)
where H denotes the Weingarten map introduced on Tp (H) in Section 2.6. The image of is in Tp (H) because the image of H is. Explicitly, one has [cf. Eq. (2.46)] ∀v ∈ Tp (M),
(v) = ∇(v) l .
(5.2)
In index notation v = (v) ∇ = ( + k )v ∇ ,
(5.3)
hence the matrix of : = ∇ + k ∇
.
(5.4)
We have already noticed that l is an eigenvector of the Weingarten map, with the eigenvalue (the non-affinity coefficient) [cf. Eq. (2.52)]. Since (k) = 0, k constitutes another eigenvector of the (extended) Weingarten map, with the eigenvalue zero: (l) = l
and
(k) = 0
.
(5.5)
Similarly, we make use of the projector to extend the definition of the second fundamental form of H with respect to the normal l by [cf. the definition (4.58) of ∗ ] := ∗ H
,
(5.6)
where H denotes the second fundamental form of H with respect to l introduced in Section 2.7. Explicitly, writes : Tp (M) × Tp (M) −→ R (u, v) −→ H ((u), (v)) .
(5.7)
Since H is symmetric, the bilinear form defined above is symmetric. Moreover, from the relation (4.57), we have (u) = q (u) − k, ul. Since l is a degeneracy direction of H [cf. Eq. (2.56)], we get ∀(u, v) ∈ Tp (M) × Tp (M),
(u, v) = H ( q(u), q (v)) .
(5.8)
Replacing H by its definition (2.53), we get ∀ (u, v) ∈ Tp (M) × Tp (M),
(u, v) = q (u) · ∇q (v) l = ∇l( q(u), q (v)) .
(5.9)
We write this relation as = q ∗ ∇l
,
(5.10)
where q ∗ is the operator on multilinear forms induced by the projector q , in a manner similar to ∗ [cf. Eq. (4.58)]: for any n-linear form T on Tp (M) or on Tp (St ), q ∗ T is the n-linear form on Tp (M) defined by q ∗ T : Tp (M)n −→ R (v1 , . . . , vn ) −→ T( q(v1 ), . . . , q (vn ))
.
(5.11)
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In index notation: (
q ∗ T )1 ...n = T 1 ... n q 1 1 · · · q n n ,
(5.12)
so that Eq. (5.10) writes (taking into account the symmetry of ) = ∇ q q
.
(5.13)
The identity (5.10) strengthens Eq. (5.6): not only “acts only” in H (in the sense that starts by a projection onto H), but it “acts only” in the submanifold St of H. The bilinear form is degenerate, with at least two degeneracy directions : l [see Eq. (2.56)] and k (since (k) = 0): (l, .) = 0
(k, .) = 0
and
.
(5.14)
From Eq. (4.31) we conclude that any vector in the plane orthogonal to St is also a degeneracy direction for : ∀v ∈ Tp (St )⊥ ,
(v, .) = 0 .
(5.15)
5.2. Expression of ∇l: rotation 1-form and Há´jiˇcek 1-form A quantity which appears very often in our study is the spacetime covariant derivative of the null normal: ∇l. Let us recall that, thanks to some null foliation (Hu ), we have extended the definition of l to an open neighborhood of H (cf. Section 2.3). Consequently the covariant derivatives ∇l and ∇l are well defined. ∇l is a bilinear form on Tp (M). Let us express it in terms of the bilinear form which we have just extended to the whole space Tp (M). For two arbitrary vectors u and v of Tp (M), by combining definition (5.7) of with the definition (2.53) of H and making use of expression (4.51) of , one has (u, v) = (u) · ∇(v) l = (u + l, uk) · ∇(v) l = u · ∇(v) l + l, uk · ∇(v) l =u · ∇v+l,vk l + l, uk · (v) =(v)
= u · ∇v l + l, vu · ∇k l + l, uk · (v) = ∇l(u, v) + l, v∇k l, u + l, uk · (v) .
(5.16)
In this expression appears the 1-form : Tp (M) −→ R v −→ −k · (v)
,
(5.17)
which we call the rotation 1-form (for reasons which will become clear later; see Section 8.6.1). Relation (5.16) then reads (u, v) = ∇l(u, v) + l, v∇k l, u − , vl, u .
(5.18)
Since this equation is valid whatever u and v in Tp (M), we obtain the relation we were looking for ∇l = + l ⊗ − ∇k l ⊗ l
.
(5.19)
Taking into account the symmetry of , the ‘index’ version of the above relation is [see Eq. (1.9)]: ∇ = + − k ∇ .
(5.20)
An equivalent form of Eq. (5.19), obtained via the standard metric duality [or by raising the last index of Eq. (5.20)], gives the covariant derivative of the vector field l
+ l ⊗ − ∇k l ⊗ l , ∇l =
(5.21)
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is the endomorphism canonically associated with the bilinear form by the metric g [see the notation (1.12)]. where Its components are = g and it is related to the Weingarten map by
:= q ◦ ◦ q = q ◦ .
(5.22)
By combining Eqs. (5.2), (5.21) and using the fact that , k = 0, so that , (v) = , v for any v ∈ Tp (M), we
and the rotation 1-form : get a simple expression relating the extended Weingarten map to the endomorphism
+ , .l =
.
(5.23)
Let us now discuss further the rotation 1-form . First, from the expressions (5.2) for and (4.51) for , one has ∀ v ∈ Tp (M),
, v = − k · ∇(v) l = −k · ∇v+l,vk l = − k · ∇v l − l, vk · ∇k l .
(5.24)
Hence = −k · ∇l − (k · ∇k l)l .
(5.25)
Next, for any vector v ∈ Tp (M) we have , v = −k, (v). Since the image of the extended Weingarten map is in Tp (H) and the action of the 1-forms k and −dt coincide on Tp (H) [cf. Eq. (4.34)], we get the following alternative expressions for : ∀v ∈ Tp (M),
, v = dt, (v) = dt, ∇(v) l ,
(5.26)
which we can write in terms of the function composition operator ◦ as = dt ◦
.
(5.27)
Besides, from the very definition (5.17) of , the eigenvector expressions (5.5) lead immediately to the following action on the null vectors l and k: , l =
and
, k = 0
.
(5.28)
The pull-back of the rotation 1-form to the 2-surfaces St by the inclusion mapping of St into M is called the Há´jiˇcek 1-form and is denoted by the capital letter . Following the four-dimensional point of view adopted in this article, we can extend the definition of to all vectors in Tp (M) thanks to the orthogonal projector q and set [see definition (5.11)] := ◦ q
or
:= q ∗
.
(5.29)
Replacing by its definition (5.17) leads to ∀v ∈ Tp (M),
, v = −k · ∇q (v) l
.
(5.30)
The 1-form has been introduced by Há´"iˇcek [82,84] in the special case = 0 (non-expanding horizons, to be discussed in Section 7) and in the general case by Damour [60,61]. is considered by Há´"iˇcek as a “gravimagnetic field”, whereas it is viewed by Damour as a surface momentum density, as we shall see in Section 6.3. The form has also been used in the subsequent membrane paradigm formulation of Price and Thorne [141]. Actually, restricting the action of to Tp (St ), on which q is the identity operator, and using Eq. (5.26), we get ∀v ∈ Tp (St ),
, v = dt, ∇v l ,
(5.31)
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This expression agrees with Eq. (2.6) of Ref. [141], used by Price and Thorne as the definition of the Há´"iˇcek 1-form. By construction (because of the orthonormal projector q onto St ), the Há´"iˇcek 1-form vanishes for any vector orthogonal to the 2-surface St : ∀v ∈ Tp (St )⊥ ,
, v = 0 .
(5.32)
In particular, it vanishes on the null dyad (l, k): , l = 0
and
, k = 0
.
(5.33)
Actually the 1-form can be viewed as a 1-form intrinsic to the 2-surface St , independently of the fact that St is a submanifold of H or t . It describes some part of the extrinsic geometry of St as a submanifold of (M, g) and is called generically a normal fundamental form of the 2-surface St [93,67,76]. The remaining part of the extrinsic geometry of St is described by the second fundamental tensor K discussed in Remark 5.4 below. We have, thanks to the expression (4.57) for q , , v = , q (v) = , (v) + k, vl = , (v) +k, v , l .
∀ v ∈ Tp (M),
=,v
(5.34)
=
Hence it follows the simple relation between the rotation 1-form , the Há´"iˇcek 1-form and the “transverse” 1-form k: = − k
.
(5.35)
5.3. Frobenius identities From expression (5.19) of the spacetime covariant derivative ∇l, we can compute the exterior derivative of H’s normal 1-form l following Eq. (1.24). We get, taking into account the symmetry of , dl = ⊗ l − l ⊗ − l ⊗ ∇k l + ∇k l ⊗ l = ∧ l + ∇k l ∧ l , dl = ( + ∇k l) ∧ l .
(5.36)
The fact that the exterior derivative of l is the exterior product of some 1-form with l itself reflects the fact that l is normal to some hypersurface (H): this is the dual formulation of Frobenius theorem already noticed in Section 2.4. Actually Eq. (5.36) has the same structure as Eq. (2.17). Let us rewrite the latter by expressing is terms of the lapse function N and the factor M [Eq. (4.17)]: dl = d ln(MN ) ∧ l
.
(5.37)
Let us now evaluate the exterior derivative of the 1-form k dual to the ingoing null vector k. First of all, the definition of k is extended to an open neighborhood of H in M as the vector field which satisfies (i) k is an ingoing null normal to the 2-surface St,u [cf. Eq. (4.2)] and (ii) k · l = −1. The exterior derivative dk is then well defined. Starting from expression (4.32) for k and using dd = 0, we have immediately [cf. formula (1.22)] 1 M 1 M dk = − d (5.38) ∧ du = − d ∧l , 2 N 2MN N where we have used l = MN du [cf. Eq. (2.14)]. Hence 1 N ∧l dk = d ln 2N 2 M
.
(5.39)
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Remark 5.1. Since a priori the 1-form d ln(N/M) is not of the form k + l (in which case Eq. (5.39) would write dk = /(2N 2 )k ∧ l), we deduce from the (dual formulation of) of Frobenius theorem (see e.g. Theorem B.3.2 in Wald’s textbook [167]) and Eq. (5.39) that the hyperplane normal to k is not in general integrable into some hypersurface. On the other side, the Frobenius theorem and relations (5.37) and (5.39) imply that the 2-planes normal to both l and k are integrable into a 2-surface: it is St [cf. the property (4.31)]. 5.4. Another expression of the rotation 1-form Let us show that the Frobenius identity (5.39) leads to the identification of the rotation 1-form with the covariant derivative of the 1-form k along the vector l. Starting from the definition of [Eq. (5.17)], any vector v ∈ Tp (M) satisfies , v = − k · ∇(v) l = l · ∇(v) k = ∇(v) k, l = ∇k · (v), l = −dk · (v) + (v) · ∇k, l ⎡ ⎤ N N ⎦ 1 ⎣ ∇(v) ln l − l, (v) d ln , l + (v) · ∇k, l = 2N 2 M M =0 1 N = l, l +∇l k, (v) ∇(v) ln 2N 2 M =0
= ∇l k, v + (l · v)k = ∇l k, v + (l · v) ∇l k, k =0
= ∇l k, v ,
(5.40)
where the relation k · l = −1 has been used in the first line, Eq. (1.24) to obtain the second line, and Eq. (5.39) to get the third line. Therefore = ∇l k
.
(5.41)
From Eq. (4.34), k can be written as k = −∗ dt on H. If we make use of this, together with expression (4.57) for , the pull-back of the above relation on H results in ∗ = −∗ ∇l dt
,
(5.42)
which provides some nice perspective on , alternative to Eq. (5.27). Combining Eqs. (5.41) and (5.28) results in the simple relation ∇k(l, l) = .
(5.43)
Another consequence of the Frobenius identity (5.39) is ⎡ ⎤ 1 ⎢ N N ⎥ k · dk = l − l, k d ln ⎣∇k ln ⎦ 2 2N M M =−1 1 N N ∇k · k − k · ∇k = 2 ∇k ln l + d ln . 2N M M
(5.44)
=0
Hence [cf. Eq. (4.60)] 1 N ∗ ∇k k = d ln 2N 2 M
.
(5.45)
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Since ∇l(k, k) = k · ∇k l = −l · ∇k k = −∇k k, l and (l) = l, we deduce from the above relation that 1 N . ∇l(k, k) = − 2 ∇l ln 2N M
(5.46)
Let us now evaluate ∇k l. Contracting the Frobenius relation (5.37) for l with the vector k yields k · dl = ∇k ln(MN)l − l, k d ln(MN) , ∗
=−1
∇l · k − k · ∇l = d ln(MN) , ∇k l = k · ∇l + ∗ d ln(MN) .
(5.47)
Substituting Eq. (5.25) for k · ∇l and using Eq. (5.46) gives 1 N ∇ ln l + ∗ d ln(MN) l 2 2N M
∇k l = − +
.
(5.48)
From this relation and Eqs. (5.46) and (4.60), we get the following expression for the rotation 1-form: = ∗ (d ln(MN) − ∇k l) ,
(5.49)
which clearly shows that the action of vanishes in the direction k. 5.5. Deformation rate of the 2-surfaces St The choice of l as the tangent vector to H’s null generators corresponding to the parameter t (cf. Section 4.2) makes it the natural vector field to describe the evolution of H’s fields with respect to t. Following Damour [60,61], we define the tensor of deformation rate with respect to l of the 2-surface St as half the Lie derivative of St ’s metric q along the vector field l: Q :=
1S 2 Ll q
,
(5.50)
where q is considered as a bilinear form field on St and S Ll is the Lie derivative intrinsic to (St ) which arises from the Lie-dragging of St by l (cf. Section 4.2 and Fig. 9). The precise definition of S Ll is given in Appendix A. The relation with the Lie derivative along l within the manifold M, Ll , is given in 4-dimensional form by Eq. (A.20) q ∗ S Ll q = q ∗ Ll q ,
(5.51)
where we have used q ∗ q = q. Thus Eq. (5.50) becomes q ∗ Q = 21 q ∗ Ll q ,
(5.52)
where q is now considered as a bilinear form on M [as given by Eq. (4.24) or Eq. (4.48)]. Let us evaluate the fourdimensional) Lie derivative in the right-hand side of the above equation, by substituting Eq. (4.48) for q: Ll q = Ll (g + l ⊗ k + k ⊗ l) = Ll g + Ll l ⊗ k + l ⊗ Ll k + Ll k ⊗ l + k ⊗ Ll l .
(5.53)
Since q ∗ l = 0 and q ∗ k = 0, only the term Ll g remains in the right-hand side when applying the operator q ∗ , so that Eq. (5.52) becomes q ∗ Q = 21 q ∗ Ll g .
(5.54)
Now Ll g is the Killing operator applied to the 1-form l : Ll g = ∇ + ∇ . Then from Eq. (5.13) and taking into account the symmetry of , we get q ∗ Q = .
(5.55)
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Replacing Q by its definition (5.50), we conclude that the second fundamental form of H is related to the deformation rate of the 2-surface St by = 21 q ∗ S Ll q = 21 q ∗ Ll q
,
(5.56)
where the second equality follows from Eq. (5.51). Let us consider a coordinate system (x ) = (t, x i ). Then, according to Eq. (4.68), l = t + V + (N − b)s, where t is the coordinate time vector associated with (x ), V = q (l − t), and b is the component of the shift vector of (x ) along the spatial normal s to St . Eq. (5.56) can then be written as = 21 (Lt q + LV q + L(N−b)s q )q q
= 21 {Lt q + V ∇ q + q ∇ V + q ∇ V + (N − b)s ∇ q + q ∇ [(N − b)s ] + q ∇ [(N − b)s ]}q q
= 21 {(Lt q )q q + q q ∇ V + q q ∇ V + (N − b)(q q ∇ s + q q ∇ s )} = 21 [Lt q + ∇ V + ∇ V + (N − b)(∇ s + ∇ s )]q q ,
(5.57)
where the last but one equality results from the identities q s = 0 and q q ∇ q = 0. This last identity follows immediately from Eq. (4.48). Now, similarly to Eq. (3.8) and thanks to the fact that V ∈ T(St ), q q ∇ V = 2 D V ,
(5.58)
where 2 D denotes the covariant derivative in the surface St compatible with the induced metric q. More generally the relation between 2 D derivatives and ∇ derivatives is given by a formula analogous to Eq. (3.8), with the projector simply replaced by the projector q : 2D
T
1 ...p
1 ...q
= q 1 1 · · · q p p q 1 · · · q q q ∇ T 1 ... p 1 ... q 1 q
,
(5.59)
where T is any tensor of type pq lying in St [i.e. such that its contraction with the normal vectors n and s (or l and k) on any of its indices vanishes]. On the other side (∇ s + ∇ s )q q = H + H = 2H ,
(5.60)
where H is the extrinsic curvature of the surface St considered as a hypersurface embedded in the Riemannian space (t , ). H is a symmetric bilinear form which vanishes in the directions orthogonal to St . It will be discussed in a greater extent in Section 10.3.1. In particular formula (5.60) is a direct consequence of Eq. (10.32) established in that section. Thanks to Eqs. (5.58) and (5.60), Eq. (5.57) becomes = 21 [(Lt q )q q + 2 D V + 2 D V ] + (N − b)H
,
(5.61)
or, in index-free notation (cf. the definition (1.27) of the Killing operator): = 21 [ q∗ Lt q + Kil(2 D, V)] + (N − b)H
.
(5.62)
In particular, if (x ) is a coordinate system adapted to H, then N − b = 0 [Eq. (4.79)] and q a = a , so that when restricting Eq. (5.61) to St (i.e. = a ∈ {2, 3}, = b ∈ {2, 3}) one obtains 1 ab = 2
jqab 2 + D a Vb + 2 D b Va jt
which agrees with Eq. (I.52b) of Damour [60].
,
(5.63)
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5.6. Expansion scalar and shear tensor of the 2-surfaces St Let us split the second fundamental form (now considered as the deformation rate of the 2-surfaces St ) into a trace part and a traceless part with respect to St ’s metric q = 21 q +
,
(5.64)
where
= = g = q = q ab ab = a := tr
a
(5.65)
canonically associated with by the metric g [see also Eq. (5.22)] and is the is the trace of the endomorphism traceless part of := − 21 q ,
(5.66)
which satisfies = q = a a = 0 .
(5.67)
The trace is called the expansion scalar of St and the shear tensor of St . The expansion is linked to the divergence of l; indeed taking the trace of Eq. (5.21) results in ∇ · l = + , l − l, ∇k l , =
(5.68)
=0
hence ∇·l=+
.
(5.69)
Another relation is obtained by combining = q with the expression (5.9) for : = q q q ∇ = q q ∇ = q ∇ ,
(5.70)
i.e. = q ∇
.
(5.71)
Another expression of is obtained as follows. Let (x ) be a coordinate system adapted to H; (x a )a=2,3 is then a coordinate system on St . We have = q ab ab [cf. Eq. (5.65)] and let us use Eq. (5.56) restricted to T(St ), i.e. under the form ab = 1/2S Ll qab . We get = 21 q ab S Ll qab = 21 S Ll ln q ,
(5.72)
where q is the determinant of the components qab of the metric q with respect to the coordinates (x a ) in St : q := det qab
.
(5.73)
The second equality in Eq. (5.72) follows from the standard formula for the variation of a determinant. Hence we have = S Ll ln
√ q
.
(5.74)
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This relation justifies the name of expansion scalar given to , for √ 2 = qdx 2 ∧ dx 3 .
√
205
q is related to the surface element 2 of St by (5.75)
Another expression of is obtained by contracting Eq. (5.63) with q ab : =
j √ ln q + 2 D a V a jt
.
(5.76)
More generally, contracting Eq. (5.61) with q leads to = q Lt q + 2 D a V a + (N − b)H
,
(5.77)
where H is twice St ’s mean curvature within (t , ) [Eq. (10.41) below]. Remark 5.2. Eqs. (5.74) and (5.76), by relating to the rate of expansion of the 2-surfaces St , might suggest that the scalar depends quite sensitively upon the foliation of spacetime by the spacelike hypersurfaces t , since the surfaces St are defined by this foliation. Actually the dependence is pretty weak: as shown by Eq. (5.69), depends only upon the null normal l to H (since depends only upon l). Hence the dependence of with respect to foliation (St ) is only through the normalization of l induced by the (St ) slicing and not on the precise shape of this slicing. 5.7. Transversal deformation rate By analogy with the expression (5.56) of , we define the transversal deformation rate of the 2-surface St as the projection onto T(St ) of the Lie derivative of St ’s metric q along the null transverse vector k: := 21 q ∗ Lk q
.
(5.78)
Remark 5.3. In the above definition q is considered as the four-dimensional bilinear form given by Eq. (4.24) or Eq. (4.48), rather than as the two-dimensional metric of St , and Lk q is its Lie derivative within the four-manifold M. Indeed since the vector field k does not Lie drag the surfaces (St ), we do not have an object such as the two-dimensional Lie derivative “S Lk ” (the analog of S Ll ) which could have been applied to the two-metric of St in the strict sense. From its definition, it is obvious that is a symmetric bilinear form. Replacing q by its expression (4.48), we get = 21 q ∗ Lk (g + l ⊗ k + k ⊗ l)
= 21 q ∗ (Lk g + Lk l ⊗ k + l ⊗ Lk k + Lk k ⊗ l + k ⊗ Lk l) .
(5.79)
Since q ∗ l = 0 and q ∗ k = 0, only the term Lk g remains in the right-hand side after the operator q ∗ has been applied. Now Lk g is nothing but the Killing operator applied to the 1-form k, so that the above equation becomes 1 1 1 = q ∗ Lk g = q ∗ Kil(∇, k) = q ∗ ∇k + dk 2 2 2 1 N 1 ∗ N ∗ ∗ = q ∇k + q d ln ∧ l = q ∇k + ∧ q ∗ l , d ln 2 2 4N M 4N M =0
where use has been made of the Frobenius identity (5.39) to get the second line. We conclude that = q ∗ ∇k
,
which is an expression completely analogous to expression (5.10) for in terms of ∇l.
(5.80)
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Remark 5.4. The metric q induced by g on the 2-surface St is a Riemannian metric (i.e. positive definite) (cf. Section 4.4). It is called the first fundamental form of St and describes fully the intrinsic geometry of St . The way St is embedded in the spacetime (M, g) constitutes the extrinsic geometry of St . For a non-null hypersurface of M, this extrinsic geometry is fully described by a single bilinear form, the so-called second fundamental form (for instance K for the hypersurface t ). For the two-dimensional surface St , a part of the extrinsic geometry is described by a normal fundamental form, like the Há´"iˇcek 1-form as discussed in Section 5.2. The remaining part is described by a type (1,2) tensor: the second fundamental tensor K [41,43] (also called shape tensor [149]), which relates the covariant derivative of a vector tangent to St taken with the spacetime connection ∇ to that taken with the connection 2 D in St compatible with the induced metric q: ∀(u, v) ∈ T(St )2 ,
∇u v = 2 D u v + K(u, v) .
(5.81)
It is easy to see that K is related to the spacetime derivative of q by K = q q ∇ q .
(5.82)
K is tangent to St with respect to the indices and and orthogonal to St with respect to the index . Moreover, it is symmetric in and [although this is not obvious on Eq. (5.82)]. From Eqs. (4.49), (5.13) and (5.80), we have K = k + .
(5.83)
Accordingly, the bilinear forms and can be viewed as two facets of the same object: the second fundamental tensor K: = − K
and = −k K .
(5.84)
By substituting Eq. (4.49) for the projector q in Eq. (5.80), using the identity ∇ k = −k ∇ (which follows from l · k = −1), expressing ∇ via Eq. (5.21) and using Eqs. (5.35) and (5.41), we get the following expression for the spacetime derivative of the 1-form k, in terms of and the Há´"iˇcek 1-form : ∇ k = − k − k ∇ k − k ,
(5.85)
or, taking into account the symmetry of , ∇k = − k ⊗ − ∇k k ⊗ l − ⊗ k
.
(5.86)
Similarly to the definition of the expansion scalar as the trace of the deformation rate [cf. Eqs. (5.65) and (5.71)], we define the transversal expansion scalar (k) as the trace of :
=
= g = q = q ab ab = q ∇ k (k) := tr
.
(5.87)
Remark 5.5. The reader will have noticed a certain dissymmetry in our notations, since we use (k) for the expansion of the null vector k and merely the expansion of the null vector l. From the point of view of the two-dimensional spacelike surface St , l and k play perfectly symmetric roles, l (resp. k) being the unique – up to some rescaling – outgoing (resp. ingoing) null normal to St . However, l is in addition normal to the null hypersurface H, whereas k has not any specific relation to H. In particular, there is not a unique transverse null direction to H, so that k is defined only thanks to the extra-structure (St ). The dissymmetry in our notations accounts therefore for the privileged status of l with respect to k. Example 5.6 (, , and for a Minkowski light cone). Let us proceed with Example 4.5, namely a light cone in Minkowski spacetime, sliced according to the standard Minkowskian time coordinate t. Comparing the components of ∇l given by Eq. (2.29) with those of q given by Eq. (4.37), we realize that 1 ∇l = q . r
(5.88)
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207
The second fundamental form = q ∗ ∇l [Eq. (5.10)] follows then immediately: 1 = q . r
(5.89)
We deduce from this relation and Eq. (5.64) that the expansion scalar is 2 r and the shear tensor vanishes identically: =
(5.90)
=0 .
(5.91)
Note that > 0, in accordance with the fact that the light cone is expanding. From expression (5.88) and the orthogonality of q with k, we deduce by means of Eq. (5.25) that the rotation 1-form vanishes identically: =0 .
(5.92)
Consequently, its pull-back on St , the Há´"iˇcek 1-form, vanishes as well: =0 .
(5.93)
Similarly, from expression (4.36) for k, we get ∇k = −1/(2r)q. From this relation and Eq. (5.80), we deduce that the transversal deformation rate has the simple expression 1 q, 2r on which we read immediately the transversal expansion scalar: =−
(k) = −
1 . r
(5.94)
(5.95)
Example 5.7 (, , and associated with the Eddington–Finkelstein slicing of Schwarzschild horizon). Let us continue the Example 4.6 about the event horizon of Schwarzschild spacetime, with the 3 + 1 slicing provided by Eddington–Finkelstein coordinates. The second fundamental form is obtained from Eqs. (5.13), (2.42) and (4.40): r − 2m r − 2m (5.96) r, r sin2 . = diag 0, 0, r + 2m r + 2m Accordingly, the expansion scalar is =
2 r − 2m r r + 2m
(5.97)
and the shear tensor vanishes identically = 0 . The transversal deformation rate is deduced from Eq. (5.80) and expression (4.39) for k: r + 2m 2 r + 2m ,− sin , = diag 0, 0, − 2 2
(5.98)
(5.99)
so that the transversal expansion scalar is 1 2m (k) = − − 2 . r r
(5.100)
The rotation 1-form is obtained from Eq. (5.25) combined with expression (2.42) for ∇l and expression (4.39) for k: 2m 2m , , 0, 0 . (5.101) = r(r + 2m) r(r + 2m)
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We deduce immediately from this expression and Eq. (5.29) that the Há´"iˇcek 1-form vanishes identically: =0 .
(5.102)
As a check, we verify that, from the obtained values for , , and k, Eq. (5.35) is satisfied. It is of course instructive to specify the above results on the event horizon H (r = 2m): H
= 0,
H
= 0,
1 q, 2m 1 H k, =− 4m H
=−
H
=0 , H
(k) = −
(5.103)
1 , m
(5.104)
H
=0 .
(5.105)
Note that for a rotating black hole, described by the Kerr metric, is no longer zero, as shown in Appendix D. 5.8. Behavior under rescaling of the null normal As stressed in Section 2, from the null structure only, the normal l to the hypersurface H is defined up to some normalization factor (cf. Remark 2.2), i.e. one can change l to l = l ,
(5.106)
where is any strictly positive scalar field on H ( > 0 ensures that l is future oriented). In the present framework, the extra-structure on H induced by the spacelike foliation t of the 3 + 1 formalism provides a way to normalize l: we have demanded l to be the tangent vector corresponding to the parametrization by t of the null geodesics generating H [cf. Eq. (4.5)], or equivalently that l be a dual vector to the gradient dt of the t field [cf. Eq. (4.6)]. It is however instructive to examine how the various quantities introduced so far change under a rescaling of the type (5.106). In Section 2, we have already exhibited the behavior of the non-affinity parameter [cf. Eq. (2.26)], as well as of the Weingarten map and the second fundamental form, both restricted to H [cf. Eqs. (2.48) and (2.57)]. In view of Eq. (4.28) the scaling properties of the transverse null vector k are simply k = −1 k. From the expression (4.55) of the projector onto H along k, in conjunction with l = l and k = −1 k, we get that is invariant under the rescaling (5.106): = .
(5.107)
This is not surprising since |Tp (H) is the identity and therefore does not depend upon l. Similarly the orthogonal projector q onto St does not depend upon l [this is obvious from its definition and is clear in expression (4.49)] so that q = q .
(5.108)
From its definition (5.1) and the scaling properties (2.48) and (5.107), we get the following scaling behavior of the extended Weingarten map = + d, (·)l ,
(5.109)
where the notation d, (·)l stands for the endomorphism Tp (M) −→ Tp (M), v −→ d, (v)l. From its definition (5.6) and the scaling properties (2.57) and (5.107), we get the following scaling behavior of the extended second fundamental form of H with respect to l: = .
(5.110)
The scaling property of the rotation 1-form is deduced from its definition (5.17) and the scaling law (5.109) for : ∀ v ∈ Tp (M),
, v = − k · (v) = −−1 k · [(v) + d, (v)l] = − k · (v) − −1 d, (v) k·l = , v + d ln , (v) .
=−1
(5.111)
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Table 1 Behavior under a rescaling l → l = l of H’s null normal
l = l = ( + ∇l ln )
=
k = k =
=
=
1
= 1
q = q
= + d, (·)l = + ∗ d ln = + 2 D ln
Hence = + ∗ d ln .
(5.112)
Since = q ∗ [Eq. (5.29)], the scaling law for the Há´"iˇcek 1-form is immediate: = + 2 D ln .
(5.113)
The scaling properties of the expansion scalar and the shear tensor are deduced from that of via their definition (5.65) and (5.66): = and
= .
(5.114)
Finally the scaling law for the transversal expansion rate is easily deduced from the Eq. (5.80) and the scaling laws k = −1 k and (5.108): = −1 .
(5.115)
For further reference, the various scaling laws are summarized in Table 1. 6. Dynamics of null hypersurfaces In the previous section, we have considered only first order derivatives of the null vector fields l and k, as well as of the metric q. In the present section we consider second order derivatives of these fields. Some of these second order derivatives are written as Lie derivatives along l of the first order quantities, like the second fundamental form , the Há´"iˇcek 1-form and the transversal deformation rate . The obtained equations can be then qualified as evolution equations along the future directed null normal l (cf. the discussion at the beginning of Section 5.5). Some other second order derivatives of l and k are rearranged to let appear the spacetime Riemann tensor, via the Ricci identity (1.14). The totality of the components of the Riemann tensor with respect to a tetrad adapted to our problem, i.e. involving l, k, and two vectors tangent to St , are derived in Appendix B. Here we will focus only on those components related to the evolution of , and . Some of the obtained evolution equations involve the Ricci part of the Riemann tensor. At this point, the Einstein equation enters into scene in contrast with all results from previous sections (except Section 3.6), which are independent of whether the spacetime metric g is a solution or not of Einstein equation. This concerns the evolution equations for the expansion scalar , the Há´"iˇcek 1-form and the transversal expansion rate . On the contrary the evolution equation for the shear tensor involves only the traceless part of the Riemann tensor, i.e. the Weyl tensor, and consequently is independent from the Einstein equation. 6.1. Null Codazzi equation Let us start by deriving the null analog of the contracted Codazzi equation of the spacelike 3 + 1 formalism, i.e. Eq. (3.34) presented in Section 3.5. The idea is to obtain an equation involving the quantity R , which is similar
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to the left-hand side of Eq. (3.34) with the normal n replaced by the normal l, and the projector replaced by the projector . Remark 6.1. We must point out that, from the very fact that l is simultaneously normal and tangent to H, the standard classification in terms Codazzi and Gauss equations employed in Section 3.5, is not completely adapted to the present case. In particular, the trace of R , associated with the contracted Codazzi equation, can also be interpreted as a component of the null analog of the contracted Gauss equation, as we shall see in Section 6.4. The starting point for the null contracted Codazzi equation is the Ricci identity (1.14) applied to the null normal l. Contracting this identity on the indices and , we get: ∇ ∇ − ∇ ∇ = R ,
(6.1)
where R is the Ricci tensor of the connection ∇ [cf. Eq. (1.18)]. Substituting Eq. (5.21) for ∇ and Eq. (5.69) for ∇ , yields ∇ [ + − k ∇ ] − ∇ ( + ) = R .
(6.2)
Expanding the left-hand side and using again Eqs. (5.21) and (5.69) leads to R = ∇ + ∇ + ( + ) − ∇ ( + ) − k ∇
− ( k ∇ + ∇ k ∇ + k ∇ ∇ ) .
(6.3)
The null contracted Codazzi equation is the contraction of this equation with the projector onto H. A difference with the spacelike case is that this projection can be divided in two pieces: a projection along l itself, since the normal l is also tangent to H, and a projection onto the 2-surfaces St . This is clear if one expresses the projector in terms of the orthogonal projector q via Eq. (4.57): R = −R k + R q .
(6.4)
We will examine the two parts successively: the first one, R , will provide the null Raychaudhuri equation (Section 6.2), whereas the second one, R q will lead to an evolution equation for the Há´"iˇcek 1-form which is analogous to a two-dimensional Navier–Stokes equation (Section 6.3). 6.2. Null Raychaudhuri equation The first part of the null contracted Codazzi equation is the one along l. It is obtained by contracting Eq. (6.3) with : R = ∇ + ∇ + ( + ) − ∇ ( + ) .
(6.5)
Taking into account the identities = 0 [Eq. (5.14)], = [Eq. (5.28)], ∇ = [Eq. (2.21)] and expression (5.21) for ∇ , we get R = − + − ∇ .
(6.6)
As shown in Appendix B, this relation can also been obtained by computing the components of the Ricci tensor from the curvature 2-forms and Cartan’s structure equations [cf. Eq. (B.39)]. We may express in terms of the shear and the expansion scalar , thanks to Eqs. (5.64) and (5.67): = + 21 2 = ab ab + 21 2 ,
(6.7)
to get finally ∇l − + 21 2 + ab ab + R(l, l) = 0
.
(6.8)
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This is the well-known Raychaudhuri equation for a null congruence with vanishing vorticity or twist, i.e. a congruence which is orthogonal to some hypersurface (see e.g. Eq. (4.35) in Ref. [90]11 or Eq. (2.21) in Ref. [40]). If one takes into account the Einstein equation (3.35), the Ricci tensor R can be replaced by the stress-energy tensor T (owing to the fact that l is null): ∇l − + 21 2 + ab ab + 8T(l, l) = 0
.
(6.9)
6.3. Damour–Navier–Stokes equation Let us now consider the second part of the Codazzi equation R = · · ·, i.e. the part lying in the 2-surface St [second term in the right-hand side of Eq. (6.4)]. It is obtained by contracting Eq. (6.3) with q : R q = q ∇ + q ∇ + ( + ) − 2 D ( + ) − k ∇ .
(6.10)
The first term on the right-hand side is related to the divergence of with respect to the connection 2 D in St by q ∇ = 2 D + (k ∇ + ∇ k ) = 2 D + (k ∇ + ) .
(6.11)
The first line results from relation (5.59) between the derivatives 2 D and ∇ for objects living on St , whereas the second line follows from Eqs. (5.41) and (5.29). Besides, the second term on the right-hand side of Eq. (6.10) can be expressed as [cf. Eq. (5.35)] q ∇ = q ∇ ( − k ) = q ( ∇ − ∇ k ) = q (Ll − ∇ − ) = q Ll − − ,
(6.12)
where, to get the last line, use has been made of Eq. (5.13) to let appear and of Eqs. (5.41) and (5.29) to let appear . Inserting expressions (6.11) and (6.12) in Eq. (6.10) results immediately in R q = q Ll + − 2 D ( + ) + 2 D .
(6.13)
An alternative derivation of this relation, based on Cartan’s structure equations, is given in Appendix B [cf. Eq. (B.47)]. Expressing in terms of the expansion scalar and the shear tensor [Eq. (5.64)], we get
2 (6.14) + 2 D . R q = q Ll + − D + 2 Taking into account the Einstein equation (3.35), the Ricci tensor can be replaced by the stress-energy tensor (owing to the fact that g q = 0) to write Eq. (6.14) as an evolution equation for the Há´"iˇcek 1-form: q Ll + = 8T q + 2 D + − 2 D 2
q ∗ L
l
+ = 8 q∗ T · l + 2 D
+ 2
.
(6.15)
− 2D ·
.
(6.16)
The components = a ∈ {2, 3} of this equation agree with Eq. (I.30b) of Damour [60]. Eq. (6.15) can also be compared with Eq. (2.14) of Price and Thorne [141], after one has noticed that their operator Dt¯ acting on the Há´"iˇcek 1-form is Dt¯ = q ∇ and therefore is related to our Lie derivative along l by the relation Dt¯ = q Ll − , 11 Eq. (4.35) in Ref. [90] assumes = 0.
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which can be deduced from (our) Eq. (6.12). Then the components = a ∈ {2, 3} of (our) Eq. (6.15) coincide with their Eq. (2.14), called by them the “Há´"iˇcek equation”. Let (x ) be a coordinate system adapted to H; (x a )a=2,3 is then a coordinate system on St . We can write Ll = Lt + LV , where t is the coordinate time vector associated with (x ) and V ∈ T(St ) is the surface velocity of H with respect to (x ) [cf. Eq. (4.80)]. The projection of this relation onto St gives q a Ll =
ja + V b2 D b a + b 2 D a V b . jt
(6.17)
Inserting this relation into Eq. (6.15) yields ja 1 + V b2 D b a + b 2 D a V b + a = 8q a T + 2 D a − 2 D b b a + 2 D a jt 2
,
(6.18)
with, of course, T = 0 in the vacuum case. Noticing that fa := −q a T
(6.19)
is a force surface density (momentum per unit surface of St and per unit coordinate time t), Damour [60,61] has interpreted Eq. (6.18) as a two-dimensional Navier–Stokes equation for a viscous “fluid”. The Há´"iˇcek 1-form is then interpreted as a momentum surface density (up to a factor −8): a := −
1 a . 8
(6.20)
V a represents then the (two-dimensional) velocity of the “fluid”, /(8) the “fluid” pressure, 1/(16) the shear viscosity (ab is then the shear tensor) and −1/(16) the bulk viscosity. This last fact holds for is the divergence of the velocity field in the stationary case: consider Eq. (5.76) with j/jt = 0. We refer the reader to Chapter VI of the Membrane Paradigm book [165] for an extended discussion of this “viscous fluid” viewpoint. 6.4. Tidal-force equation The null Raychaudhuri equation (6.8) has provided an evolution equation for the trace of the second fundamental form . Let us now derive an evolution equation for the traceless part of , i.e. the shear tensor . For this purpose we evaluate ∇ (∇ ) in two ways. Firstly, we express it in terms of the Riemann tensor by means of the Ricci identity (1.14): ∇ (∇ ) = (R + ∇ ∇ ) .
(6.21)
Making repeated use of Eq. (5.21) to expand ∇ and employing = 0 we find ∇ (∇ )= R − + − [k ∇ −k (∇ ) ]+ [ ∇ + ] .
(6.22)
On the other hand, using directly Eq. (5.21) to expand ∇ we find ∇ (∇ ) = ∇ − [(k ∇ ) + ∇ (k ∇ )] + [ + ∇ ] .
(6.23)
From both expressions for ∇ (∇ ), and projecting on St we obtain q q ∇ = − − q q R ,
(6.24)
− q ∗ Riem(l, ., l, .) . q ∗ ∇l = − ·
(6.25)
i.e.
Now, expressing the Lie derivative Ll in terms of the covariant derivative ∇ and using = q ∗ ∇l [Eq. (5.10)], we find the relation
, q ∗ Ll = q ∗ ∇l + 2 ·
(6.26)
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so that Eq. (6.25) becomes
− q ∗ Riem(l, ., l, .) q ∗ Ll = + ·
.
(6.27)
An alternative derivation of this relation, based on Cartan’s structure equations, is given in Appendix B [cf. Eq. (B.32)]. The equation equivalent to (6.27) in the framework of the quotient formalism (cf. Remark 2.8) is called a ‘Ricatti equation’ by Galloway [73], by analogy with the classical Ricatti ODE: y = a(x)y 2 + b(x)y + c(x). See also Refs. [101,111]. Expressing the four-dimensional Riemann tensor in terms of the Weyl tensor C (its traceless part) and the Ricci tensor R via Eq. (1.19), Eq. (6.27) becomes
− q ∗ C(l, ., l, .) − 1 R(l, l)q q ∗ Ll = + · 2
(6.28)
or q q (Ll ) = + − q q C − 21 (R )q
,
(6.29)
where we have made use of q · l = 0 and l · l = 0. Taking the trace of Eq. (6.28) and making use of Eq. (6.7), results immediately in an evolution equation for the expansion scalar , which is nothing but the Raychaudhuri Eq. (6.8). From the components of the Riemann tensor appearing in Eq. (6.27), Eq. (6.28) could have been considered as the projection on St of the null analog of the 3 + 1 Ricci equation (3.31). Thus the Raychaudhuri equation (6.8) can be derived either from the null Codazzi equation as in Section 6.2, or from the null Ricci equation. This reflects the fact that the Gauss–Codazzi–Ricci terminology is not well adapted to the null case, as anticipated in Remark 6.1. On the other hand, the traceless part of Eq. (6.28) results in the evolution equation for the shear tensor: q ∗ Ll = + ab ab q − q ∗ C(l, ., l, .)
(6.30)
or q q (Ll ) = + q − q q C
,
(6.31)
where we have used the fact that for a two-dimensional symmetric tensor we have = 21 q . In particular,
we find the equivalent expression from q ∗ Ll = q ∗ ∇ + 2 · , q q ∇ − ( − ) = −q q C ,
(6.32)
which coincides with Eq. (2.13) of Price and Thorne [141], once we identify q q ∇ with Dt¯ in that reference. This equation is denominated tidal equation in Ref. [141], since the term in the right-hand side is directly related to the driving force responsible for the relative acceleration between two null geodesics via the geodesic deviation equation (see e.g. Ref. [167]). In other words, this force is responsible for the tidal forces on the 2-surface St . The tidal equation (6.30) and the null Raychaudhuri equation (6.8) are part of the so-called optical scalar equations derived by Sachs within the Newman–Penrose formalism [143]. 6.5. Evolution of the transversal deformation rate Let us consider now an equation that can be seen as the null analog of the contracted Gauss (3.32) combined with the Ricci (3.31). It is obtained by projecting the spacetime Ricci tensor R onto the hypersurface H. The difference with the spacelike case of Section 3.5 is that this projection is not an orthogonal one, but instead is performed via the projector along the transverse direction k.
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Remark 6.2. The projection ∗ R of the spacetime Ricci tensor onto H [as defined by Eq. (4.58)] can be decomposed in the following way, thanks to Eq. (4.57): R = R q q − R q k − R q k + R k k . Codazzi 2 Codazzi 1 Codazzi 2
(6.33)
We notice that among the four parts of this decomposition, three of them are parts of the Codazzi equation and have been already considered as the Raychaudhuri equation (Codazzi 1, Section 6.2) or the Damour–Navier–Stokes equation (Codazzi 2, Section 6.3). Thus the only new piece of information contained in the null analog of the contracted Gauss equation is q ∗ R, namely the (orthogonal) projection of the Ricci tensor onto the 2-surfaces St . This contrasts with the spacelike case of the standard 3 + 1 formalism, where the contracted Gauss equation (3.32) is totally independent of the Codazzi equation (3.30). In order to evaluate q ∗ R, let us start by the contracted Ricci identity applied to the connection 2 D induced by the spacetime connection ∇ onto the 2-surfaces St : 2
D 2 D v − 2 D 2 D v =2 R v ,
(6.34)
where v is any vector field in T(St ) and 2 R is the Ricci tensor associated with 2 D. Expressing each 2 D derivative in terms of the spacetime derivative ∇ via Eq. (5.59) and substituting Eq. (4.49) for the projector q leads to 2
R v = [q (k + (k) ) − k − ]∇ v + q q (∇ ∇ v − ∇ ∇ v ) ,
(6.35)
where use has been made of Eqs. (5.13), (5.20), (5.65), (5.80), (5.85) and (5.87) to let appear , , and (k) . Now the four-dimensional Ricci identity (1.14) applied to the vector field v yields q q (∇ ∇ v − ∇ ∇ v ) = q q R v = q q R q v ,
(6.36)
where R denotes the spacetime Riemann curvature tensor. Moreover, l · v = 0 and k · v = 0 [since v ∈ T(St )], so that we can transform Eq. (6.35) into 2
R v = [− − (k) + + ]v + q q q R v .
(6.37)
Since this identity is valid for any vector v ∈ T(St ), we deduce the following expression of the Ricci tensor of the two-dimensional Riemannian spaces (St , q) in terms of the Riemann tensor of (M, g), the second fundamental form of H and the transversal deformation rate : 2
R = q q q R − − (k) + + .
(6.38)
Let us now express the term q q q R in terms of the spacetime Ricci tensor R . We have, using the symmetries of the Riemann tensor and the Ricci identity (1.14) for the vector field k q q q R = q q ( + k + k )R
= q q (R − R k − R k )
= q q [R − (∇ ∇ k − ∇ ∇ k ) − (∇ ∇ k − ∇ ∇ k )]
= q q [R − ∇ ∇ k +∇ −∇ ∇ k − ∇ ∇ k +∇ −∇ ∇ k ] ,
(6.39)
where use has been made of the relation ∇ k = [Eq. (5.41)]. After expanding the gradient of k by means of Eq. (5.86) and the gradient of l by means of Eq. (5.21), we arrive at q q q R = q q (R − 2 ∇ + ∇ + ∇ ) + 2 − − .
(6.40)
Now, by means of Eqs. (5.35) and (5.59), q q (∇ + ∇ ) = 2 D + 2 D − 2 .
(6.41)
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Inserting this relation along with ∇ = Ll − ∇ − ∇ into Eq. (6.40) results in q q q R = q q (R − 2Ll ) + 2 D + 2 D + 2 − 2 + + .
(6.42)
Replacing into Eq. (6.38) leads to the following evolution equation for : 1 1 1 q q Ll = (2 D + 2 D ) + − 2 R + q q R 2 2 2 (k) − + − + + . 2 2 or (k) 1 1 1
+·
. q ∗ Ll = Kil(2 D, ) + ⊗ − 2 R + q ∗ R − + − +· 2 2 2 2 2
(6.43)
(6.44)
Remark 6.3. It is legitimate to compare Eq. (6.44) with Eq. (B.57) derived in Appendix B by means of Cartan’s structure equations, since both equations involve q ∗ Ll , q ∗ R and 2 R. The major difference is that Eq. (B.57) involves in addition the Lie derivative Lk . Actually Eq. (B.57) is completely symmetric between l and k (and hence between and ). This reflects the fact that q ∗ R and 2 R depend only upon the 2-surface St and, from the point of view of St alone, l and k are on the same footing, being respectively the outgoing and ingoing null normals to St . However, in the derivation of Eq. (6.44), we have broken this symmetry, which is apparent in Eq. (6.38), at the step (6.39) by rearranging terms in order to consider the Ricci identity for the vector k only. Actually by a direct computation (substituting Eq. (5.19) for and permuting the derivatives of l via Ricci identity), one gets the following relation between the two Lie derivatives: q ∗ Lk = q ∗ Ll + 2 D 2 D + 2 D ⊗ 2 D − ⊗ 2 D − 2 D ⊗ − Kil(2 D, ) + N −2 ∇l + . (6.45) Substituting this expression for q ∗ Lk into Eq. (B.57), we recover Eq. (6.44). If we take into account Einstein equation (3.35), the four-dimensional Ricci term can be written q ∗ R = 8( q∗ T −
is the trace of the energy–momentum tensor T. The evolution equation for becomes then 1/2T q), where T = tr T 1 1 T Kil(2 D, ) + ⊗ − 2 R + 4 q ∗ T − q 2 2 2 (k)
+·
. − + − +· 2 2
q ∗ Ll =
(6.46)
Example 6.4 (“Dynamics” of Minkowski light cone). As a check of the above dynamical equations, let us specify them to the case where H is a light cone in Minkowski spacetime, as considered in Examples 2.4, 4.5, and 5.6. Since = 0, = 0 and T = 0 for this case [Eqs. (2.30) and (5.91)], the null Raychaudhuri equation (6.9) reduces to ∇l + 21 2 = 0 .
(6.47)
Using the values = (1, x/r, y/r, z/r) and = 2/r given respectively by Eqs. (2.28) and (5.90), we check that the above equation is satisfied. Besides, since = 0 in this case [Eq. (5.93)], the Damour–Navier–Stokes equation (6.16) reduces to 2
D = 0 .
(6.48)
= 0, i.e. the Since = 2/r is a function of r only and r is constant on St (being equal to t), we have Damour–Navier–Stokes equation is fulfilled. The tidal force equation (6.30) is trivially satisfied in the present case since both the shear tensor and the Weyl tensor C vanish. On the other hand, the evolution equation for , Eq. (6.46), reduces somewhat, but still contains many non-vanishing terms: 2 D
q ∗ Ll = −
(k) 12
+·
. R− − +· 2 2 2
(6.49)
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Using the value of given by Eq. (5.94) allows to write the left-hand side as ⎤ ⎡ ⎥ 1⎢ 1 1 1 q ∗ Ll = q ∗ Ll − q = − ⎢ Ll q + q ∗ Ll q⎥ ⎣ 2r 2 r r ⎦
=2= 2 q
r 1 j 2 1 1 = − + 2 q=− 2q ,
2 jx r r 2r
(6.50)
where we have used expressions (5.89) for and (2.28) for . Besides, since St is a two-dimensional manifold, the Ricci tensor 2 R which appears on the right-hand side of Eq. (6.49) is expressible in terms of the Ricci scalar 2 R as 2 R = 1 2 Rq. Moreover, S being a metric 2-sphere of radius r, 2 R = 2/r 2 . Thus t 2 2
R=
1 q. r2
(6.51)
Inserting Eqs. (6.50) and (6.51) as well as the values of , , and (k) obtained in Example 5.6 into Eq. (6.49), and using q · q = q, leads to “0 = 0”, as it should be. Example 6.5 (Dynamics of Schwarzschild horizon). In view of the values obtained in Example 5.7 for , , and corresponding to the Eddington–Finkelstein slicing of the event horizon of Schwarzschild spacetime, let us specify the dynamical equations obtained above to that case. First of all, the Ricci tensor and the stress-energy tensor vanish identically, since we are dealing with a vacuum solution of Einstein equation: R = 0 and T = 0. Taking into account H
H
= 0 and = 0 [Eq. (5.103)], the null Raychaudhuri equation (6.9) is then trivially satisfied on H. Similarly, since H
H
= 0 [Eq. (5.105)] and = 1/(4m) is a constant [Eq. (2.45)], the Damour–Navier–Stokes equation (6.16) is trivially satisfied on H. On the other side, the tidal force equation (6.30) reduces to q ∗ C(l, ., l, .) = 0 .
(6.52)
This constraint on the Weyl tensor is satisfied by the Schwarzschild solution, as a consequence of being of Petrov type D and l a principal null direction (see e.g. Proposition 5.5.5 in Ref. [130]). Finally Eq. (6.46) giving the evolution of H
H
the transversal deformation rate reduces to (since T = 0, = 0 and = 0) H
q ∗ Ll = − 21 2 R − .
(6.53) H
H
H
Now from Eq. (5.104), we have q ∗ Ll = −(2m)−1 q ∗ Ll q = −m−1 = 0, hence H
q ∗ Ll = 0 .
(6.54)
H
On the other side, = 1/(4m) [Eq. (2.45)] and expression (5.104) for leads to H
= −
1 q. 8m2
(6.55) H
Besides, since St is a metric 2-sphere, as in Example 6.4 above, Eq. (6.51) holds. Since r = 2m, it yields 2
H
R=
1 q. 4m2
(6.56)
Gathering Eqs. (6.54)–(6.56), we check that Eq. (6.53) is satisfied. 7. Non-expanding horizons All results presented in the previous sections apply to any null hypersurface and are not specific to the event horizon of a black hole. For instance, they are perfectly valid for a light cone in Minkowski spacetime, as illustrated by Examples
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2.4, 4.5, 5.6 and 6.4. In this section, we move on the way to (quasi-equilibrium) black holes by requiring the null hypersurface H to be non-expanding, in the sense that the expansion scalar defined in Section 5.6 is vanishing. Indeed, one should remind that measures the rate of variation of the surface element of the spatial 2-surfaces St foliating H [cf. Eqs. (5.74) or (5.76)]. We have seen in Example 5.7 that = 0 for the event horizon of a Schwarzschild black hole [cf. Eq. (5.103)]. On the contrary, in any weak gravitational field, the null hypersurfaces with compact spacelike sections are always expanding or contracting (cf. Example 5.6 for the light cone in flat spacetime). Therefore the existence of a non-expanding null hypersurface H with compact sections St is a signature of a very strong gravitational field. As we shall detail below, non-expanding horizons are closely related to the concept of trapped surfaces introduced by Penrose in 1965 [132] and the associated notion of apparent horizon [90]. They constitute the first structure in the hierarchy recently introduced by Ashtekar et al. [10–13,15,18] which leads to isolated horizons. Contrary to event horizons, isolated horizons constitute a local concept. Moreover, contrary to Killing horizons—which are also local —, isolated horizons are well defined even in the absence of any spacetime symmetry. 7.1. Definition and basic properties 7.1.1. Definition Following Há´"iˇcek [82,83] and Ashtekar et al. [15,13], we say that the null hypersurface H is a non-expanding horizon (NEH)12 if, and only if, the following properties hold13 (1) H has the topology of R × S2 ; (2) the expansion scalar introduced in Section 5.6 vanishes on H: H
=0
;
(7.1)
(3) the matter stress-energy tensor T obeys the null dominant energy condition on H, namely the “energy–momentum current density” vector
·l W := −T
(7.2)
is future directed timelike or null on H. Let us recall that, although can be viewed as the rate of variation of the surface element of the spatial 2-surfaces St foliating H [cf. Eqs. (5.74) or (5.76)], it does not depend upon St but only on l (cf. Remark 5.2). Moreover, thanks to the behavior → = [Eq. (5.114)] under the rescaling l → l = l, the vanishing of does not depend upon the choice of a specific null normal l. Similarly the property (3) does not depend upon the choice of the null normal l, provided that it is future directed. In other words, the property of being a NEH is an intrinsic property of the null hypersurface H. In particular it does not depend upon the spacetime foliation by the hypersurfaces (t ). Remark 7.1. The topology requirement on H is very important in the definition of an NEH, in order to capture the notion of black hole. Without it, we could for instance consider for H a null hyperplane of Minkowski spacetime. Indeed let t − x = 0 be the equation of this hyperplane in usual Minkowski coordinates (x ) = (t, x, y, z). The components of the null normal l with respect to these coordinates are then = (1, 1, 0, 0), so that ∇l = 0. Consequently, H fulfills condition (2) in the above definition: = 0, although H has nothing to do with a black hole. Remark 7.2. The null dominant energy condition (3) is trivially fulfilled in vacuum spacetimes. Moreover, in the non-vacuum case, this is a very weak condition, which is satisfied by any electromagnetic field or reasonable matter model (e.g. perfect fluid). In particular, it is implied by the much stronger dominant energy condition, which says that energy cannot travel faster than light (see e.g. the textbook [90, p. 91], or [167, p. 219]). 12 Há´"iˇcek [83] used the term “perfect horizon” instead of “non-expanding horizon”.
13 In this review we are working with metrics satisfying Einstein equation on the whole spacetime M, and in particular on H (this has been fully employed in Section 6). In more general contexts, the NEH definition must also include the enforcing of the Einstein equation on H.
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7.1.2. Link with trapped surfaces and apparent horizons Let us first recall that a trapped surface has been defined by Penrose [132] as a closed (i.e. compact without boundary) spacelike 2-surface S such that the two systems of null geodesics emerging orthogonally from S converge locally at S, i.e. they have non-positive scalar expansions (see also the definition p. 262 of Ref. [90] and Ref. [62] for an early characterization of black holes by trapped surfaces). In the present context, demanding that the spacelike 2-surface St = H ∩ t is a trapped surface is equivalent to 0
and (k) 0 ,
(7.3)
where (k) is defined by Eq. (5.87). The sub-case = 0 or (k) = 0 is referred to as a marginally trapped surface (or simply marginal surface by Hayward [93]). Penrose’s definition is purely local since it involves only quantities defined on the surface S. On the contrary, Hawking [89] (see also Ref. [90, p. 319]) has introduced the concept of outer trapped surface, the definition of which relies on a global property of spacetime, namely asymptotic flatness: an outer trapped surface is an orientable compact spacelike 2-surface S contained in the future development of a partial Cauchy hypersurface 0 and which is such that the outgoing null geodesics emerging orthogonally from S converge locally at S. This requires the definition of outgoing null geodesics, which is based on the assumption of asymptotic flatness. In present context, demanding that the spacelike 2-surface St = H ∩ t is an outer trapped surface is equivalent to (1) the spacelike hypersurface t is asymptotically flat (more precisely strongly future asymptotically predictable and simply connected, cf. Ref. [89, p. 26], or Ref. [90, p. 319]) and the scalar field u defining H has been chosen so that the exterior of St (defined by u > 1, cf. Section 4.2) contains the asymptotically flat region, so that l is an outgoing null normal in the sense of Hawking; (2) the expansion scalar of l is negative or null: 0 .
(7.4)
The sub-case = 0 is referred to as a marginally outer trapped surface. Note that the above definition does not assume anything on (k) , contrary to Penrose’s one. A related concept, also introduced by Hawking [89] and widely used in numerical relativity (see e.g. [125,172,55,162, 81,145,146] and Section 6.1 of Ref. [25] for a review), is that of apparent horizon: it is defined as a 2-surface A inside a Cauchy spacelike hypersurface such that A is a connected component of the outer boundary of the trapped region of . By trapped region, it is meant the set of points p ∈ through which there is an outer trapped surface lying in .14 From Proposition 9.2.9 of Hawking and Ellis [90], an apparent horizon is a marginally outer trapped surface (but see Section 1.6 of Ref. [48] for an update and refinements). In view of the above definitions, let us make explicit the relations with a NEH: if H is a NEH in an asymptotically flat spacetime, then each slice St is a marginally outer trapped surface. If, in addition, (k) 0, then St is a marginally trapped surface. In general, k being the inward null normal to St , (k) is always negative. However there exist some pathological situations for which (k) > 0 at some points of St [74]. Hence a NEH can be constructed by stacking marginally outer trapped surfaces. In particular, it can be obtained by stacking apparent horizons. However, it must be pointed out that contrary to the black hole event horizon, nothing guarantees that the world tube formed by a sequence of apparent horizons is smooth. It can even be discontinuous (cf. Fig. 60 in Ref. [90] picturing the merger of two black holes)! Moreover, even when it is smooth, the world tube of apparent horizons is generally spacelike and not null (Ref. [90, p. 323]). It is only in some equilibrium state that it can be null. Note that this notion of equilibrium needs only to be local: non-expanding horizons can exist in non-stationary spacetimes [47,18]. It is also worth to relate NEHs to the concept of trapping horizon introduced in 1994 by Hayward [93] (see also [95]) and aimed at providing a local description of a black hole. A trapping horizon is defined as a hypersurface of M foliated by spacelike 2-surfaces S such that the expansion scalar (l) of one of the two families of null geodesics 14 Note that this definition of apparent horizon, which is Hawking’s original one [89,90] and which is commonly used in the numerical relativity community, is different from that given in the recent study [65] devoted to the use of isolated horizons in numerical relativity, which requires in addition (k) < 0.
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orthogonal to S vanishes. A trapping horizon can be either spacelike or null. It follows immediately from the above definition that NEHs are null trapping horizons. 7.1.3. Vanishing of the second fundamental form Let us show that on a NEH, not only the trace of the second fundamental form vanishes, but also as a whole. Setting = 0 in the null Raychaudhuri equation (6.9) leads to ab ab + 8T(l, l) = 0 .
(7.5)
Besides, q being a positive definite metric on St , one has ab ab 0 .
(7.6)
Moreover, the null dominant energy condition (condition (3) in the definition of a NEH) implies T(l, l)0 .
(7.7)
The three relations (7.5)–(7.7) imply ab ab = 0
(7.8)
T(l, l) = 0 .
(7.9)
and
Note that this last constraint is trivially satisfied in the vacuum case (T=0). Invoking again the positive definite character of q and the symmetry of ab , Eq. (7.8) implies that ab = 0, i.e. the vanishing of the shear tensor: =0 .
(7.10)
Since we had already = 0, we conclude that for a NEH, not only the scalar expansion vanishes, but also the full tensor of deformation rate [cf. the decomposition (5.64)]: =0
.
(7.11)
From Eq. (5.56), this implies SL
lq = 0
,
(7.12)
which means that the Riemannian metric of the 2-surfaces St is invariant as t evolves. Remark 7.3. The vanishing of the second fundamental form does not imply the vanishing of H’s Weingarten map , as it would do if the hypersurface H was not null: Eq. (5.23) shows clearly that the vanishing of would require = 0 in addition to = 0. On the contrary, for the spatial hypersurface t , the Weingarten map K and the second fundamental form −K are related by K = −g K [cf. Eqs. (3.12) and (3.14)], so that K = 0 ⇒ K = 0. 7.2. Induced affine connection on H Since is the pull-back of the bilinear form ∇l onto H [Eq. (2.55)], its vanishing is equivalent to ∗ ∇l = 0 .
(7.13)
An important consequence of this is that ∀(u, v) ∈ T(H) × T(H),
· v) − l · ∇u v = ∇u (l =0
v · ∇u l = 0 .
∗
∇l(v,u)=0
(7.14)
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This means that for any vectors u and v tangent to H, ∇u v is also a vector tangent to H. Therefore ∇ gives birth to an ˆ to distinguish it from the connection on M: affine connection intrinsic to H, which we will denote ∇ ∀(u, v) ∈ T(H) × T(H),
ˆ u v := ∇u v ∇
.
(7.15)
ˆ the connection induced on H by the spacetime connection ∇. We naturally call ∇ Remark 7.4. More generally, i.e. when H is not necessarily a NEH, the vector k, considered as a rigging vector (cf. Remark 4.8), provides a torsion-free connection on H via the projector along k: ∀(u, v) ∈ T(H) × T(H),
ˆ u v := (∇u v) . ∇
(7.16)
This connection is called the rigged connection [118]. By expressing via Eq. (4.55) it is easy to see that ∀(u, v) ∈ T(H) × T(H),
ˆ u v = ∇u v − (u, v)k . ∇
(7.17)
ˆ is independent of k, i.e. of the choice of the slicing (St ): it We see then clearly that in the NEH case ( = 0), ∇ becomes a connection intrinsic to H. As a consequence of ∇u u ∈ T(H), whatever u ∈ T(H), and the fact that a geodesic passing through a given point is completely determined by its derivative at that point, it follows that any geodesic curve of M which starts a some point p ∈ H and is tangent to H at p remains within H for all points. For this reason, H is called a totally geodesic hypersurface of M. This explains why in Há´"iˇcek’s study [82], non-expanding horizons are called “TGNH” for “totally geodesic null hypersurfaces”. ˆ can be extended to 1-forms on T(H) by means of the Leibnitz rule: given a 1-form field The definition of ∇ ˆ is defined by ∈ T∗ (H), the bilinear form ∇ ∀ (u, v) ∈ T(H) × T(H),
ˆ ˆ v , u ∇(u, v) := ∇ ˆ v u . ˆ v , u − , ∇ := ∇
(7.18)
Now, thanks to Eq. (7.15), ˆ ∇(u, v) = ∇v , u − , ∇v u = ∇v , (u) − , (∇v u) = ∇v ∗ , u − ∗ , ∇v u = ∇(∗ )(u, v) ,
(7.19)
where ∗ ∈ T∗ (M) is the extension of to T(M) provided by the projector onto H [cf. Eq. (4.58)]. Since the ˆ above equation is valid for any pair of vectors (u, v) in T(H), we conclude that the ∇-derivative of the 1-form is the pull-back onto H of the spacetime covariant derivative of ∗ : ˆ = ∗ ∇(∗ ) . ∇
(7.20)
ˆ The above relation is extended to any multilinear form A on T(H), in order to define ∇A: ˆ = ∗ ∇(∗ A) ∇A
.
(7.21)
ˆ of a multilinear form A on T(H) is the pull-back via the embedding of In words: the intrinsic covariant derivative ∇A H in M of the ambient spacetime covariant derivative of the extension of A to T(M), the extension being provided by the projector onto T(H). In particular, for the bilinear form q constituting the (degenerate) metric on H: ˆ = ∗ ∇q , ∇q
(7.22)
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where we have used ∗ q = q. Then ∀ (u, v, w) ∈ T(H)3 ,
ˆ ∇q(u, v, w) = ∇q(u, v, w) = ∇w q(u, v) = ∇w l, uk, v + k, u∇w l, v = (u, w)k, v + (v, w)k, u =0 ,
(7.23)
where we have used = ∗ ∇l [Eq. (2.55)] to let appear and the property = 0 [Eq. (7.11)] characterizing NEHs. Hence ˆ =0 ∇q
.
(7.24)
ˆ is compatible with the metric q on H. We thus conclude that the induced connection ∇ Remark 7.5. Since the metric q on H is degenerate, there is a priori no unique affine connection compatible with it ¯ such that ∇q ¯ = 0). The non-expanding character of H allows then a canonical choice (i.e. a torsion-free connection ∇ ˆ which coincides with the ambient spacetime connection. The couple for such a connection, namely the connection ∇ ˆ (q, ∇) defines an intrinsic geometry of H. This geometrical structure, which was first exhibited by Há´"iˇcek [82,83], is ˆ are largely two independent however different from that for a spacelike or timelike hypersurface, in so far as q and ∇ ˆ entities [apart from the relation (7.24)]: for instance the components ∇A with respect to a given coordinate system (x A ) on H are not deduced from the components qAB by means of some Christoffel symbols. ˆ The ∇-derivative of the null normal l (considered as a vector field in T(H)) takes a simple form, obtained by
= 0 in Eq. (5.21) and using ∗ l = 0: setting ˆ =l⊗ . ∇l
(7.25)
7.3. Damour–Navier–Stokes equation in NEHs The vanishing of and means that for a NEH, all the “viscous” terms of Damour–Navier–Stokes equation [Eq. (6.16)] disappear, so that one is left with q ∗ Ll = 8 q∗ T · l + 2 D .
(7.26)
In this equation it appears the orthogonal projection onto the spatial 2-surfaces St of the “energy–momentum current density” vector W defined by Eq. (7.2). The orthogonal projection q (W) on the 2-surfaces St is the force surface density denoted by f in Eq. (6.19). For a NEH, Eq. (7.9) holds and yields l·W=0 .
(7.27)
This means that W is tangent to H. Then W cannot be timelike (for H is a null hypersurface). From the null dominant energy condition (hypothesis (3) in Section 7.1.1), it cannot be spacelike. It is then necessarily null. Moreover, being tangent to H, it must be collinear to l: W = wl ,
(7.28)
where w is some positive scalar field on H. Note that in the vacuum case, this relation is trivially fulfilled with w = 0. An immediate consequence of (7.28) is the vanishing of the force surface density, since q (l) = 0: q (W) = 0 .
(7.29)
Accordingly, the Damour–Navier–Stokes Eq. (7.26) simplifies to q ∗ Ll = 2 D
,
i.e. the only term left in the right-hand side is the “pressure” gradient 2 D.
(7.30)
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Finally, we note that Eq. (7.28) can be recast by using Einstein equation (3.35) into R(l, .) = ( 21 R − 8w)l ,
(7.31)
which implies ∗ R(l, ·) = 0 .
(7.32)
7.4. Evolution of the transversal deformation rate in NEHs After having considered the non-expanding limit of the Raychaudhuri and Damour–Navier–Stokes equations, let us now turn to the evolution equation for the transversal deformation rate , namely Eq. (6.46). Setting = 0 in it, we get q ∗ L
1 12 T 2 ∗ R + 4 q T − q − l = Kil( D, ) + ⊗ − 2 2 2
.
(7.33)
As a check, we verify that this equation agrees with Eq. (3.9) of Ashtekar et al. [13], after the proper changes of notation c 2
˜ a to 2 D , ˜ have been performed: Ashtekar’s S˜ab corresponds to our ab , D a ˜ a to a , Rab to Rab and q˜ a to qa . Note also that objects in Ashtekar et al. are generally defined only on H (or in an appropriate quotient space of it), whereas we are considering four-dimensional objects. 7.5. Weingarten map and rotation 1-form on a NEH We have already noticed (Remark 7.3) that the vanishing of the second fundamental form on a non-expanding horizon does not imply the vanishing of the Weingarten map , because H is a null hypersurface. The expression of when = 0 however simplifies [cf. Eq. (5.23)]: = , .l
.
(7.34)
Remark 7.6. Eq. (7.34) shows that, on a NEH, all the information about the Weingarten map is actually encoded in the rotation 1-form . Restricted to Tp (H), Eq. (7.34) implies ∀v ∈ Tp (H),
ˆ v l = , vl . ∇
(7.35)
Actually this last relation is that used by Ashtekar et al. [15,13,18] to define for a NEH as a 1-form in T∗ (H). It is clear from Eq. (7.35) that, on a NEH, depends only upon l (more precisely upon the normalization of l) and not directly upon the 2-surfaces St induced by the 3 + 1 slicing. On the contrary, the Há´"iˇcek 1-form depends directly upon St , since its definition (5.29) involves the orthogonal projector q onto St : = q ∗ . We have seen that, for a NEH, the degenerate metric q does not vary along l [Eq. (7.12)]. It is then interesting to investigate the evolution of along l, i.e. to evaluate Ll . Since l ∈ T(H), we may a priori consider two Lie derivatives: the Lie derivative of along l within the manifold M, denoted by Ll , and the Lie derivative of (or more precisely of the pull-back ∗ of onto H) along l within the manifold H, that we will denote by HLl . The relation between these two Lie derivatives is given in Appendix A. In particular Eq. (A.3) gives ∗ HLl = ∗ Ll (∗ ) .
(7.36)
Since , k = 0 [Eq. (5.28)], we have ∗ = , so that Eq. (7.36) results in ∗ HLl = ∗ Ll = ∗ Ll ( − k) = ∗ (Ll − ∇l k − Ll k) ,
(7.37)
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where use has been made of Eq. (5.35). Now, by Cartan identity (1.26), Ll k = l · dk + dk, l = l · dk (since k, l = −1). Using the Frobenius relation (5.39) to express dk, we get 1 N ∇ ln (7.38) l. Ll k = l 2 2N M It is then obvious that ∗ Ll k = 0
(7.39)
∗
since l = 0 [Eq. (4.64)], so that Eq. (7.37) reduces to ∗ HLl = ∗ Ll − ∇l k ,
(7.40)
where use has been made of the property ∗ k = k [Eq. (4.64)]. Using relation (4.57), we have ∗ Ll = q ∗ Ll , hence ∗ HLl = q ∗ Ll − ∇l k .
(7.41)
Substituting Eq. (7.30) for q ∗ Ll , we obtain ∗ HLl = 2 D − ∇l k .
(7.42)
2 D = q
∗ ∇
[cf. Eq. (5.59)] by substituting + k, .l for q [Eq. (4.57)], we realize that the Expanding the relation right-hand side of the above equation is nothing but the projection on H of the spacetime gradient of : ∗ HLl = ∗ ∇ .
(7.43)
We can rewrite this four-dimensional equation as a three-dimensional equation entirely within H, by means of the ˆ introduced in Section 7.2: induced connection ∇ HL
ˆ l = ∇
.
(7.44)
This simple relation, which is of course valid only for a non-expanding horizon, has been obtained by Ashtekar et al. [13] [cf. their Eq. (2.11)]. Its orthogonal projection onto the 2-surfaces St foliating H is the reduced Damour–Navier– Stokes (7.30). 7.6. Rotation 2-form and Weyl tensor 7.6.1. The rotation 2-form as an invariant on H In Remark 7.6, we have noticed that for a NEH, the rotation 1-form is “almost” intrinsic to H, in the sense that it does not depend upon the specific spacelike slicing St of H but only on the normalization of l. On the other side, considering as a 1-form in T∗ (H) (more precisely considering the pull-back 1-form ∗ ), its exterior derivative within the manifold H, which we denote by H d, is fully intrinsic to H. It is indeed invariant under a rescaling of the null normal l, as we are going to show. Consider a rescaling l = l of the null normal, as in Section 5.8. Then varies according to Eq. (5.112), which we can write [via Eq. (4.60)], = + d ln + (∇k ln )l .
(7.45)
Taking the exterior derivative (within M) of this relation and using dd = 0, as well as dl = d ∧ l [Eq. (2.17)], yields d = d + [d(∇k ln ) + ∇k ln d ] ∧ l .
(7.46)
Let us consider the pull-back of this relation onto H [cf. Eq. (2.7)]. First of all, we have that the external differential is natural with respect to the pull-back15 ∗ d = H d .
(7.47)
15 More precisely, we should write ∗ d = H d(∗ ), but the above remark about “leaving essentially” in H allows us not to distinguish between ∗ and .
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It is straightforward to establish it by means of a coordinate system (x ) adapted to H (cf. Section 4.8): ∀ (u, v) ∈ Tp (H)2 ,
∗ d(u, v) = d(∗ u, ∗ v) = (j − j )(∗ u) (∗ v) = (jA B − jB A )uA v B = H d(u, v) .
(7.48)
Taking into account Eq. (7.47) and the similar relation for d , as well as ∗ l = 0, the pull-back of Eq. (7.46) onto H results in H d
= H d
,
(7.49)
which shows the independence of the 2-form H d with respect to the choice of the null normal l. We call H d the rotation 2-form of the null hypersurface H. 7.6.2. Expression of the rotation 2-form Let us compute the rotation 2-form H d from Eq. (7.47), i.e. by performing the pull-back of the four-dimensional exterior derivative d. The latter is given by the Cartan structure equation (B.23) derived in Appendix B. Indeed, is closely related to the connection 1-form 0 0 associated with the tetrad (l, k, e2 , e3 ), where (e2 , e3 ) is any orthonormal basis of Tp (St ) [cf. Eq. (B.6)]. When vanishes on H (NEH), Eq. (B.23) simplifies to d = Riem(l, k, ., .) + A ∧ l ,
(7.50)
where A is a 1-form, the precise expression of which is given by Eq. (B.23) and is not required here. The pull-back of Eq. (7.50) on H yields [taking into account Eq. (7.47) and ∗ l = 0] H
d = ∗ Riem(l, k, ., .) .
(7.51)
Let us consider two vectors u and v tangent to H. In the tetrad (e ) = (l, k, e2 , e3 ) of Appendix B, they do not have any component along k and expand as u = u0 l + ua ea
and v = v 0 l + v a ea .
(7.52)
Then Eq. (7.51) leads to H
d(u, v) = Riem(l, k, ua ea + u0 l, v b eb + v 0 l) = Riem(l, k, ua ea , v b eb ) + (u0 v a − v 0 ua )Riem(l, k, l, ea ) ,
(7.53)
where we have taken into account the antisymmetry of the Riemann tensor with respect to its last two arguments. Let us evaluate the last term in the above equation. By virtue of the symmetry property of the Riemann tensor with respect to the permutation of the first pair of indices with the second one, we can write Riem(l, k, l, ea ) = Riem(l, ea , l, k). Then we may express Riem(l, ea , l, k) by plugging the vector pair (l, k) in Cartan’s structure equation (B.27) derived in Appendix B. Notice that, since we are dealing with a NEH, we can set to zero all the terms involving ab in the right-hand side of Eq. (B.27), but not the term in the left-hand side, since the derivative of in directions transverse to H is a priori not zero. However, d(ab eb ) = dab ∧ eb + ab deb = dab ∧ eb and eb , l = 0 and eb , k = 0, so that the left-hand side of Eq. (B.27) vanishes when applied to (l, k). Consequently, one is left with Riem(l, k, l, ea ) = − ∇l (∇ea − a ) − b a0 (∇eb − b ) = ∇l ( − 2 D ), ea ,
(7.54)
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where we have used − d , l = − ∇l = 0 [Eq. (2.22)] to get the first line and = a ea and 2 D = (∇ea )ea to get the second one. Now ∇l 2 D , ea is the component along ea of the 1-form q ∗ ∇l 2 D . Let us evaluate the latter (using index notation): q ∇ (2 D ) = q ∇ (q ∇ )
= q [∇ (k + k )∇ + q ∇ ∇ ]
= q { ∇ k ∇ + q [∇ ( ∇ ) − ∇ ∇ ]} = q { + q [∇ − ( + )∇ ]}
= + 2 D − ∇ −
= 2 D − ∇ .
(7.55)
For a NEH, the term involving vanishes, so that one is left with q ∗ ∇l 2 D = 2 D .
(7.56)
Now, let us use the Damour–Navier–Stokes Eq. (7.30) to replace 2 D and get q ∗ ∇l 2 D = q ∗ Ll .
(7.57)
Substituting this last relation for ∇l 2 D into Eq. (7.54) yields Riem(l, k, l, ea ) = ∇l − Ll , ea .
(7.58)
= 0) Now, by expressing the Lie derivative in terms of the connection ∇, one has immediately the relation (using
=0 . q ∗ (∇l − Ll ) = − ·
(7.59)
Thus we conclude that, for a NEH, Riem(l, k, l, ea ) = 0 .
(7.60)
Consequently, there remains only one term in the right-hand side of Eq. (7.53), which we can write, taking into account that the orthogonal projections of u and v onto Tp (St ) are expressible as q (u) = ua ea and q (v) = v a ea , H
d(u, v) = Riem(l, k, q (u), q (v)) ,
(7.61)
Since u and v are any vectors in Tp (H), we conclude that the following identity between 2-forms holds: H d
= q ∗ Riem(l, k, ., .)
.
(7.62)
This relation considerably strengthens Eq. (7.51): the presence of the operator q ∗ means that the 2-form H d acts only within the subspace Tp (St ) of Tp (H). Now since the vector space Tp (St ) is of dimension two, the space of 2-forms on it is of dimension only one and is generated by e2 ∧ e3 . Thus, because of the antisymmetry in the last two indices of the Riemann tensor, Eq. (7.62) implies H d = ae2 ∧ e3 , with the coefficient a being simply Riem(l, k, e2 , e3 ). Moreover, since (e2 , e3 ) is an orthonormal basis of Tp (St ) (e2 and e3 can be permuted if necessary to match the volume orientation) we have e2 ∧ e3 = 2 ,
(7.63)
where 2 is the surface element of St induced by the spacetime metric [cf. Eq. (5.75)]. Consequently, we have H d
= a 2
with a := Riem(l, k, e2 , e3 ) .
(7.64)
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Actually, the coefficient a can be completely expressed in terms of the Weyl tensor C: replacing the Riemann tensor in Eq. (7.64) by its decomposition (1.19) in terms of the Ricci and Weyl tensors yields, thanks to Eq. (7.32) a = C(l, k, e2 , e3 ).
(7.65)
In order to make the link with previous results in the literature, let us express a in terms of the complex Weyl scalars ¯ by inverting of the Newman–Penrose formalism introduced in Section 4.6.2. Expanding e2 and e3 in terms of m and m Eqs. (4.42) and (4.43), and using the cyclic property in the last three slots of the Weyl tensor (inherited from property (1.16) of the Riemann tensor), we get 1 ¯ k) − C(l, m, ¯ m, k)] . a = [C(l, m, m, i
(7.66)
From definition (4.50) of the Weyl scalars n , the coefficient a is written in terms of the imaginary part of 2 , so that H d
= 2 Im 2 2
.
(7.67)
This relation has been firstly derived in the seventies by Há´"iˇcek (cf. Eq. (23) in Ref. [82]) and special emphasis has been put on it by Ashtekar et al. [15,13,18]. More precisely, Há´"iˇcek has derived Eq. (7.67) in the case = 0, for which coincides with . However, since (i) H d does not depend on the normalization of l and (ii) it is always possible to rescale l to ensure = 0 [cf. Eq. (2.26)], the demonstration of Há´"iˇcek is fully general. Remark 7.7. Let us compute the Lie derivative of along l within the manifold H from Eq. (7.64), by means of Cartan identity (1.26): H
ˆ . Ll = l · H d + H d, l = a 2 (l, .) +H d = ∇
(7.68)
=0
Thus we recover the evolution equation (7.44) as a consequence of Eq. (7.64). 7.6.3. Other components of the Weyl tensor We have just shown that the component C(l, k, e2 , e3 ) = 2 Im 2 of the Weyl tensor with respect to the tetrad (l, k, e2 , e3 ) provides the proportionality between the 2-forms H d and 2 . Let us now investigate some other components of the Weyl tensor. Setting = 0 in the evolution equation (6.27) for yields q ∗ Riem(l, ., l, .) = 0 .
(7.69)
Making use of property (7.32), we rewrite this expression as C(l, ea , l, eb ) = 0 .
∀a, b ∈ {2, 3},
(7.70)
Moreover, from Eq. (7.60) and again Eq. (7.32), we get ∀a ∈ {2, 3},
C(l, ea , l, k) = 0 ,
(7.71)
where we have made use of the symmetries of the Weyl tensor. From Eqs. (4.42) and (4.43) together with (4.50), the above relations imply the vanishing of two of the complex Weyl scalars: 0 = 1 = 0
.
(7.72)
This means that the NEH structure sets strong constraints on the Weyl tensor evaluated at H. These constraints are physically relevant. On the one hand, the Weyl components 0 and 1 are associated with the ingoing transversal and longitudinal parts of the gravitational field [159]. Their vanishing is consistent with the quasi-equilibrium situation modelled by NEHs, since no dynamical gravitational degrees of freedom fall into the black hole by crossing the horizon. On the other hand, the change 2 of the Weyl scalar 2 under a Lorentz transformation (i.e. either a boost or a rotation)
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of the tetrad (l, k, e2 , e3 ), turns out to be a linear combination of scalars 0 and 1 [45]. Consequently, as long as we choose the first vector in the null tetrad to be the l normal to H, 2 is an invariant. In particular this means that the value of 2 does not depend on the chosen null normal, therefore guaranteeing its invariance under the choice of the spacelike slicing. Finally, we point out that the vanishing of 0 and 1 could have been obtained directly as an application of the Goldberg–Sachs theorem, which establishes the equivalence between 0 = 1 = 0 and the existence of a geodesic ( = 0), shear-free null vector l (see [45]; as we mentioned above and will discuss in more detail in Section 8.2, a NEH always admits a null normal with vanishing non-affinity coefficient ). In particular, this means that the Weyl tensor is of Petrov type II on H [10,11]. 7.7. NEH-constraints and free data on a NEH 7.7.1. Constraints of the NEH structure Let us determine which part of the geometry of a NEH can be freely specified. As we shall see, such a part is essentially given by fields living on the spatial slices St of the horizon, that can be considered as initial data. This is specially important in the present setting, since it perfectly matches with the 3 + 1 point of view we have adopted. ˆ However, in order to build As discussed in Remark 7.5, the geometry of a NEH is characterized by the pair (q, ∇). ˆ if H is a null hypersurface within a spacetime a NEH one cannot make completely arbitrary choices for q and ∇ ˆ and the Ricci tensor R must satisfy certain relations, as established satisfying Einstein equation. The reason is that q, ∇ ˆ and R on H will be referred to as in the previous sections. Following [13,111] any geometrical identity involving q, ∇ a constraint of the NEH structure (or NEH-constraint). Such geometrical identities can be obtained by evaluating the ˆ along the integral lines of a null normal l. change of q and ∇ Regarding q, the NEH condition (7.12) directly provides the constraint S Ll q = 0. In order to cope with the ˆ we follow an analysis which dwells directly on the spatial slicing of H. constraints associated with the evolution of ∇, As explained in Section 4.2, the foliation (St ) of H is preserved by the flow of l due to the normalization (4.6). The pull-back of the 1-form k on H, ∗ k, is also preserved by the flow of l: H
Ll ∗ k = 0 .
(7.73)
This follows directly from Eq. (7.39) and relation (A.4) between HLl and Ll . Eq. (7.73) can also be obtained by simply noticing that, according to Eq. (4.34), ∗ k is (minus) the pull-back to H of the differential of t. ˆ [this is also performed in Following Ref. [111], let us determine the different objects composing H’s connection ∇ ˆ can be decomposed in the Appendix B; see Eqs. (B.6)–(B.10)]. Considering an arbitrary vector field v ∈ T(H), ∇v following parts: q a q b ∇ˆ v = 2 D a (q b v ) q a k ∇ˆ v = 2 D a (v k ) − q a v ∇ k ∇ˆ v = Ll v + v
(spatial.spatial) , (spatial.null) , (null.arbitrary) .
(7.74)
ˆ on each From this decomposition and taking into account expression (5.86) for the gradient of k, we note that ∇v slice St can be reconstructed if (2 D, , ) are known on St . Likewise, the invariance of the foliation (St ) under l ˆ in terms of the evolution of (2 D, , ) or, equivalently, of (q, , , ). Since we permits to express the evolution of ∇ want to emphasize the 3 + 1 point of view, we adopt the second set of variables, which are fields intrinsic to St . The NEH-constraints are therefore given by the previously derived equations (7.12), (7.30) [or (7.44)] and (7.33): SL q = 0 l q ∗ Ll = 2 D
ˆ (HLl = ∇) 1 1 T q ∗ Ll = Kil(2 D, ) + ⊗ − 2 R + 4 q ∗ T − q − 2 2 2
.
(7.75)
Notice that the geometry on H does not enforce any evolution equation for the non-affinity parameter . In fact, the NEH geometry constrains neither the value of on St nor its evolution. This is a consequence of the freedom to rescale l in the NEH structure (see below in relation with the gauge ambiguity in the choice of initial free data).
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Fig. 15. Reconstruction of the NEH from the free data. Left: choice of free data on St . Center: free data as fields on H. Right: evolution of the free data to a infinitesimally close surface St+ t .
ˆ and R. In fact, components Remark 7.8. We have defined the constraints on the NEH structure as identities relating q, ∇ of the Ricci tensor parallel to H actually constrain the null geometry via Einstein equation. However, we have derived in previous subsections other kind of geometric relations like (7.67) or conditions like (7.72). They involve some of the components of the Weyl tensor, i.e. the part of the Riemann that is not fixed by Einstein equation. In particular, once ˆ is given, the components 0 , 1 and the imaginary part of 2 are fixed. In this sense, they could the geometry (q, ∇) be considered rather as constraints for the 4-geometry containing a NEH. 7.7.2. Reconstruction of H from data on St . Free data Let us describe how a NEH may be reconstructed from data on an initial spatial slice St . We proceed in three steps (see Fig. 15). Firstly, a free choice for the values of (q, , , ) on St is made, considering them as objects intrinsic to St . Secondly, these objects are regarded as fields living in H by imposing, on the slice St , the vanishing of their corresponding null components: l · q = 0,
l · = 0,
l·=0 .
(7.76)
Finally, the value of these fields is calculated in an infinitesimally close slice St+ t by employing Eqs. (7.75). The ˆ = l ⊗ [Eq. (7.25)], which value of on St+ t can be chosen freely again. We note by passing that the expression ∇l can be seen as a constraint on the NEH geometry, is automatically satisfied by following the above procedure, since l is torsion free (being normal to H) and we construct a vanishing . It is not an independent constraint. We conclude that the free data for the null NEH geometry are given by NEH-free data: (q|St , |St , |St , |H )
.
(7.77)
That is, the initial data q, and on St , together with the function on H, can be freely specified. This is in contrast with the 3 + 1 Cauchy problem, where the initial data on the spatial slice t are in fact constrained (cf. Section 3.6). Once the free data are given, the full geometry in H can be reconstructed by evolving these quantities along l, this null normal being determined through Eq. (4.13) by the additional structure provided by the slicing (t ). Therefore, for a ˆ their initial data (q|S , |S , |S , |H ) are associated with a particular l = N (n + s) given null geometry (q, ∇), t t t (see below). Gauge freedom in the choice of the free data. Even though each specific choice of data in (7.77), together with a slicing (t ), fixes the NEH geometry, there are different choices that actually define the same NEH structure: there exists a degeneracy in the free data that can be referred to as a gauge freedom. This degeneracy presents two aspects, the first one linked to the choice of the null normal, and the second one to the slicing of H once l is fixed (see Fig. 16): (a) If we start from a fixed slice S0 , our choice of (q, , , ) is, as mentioned above, associated with a particular null vector l. However, the NEH geometry is not changed under a rescaling of the null normal, l = l. Under this rescaling, the fields (q, , , ) change according to the transformations given in Table 1. The resulting intrinsic
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Fig. 16. On the left the active aspect of the gauge freedom, linked to the choice of the null normal, and on the right the passive aspect, associated with the choice of a initial slice.
ˆ is the same, even though the slicings (St ) and (St ) of H induced by the transport of S0 null geometry (q, ∇) along l and l will in general differ. We will call this ambiguity the active aspect of the gauge freedom. It is associated with the fact that different slicings (t ) fix (in general) different null normals l via Eq. (4.13), and it is a natural gauge freedom when one actually constructs H starting from a slice S0 as in a Cauchy problem. (b) Even though a slicing (t ) fixes l the reverse is not true, since this null normal is compatible with different spatial slicings. Keeping l fixed, we can consider initial slices S0 and S0 in H belonging to different slicings (St ) and ˆ = ∇t ˆ − ∇g, ˆ (St ) of H compatible with l via (4.6). The corresponding level functions t and t are related by ∇t H 16 l, and consequently , but induces via where Ll g = 0. This change in the slicing does not affect q, Eq. (4.34) the transformations (cf. Eq. (3.10) in [13]) ˆ k → k + ∇g,
ˆ → − ∇g,
ˆ ∇g ˆ . →+∇
(7.78)
This aspect of the gauge freedom, that we can refer to as passive and which corresponds to a coordinate choice in H, is natural when H is a hypersurface existing a priori, its slicings only entering in a second step once l is fixed. It is the aspect discussed in Refs. [13,108]. In brief, from a 3 + 1 perspective, in both cases the underlying source of degeneracy in the free data is related to the choice of a specific slicing of H. When there is no canonical manner of fixing neither the initial slice nor the null vector l, as it is the case in a purely intrinsic formulation of the geometry of H, both sources of gauge freedom are simultaneously present17 (see discussion in Section VI.A.2 of Ref. [111]). 7.7.3. Evolution of ∇ˆ from an intrinsic null perspective Following the 3 + 1 approach adopted in this article, the previous discussion on the NEH-constraints has made ˆ can be described in a more explicit use of a spatial slicing of H. However, the time evolution of the connection ∇ intrinsic manner. In order to prepare the notion of (strongly) isolated horizon in Section 9.1, we briefly comment on it. ˆ along l can be written (in components) as follows (see Refs. [13,111]) The evolution of the connection ∇ [HLl , ∇ˆ ] = −N
(7.79)
16 Note that although the metric q (as an object acting on T(H)) does not change, the projector q ˆ
on the slices St does change: q → q − l∇g, as follows from Eq. (4.57). 17 We have discussed here the gauge freedom in the determination of intrinsic geometrical objects on S . Of course, regarding their specific 0 coordinate expression, there exists an additional underlying freedom related to the choice of coordinate system on S0 .
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for any 1-form ∈ T∗ (H), where N = ∇ˆ ( ) + + 21 (R − 2 R ) .
(7.80)
Together with the evolution for q in Eq. (7.12), these evolution equations constitute geometrical identities involving ˆ and R, i.e. they provide the NEH constraints. only q, ∇ Following the procedure in the isolated horizon literature [13,111], we introduce the following tensor on H: S := ∗ ∇k ,
(7.81) ∗ k
where k is associated with a foliation compatible with l. According to Eq. (4.34), is exact on H implying that S is symmetric. In addition, q ∗ S = and l · S = (see Eqs. (5.80) and (5.86), respectively). From the discussion in ˆ is given by that of the pair (q, S). Making equal to k in Sections 7.7.1 and 7.7.2, it follows that the evolution of ∇ Eq. (7.79), one obtains H
Ll S = N ,
(7.82)
thus providing the evolution equation for S. The NEH-constraints are now S
Ll q = 0 ,
H
Ll S = ∇ˆ ( ) + +
(7.83) 1 2 (R
− R ) . 2
(7.84)
Contracting the second one with l, Eq. (7.44) for the evolution of is recovered, whereas the projection onto a slice St leads to the evolution (7.33) for . 8. Isolated horizons I: weakly isolated horizons 8.1. Introduction The NEH notion introduced in the previous section, represents a first step toward the quasi-local characterization of a black hole horizon in equilibrium. As we have seen, this is achieved essentially by imposing a condition on the degenerate metric q, namely to be time-independent [Eq. (7.12)]. This minimal geometrical condition captures some fundamental features of a black hole in quasi-equilibrium. On the one hand, it is sufficiently flexible so as to accommodate a variety of interesting physical scenarios. But on the other hand, such a structure is not tight enough to determine some geometrical and physical properties of a black hole horizon. From the point of view of the geometry of H as a hypersurface in M, the NEH notion by itself provides a limited set of tools to extract information about the spacetime containing the black hole. For instance, it does not pick up any particular normalization of the null normal nor suggest any concrete foliation of H, something that was accomplished in the previous sections by using the additional structure (t ). Since in the spacetime construction as a Cauchy problem, the 3 + 1 foliation is actually dynamically determined, it would be very useful to dispose of a slicing (St ) motivated from an intrinsic analysis of H for employing it as an inner boundary condition for (t ). Regarding the determination of physical black hole parameters, a point of evident astrophysical interest, a NEH does not determine any prescription for the mass, the angular momentum or, more generally, for the black hole multipole moments. In order to address these issues, i.e. the discussion of the horizon properties from a point of view intrinsic to H, we need a finer characterization of the notion of quasi-equilibrium. Following Ashtekar et al. [13], this demands the introduction of additional structures on H. After imposing the degenerate metric q to be time-independent, a natural way to proceed in order to further constrain the horizon geometry consists in extending this condition to the rest of the ˆ This strategy directly leads to the introduction of geometrical objects on H, in particular to the induced connection ∇. a hierarchy of quasi-equilibrium structures on the horizon, which turns out to be very useful for keeping control of the physical and geometrical hypotheses actually assumed. A particularly clear synthesis of the resulting formalism can be found in Refs. [111,107]. ˆ or equivalently (q, , , ) (see Section 7.7), characterizes the geometry of a NEH (cf. The pair of fields (q, ∇), Remark 7.5). From a 3 + 1 perspective, and following the strategy previously outlined, a natural manner of obtaining different degrees of quasi-equilibrium for the horizon would consist in imposing the time independence of different
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combinations of the fields (q, , , ). However, the resulting notions of horizon quasi-equilibrium would be slicingdependent. Even though in Section 9.2 we will revisit this idea, we rather proceed here by constructing the new structures on H on the grounds of fields intrinsic to H. Once the time independence of q has been used to define a NEH, this ˆ or, in a intermediate step, on its component . This defines means imposing time independence on the full connection ∇ the three levels in the (intrinsic) isolated horizon hierarchy. We separate the study of the isolated horizons in two sections. In the present one, we focus on the intermediate level resulting from the introduction, in a consistent way, of a time-independent rotation 1-form . This second level in the isolated horizon hierarchy leads to the notion of weakly isolated horizon (WIH). After introducing in Section 8.2 its definition and straightforward consequences, Sections 8.3, 8.4 and 8.5 are devoted to different implications on the null geometry of H. Finally Section 8.6 briefly presents how a WIH structure permits to determine the physical parameters associated with the black hole horizon. The stronger level in the isolated horizon hierarchy, which results from a full ˆ will be discussed in Section 9, where we will also comment on other developments naturally time-independent ∇, related to the isolated horizon structures. 8.2. Basic properties of weakly isolated horizons 8.2.1. Definition As already noticed, the NEH notion is independent of the rescaling of the null normal l. On the contrary, imposing ˆ along l, cf. Eq. (7.74)] under the flow of the null normal, does the constancy of the rotation 1-form [the part of ∇ depend on the actual choice for l. This is a consequence of the transformation of under a rescaling of l (cf. Table 1): l→l
−→ + ∗ d ln .
(8.1)
If we consider a null normal l such that HLl =0 then, after a rescaling of l by some non-constant , the new rotation 1-form will not be time-independent in general. In order to make sense of condition HLl = 0, we must restrict the set of the null normals for which it actually applies. From Eq. (8.1) we conclude that if is invariant under a given null normal l, then it is also invariant under any constant rescaling of it (this could be relaxed to time-independent functions). This is formalized by the following definitions [15,13]: (i) Two null normals l and l are said to be related to each other if and only if l = cl with c a positive constant. This defines an equivalence relation whose equivalence classes are denoted by [l]. (ii) A weakly isolated horizon (WIH) (H, [l]) is a NEH H endowed with an equivalence class [l] of null normals such that HL
l=0
.
(8.2)
We comment on some consequences of this definition. 1. Extremal and non-extremal WIH. As a consequence of the transformation rule for under a rescaling of l (cf. Table 1), two null normals l and l belonging to the same WIH class have non-affinity coefficients (l) and (l ) related by18 (l ) = c(l) ,
(8.3)
where c is the constant linking the two null vectors: l = cl. This implies that there is no canonical value of the non-affinity coefficient on a given WIH. This reflects the absence of canonical representative in the class [l]. We have already presented a solution to this point, relying on the 3 + 1 spacelike slicing in Section 4.2. We will present another solution, intrinsic to H, in Section 8.6. The transformation law (8.3) means that, on a given NEH, the WIH structures are naturally divided in two types: those with vanishing (l) , that will be referred as extremal WIHs, and 18 In this section, since we will deal with different null normals at the same time, we make explicit the dependence of the non-affinity coefficient
on l.
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those with (l) = 0, non-extremal WIHs. This terminology arises from the Kerr spacetime, where one can always choose the null normal l to let it coincide with a Killing vector field. Then (l) is nothing but the surface gravity of the black hole [cf. Eq. (2.44) in Example 2.5 for the non-rotating case and Eq. (D.30) in Appendix D for the general case]. Extreme Kerr black holes are those for which the surface gravity vanishes. As shown by Eq. (D.30), this corresponds to the angular momentum parameter a/m = 1. They are not expected to exist in the Universe, because it is not possible to make a black hole rotate faster than a/m = 0.998 by standard astrophysical processes (i.e. infall of matter from an accretion disk [164]). In this article, we will focus on the non-extremal case, and refer the reader to [13,111] for the general case. 2. Constancy of the surface gravity on H. From the NEH Eq. (7.67) l · H d = 0 ,
(8.4)
and using the Cartan identity (1.26) on H: 0=HLl = l · H d + H dl, = H d(l) ⇔
ˆ (l) = 0 ∇
.
(8.5)
Therefore, the non-affinity coefficient of a given l is a constant on H. This property, that is referred to as the zeroth law of black hole mechanics, characterizes [l] as associated with a WIH structure. It will be discussed further in Remark 8.3 below. 3. Any NEH admits a WIH structure. We have just seen that the class [l] associated with l is a WIH structure if and only if (l) is a constant on H. In Sections 7.7 and 7.7.2 we established that a given NEH geometry is determined by the set of fields (q, , (l) , ), where (l) is an arbitrary function, and also by any other set obtained from this one by applying the transformations in Table 1. This was called the active aspect of the gauge freedom in the NEH-free data, and it simply corresponds to a rescaling by of the null normal l. Therefore if, according to Eq. (2.26), we choose satisfying = ∇l + (l)
(8.6)
with constant on H, then [l ] given by l = l constitutes a WIH class (in particular, making = 0 shows that any NEH admits an extremal horizon; Section III.A of Ref. [13] firstly shows the existence of an extremal WIH for any NEH, and then constructs from it a family of non-extremal ones). As a consequence, the addition of a WIH does not represent an actual constraint on the null geometry of H. It rather distinguishes certain classes of null normals. 4. Infinite freedom of the WIH structure. Not only it is always possible to choose a WIH structure on a NEH, but there exists actually an infinite number of non-equivalent WIHs. Reasoning for the non-extremal case, if l is such that (l) is a non-vanishing constant on H and t is a coordinate on H compatible with l, i.e. HLl t = 1, then the class [l ] associated with the vector l defined by l = (1 + Be−(l) t )l
(8.7)
H
(8.8)
Ll B = 0 ,
is distinct from the class [l] (for := 1 + Be−(l) t is not constant) and also defines a WIH. This follows from (l ) = (l) = const. Using the slicing (St ) induced on H from the 3 + 1 decomposition, the functions B in Eq. (8.7) have actually support on the sections St and parametrize the different WIH structures. In an analogous manner for the extremal case, (l) = 0, the non-equivalent rescaled null normal l = Al with A a non-constant function on St , is also associated with a non-equivalent extremal WIH. 8.2.2. Link with the 3 + 1 slicing Since a WIH structure does not further constrain the geometry of H, from a quasi-equilibrium point of view a WIH is not more isolated than a NEH. However, this notion presents a remarkable richness as a structural tool. In this sense its interest is two-fold, both from a geometrical and physical point of view. In addition, this concept provides a natural framework to discuss the interplay between the horizon H and the spatial 3 + 1 foliation (t ), something specially relevant in our approach. In this last sense, the issue of the compatibility between the structures intrinsically defined on H and the additional one provided by the 3 + 1 slicing of the spacetime, is naturally posed.
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Given a WIH (H, [l]), a 3 + 1 slicing (t ) is called WIH-compatible if there exists a representative l in [l] such that the associated level function t evaluated on the horizon H constitutes a natural coordinate for l, i.e. Ll t = 1 [99]. Note that the representative l is then nothing but the null normal associated with the slicing (St ) of H by the normalization (4.6). Given independently a NEH H and a 3 + 1 slicing of M, there is no guarantee that there exists a WIH structure on the NEH such that (t ) is WIH-compatible. If such a WIH exists, l is tied to the t function and therefore to the slicing. If not, no relation exists between that WIH and (t ). Therefore, even though the choice of a specific WIH structure on our NEH does not affect the intrinsic geometry of the horizon, demanding the 3 + 1-slicing to be WIH-compatible represents an actual restriction on the 3 + 1 description of spacetime (see Section 11.3.1). And this fact is crucial in our approach: some of the possible spacetimes slicings are directly ruled out. We have noticed in Remark 7.6 that, on a NEH, the Há´"iˇcek 1-form (more concretely its divergence) is an object directly depending upon the 3 + 1 slicing, whereas the rotation 1-form depends only upon the normalization of l. Let us then investigate the consequences of the WIH condition on . From Eq. (7.40), we have H
Ll =HLl − (HLl (l) )∗ k ,
(8.9) ∗
(H) (identifying them with their pull-back ∗ and ∗ ).
where we have considered both and as a 1-forms in T In view of the above relation, we deduce from Eqs. (8.2) and (8.5) that ((H, [l]) is a WIH) ⇐⇒
HL = 0 l HL l (l) = 0
.
(8.10)
Note that HLl (l) = 0 is listed here, along with HLl = 0, as a sufficient condition to have a WIH, but once the WIH structure holds, one has actually the much stronger property of constancy of (l) on all H [zeroth law, Eq. (8.5)]. 8.2.3. WIH-symmetries The discussion of the physical parameters of the black hole horizon in Section 8.6 demands the introduction of the notion of a symmetry related to a WIH horizon. We present a brief account of it, with emphasis in the non-extremal case and refer the reader to Refs. [12,107] for details and extensions. A symmetry of a WIH is a diffeomorphism of H preserving the relevant structures of the WIH. Infinitesimally this is captured as follows: a vector field W tangent to H is said to be an infinitesimal WIH-symmetry of (H, [l]), if it preserves the equivalence class of null normals, the metric q and the 1-form , namely H
LW l = const · l,
H
LW q = 0
and
H
LW = 0 .
(8.11)
In the considered non-extremal case, the general form of such a WIH symmetry is given by (see Section III in [12]) W = cW l + bW S ,
(8.12)
where cW and bW are constant on H and the vector field S, satisfying l · S = 0, is an isometry on each section (St , q). From this general form of an infinitesimal symmetry, and according to the number of independent generators in the associated Lie algebra of symmetries, one can distinguish different universality classes of WIH-symmetries. Since l ∈ [l] is an infinitesimal symmetry by construction, the Lie algebra (and therefore the Lie group) of WIH-symmetries is always at least one-dimensional: (a) Class I. The symmetry Lie algebra is generated by l together with the infinitesimal rotations acting on the 2-sphere St . The resulting group is the direct product of SO(3) and the translations in the l direction. This case corresponds to the horizon of a non-rotating black hole. (b) Class II. The symmetry group is now the direct product of the translations along l and an axial SO(2) symmetry on St . It corresponds to an axisymmetric horizon and represents the most interesting physical case, since it corresponds to a black hole with well-defined non-vanishing angular momentum (see Section 8.6). (c) Class III. The symmetry group is one-dimensional (translations along l). It corresponds to the general distorted case. Note that I is a special case of II, and the latter is a special case of III.
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8.3. Initial (free) data of a WIH As commented in the point 3, after the WIH definition, the geometry of a WIH as a null hypersurface is that of a NEH. In particular, the free data of a WIH are essentially those presented in Section 7.7 for a NEH. The only difference is the choice of a null normal l such that the zeroth law (8.5) is satisfied and, consequently, (l) is constant on H, o : WIH-free initial data :
(q|St , |St , |St , o )
.
(8.13)
We note that, since in a WIH the null normal [l] is fixed up to a constant, the gauge freedom in the WIH-free data concerns mainly what we called the passive aspect in Section 7.7.2 . The active one reduces to constant rescalings of (l) and . Once these free data are fixed on St , the reconstruction of the WIH on H proceeds as in Section 7.7.2. The only subtlety now enters in the third step represented in Fig. 15, when the fields on the slice St are transported to the next slice. In the construction of the WIH not only the metric q is Lie dragged by l, but also the Há´"iˇcek form and the non-affinity coefficient o , as shown by Eq. (8.10). Now, the only field evolving in time is . In view of Eq. (7.33) and the time-independence of q (hence of 2 D and 2 R), and , the time dependence of can be explicitly integrated if we assume that the projection of the four-dimensional Ricci tensor (or the matter stress-energy tensor via the Einstein equation) is time-independent, i.e. ∗ Ll R = 0 .
(8.14)
In fact, this condition is actually well-defined on a NEH, i.e. it does not depend on the choice of the null normal l. This follows from property (7.32). Condition (8.14) is rather mild and is obviously satisfied in a vacuum spacetime. If we deal with a WIH built on a NEH that fulfills (8.14), then in the evolution of dictated by Eq. (7.33) the only terms which depend on time are and HLl . In this situation, and assuming a non-extremal WIH ((l) = 0), this equation can be straightforwardly integrated [13], resulting in 1 1 12 T −(l) t 0 2 ∗ =e + Kil( D, ) + ⊗ − R + 4 q T − q , (8.15) (l) 2 2 2 where 0 is the integration constant, a time-independent symmetric tensor, and t is a coordinate on H compatible with l via Eq. (4.6), i.e. Ll t = 1. 8.4. Preferred WIH class [l] We have seen that a NEH admits an infinite number of WIH structures which, in the non-extremal case and according to (8.7), are parametrized by functions B defined on St . The question about the existence of a natural choice among them is naturally posed. In case we dispose of an a priori slicing (t ) of M, a slicing (St ) of the horizon is determined independently of the geometrical structures defined on H, as discussed in Section 4. Such a slicing fixes the null normal l via the normalization (4.6) [or, equivalently, the slicing fixes the lapse N and N determines l by Eq. (4.13)]. If the slicing is a WIH-compatible one, a particular class [l] is chosen. If not, such a slicing does not help in making such a choice. More interesting is, however, the opposite situation, which occurs whenever the 3 + 1 slicing (t ) is determined in a dynamical way. In such a context, an intrinsic determination of a preferred WIH class helps in the very construction of the 3 + 1 slicing. In fact, if such a WIH class is provided together with a definite initial cross-section S0 , then the slicing (St ) of H is completely determined,19 and can be used as a boundary condition to fix (t ). From an intrinsic point of view, fixing the class [l] reduces to choosing the function B in Eq. (8.7). Such a choice can be made by imposing a condition on a scalar definable in terms of the fields defining the WIH geometry. Following [13], an appropriate scalar in this sense is provided by the trace of HLl which, on a NEH, corresponds to the 19 We understand here the slicing (S ) of H as the set of slices S . The global constant ambiguity in the WIH class affects the rate at which t t the null generator on H traverses this ensemble (thus determining the associated lapse N on H up to constant), but does not change the ensemble itself.
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Lie derivative in the l direction of the expansion (k) associated with the ingoing null normal k [cf. Eq. (5.87)]. Contracting Eq. (7.33) with q yields Ll (k) = 2 D + −
12 2 R
+ 21 q R − (k) .
(8.16)
We start with a WIH with null normal l and free data (q, , , (l) ). Firstly we notice that, if condition (8.14) holds, the value of Ll (k) at t = 0 resulting from Eq. (8.15) satisfies (where 0 are constant) Ll (k) |t=0 = −(l) (q 0 ) .
(8.17)
Now we search a transformation of (q, , , (l) ) to new free data corresponding to another WIH built on the same NEH, such that the new (q , , , (l ) ) make Ll (k ) to vanish via the “primed” Eq. (8.16). In the non-extremal case, this is achieved by a rescaling l = l with = (1 + Be−(l) t ) ,
(8.18)
with t a coordinate compatible with l, i.e. an active transformation of the free data. Up to some caveats we shall mention below, this permits to fix the function B and therefore the WIH class (we adapt the discussion in [13,108], where it is carried out in terms of the tensor N introduced in Section 7.7.3). According to Table 1 and using the coefficient given by Eq. (8.18), the transformations of the objects in the righthand side of Eq. (8.16) are parametrized by the functions B. We make explicit these transformations in a first order expansion in the parameter B, and evaluate the transformed fields on the 2-surface S0 (i.e. we set t = 0): 2
D → 2D = 2D ,
(8.19)
2D
B → = + ≈ + 2 D B , 1+B (l) → (l ) = (l) , (k) ≈ (k) − B(k) . (k) → (k ) = 1+B
(8.20) (8.21) (8.22)
Introducing these transformed fields in the (transformed) Eq. (8.16) and gathering together the terms that expand Ll (k) by using again the “non-primed” Eq. (8.16), we obtain (2 D
2
D + 2 2 D + D + −
12 2 R
+ 21 q R )B = − Ll (k) t=0 .
(8.23)
Following Ashtekar et al. [13] we denote the operator acting on B as M. Making use of Eq. (8.17), we finally find the following condition on B: MB = (l) (q 0 )
.
(8.24)
A NEH is called generic if it satisfies condition (8.14) and if it admits a null normal l with constant (l) = 0 such that its associated M is invertible [13]. If a NEH is generic, Eq. (8.24) admits a unique solution B which is inserted in the rescaling (8.18) of l for finding the null vector l which satisfies Ll (k ) = 0. The main point to retain from this discussion is the fact that the condition Ll (k) = 0, together with (l) = const, fixes a unique WIH structure on the NEH (cf. [13]). 8.5. Good slicings of a non-extremal WIH Fixing the WIH class determines the foliation of H if an initial cross-section is provided. This is particularly interesting for the construction, from a Cauchy slice, of a spacetime containing a WIH. However, in more general problems in which no initial slice is singled out, simply demanding the slicing of H to be compatible with the chosen WIH class, is not enough to fix (St ) (this corresponds to the passive aspect of the gauge freedom discussed in
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Section 7.7.2). It is worthwhile to consider if a particular slicing associated with a WIH class can be chosen in a natural way. Even though in Section 11.3 we will provide a more intuitive presentation of this issue in terms of the 3 + 1 decomposition, we briefly show here the intrinsic approach followed in Refs. [13,108,111]. The foliation (St ) is fixed by providing a function t on H, that can be seen as the restriction to H of a scalar field defined in the whole spacetime M, and whose inverse images do foliate H. According to Eq. (4.34), fixing such a function t entails a specific choice of k [see Remark 4.2 for further insight on the relation between k and the foliation (St )]. If we fix a WIH class, for instance following the procedure explained in the previous section, the rotational 1-form is completely determined in this class, according to transformation rule in Table 1. Consequently, for a non-extremal horizon = const = 0, and taking into account Eq. (5.35), fixing k (or equivalently the slicing on H) translates into specifying the Há´"iˇcek form . Fixing (St ) (H, [l]) non-extremal WIH
! ⇔ fixing Há´"iˇcek 1-form
.
(8.25)
This will be the starting point of the discussion in Section 11.3.3. Here we briefly comment on an intrinsic procedure to fix . Firstly we note that, on a WIH with k satisfying HLl k = 0 (as it is the case), we have Ll = 0. In addition , l = 0, so projects to the sphere obtained as the quotient of H by the trajectories of the vector field l [cf. Remark 2.8 for a brief comment on this construction, but in terms of vectors in Tp (H)]. In general, a 1-form on a sphere S 2 can always be decomposed in = div-free + exact ,
(8.26)
where 2 D · div-free = 0 and exact = 2 Df for some function f on S 2 . This is a specific case of the general expression, known as Hodge decomposition for p-forms defined on a compact manifold provided with a non-degenerate metric (see for instance Refs. [46] or [124]). The divergence-free part of the Há´"iˇcek 1-form is determined by Eq. (7.67) that, together with (l) = const and Eq. (5.39), implies ddiv-free = 2 Im 2 2 .
(8.27)
Again in the context of the Hodge decomposition, the divergence-free part can always be written as div-free = 2 Dh·2
(8.28)
for a certain function h. In terms of h, Eq. (8.27) results in the Laplace equation on the sphere 2
h = 2 Im 2 ,
(8.29)
which completely fixes div-free . In order to consider the exact part of , we take the divergence of Eq. (8.26), resulting in 2
f = 2 D · exact = 2 D · .
(8.30)
Whereas the divergence-free part of is fixed by the WIH geometry, its divergence is not constrained by the null geometry. Therefore, in order to fix the exact part we must make a choice for the value 2 D · . Therefore f encodes the passive gauge freedom in the determination of the foliation (St ). A natural condition [13] consists simply in choosing f = 0, i.e. 2 D · = 0, which implies the vanishing of exact . However, such a choice does not lead to the usual foliations in the case of a rotating Kerr metric [108,18]. A choice that permits to recover the Kerr–Schild slicing of the horizon (cf. Appendix D), and which is motivated by the extremal Kerr black hole, is given by the Pawlowski gauge [108,18]: 2
D · = − 13 2 ln |2 | ,
(8.31)
where the complex Weyl scalar 2 has been defined by Eq. (4.50). Remark 8.1. As we saw in Section 7.7, the discussion of the free data associated with the null geometry involves a slicing of H. Since in this article we are working with the additional structure provided by the slicing (St ), it was
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appropriate for us to carry out such analysis in the context of a NEH. However, if no extra structure is added to that intrinsically defined on H, a WIH is needed in order to define a slicing of H (in the non-extremal case) as shown above. In such an approach, this Section 8 on WIH would probably offer a more natural setting for the general discussion on the free data, as done in Ref. [13]. 8.6. Physical parameters of the horizon Having discussed the applications of the WIH structure for analyzing the geometry of H as a hypersurface in M, let us turn our attention to the determination of the physical parameters associated with the black hole horizon.20 The introduction of a WIH structure on H permits to associate a quasi-local notion of mass and angular momentum with the black hole, independently of its environment. Such quasi-local notions are of fundamental astrophysical relevance for the study of black holes. Regarding the mass, the ADM mass of an asymptotically flat spacetime (see the textbooks [167] or [139] for a brief presentation of the different notions of mass in general relativity) accounts for the total mass included in a spacelike slice t . However, in a multi-component system it does not allow to determine which part is properly associated with the black hole and which part corresponds to the binding energy or the gravitational radiation. Remark 8.2. There exist in the literature other quasi-local approaches to prescribe the physical parameters associated with spatially bounded regions. See in this sense the review [158]. Let us highlight Brown and York work [35], where a review of the existing literature can also be found, and Refs. [92,116] for recent developments in the notion of quasi-local mass. Here we simply present the approach followed in Refs. [15,12], developed in the framework of quasiequilibrium black hole horizons modeled by null surfaces. For an extension of the discussion to the dynamical regime, see Ref. [17]. The strategy to determine the quasi-local parameters is also geometrical, but relying on techniques which are rather different from the ones introduced in the present article, where we have focused on the characterization of the geometry of H as a hypersurface embedded in spacetime. The setting for the discussion of the physical parameters is provided by the so-called Hamiltonian or symplectic techniques (see for instance Refs. [1,8,80] for general presentations). As in standard classical mechanics, physical parameters are characterized as quantities conserved under certain transformations, which in the present case are related to symmetries of the horizon (see Section 8.2). More specifically, one considers the phase space of solutions to the Einstein equation containing a WIH (H, [l]) in its interior. That is, each point of is a Lorentzian manifold (M, g) endowed with a WIH (H, [l]). Diffeomorphisms of M preserving H and such that their restriction to H implement a WIH-symmetry, induce canonical transformations on (when some additional non-trivial conditions are fulfilled; see Appendix C). The functions on generating these canonical transformations are identified with the physical quantities. A systematic discussion of these tools lays beyond the scope of this article. A brief account of them, organized in terms of (relevant) examples rather than a formal presentation, can be found in Appendix C. 8.6.1. Angular momentum Following Ashtekar et al. [13], we restrict ourselves to those horizons H which admit a WIH of class II (see Section 8.2). Therefore, there exists an axial vector field on H which is a SO(2) isometry of the induced metric q and is normalized in order to have a 2 affine length. Noting that this vector field presents in fact the standard form (8.12) (with c = 0, b = 1), a conserved quantity in associated with the horizon H can be defined (see Appendix C). This quantity, denoted as JH and identified with the angular momentum of the horizon, has the explicit form21 JH = −
1 " 1 " 1 " , 2 = − , 2 = − f Im 2 2 S S t t 8G 8G 4G St
,
(8.32)
20 As commented in the Introduction, in this review we do not discuss the electromagnetic properties of a black hole. In particular, in this section we restrain ourselves to solutions without matter. For a more general study (incorporating electromagnetic and Yang–Mills fields) see Refs. [10,12]. 21 In the expressions for the physical parameters we reintroduced explicitly the Newton constant G.
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where the second equality holds thanks to ∈ Tp (St ), and in the third equality we have used the fact that, since
· 2 [analogue to Eq. (8.28)]. Using then
is a divergence-free vector, there exists a function f such that =2 Df Eq. (7.67) and an integration by parts leads to the third form for JH . We note that this expression justifies the rotation 1-form terminology introduced in Section 7.5 for . If the vector can be extended to a stationary and axially symmetric neighborhood of H in M, representing the corresponding rotational Killing symmetry, then expression (8.32) can be shown to be equivalent to the Komar angular momentum [106,12] (see also expression (10.21) and, for instance, [139]). We point out that the integral (8.32) is well defined even if is not a WIH-symmetry (in fact, the divergence-free property is enough to guarantee the independence of (8.32) from the cross-section St of H [14,18]). However, in the absence of a symmetry, it is not so clear how to associate physically this value with a physical parameter. 8.6.2. Mass The definition of H’s mass is related to the choice of an evolution vector t. In order to have simultaneously a notion of angular momentum, we restrict ourselves again to horizons of class II and choose a fixed axial symmetry
on H. Since we want the restriction of t on H to generate a WIH-symmetry of (H, [l]), we demand, according to expression (8.12), (t + (t) )|H ∈ [l] ,
(8.33)
where (t) is a constant on H. Once these boundary conditions for t are set, the determination of the expression for the mass proceeds in two steps. 1. First Law of Thermodynamics t in the phase space , it can be shown As a result of demanding t to be associated with a conserved quantity EH t [12] that the function EH must depend only on two parameters defined entirely in terms of the horizon geometry: the " area, aH = St 2 , of the 2-slice St (constant, as a consequence of the NEH geometry) and the angular momentum t with respect to these parameters must satisfy [12] JH defined in Eq. (8.32). In fact, the variation of EH t
EH =
(t) (aH , JH )
aH + (t) (aH , JH ) JH . 8G
(8.34)
t is an This expression can be interpreted as a first law of black hole mechanics (see Remark 8.3 below), where EH energy function associated with the horizon.22 Note that (t) (aH , JH ) and (t) (aH , JH ) are constant on a given horizon H, where aH and JH have a definite value; in Eq. (8.34), aH and JH are rather parameters in the phase space (see Appendix C). However, this result does not suffice to prescribe a specific expression for the mass of the black hole. In fact, since in condition (8.33) we have not made an explicit choice for the representative l ∈ [l], the evolution vector t has not been t are not fixed. However, completely specified. Therefore, the functional forms of (t) (aH , JH ), (t) (aH , JH ) and EH once their dependences on aH and JH are specified, they turn out to be the same for every spacetime in , no matter how distorted is the WIH or how dynamical is the neighboring spacetime. This is a non-trivial result. 2. Normalization of the Energy function The second step consists precisely in fixing the functional forms of the physical parameters. In the space of solutions to the Einstein equation containing a WIH, there exists a subspace constituted by stationary spacetimes (the Kerr family, in fact parametrized by the area and angular momentum), where the existence of an exact rotational spacetime Killing symmetry tKerr provides a natural choice for the representative in [l]. This fixes the evolution vector t on H as well as the functional dependence of (t) and (t) for this family. If we impose the functional forms of the physical parameters, forms which are the same for any spacetime in , to coincide with those of the Kerr family when we restrict to its submanifold of stationary solutions, this completely determines their dependence on aH and JH (the biparametric nature of the Kerr family is crucial for this). This is not an arbitrary choice but a consistent normalization. 22 In Eq. (8.34), = t − q∗ t, , and a more precise meaning for the symbol in this context can be found in Appendix C. ( t)
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Defining the areal radius of the horizon RH by 2 := RH
1 " 2 aH = 4 4 St
,
(8.35)
the horizon black hole physical parameters can be expressed as MH (RH , JH ) := MKerr (RH , JH ) =
4 + 4G2 J 2 RH H
, 2GR H 4 − 4G2 J 2 RH H H (RH , JH ) := Kerr (RH , JH ) = , 3 4 2RH RH + 4GJ 2H 2GJ H H (RH , JH ) := Kerr (RH , JH ) = . 4 + 4GJ 2 RH RH H
(8.36)
8.6.3. Final remarks The following results can be retained from the previous discussion: (a) Explicit quasi-local expressions for the physical parameters associated with the black hole. Their determination proceeds by firstly calculating RH and JH from the geometry of H,23 via evaluation of expressions (8.32) and (8.35). These values are then plugged into (8.36). (b) Even though we need a WIH structure on H in order to derive the physical parameters, the final expressions only depend on the NEH geometry, and not on the specific chosen WIH. This is straightforward for the radius RH , since it only depends on the 2-metric induced on St . Regarding the angular momentum, the value of JH through Eq. (8.32) does not depend on the null normal l chosen on the NEH. Given a null normal l, a different one l is related to l by the rescaling l = l for some function on H. Using the transformation rule for in Table 1, the and J , calculated respectively with l and l is given by difference between JH H # # 1 1 JH − JH = − 2 D(ln ), 2 = (ln )d( · 2 ) = 0 , (8.37) 8G S 8G S where we have firstly integrated by parts and then used that , being an isometry of q, is a divergence-free vector (or straightforwardly, d( · 2 ) = L 2 − · d(2 ) = 0, since L q = 0). Therefore, it makes sense to refer to JH as the angular momentum of a NEH and, in fact, its very notation makes sense. (c) A by-product of the Hamiltonian analysis with implications for the null geometry of a non-extremal WIH (H, [l]), is the singularization of a specific null normal l0 in [l]. In any non-extremal WIH class [l] there is a unique representative such that its associated non-affinity coefficient coincides with the surface gravity of the Kerr family. In terms of an arbitrary null normal l in [l], and according to transformations in Table 1, l0 =
H (RH , JH ) l. (l)
(8.38)
The choice of a physical normalization for the null normal permits, on the one hand, to refer to its non-affinity coefficient 0 = H (RH , JH ) as the horizon surface gravity. On the other hand, such a normalization can be conveniently exploited for the determination of the horizon slicing as discussed in Sections 8.4 and 8.5. See in this sense Section 11.3. More generally, expression (8.33) can be used to set boundary conditions for certain fields on H (see Section 11). 23 In this article we are not including the electro-magnetic field. In the context of the Einstein–Maxwell theory, the resulting expressions for the horizon angular momentum and mass include an additional term corresponding to the electromagnetic field (see Ref. [12]). In the even more general Einstein–Yang–Mills case, implications on the mass of the solitonic solutions in the theory follow from the analysis of the first law in the isolated horizon framework [18].
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Remark 8.3. A major motivation for introducing the WIH structure on H is the extension of the black hole mechanics laws beyond the situation in which the horizon is embedded in a stationary spacetime [20]. The thermodynamical aspects of black hole horizons represent a cornerstone in understanding the physics of Gravity, both at the classical and the quantum level [168,131] (for the case of black hole binaries, see Ref. [71]). In this article, we have focused on those applications of isolated horizons to the null geometry of a hypersurface representing a black hole horizon in quasi-equilibrium inside a generally dynamical spacetime, and mainly aiming at their astrophysical applications in numerical relativity. However, the implications of this formalism go far beyond this aspect and, in particular, its results in black hole mechanics offer a link to the applications in quantum gravity. See the review [18] for a detailed account. We have seen how the constancy of the surface gravity (l) on the horizon, the zeroth law, follows from the WIH definition [see Eq. (8.5)], whereas the first law results from the introduction of a consistent notion of energy associated with the horizon [Eq. (8.34)]. In order to discuss the second law, linked to the increasing law of the area, one should go beyond the quasi-equilibrium regime and enter into the properly dynamical one (see Section 9.3 for a brief outline of this regime). As a by-product, dynamical horizons provide another version of the first law [17], associated with the evolution (a process) of a single system—Clausius–Kelvin’ sense—, whereas Eq. (8.34) dwells on (horizon) equilibrium states—Gibbs’ sense—in the phase space . 9. Isolated horizons II: (strongly) isolated horizons and further developments 9.1. Strongly isolated horizons After introducing the NEHs and WIHs, the third and final level in the isolated horizon hierarchy of intrinsic structures capturing the concept of black hole horizon in quasi-equilibrium, is provided by the notion of strongly isolated horizon, or simply isolated horizon (IH). We continue the strategy outlined at the beginning of Section 8. Consequently, starting ˆ to be time-independent. from a NEH, we demand the full connection ∇ Following Ashtekar et al. [15], a Strongly IH is defined as a NEH, provided with a WIH-equivalence class [l] such that ˆ =0 [HLl , ∇]
.
(9.1)
The consequences of imposing this structure on H can be analyzed in terms of the constraints and free data of the ˆ implies the vanishing of HLl S in null geometry. From the discussion in Section 7.7.3, the time independence of ∇ Eq. (7.84), that is ∇ˆ ( ) + + 21 (R − 2 R ) = 0 .
(9.2)
ˆ the time independence of S implies the WIH condition HLl = 0 (i.e. In terms of the decomposition (7.74) of ∇, H
∗ Ll . That is, an IH is characterized by the constraints l = 0, Ll (l) = 0) together with the vanishing of q
HL
HL
H
∗ Ll = 0 l q= Ll = q
,
(9.3)
where the only difference with respect to the WIH case discussed in Section 8.3 is the time independence of . From Eq. (7.33), it follows 1 1 T = Kil(2 D, ) + ⊗ − 2 R + 4 q ∗ T − q . (9.4) 2 2 2 In contrast with the NEH and WIH cases, where the initial data fields (q, , o , ) can be freely specified on a given cross-section, in the IH case Eq. (9.4) sets a constraint on the IH initial data. We note by passing that this is in complete analogy with the 3 + 1 spacetime case where initial data (, K) are constraint by Eqs. (3.37) and (3.38). We comment on the non-extremal case and, for completeness at the level of basic definitions, also on the extremal case. Regarding the non-extremal case o = 0, the IH constraint can be straightforwardly solved. In fact, once the
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fields q and and the constant o are freely chosen, the field is fixed by Eq. (9.4). Therefore, the free data are given in this case by non-extremal IH-free data : (q|St , |St , o = 0)
,
(9.5)
ˆ is then given by Eq. (9.4). The needed for reconstructing the full connection ∇ In the extremal case, o = 0, the situation changes. The vanishing of the left-hand side in Eq. (9.4) leaves as a free field on the initial cross-section St . The right-hand side becomes a constraint on (q|St , |St ) 1 12 T 2 ∗ Kil( D, ) + ⊗ − R + 4 q T − q = 0 . (9.6) 2 2 2 The initial data are given in this case by extremal IH-initial data : (q|St , |St , |St , o = 0) where q|St and |St satisfy Eq. (9.6)
.
(9.7)
9.1.1. General comments on the IH structure ˆ are timeIn simple terms, an isolated horizon is a NEH in which all the objects defining the null geometry (q, ∇) independent. It represents the maximum degree of stationarity for the horizon defined in a quasi-local manner. However, the notion of IH is less restrictive than that of a Killing horizon, which involves also the stationarity of the neighboring spacetime. In fact, a Killing horizon is a particular case of an isolated horizon, but the reverse is not true. One can have an IH such that no spacetime Killing vector can be found in any neighborhood of the horizon. Consequently, an IH permits to model situations with a stationary horizon inside a truly dynamical spacetime. Interestingly, non-trivial exact examples of this situation are provided in [110], in the context of a local analysis, and globally by the Robinson–Trautman spacetime (see [47]). This flexibility of the IH structure is important for its applications in dynamical astrophysical situations. In contrast with a WIH structure, an IH represents an actual restriction on the geometry of a NEH. In other words, if we start with an arbitrary NEH, it is not guaranteed that a null normal l can be found such that condition (9.1) is satisfied. An IH is in particular a WIH. Reasoning in terms of initial data as in point 3. of Section (8.2), given an arbitrary l on a NEH with associated data (q|St , |St , St , (l) ), a function can always be found such that the fields transformed under the rescaling l → l = l correspond to WIH free data (cf. Table 1). That is, (l ) = const (let us assume (l ) = 0 for definiteness). If the transformed fields satisfy (9.4), they correspond to the initial data of an IH. If not, the remaining freedom in these data corresponds to a new transformation l → l = l with = 1 + Be−(l ) t ,
(9.8)
where B is a function on St and H Ll t = 1. Substituting the transformed fields into (9.4) leads to three independent equations for a single variable B. If the system has no solution, this means that the NEH does not admit any IH. In general the choice of B only permits to cancel a scalar obtained from HLl (this was in fact the strategy in Section 8.4 to fix the WIH class). A similar argument applies in the extremal case. An analysis of the necessary conditions for a NEH to admit an IH can be found in A.2. of Ref. [13]. A posteriori analysis of black hole spacetimes. The interest of applying the geometrical tools discussed so far in numerical relativity is twofold. On the one hand, they can be used to set constraints on the fields entering in the numerical construction of a spacetime. This will be discussed in some detail in Section 11, mainly involving the NEH and WIH structures. On the other hand, they can be employed to extract physics in an invariant manner out of already constructed spacetimes. In fact, even though it could happen (but it remains to be studied) that the IH level is not flexible enough in order to accommodate astrophysically realistic initial situations, it is very well suited for the a posteriori analysis of the dynamical evolution toward stationarity after a stellar collapse or a black hole merger. In this sense, we briefly comment on the possibility of constructing a coordinate system in an invariant way for a neighborhood of the horizon H (in fact, only WIH notions are involved). This can be specially relevant for comparing results between different numerical simulations. Once a WIH class is fixed (using for instance results in Section 8.4;
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see also Section 11.3), we can choose a vector l in [l] [for instance via Eq. (8.38)] and a compatible slicing of H (see Section 8.5). Then, coordinates (t, , ) can be chosen on H, up to a re-parametrization of (, ) on the cross-sections St . In order to construct the additional coordinate outside the horizon, we choose the only vector k normal to St that satisfies k·l=−1. The affine parameter r of the only geodesic passing through a given (generic) point in H (with r =r0 and derivative −k on that point) provides a coordinate in a neighborhood of H. The rest of the coordinates (t, , ) are Lie-dragged along these geodesics. An analogous procedure can be followed in order to construct invariantly a tetrad in a neighborhood of H. Details of this construction can be found in [13,11,108]. In particular, a near horizon expansion of the metric in such a (null) tetrad can be found in [11]. Another example of a posteriori extraction of physics is provided in [65,108]. This reference presents an algorithm for assessing the existence of the horizon axial symmetry entering in the expression for the angular momentum (8.32). In case this symmetry actually exists, is explicitly reconstructed permitting the coordinate-independent assignation of mass and angular momentum to the horizon (the algorithm can also be used to look for approximate symmetries in case of small departs from axisymmetry). 9.1.2. Multipole moments In this section we address an important point from the point of view of applications and whose physical interpretation is quite straightforward. As in Section 8.6, we only consider the horizon vacuum case (see Ref. [14] for the inclusion of electromagnetic fields). In analogy with the source multipoles of an extended object in Newtonian gravity, which encode the distribution of matter of the source, the geometry of an IH permits to define a set of mass and angular momentum multipoles which characterize the black hole (whose horizon is in quasi-equilibrium) as a source of gravitational field (see [14]). As we saw in Section 8.6, a meaningful notion of angular momentum can be associated with the horizon if we impose its transversal sections St to admit an axial symmetry , i.e. if the horizon is of class II in the terminology of Section 8.6 (the requirement on the Killing character of can be relaxed to a divergence-free condition; see in this sense the discussion following Eq. (3.18) in Ref. [14]). In the same spirit, the discussion here applies only to class II horizons (and its subclass I). We limit ourselves again to the non-extremal case. As we have shown in Section 9.1, the geometry of a non-extremal IH is determined by the free data (q, ) on a cross-section St (the different constant values of change the representative of [l], but not the IH geometry). We must therefore characterize these two geometrical objects. The main idea is to identify two scalar functions, associated respectively with q and , such that they encode the geometrical information of these initial data [remember that only the divergence-free part of is a geometrical object in the sense of being independent of the cross section; see discussion in Section 8.5 or transformation rules (7.78)] Multipoles are then given by the coefficients in the expansion of these scalars in a spherical harmonic basis. This can actually be achieved in an invariant manner. In this section we simply present a brief account of the results in [14], referring the reader to this reference for details. The metric q. On a sphere S 2 , the geometrical content of a metric q can be encoded in a scalar function, such as its scalar curvature 2 R. The crucial remark is that, given a metric on S 2 with an axial symmetry (i.e. is a Killing vector on S 2 with closed orbits and vanishing exactly on two points), a particular coordinate system (, ) can be constructed in an invariant manner,24 where the 2-metric is written as q = (RH )2 (f −1 sin2 d ⊗ d + f d ⊗ d) ,
(9.9)
with f = q( , )/(RH )2 and RH is given by Eq. (8.35). The function f is related to the Ricci scalar 2 R by 2
R =−
1
d2 f
(RH )2 d(cos )2
.
(9.10)
Therefore, from the knowledge of 2 R the metric q can be reconstructed by using Eqs. (9.9) and (9.10). The round metric q0 q0 = (RH )2 (d ⊗ d + sin2 d ⊗ d)
(9.11)
24 This coordinate system plays a fundamental role in the discussion of IH multipoles. However, the technical details of its construction go beyond the scope of this section, mainly focused in presenting the final expressions for the multipoles. We refer the reader to Ref. [14] for a complete presentation.
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is obtained by making f = sin2 and has the same volume element, d2 V ≡ 2 0 = (RH )2 sin2 d ∧ d, than the physical metric q. The Há´jiˇcek 1-form As shown in Section 8.5 the divergence-free part div-free of the Há´"iˇcek 1-form is completely characterized by Im 2 , whereas its exact part exact is a gauge term related to the foliation in H, but without affecting the intrinsic geometry of the horizon. Therefore, the geometry of the IH is encoded in the pair (2 R, Im 2 ) (together with the radius RH ). In order to characterize these fields, and taking advantage of the invariant coordinate system previously introduced, an expansion in spherical harmonics Ylm (, ) can be performed. Due to the axial symmetry, the only functions entering into the decomposition are Yl0 (), which do not depend on . Since the volume element 2 , corresponding to q, coincides with the volume 2 0 = d2 V associated with the round metric q0 , the spherical harmonics are the standard ones, normalized according to # Yn0 ()Ym0 () d2 V = (RH )2 nm . (9.12) St
We define the two series of numbers In , Ln as # 1 2 RYn0 () d2 V , In := 4 St # Ln := − Im 2 Yn0 () d2 V . St
(9.13) (9.14)
These are geometrical dimensionless quantities that measure, respectively, the distortions and rotations of the horizon with respect to the round metric. We see explicitly that even the strongest notion of quasi-equilibrium that we have introduced, i.e. the IH structure, is rich and flexible enough so as to model physically interesting scenarios. If the two series In and Ln are given, the full isolated horizon geometry can be reconstructed from 2
R=
4
∞
(RH )2
n=0
Im 2 = −
In Yn0 () ,
1
∞
(RH )2
n=0
Ln Yn0 () .
(9.15) (9.16)
We note that in a NEH, the information about q and is also encoded invariantly in (2 R, Im 2 ) [together with the invariant coordinate system where q takes the form (9.9)]. Therefore, it makes sense to define In and Ln for a NEH. In this case, however, the full geometry cannot be reconstructed since the information on is missing. In order to obtain physical magnitudes that one can associate with the mass and rotation multipoles of the horizon, one must rescale In and Ln with dimensionful parameters. Motivated by heuristic considerations based in the analogy with magnetostatics and electrostatics in flat spacetime (see [14]), together with the results for the angular momentum JH and mass MH presented in Section 8.6, rotation multipoles are defined as [14] # n+1 R n+1 4 RH (2 2 D Pn (cos )) d2 V , (9.17) Ln = − H Jn := 2n + 1 4G 8G St resulting J0 = 0 and J1 = JH . The mass multipoles are then introduced as n n # 4 MH RH 4 MH RH (2 RYn0 ()) d2 V , In = Mn := 2n + 1 2 2n + 1 2 St
(9.18)
where MH is given by expression (8.36), resulting M0 = MH and M1 = 0 (centre of mass frame). These source multipoles present a vast domain of applications [14,18] ranging from the description of the motion of a black hole inside a strong external gravitational field, the study of the effects on the black hole induced by a companion or the invariant comparison at sufficiently late times of the numerical simulations of black hole spacetimes having suffered a strongly dynamical process (numerical simulations in Refs. [19,27] show that the isolated horizon notion becomes a good approximation quite fastly after the black hole formation).
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9.2. 3 + 1 slicing and the hierarchy of isolated horizons The stratification of the IH hierarchy in NEH, WIH and IH can be defined in terms of the null geometry because the ˆ are intrinsic to H. structures on which the time independence is imposed, i.e. q, and ∇, However, as indicated at the beginning of Section 8, when a 3+1 perspective is adopted by introducing the additional structure provided by the spatial slicing (t ), new objects which are not intrinsic to H enter into scene. This is the case of the Há´"iˇcek 1-form and . The geometry of H can now be defined in terms of the initial values of these fields on a given slice St . In this context, it seems natural to introduce progressive levels of horizon quasi-equilibrium by demanding the time independence of different combinations of the fields (q, , , ). This is a practical manner to proceed in the actual construction of the spacetime from a given initial slice. In accordance with the discussion of NEH initial data in Section 7.7.2, we introduce the non-affinity coefficient as an initial data, even if this parameter can be gauged away by a rescaling of the null normal. As we will see in Section 11, such a rescaling actually contain relevant information on the 3 + 1 description, in particular on the lapse, when a WIH-compatible slicing is chosen. In addition, we split tensor in its trace and traceless (shear) parts = 21 (k) q + (k) .
(9.19)
Given a NEH and a specific null normal l, the fields capable of changing in time are (, , (k) , (k) ). Let us denote by the letters A, B, C, D the four conditions A : HLl = 2 D = 0; B : HLl = 0 ; C : HLl (k) = 0; D : HLl (k) = 0 .
(9.20)
A NEH endowed with a null normal l will be called a (A, B, . . .)-horizon if conditions A, B, . . . are satisfied. As an example, a (A, B)-horizon is simply a WIH, whereas a (A, B, C, D)-horizon is a (strongly) IH. It is important to underline that this terminology makes no sense from a point of view intrinsic to H. Even more, due to the gauge freedom in the set of initial data, some of these (A, B, . . .)-horizons actually correspond to the main intrinsic object (for instance, a (B)-horizon is simply a NEH where we have chosen l in such a way that is time-independent, although it can depend on the angular variables on St ). The only aim of such a decomposition of the horizon quasi-equilibrium conditions, is to classify the different potential constraints on the null geometry of H that one would straightforwardly find in the 3 + 1 spacetime construction. In any case, it turns out to be useful for keeping track of the structures that are actually imposed in the construction of the horizon (see Section 11). 9.3. Departure from equilibrium: dynamical horizons This article deals with the properties of a black hole horizon in equilibrium, following a quasi-local perspective. The basic idea is to consider an apparent horizon S in a spatial slice t , and then assume that this apparent horizon evolves smoothly into other apparent horizons (see discussion in Section 7.1.2). The hypersurface H defined in this way constitutes the quasi-local characterization of the black hole. The key element associated with the quasi-equilibrium of this apparent-horizon world-tube is the null character of H. In this section we briefly indicate how these quasi-local ideas have been extended in the literature to the regime in which the black hole horizon is dynamical (see Refs. [18,31] for recent reviews). The horizon is in quasi-equilibrium if neither matter nor radiation actually cross it [31,32]. Motivated by Hawking’s black hole area theorem [87] for event horizons, the world-tube H corresponds to a quasi-equilibrium situation if the volume element of the apparent horizons remains constant (this implies that the area is constant, but the converse is not true in general). On the contrary, the dynamical case corresponds to an increasing area along the evolution of the world-tube. These considerations on the rate of change of the area, translate into the metric type of H as follows. Under the physically reasonable assumption Lk (l) < 0 (a necessary condition for the spheres inside the apparent horizon to be future-trapped), a vector z tangent to H and normal to the apparent horizon sections St is either null or spacelike (see [93,65,108]). The volume element of the apparent horizons is constant, corresponding to the quasi-equilibrium case, if and only if z is everywhere null on H, i.e. if H is null hypersurface [108,65] (cf. Eq. (5.74)). If z is everywhere
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spacelike, and (k) < 0, then the area of the horizons actually increases. In the intermediate cases, with (k) < 0, the area is never decreasing. The properly dynamical regime can be quasi-locally characterized by the notion of dynamical horizon introduced by Ashtekar et al. [16,17]. A dynamical horizon is a spacelike hypersurface in spacetime that is foliated by a family of spheres (St ) and such that, on each St , the expansion (l) associated with the outgoing null normal l vanishes and the expansion (k) associated with the ingoing null normal k is strictly negative. This is a particular case of the previous notion of trapping horizon introduced by Hayward [93] and commented at the end of Section 7.1.2. In particular, it is directly related to future outer trapping horizons, whose definition coincides with that of dynamical horizons once the spacelike nature of H has been substituted by the condition Lk (l) < 0. This last condition can be crucial to ensure that trapped surfaces (i.e. having both (l) < 0 and (k) < 0) exist inside H [148]. A future outer trapping horizon can be a null or spacelike hypersurface (more generally, the vector z can be either null or spacelike), potentially permitting a clearer description of the transition from the equilibrium to the dynamical situation [33]. Let us note that the Damour–Navier–Stokes equation discussed in Section 6.3 has been recently extended to future outer trapping horizons and dynamical horizons [76]. We can think of implementing Hayward’s dual null construction, since in our approach we have extended k outside H. For doing this, we demand the null field k to be normal to a null hypersurface H , in such a way that S = H ∩ H . From Frobenius’s identity (see Section 5.3; in particular Eq. (5.39) and Remark 5.1), it follows d ln
N M
= k + l .
(9.21)
On the one hand, this can be used as a constraint between the lapse N and M [the latter being a function of the 3-metric , as a consequence of Eq. (4.19)]. On the other hand, since k is null and hypersurface-normal, it is pre-geodesic (Section 2.5). This can be checked explicitly in Eq. (5.44) by noting that, due to (9.21), we have ∇k ln(N/M)=k·d ln(N/M)=−, so ∇k k = (/N 2 )k. Consequently, H provides the surface t = const parametrized by (r, , ) in the invariant construction of the coordinate system on a neighborhood of H, presented at the end of Section 9.1.1.
10. Expressions in terms of the 3 + 1 fields 10.1. Introduction Hitherto we have used the 3 + 1 foliation of spacetime by the spacelike hypersurfaces (t ) only (i) to set the normalization of the normal l to the null hypersurface H (by demanding that l is the tangent vector of the null generators of H when parametrizing the latter by t [Eq. (4.5)]), and (ii) to introduce the ingoing null vector k and the associated projector onto H, . In the present section, we move forward in our “3 + 1 perspective” by expressing all the fields intrinsic to the null hypersurface H, such as the second fundamental form or the rotation 1-form , in terms of the 3 + 1 basics objects, like the extrinsic curvature tensor K, the lapse function N or the timelike unit normal n. In this process, we benefit from the four-dimensional point of view adopted in defining , , and other objects relative to H, thanks to the projector . 10.2. 3 + 1 decompositions 10.2.1. 3 + 1 expression of H’s fields We have already obtained the 3 + 1 decomposition of the null normal l [Eq. (4.13)]. By inserting it into Eq. (5.13), we get = ∇ q q = ∇ [N (n + s )]q q
= N (∇ n + ∇ s )q q = N (∇ n + ∇ s ) q q = N (−K + D s )q q ,
(10.1)
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where we have used n q = 0 and s q = 0 to get the second line and Eqs. (3.15) and (3.8) to get the third one. Hence the 3 + 1 expression of the second fundamental form: = N (D s − K )q q
,
(10.2)
or equivalently (taking into account the symmetry of and K): = N q ∗ (Ds − K)
.
(10.3)
Contracting Eq. (10.2) with q gives the expansion scalar [cf. Eq. (5.65)]: = N q (D s − K ) = N ( − s s )(D s − K ) = N(D s − K − s s D s +K s s ) ,
(10.4)
=0
hence = N (Di s i + Kij s i s j − K)
,
(10.5)
or equivalently = N (D · s + K(s, s) − K)
.
(10.6)
We then deduce the 3 + 1 expression of the non-affinity parameter via Eq. (5.69): = ∇ · l − = ∇ [N (n + s )] −
= ∇ ln N + N (−K + ∇ s ) − N (Di s i + Kij s i s j − K) = ∇ ln N + N (∇ s − Di s i − Kij s i s j ) .
(10.7)
Now, by taking the trace of D s = ∇ s [cf. Eq. (3.8)], we get the following relation between the 3-dimensional and 4-dimensional divergences of the vector s Di s i = ∇ s = ∇ s + n n ∇ s = ∇ s − n s ∇ n = ∇ s − s D ln N ,
(10.8)
where we have used n s = 0 and Eq. (3.20). Substituting Eq. (10.8) for Di s i into Eq. (10.7) leads to = ∇ ln N + s i Di N − N K ij s i s j
,
(10.9)
or equivalently, = ∇l ln N + Ds N − N K(s, s)
.
(10.10)
To compute the 3 + 1 expression of the rotation 1-form, the easiest manner is to start from expression (5.41) for and to replace in it k from Eq. (4.29): 1 1
n − = ∇ k = ∇
N 2N 2 1 1 1 1 = − 2 ∇ N n + (−K − n D ln N ) + 3 ∇ N − N N N 2N 2 1
= D ln N − K s + ∇ ln N − s − n , (10.11) N 2 2N
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where we have used Eqs. (3.18) and (2.21) to get the second line. Substituting Eq. (10.9) for in the s term yields 1 = D ln N − K s + (n ∇ ln N + Kij s i s j )s − n 2 2N
,
(10.12)
or equivalently, 1 = D ln N − K(s, .) + [∇n ln N + K(s, s)]s − n 2 2N
.
(10.13)
The 3 + 1 expression of the Há´"iˇcek 1-form is then immediately deduced from = q along with n q = 0 and s q = 0 = 2 D ln N − K s q
,
(10.14)
or equivalently, = 2 D ln N − q ∗ K(s, .)
.
(10.15)
In Appendix D, the above formulæ are evaluated in the specific case where H is the event horizon of a Kerr black hole, with a 3 + 1 slicing linked to Kerr coordinates. In particular the standard value of (called surface gravity in that case) is recovered from Eq. (10.10). Moreover the Há´"iˇcek 1-form computed from Eq. (10.15), once plugged into formula (8.32), leads to the angular momentum JH = am (where a and m are the standard parameters of the Kerr solution), as expected. Let us now derive the 3 + 1 expression of the transversal deformation rate . Similarly to the computation leading to Eq. (10.1), we have, thanks to Eqs. (5.80) and the 3 + 1 expression (4.27) of k, 1
= ∇ k q q = ∇
(n − s ) q q 2N 1 (∇ n − ∇ s )q q + 0 . = (10.16) 2N Hence = −
1 (D s + K )q q 2N
,
(10.17)
or equivalently (taking into account the symmetry of and K): =−
1 ∗ q (Ds + K) 2N
.
(10.18)
Contracting (10.17) with q gives the transversal expansion scalar [cf. Eq. (5.87)]: (k) = −
1 (Di s i − Kij s i s j + K) 2N
,
(10.19)
or equivalently (k) = −
1 (D · s − K(s, s) + K) 2N
.
(10.20)
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10.2.2. 3 + 1 expression of physical parameters In Section 8.6 physical parameters have been associated with H, when the latter constitutes a NEH. More specifically, we proceeded by firstly characterizing the radius RH and the angular momentum JH in terms of geometrical objects on H through Eqs. (8.35) and (8.32), respectively, and then we introduced expressions for the mass, surface gravity and angular velocity through (8.36). In an analogous manner, in Section 9.1.2 mass and angular momentum multipoles Mn and Jn have been expressed in terms of geometrical multipoles In and Ln through Eqs. (9.17)–(9.18) and Eqs. (9.13)–(9.14). Regarding their expressions in terms of 3 + 1 fields, RH and In are already in an appropriate form, since they are completely defined in terms of the metric q living on St ⊂ t . In order to obtain a 3 + 1 expression for the angular momentum we make use of Eq. (10.14), obtaining JH =
# 1 s K d2 V . 8G St
(10.21)
Regarding Ln and Jn , the relation H d = 2 Im 2 2 , which follows from (7.67) and (7.73), permits to express Im 2 =
1 2 2 D . 2
Making use again of (10.14), Jn reads n+1 # 4 RH 2 D (2 s K )Yn0 () d2 V . Jn = 2n + 1 8G St
(10.22)
(10.23)
10.3. 2+1 decomposition As discussed in Section 4.2, each spatial hypersurface t can be foliated in the vicinity of H by a family of 2-surfaces (St,u ) defined by u = const and such that the intersection St of t with the null hypersurface H is the element u = 1 of this family. The foliation (St,u ) induces an orthogonal 2+1 decomposition of the three-dimensional Riemannian manifold (t , ), in the same manner that the foliation (t ) induces an orthogonal 3 + 1 decomposition of the fourdimensional Lorentzian manifold (M, g), as presented in Section 3. The 2+1 equivalent of the unit normal vector n is then s and the 2+1 equivalent of the lapse function N is the scalar field M defined by Eq. (4.17). Indeed we have shown the relation s = MDu [Eq. (4.19)], which is similar to relation (3.1) between n and dt. The only difference is a sign factor, owing to the fact that n is timelike, whereas s is spacelike. 10.3.1. Extrinsic curvature of the surfaces St The second fundamental form (or extrinsic curvature) of St , as a hypersurface of (t , ), is the bilinear form R H0 : Tp (t ) × Tp (t ) −→ (u, v) −→ u · Dq (v) s .
(10.24)
Notice the similarity with Eq. (3.13) defining the extrinsic curvature K of t and with Eq. (5.9) defining the second fundamental form of H. Following our four-dimensional point of view, we extend the definition of H0 to Tp (M) × Tp (M), via the mapping ∗ [cf. Eq. (3.5)]: H := ∗ H0 .
(10.25)
Then, for any pair of vectors (u, v) in Tp (M), H(u, v) = (u) · Dq (v) s. Actually the projector in front of u is not necessary since Dq (v) s is tangent to t , so that we can write R H : Tp (M) × Tp (M) −→ (u, v) −→ u · Dq (v) s .
(10.26)
In index notation, H = D s q .
(10.27)
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We have, for any (u, v) ∈ Tp (M) × Tp (M), H( q(u), q (v)) = q (u) · Dq (v) s = ( (u) − s, us) · Dq (v) s = H(u, v) ,
(10.28) (10.29)
for s · s = 1 implies s · Dq (v) s = 0. Combining (10.28) and (10.29), we realize that H = q ∗ Ds ,
(10.30)
or equivalently H = D s q q ,
(10.31)
which strengthens Eq. (10.27). Since Ds = ∗ ∇s [cf. Eq. (3.9)] and q ∗ ∗ = q ∗ , we obtain from Eq. (10.30) that H = q ∗ ∇s .
(10.32)
Substituting Eq. (4.16) for s in this expression leads to H = ∇ (N∇ t + M∇ u)q q
= (∇ N ∇ t + N∇ ∇ t + ∇ M∇ u + M∇ ∇ u)q q = (N∇ ∇ t + M∇ ∇ u)q q ,
(10.33)
where we have used q ∇ t = −N −1 q n = 0 and q ∇ u = e− q = 0. Since ∇ ∇ f = ∇ ∇ f for any scalar field f (vanishing of ∇’s torsion), Eq. (10.33) allows us to conclude that the bilinear form H is symmetric. This property, which is shared by the other second fundamental forms K and , arises from the orthogonality of s with respect to some surface (St ) and is a special case of what is referred to as the Weingarten theorem. Expanding the q in the definition (10.27) of H leads to H = D s q = D s ( − s s ) = D s − s D s s .
(10.34)
Let us evaluate the “acceleration” term s D s which appears in this expression. We have s D s = s ∇ s = s ∇ s ,
(10.35)
Ds s = ∗ ∇s s .
(10.36)
hence
∇s s is easily expressed in terms of the exterior derivative of s: indeed, (s · ds) = s (∇ s − ∇ s ) = s ∇ s , since s ∇ s = 0 for s has a fixed norm. Thus Eq. (10.36) becomes Ds s = ∗ (s · ds) = ∗ [s · d(N dt + Mdu)] = ∗ [s · (dN ∧ dt + dM ∧ du)] = ∗ (dN, sdt − dt, s dN + dM, s du − du, s dM) =0
∗
∗
= dN, s dt +dM, s du −M =0
−1 ∗
=M −1
dM
=M −1 s
= − ∗ d ln M + d ln M, ss = − q∗ d ln M ,
(10.37)
from which we conclude that Ds s = −2 D ln M
,
(10.38)
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which is a relation similar to Eq. (3.20). If we use it to replace s D s in Eq. (10.34) and use the symmetry of H, we get [compare with (3.18)] H = D s + s 2 D ln M
(10.39)
or equivalently, H = Ds + 2 D ln M ⊗ s
.
(10.40)
Again note the sign differences with Eq. (3.19).
The mean curvature of St , as a surface embedded in (t , ), is given by half the trace of H:
= H = g H = q H = q ab Hab = H a a . H := tr H
(10.41)
From Eq. (10.40), H is equal to the three-dimensional divergence of the unit normal to St : H =D·s .
(10.42)
Remark 10.1. We recover immediately from this expression that for a sphere in the Euclidean space R3 , the mean curvature is nothing but the inverse of the radius. Indeed Eq. (10.42) yields H = 2/R, where R is the radius of the sphere. In the Riemannian 3-manifold (t , ), H may vanish if St is a minimal surface, in the very same manner that K vanishes if t is a maximal hypersurface of spacetime. Note that minimal surfaces have been used as inner boundaries in the numerical construction of black initial data by many authors [122,114,34,49,53,50,136,64,77,79]. 10.3.2. Expressions of and in terms of H Combining Eqs. (10.30) and (10.3), we get an expression of the second fundamental form of H (associated with l) in terms of the second fundamental form of the 2-surface St embedded in t (i.e. H) and the second fundamental form of the 3-surface t embedded in M (i.e. K): = N(H − q ∗ K)
.
(10.43)
Similarly, expression (10.18) for the transversal deformation rate becomes =−
1 (H + q ∗ K) 2N
.
(10.44)
As a check of formulæ (10.43) and (10.44), we can compare them with those in Eq. (29) √ of Cook and Pfeiffer [54], after having noticed that the null normal vector l used by√these authors is l = lˆ = ( 2N )−1 l, with lˆ defined √ by Eq. (4.14); this results in a second fundamental form = ( 2N )−1 and a transversal deformation rate = 2N [cf. the scaling laws in Table 1].25 Remark 10.2. Replacing and by the expressions (10.43) and (10.44), as well as l and k by their expressions in terms of n and s [Eqs. (4.13) and (4.27)], into formula (5.83) for the second fundamental tensor of the 2-surface St (cf. Remark 5.4) results in q ∗ K) n − H s . K = −(
(10.45)
This expression has the same structure than Eq. (5.83), describing K in terms of the timelike–spacelike pair of normals (n, s), whereas Eq. (5.83) describes K in terms of the null–null pair of normals (l, k). 25 Also note that Cook and Pfeiffer [54] define and (denoted by them and ´ respectively) by = 1/2 q∗ Ll g and = 1/2 q∗ Lk g, [their Eqs. (24) and (25)], whereas we have established that = 1/2 q∗ Ll q and = 1/2 q∗ Lk q [our Eqs. (5.56) and (5.78)]. It can be seen easily that both expressions for and coincide, thanks to the operator q ∗ and the relation q = g + l ⊗ k + k ⊗ l.
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251
In Appendix D, formulæ (10.43) and (10.44) are used to evaluate and in the specific case of the event horizon of a Kerr black hole. In particular Eq. (10.43) leads to = 0 as expected for a stationary horizon. Thanks to Eq. (10.42), expressions (10.6) and (10.20) for the expansion scalars and (k) become = N (H − K + K(s, s))
(10.46)
and (k) = −
1 (H + K − K(s, s)) 2N
.
(10.47)
Remark 10.3. Eqs. (10.43) and (10.46) constitute a 2+1 writing of and . We have obtained a different (but equivalent !) 2+1 writing in Section 5.5, via Eqs. (5.62) and (5.77). 10.4. Conformal decomposition 10.4.1. Conformal 3-metric In many modern applications of the 3 + 1 formalism in numerical relativity, a conformal decomposition of the spatial metric is performed. This includes the 3 + 1 initial data problem, following the works of Lichnerowicz [113] and York and collaborators [170,129,173,137] and the time evolution schemes proposed by Shibata and Nakamura [150] and Baumgarte and Shapiro [24], as well as the recent constrained scheme based on Dirac gauge proposed by Bonazzola et al. [29]. The conformal decomposition consists in writing = 4 ˜ ,
(10.48)
where is some scalar field. Very often, is chosen so that ˜ is unimodular. As shown in Ref. [29], this can be achieved without making and ˜ tensor densities by introducing a background flat metric f on t . The conformal factor is then defined by det ij 1/12 , (10.49) = det fij where det ij (resp. det fij ) is the determinant of the components of (resp. f) with respect to a coordinate system (x i ) on t . The quotient of the two determinants is independent of the coordinates (x i ), so that is a genuine scalar field and ˜ a genuine tensor field. The unimodular character of ˜ then translates into det ˜ ij = det fij ,
(10.50)
with det fij = 1 if (x i ) are coordinates of Cartesian type. ˜ the connection on t compatible with the metric ˜ . The D-derivative and D-derivative ˜ Let us denote by D of any vector v ∈ T(t ) or any 1-form ∈ T∗ (t ) are related by Di v j = D˜ i v j + C j ki v k
and Di j = D˜ i j − C k j i k ,
(10.51)
with C k ij := 21 kl (D˜ i lj + D˜ j il − D˜ l ij ) = 2(D˜ i ln k j + D˜ j ln k i − D˜ k ln ˜ ij ) .
(10.52)
˜ as a tensor field on M by For any tensor field T on t , we define DT ˜ = ∗ t DT ˜ , DT
(10.53)
˜ is the original definition of the D-derivative ˜ where t DT of T within the manifold t , as introduced above. Actually Eq. ˜ (10.53) allows us to manipulate D-derivatives as four-dimensional objects, as we have done already for D-derivatives.
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10.4.2. Conformal decomposition of K In addition to the conformal decomposition of the spatial metric , the conformal 3 + 1 formalism is based on a conformal decomposition of t ’s extrinsic curvature K: ˜ + 1 K K =: A 3
,
(10.54)
˜ captures the traceless part of K: ij A˜ ij = ˜ ij A˜ ij = 0 and the exponent is usually chosen to be −2 or where A 4. The choice = −2 has been introduced by Lichnerowicz [113] and is called the conformal transverse-traceless decomposition of K [51,171]; it leads to an expression of the momentum constraint (3.38) which is independent of for maximal slices (K = 0) in vacuum spacetimes. It has been notably used to get the Bowen–York semi-analytical initial data for black hole spacetimes [34]. The choice = 4 is called the physical transverse–traceless decomposition ˜ in terms of the time derivative of the conformal metric, the shift vector of K [51,129] and leads to an expression of A and the lapse function which is independent of [Eq. (10.58) below]. This choice has been employed mostly in time evolution studies [150,24,29]. In the present article, we do not choose a specific value for , so that the results are valid for both of the cases above. Let us consider a coordinate system (x i ) on each hypersurface t so that (x )=(t, x i ) constitute a smooth coordinate system on M. We denote the shift vector of these coordinates by . The coordinate time vector is then t = N n + [Eq. (3.24)] and we define the time derivative of the conformal metric ˜ by ˙˜ := Lt ˜ .
(10.55)
Written in terms of tensor components with respect to (x i ), this definition becomes ˙˜ ij :=
j˜ ij jt
= −˜ ik ˜ j l
j˜ kl , jt
(10.56)
where the second equality follows from ˜ ik ˜ kj = ij . The trace of relation (3.42) between the extrinsic curvature K and the time derivative of the metric leads to the following evolution equation for the conformal factor j 1 ˜ ln − L ln = (D · − N K) , jt 6
(10.57)
whereas its traceless part gives a relation between A˜ and ˙˜ : 4− ˜= A 2N
˜ − ˜ ) Kil(D,
2 ˜ (D · )˜ − ˙˜ 3
,
(10.58)
˜ denotes the connection associated with the conformal metric ˜ and ˜ is the 1-form dual to the shift vector via where D the conformal metric: ˜ := ˜ (, .) = −4 ,
(10.59)
or in index notation: ˜ i := ˜ ij j = −4 j . Inserting Eq. (10.58) into Eq. (10.54) leads to the following expression of the extrinsic curvature: 4 ˜ + 2 (N K − D ˜ ) ˜ · )˜ − ˙˜ , K= Kil(D, 2N 3 which is independent of .
(10.60)
(10.61)
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10.4.3. Conformal geometry of the 2-surfaces St The conformal metric ˜ induces an intrinsic and an extrinsic geometry of the 2-surfaces St ⊂ t . First of all, the vector normal to St and with unit length with respect to ˜ is s˜ := 2 s .
(10.62)
We denote by s˜ the 1-form dual to it with respect to the metric ˜ 26 s˜ := ˜ (˜s, .) = −2 (s, .) = −2 s ,
(10.63)
or in components, s˜ := ˜ s˜ = −2 s .
(10.64)
Combining Eqs. (4.19) and (10.62), we get ˜ , s˜ = MDu
(10.65)
M˜ := −2 M .
(10.66)
with
Let us notice that the orthogonal projector onto the 2-surface St is the same for both metrics and ˜ : q = − s, .s = − ˜s, .˜s .
(10.67)
As a hypersurface of t endowed with the conformal metric ˜ , the first fundamental form of St is q˜ = −4 q = ˜ − s˜ ⊗ s˜ ,
(10.68)
˜ s replaced by s˜ and its second fundamental form is defined by a formula similar to Eq. (10.26), with D replaced by D, and the scalar product [simply denoted by a dot in Eq. (10.26)] taken with ˜ : ˜ : Tp (M) × Tp (M) −→ R H ˜ q (v) s˜) . (u, v) −→ ˜ (u, D
(10.69)
In index notation, one has H˜ = ˜ q D˜ s˜ = D˜ s˜ q .
(10.70)
˜ is symmetric and we have the following property: Similarly to H, H ˜ = q ∗ D˜ ˜s . H
(10.71)
˜ s˜ s˜ is given by a formula similar to Eq. (10.38) The “acceleration” D ˜ s˜ s˜ = −2 D ˜ ln M˜ D
,
(10.72)
˜ denotes the connection associated with the conformal metric q˜ in St . The 2 D-derivatives ˜ where 2 D of a tensor field can ˜ by a projection formula identical to Eq. (5.59), except for ∇ in the right-hand side replaced be expressed in terms of D ˜ It is actually easy to establish Eq. (10.72) from Eq. (10.38): the D-derivative and D-derivative ˜ by D. of the 1-form s are related by Eq. (10.51). Substituting 2 s˜j for sj [Eq. (10.64)], we get Di sj = 2 (D˜ i s˜j − 2˜si D˜ j ln + 2˜s k D˜ k ln ˜ ij ) ,
(10.73)
from which we obtain s k Dk si = s˜ k D˜ k s˜i − 2D˜ i ln + 2˜s k D˜ k ln ˜si . Substituting Eq. (10.38) for s k Dk si and replacing M by 2 M˜ then leads to Eq. (10.72). 26 This notation does not follow the underlining convention stated in Section 1.2.1 [cf. Eq. (1.10)], namely s˜ is not the dual to s˜ provided by the metric g (= on t ).
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˜ leads to Expressing q = − s˜ s˜ in Eq. (10.70) and using Eq. (10.72) together with the symmetry of H H˜ = D˜ s˜ + s˜ 2 D˜ ln M˜ ,
(10.74)
˜ = D˜ ˜ s + 2D ˜ ln M˜ ⊗ s˜ . H
(10.75)
or
˜ with respect to the conformal metric ˜ : Let us denote by H˜ the trace of H H˜ := ˜ ij H˜ ij .
(10.76)
We then have, similarly to Eq. (10.42), ˜ · s˜ . H˜ = D
(10.77)
˜ s and compare with Eq. (10.71), we get If we substitute Ds in Eq. (10.30) by its expression (10.73) in terms of D˜ ˜ + 2(D ˜ s˜ ln )˜q] H = 2 [H
.
(10.78)
The trace of this equation writes ˜ s˜ ln ) . H = −2 (H˜ + 4D
(10.79)
10.4.4. Conformal 2 + 1 decomposition of the shift vector As in Section 10.4.2, we consider a coordinate system (x i ) on t and the associated shift vector . In Section 4.8 we have already introduced the 2 + 1 orthogonal decomposition of with respect to the surface St [cf. Eq. (4.67)]: = bs − V
with V ∈ Tp (St ) .
(10.80)
Let us re-write this decomposition as ˜s − V , = b˜
(10.81)
with b˜ := −2 b = ˜ (˜s, ) = ˜s,
.
(10.82)
If (t, x i ) constitutes a coordinate system stationary with respect to H, then b = N on H [Eq. (4.79)], so that H ((x ) stationary w.r.t. H) ⇐⇒ b˜ = N −2 ,
(10.83)
H
where = means that the equality holds only on H. As a consequence of Eq. (10.81), the 1-form ˜ defined by Eq. (10.59) has the 2 + 1 decomposition ˜s − V ˜ , ˜ = b˜
(10.84)
˜ := ˜ (V, .) = −4 V . V
(10.85)
with
In index notation: ˜ si − V˜i ˜ i := b˜
with V˜i = ˜ ij V j = −4 Vi .
(10.86)
From this expression and Eq. (10.74), we get ˜ + s˜i D˜ j b˜ + s˜j D˜ i b˜ − D˜ i V˜j − D˜ j V˜i D˜ i ˜ j + D˜ j ˜ i = b(2H˜ ij − s˜i 2 D˜ j ln M˜ − s˜j 2 D˜ i ln M)
(10.87)
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255
and D˜ i i = b˜ H˜ + s˜ i D˜ i b˜ − 2 D˜ a V a − V a 2 D˜ a ln M˜ . Injecting these last two relations into expression (10.61) of the extrinsic curvature, we get 4 ˜ + s˜i D˜ j b˜ + s˜j D˜ i b˜ − D˜ i V˜j Kij = b(2H˜ ij − s˜i 2 D˜ j ln M˜ − s˜j 2 D˜ i ln M) 2N 2 k ˜ ˜ 2 ˜ a a2 ˜ ˙ ˜ ˜ ˜ ˜ ˜ −Dj Vi + (N K − bH − s˜ Dk b + D a V + V D a ln M)˜ ij − ˜ ij . 3
(10.88)
(10.89)
We deduce immediately from this expression the scalar K(s, s) which appears in formulae of Section 10.2 and 10.3: 1 1 1 (2˜s k D˜ k b˜ − 2V a 2 D˜ a ln M˜ + N K − b˜ H˜ + 2 D˜ a V a ) − ˙˜ kl s˜ k s˜ l . (10.90) K(s, s) = N 3 2 We deduce also from Eq. (10.89) that Kkl s k q l i =
2 2 ˜ ˜ ˜ 2 ˜ ( D i b − b D i ln M˜ − s˜ k 2 D˜ k V˜l q l i + H˜ ik V k − ˙˜ kl s k q l i ) 2N
and Kkl q k i q l j =
4 2 2b˜ H˜ ij − 2 D˜ i V˜j − 2 D˜ j V˜i − ˙˜ kl q k i q l j + (N K − b˜ H˜ − s˜ k D˜ k b˜ 2N 3 ˜ q˜ij . +2 D˜ a V a + V a 2 D˜ a ln M)
(10.91)
(10.92)
10.4.5. Conformal 2 + 1 decomposition of H’s fields We are now in position to give expressions of the various fields related to H’s null geometry in terms of conformal 2 + 1 quantities. First of all, replacing K(s, s) by formula (10.90) in Eq. (10.10) leads to = ∇ ln N + −2 s˜ k D˜ k N + 21 ˜˙ kl s˜ k s˜ l ˜ . + 13 (b˜ H˜ − N K + 2V a 2 D˜ a ln M˜ − 2 D˜ a V a − 2˜s k D˜ k b)
(10.93)
H H For a coordinate system stationary with respect to H, one has b˜ = N −2 [Eq. (10.83)] and l = t + V [Eq. (4.80)], so that the above expression can be written as H,sc
=
j N ln N + V a 2 D˜ a ln N + −2 s˜ k D˜ k N + (−2 H˜ − K) jt 3 1˙ k l 2 1 + ˜ kl s˜ s˜ + V a 2 D˜ a ln M˜ − 2 D˜ a V a − s˜ k D˜ k b˜ , 2 3 2
(10.94)
H,sc
where = means that the equality holds only on H and for a coordinate system stationary with respect to H. Next, replacing H by formula (10.79) and K(s, s) by formula (10.90) in Eq. (10.46) leads to = N −2 (4˜s k D˜ k ln + 23 H˜ ) − 21 ˜˙ kl s˜ k s˜ l ˜ H˜ + 2˜s k D˜ k b˜ − 2V a 2 D˜ ln M˜ + 2 D˜ V a − 2N K] . + 13 [(N−2 − b) a a
(10.95)
For a coordinate system stationary with respect to H, this expression simplifies somewhat, thanks to relation (10.83): H,sc = N −2 (4˜s k D˜ k ln + 23 H˜ ) − 21 ˜˙ kl s˜ k s˜ l + 23 (˜s k D˜ k b˜ − V a 2 D˜ a ln M˜ + 21 2 D˜ a V a − N K) .
(10.96)
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The expression of H’s second fundamental form in terms of the conformal 2 + 1 quantities is obtained by replacing H and q ∗ K in Eq. (10.43) by their expressions (10.78) and (10.92): ˜ H ˜ + = 4 {(N−2 − b)
1 2
˜ V) ˜ + 1 q ∗ ˙˜ + [2N −2 D ˜ s˜ ln Kil(2 D, 2
(10.97)
˜ q} , ˜ · V − 2D ˜ ln M)]˜ ˜ s˜ b˜ − N K − 2 D + 13 (b˜ H˜ + D V or, in index notation, ˜ H˜ ab + ab = 4 {(N−2 − b)
1 2 ˜ ˜ 2 ( D a Vb
+ 2 D˜ b V˜a ) +
1 ˙ ˜ kl 2
qka qlb
˜ + 2N −2 s˜ k D˜ k ln ]q˜ab } . + [ 13 (b˜ H˜ + s˜ k D˜ k b˜ − N K − 2 D˜ a V a − V a 2 D˜ a ln M)
(10.98)
For a coordinate system stationary with respect to H, Eq. (10.97) simplifies to H,sc
˜ V) ˜ + 1 q ∗ ˙˜ + [2N−2 D ˜ s˜ ln = 4 { 21 Kil(2 D, 2 ˜ q} ˜ s˜ b˜ − N K − 2 D ˜ · V − 2D ˜ ln M)]˜ + 13 (b˜ H˜ + D V
.
(10.99)
The shear tensor of St is deduced from Eqs. (10.95) and (10.97) by = − 1/2q [Eq. (5.64)]: =
4 2
˜ V) ˜ − (2 D ˜ · V)˜q + q ∗ ˙˜ + 1 ˜˙ (˜s, s˜)˜q Kil(2 D, 2 ! ˜ H ˜ − 1 H˜ q˜ +2(N −2 − b) , 2
(10.100)
or in index notation 4 ab = 2
1 k l 2 ˜ ˜ 2 ˜ c k l ˙ ˜ ˜ ˜ D a Vb + D b Va − ( D c V )q˜ab + ˜ kl q a q b + s˜ s˜ q˜ab + 2(N −2 − b) 2 ! 1 . × H˜ ab − H˜ q˜ab 2
2
(10.101)
For a coordinate system stationary with respect to H, this expression becomes very simple: H,sc
=
4 2
˜ V) ˜ − (2 D ˜ · V)˜q + q ∗ ˙˜ + 1 ˙˜ (˜s, s˜)˜q Kil(2 D, 2
.
(10.102)
˜ V) ˜ − (2 D ˜ · V)˜q is the conformal Killing operator associated with the metric q˜ and applied to V. Note that Kil(2 D, Let us now give the 2 + 1 conformal decomposition of the transversal deformation rate . From Eq. (10.47), its trace becomes 1 (k) = − N
$ −2
2
1 +K + N
s˜ k D˜
1 k ln + H˜ 3
1 ˙ k l 1 + ˜ s˜ s˜ + 4N kl 3
1 V a 2 D˜ a ln M˜ − 2 D˜ a V a − s˜ k D˜ k b˜ 2
%&
b˜ − −2 N
()
'
H˜ 2 (10.103)
,
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257
which for a coordinate system stationary with respect to H results in (k)
1 ˙ k l 1 1 ˜ −2 k ˜ 2 ˜ kl s˜ s˜ s˜ Dk ln + H + = − N 3 4N ! 1 1 12 ˜ a a 2 k ˜ ˜ ˜ ˜ . V D a ln M − D a V − s˜ Dk b + K+ 3 N 2
H,sc
(10.104)
Replacing H and q ∗ K in Eq. (10.44) by their expressions (10.78) and (10.92) yields 4 −2 ˜ H ˜ − 1 Kil(2 D, ˜ V) ˜ + q ∗ ˙˜ + 2−2 D ˜ s˜ ln + b) (N 2N 2 2 ! 1 ˜ q˜ . ˜ · V + 2D ˜ ln M) ˜ s˜ b˜ + 2 D + (N K − b˜ H˜ − D V 3
=−
(10.105)
The conformal 2 + 1 expression of Há´"iˇcek’s form is obtained by inserting Eq. (10.91) into Eq. (10.15): 2 ˜ ln N + [b˜ 2 D ˜ ln M˜ − 2 D ˜ b˜ − H(V, ˜ ˜ s˜ V˜ .) + q ∗ D = 2D 2N + q∗ ˜˙ (˜s, .)] .
(10.106)
H For a coordinate system stationary with respect to H, one has b˜ = N −2 [Eq. (10.83)], so that the above expression results in H,sc
=
2 12˜ ˜ + [ q∗ D ˜ ˜ s˜ V ˜ + q ∗ ˙˜ (˜s, .) − H(V, .)] D ln(2 N M) 2 2N
.
(10.107)
10.4.6. Conformal 2 + 1 expressions for , and viewed as deformation rates of St ’s metric As already noticed in Remark 10.3, we have obtained 2+1 expressions of and in Section 5.5 [cf. Eqs. (5.62) and (5.77)]. These expressions involve the time derivative of St ’s metric q, whereas the expressions derived above involve the time derivative of the conformal 3-metric ˜ . Therefore it is worth performing a conformal decomposition of the equations of Section 5.5, since they will lead to expressions letting appear time derivatives different than to those found above. Let us start from Eq. (5.62) for . The first term in the right-hand side is conformaly decomposed as q ∗ Lt q = q ∗ Lt (4 q˜ ) = 4 [ q∗ Lt q˜ + 4(Lt ln )˜q] ,
(10.108)
where we have used q ∗ q˜ = q˜ . The second term in the right-hand side of Eq. (5.62) involves 2 D a Vb + 2 D b Va . Now similarly to relation (10.51), 2
D a Vb = 2 D˜ a Vb − 2 C c ba Vc ,
(10.109)
2
C c ba := 21 q cd (2 D˜ a qdb + 2 D˜ b qad − 2 D˜ d qab ) c = 2(2 D˜ a ln q c b + 2 D˜ b ln q c a − 2 D˜ ln q˜ab ) .
(10.110)
with
Combining with Vb = 4 V˜b [Eq. (10.85)], we get 2
D a Vb = 4 [2 D˜ a V˜b + 2 2 D˜ a ln V˜b − 2 2 D˜ b ln V˜a + 2(V c2 D˜ c ln )q˜ab ] .
(10.111)
D a Vb + 2 D b Va = 4 [2 D˜ a V˜b + 2 D˜ b V˜a + 4(V c2 D˜ c ln )q˜ab ] .
(10.112)
Hence 2
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Finally the 2 + 1 split of the last term in Eq. (5.62) is nothing than Eq. (10.78). Inserting this last relation, as well as Eqs. (10.108) and (10.112) into Eq. (5.62) leads to 4 ∗ ˜ V) ˜ + 4(2 D ˜ ln )˜q { q Lt q˜ + 4(Lt ln )˜q + Kil(2 D, V 2 −2 ˜ ˜ ˜ +2(N − b)[H + 2(Ds˜ ln )˜q]} ,
=
(10.113)
or in index notation 4 ˜ {Lt q˜ q a q b + 4(Lt ln )q˜ab + 2 D˜ a V˜b + 2 D˜ b V˜a + 4(V c2 D˜ c ln )q˜ab + 2(N −2 − b) 2 × [H˜ ab + 2(˜s k D˜ k ln )q˜ab ]} . (10.114)
ab =
Contracting this equation with q ab leads immediately to an expression for the expansion scalar : = 21 q˜ Lt q˜ + 4Lt ln + 2 D˜ a V a + 4V a 2 D˜ a ln ˜ H˜ + 4˜s k D˜ k ln ) . +(N−2 − b)(
(10.115)
We then obtain the shear tensor by forming − /2q from Eqs. (10.113) and (10.115). All the terms involving ln cancel and there remains =
4 ∗ 1 ˜ V) ˜ − (2 D ˜ · V)˜q q Lt q˜ − (q˜ Lt q˜ )˜q + Kil(2 D, 2 2 ! ˜ H ˜ − 1 H˜ q˜ +2(N−2 − b) , 2
(10.116)
or, in index notation ab =
4 1 Lt q˜ q a q b − (q˜ Lt q˜ )q˜ab + 2 D˜ a V˜b + 2 D˜ b V˜a − (2 D˜ c V c )q˜ab 2 2 ! 1 ˜ −2 ˜ ˜ . +2(N − b) Hab − H q˜ab 2
(10.117)
If one uses a coordinate system stationary with respect to H, the above equations simplify somewhat, thanks to the vanishing of N −2 − b˜ [Eq. (10.83)]: H,sc
=
4 ∗ ˜ V) ˜ + 4(2 D ˜ ln )˜q] , [ q Lt q˜ + 4(Lt ln )˜q + Kil(2 D, V 2
H,sc 1 2 q˜ Lt q˜
=
H,sc
=
+ 4Lt ln + 2 D˜ a V a + 4V a 2 D˜ a ln ,
1 4 ∗ ˜ V) ˜ − (2 D ˜ · V)˜q . q Lt q˜ − (q˜ Lt q˜ )˜q + Kil(2 D, 2 2
(10.118) (10.119) (10.120)
Moreover, in the case of a coordinate system (t, x i ) adapted to H, the term q˜ Lt q˜ can be expressed in terms of the variation of determinant q˜ of the conformal 2-metric components q˜ab in this coordinate system: q˜ Lt q˜ = Lt ln q˜ .
(10.121)
Remark 10.4. Eqs. (10.113), (10.115) and (10.116) constitute 2 + 1 expressions of respectively , and in terms of Lt q˜ and (for and only) Lt ln . On the other side, Eqs. (10.97), (10.95) and (10.100), provide the same quantities
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, and in terms of Lt ˜ . The equivalence between the two sets can be established in view of the two identities: q ∗ Lt ˜ = q ∗ Lt q˜ , ˜ s˜ ln − 2 D ˜ ln + Lt ln = b˜ D V
1 6
˜ s˜ b˜ − N K − 2 D ˜ · V − 2D ˜ ln M˜ b˜ H˜ + D V
(10.122)
.
(10.123)
The first identity is an immediate consequence of q˜ = ˜ − s˜ ⊗ s˜ [Eq. (10.68)] and q ∗ s˜ = 0, whereas the second identity is nothing but the 2 + 1 split of D˜ i i in the expression of Lt ln as given by Eq. (10.57) [cf. Eq. (10.88)]. 11. Applications to the initial data and slow evolution problems Let us apply the results of previous sections to derive inner boundary conditions for the partial differential equations arising from the 3 + 1 decomposition of Einstein equation. More specifically, we consider the problem of constructing numerically, within the 3 + 1 formalism, a spacetime containing a black hole in quasi-equilibrium by employing some excision technique. By excision is meant the removal of a 2-sphere St and its interior from the initial Cauchy surface t . If we ask St to represent the apparent horizon [160,120,57] of a black hole in quasi-equilibrium,27 the quasi-local tools presented previously are very well suited to set the appropriate values, or more generally constraints, to be satisfied by the 3 + 1 fields on this inner boundary. Two most important physical problems where these boundary conditions can be naturally applied are: (i) The construction of initial data for binary black holes in quasi-circular orbits. By quasi-circular is meant orbits for which the decay due to gravitational radiation can be neglected. This is a very good approximation for sufficiently separated systems and the spacetime can then be considered as being endowed with a helical Killing vector (see e.g. Refs. [77,71] for a discussion). (ii) The slow dynamical evolution of spacetimes containing a black hole (more precisely, slow evolution of initial data containing a marginally trapped surface). For concreteness, the system of coupled elliptic equations in the minimal no-radiation approximation proposed in Ref. [144] (a gravitational analog of the magneto-hydrodynamics approximation in electromagnetism), provides an appropriate framework for implementing the isolated horizon prescriptions as inner boundary conditions. Alternatively, the fully constrained evolution scheme presented in Ref. [29] (or approximations based on it) can also prove to be useful, provided appropriate quasi-equilibrium initial free data are chosen on the initial slice.28 11.1. Conformal decomposition of the constraint equations The Hamiltonian and momentum constraint equations arising from the 3 + 1 decomposition of Einstein equation have been presented in Section 3.6 [Eqs. (3.37) and (3.38)]. Besides we have introduced in Section 10.4 a conformal decomposition of the 3-metric [Eq. (10.48)] and the extrinsic curvature K of the hypersurface t [Eq. (10.54)]. Let us examine the impact of these decompositions on the constraint equations. 11.1.1. Lichnerowicz–York equation From the conformal decomposition = 4 ˜ [Eq. (10.48)], the Ricci scalar 3 R associated with is related to the Ricci scalar 3 R˜ related to ˜ by 3 R = −4 3 R˜ − 8−5 D˜ k D˜ k , so that the Hamiltonian constraint (3.37) becomes the Lichnerowicz–York equation for : 3R ˜ K2 1 ˜ ˜ ij 2 −3 k ˜ ˜ + 2E − Dk D − + Aij A 5 = 0 8 8 12
,
(11.1)
27 One can also set the excised sphere S inside the horizon [147,4,5], instead of prescribing it to be the actual apparent horizon. In such a t case one must find numerically the apparent horizon (see [162,163] and references therein), or even the event horizon [112,63,44], and then the quasi-local techniques in this article can be used in an a posteriori analysis, rather than as inner boundary conditions. 28 If the initial data encode modes whose evolution propagate towards the horizon and eventually fall into it, the enforcing of isolated horizon conditions would lead to an ill-posed problem. Therefore initial data must be carefully chosen so as to minimize these perturbations (e.g. making them smaller than the numerical noise).
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where use has been made of the conformal decomposition (10.54) of K and A˜ ij is related to A˜ ij by means of the conformal metric: A˜ ij := ˜ ik ˜ j l A˜ kl .
(11.2)
˜ Eq. (11.1) is a non-linear equation for ; it has been first derived and For a fixed conformal metric ˜ and a fixed A, analyzed by Lichnerowicz [113] in the special case = −2, K = 0 and E = 0 or const. The negative sign of the exponent 2 − 3 in this case is crucial to guarantee the existence and uniqueness of the solution to this equation. It has been extended to the case K = 0 and discussed in great details by York [171,174]. 11.1.2. Conformal thin sandwich equations We place ourselves in the framework of the conformal thin sandwich approach to the 3 + 1 initial data problem developed by York and Pfeiffer [173,137,174,51]. Regarding the problem of binary black hole on close circular orbits, this approach has led to the most successful numerical solutions to date [79,54,7]. It is based on the transformation of the momentum constraint equation (3.38) into an elliptic equation for the shift vector .29 Indeed inserting the expression (10.61) for K into Eq. (3.38) yields D˜ k D˜ k i + 13 D˜ i D˜ k k + 3 R˜ i k k = 164 N J i + 43 N D˜ i K − D˜ k ˙˜ ij +2N −4 A˜ ik D˜ k ln(N −6 )
,
(11.3)
˜ compatible with ˜ , and where 3 R˜ i j := ˜ ik 3 R˜ ij , 3 R˜ ij being the Ricci tensor of the connection D j˜ ij ˙˜ ij := = −˜ ik ˜ j l ˙˜ kl , jt
(11.4)
where ˙˜ kl is the quantity introduced in Eq. (10.55). Notice that the quantity A˜ ij which appears in Eq. (11.3) is considered as a function of the shift vector according to 1 2 ˜ k ij ˙ ij ij i j j i ˜ ˜ A = D + D − Dk ˜ + ˜ , (11.5) 3 2N −4 which follows from Eq. (11.2), (10.58) and (11.4). When solving the initial data problem with the Hamiltonian constraint under the form of the Lichnerowicz–York equation (11.1) and the momentum constraint under the form of the vector elliptic (11.3), one can choose freely the conformal 3-metric ˜ , its time derivative ˙˜ and the trace of t ’s extrinsic curvature, K. Prescribing in addition the value of the time derivative K˙ = jK/jt, leads to the extended conformal thin sandwich formalism as presented in Ref. [137]. Indeed prescribing K˙ leads to a “constraint” on the lapse function N, which can be derived by taking the trace of the dynamical Einstein equation (3.41): K2 4 k k ˜ ˜ ˜ ˜ Dk D N + 2Dk ln D N = N 4(E + S) + 3 ! k 2 −K˙ + D˜ k K + N −4 A˜ kl A˜ kl .
(11.6)
To summarize, in the extended conformal thin sandwich framework, the free data (modulo boundary values of the ˙ on some spatial hypersurface 0 . The elliptic equations (11.1), (11.3) constraint parameters) are the fields (˜, ˙˜ , K, K) and (11.6) are to be solved for the conformal factor , the shift vector and the lapse function N. One then gets a valid initial data set (0 , , K), i.e. a data set satisfying the Hamiltonian and momentum constraints. Moreover the initial time development of these initial data will be such that K˙ takes the prescribed value. One should note that according to a recent study [138], the uniqueness of a solution (, N, ) of the extended conformal thin sandwich equations is 29 Likewise, York’s original approach [171,174] reduces the resolution of the momentum constraint to an analogous vectorial elliptic equation; see [137,135] for the discussion of the relation between both approaches.
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˙ two distinct solutions (, N, ) have been found in a spatial not guaranteed: for some choice of free data (˜, ˙˜ , K, K), slice where no sphere has been excised. However, a unique solution of the extended conformal thin sandwich has been found for problems of direct astrophysical interest, like binary neutron stars [23,78,117,166] or binary black holes [79,54]. Moreover, this method of solving the initial data problem has been recognized to have greater physical content than previous conformal formulations [113,129,171,51], because it allows a direct control on the time derivative of the conformal metric. In particular, in a quasi-equilibrium situation, it is natural to choose ˙˜ = 0 and K˙ = 0 to guarantee that the coordinates are adapted to the approximate Killing vector reflecting the quasi-equilibrium (see Refs. [52,151]). In the rest of this section, we translate the isolated horizon geometrical conditions into boundary conditions for the constrained parameters of the conformal thin sandwich formulation. We separate this analysis in NEH (Section 11.2) and WIH boundary conditions (Section 11.3). Possible boundary conditions from the complete (strong) IH structure will not be considered in this review. 11.2. Boundary conditions on a NEH As seen in Section 7, a NEH is characterized by the vanishing of its second fundamental form [see Eq. (7.11)]. According to the transformation rules in Table 1 under rescalings of l, the condition =0 is independent of the specific choice for the null normal l. Using the decomposition (5.64) of , this condition translates into the vanishing of the expansion and shear associated with l. 11.2.1. Vanishing of the expansion: = 0 Imposing this condition on a sphere St defines it as a marginally outer trapped surface in t (see Section 7.1.2). If in addition (k) 0, it corresponds to a future marginally trapped surface. In this second case, we find from Eqs. (10.46) and (10.47) K(s, s) − K =
+ N (k) 0 . 2N
(11.7)
If St is the outermost marginally trapped surface, then it is properly called an apparent horizon (see again Section 7.1.2). The condition = 0 can be expressed in a variety of forms. A convenient expression follows from Eq. (10.46) when substituting the conformal decomposition of the metric : 4˜s i D˜ i ln + D˜ i s˜ i + −2 Kij s˜ i s˜ j − 2 K = 0
.
(11.8)
If the conformal and 2 + 1 decomposition for K is included, it follows from Eq. (10.95) N −2 (4˜s k D˜ k ln + 23 H˜ ) − 21 ˙˜ kl s˜ k s˜ l ˜ H˜ + 2˜s k D˜ k b˜ − 2V a 2 D˜ ln M˜ + 2 D˜ V a − 2N K] = 0 . + 13 [(N−2 − b) a a
(11.9)
An alternative expression, that exploits the relation between the expansion and the time evolution of the volume element, follows by substituting the value for Lt provided by (10.57) into Eq. (10.115) ˜ s i + V i )D˜ i 4(i D˜ i + (N −2 − b)˜ 2 ˜ i ˜ H˜ ] = 0 . +[ 3 (Di − N K) + 2 D˜ a V a + 21 q˜ Lt q˜ + (N −2 − b)
(11.10)
Eqs. (11.8)–(11.10) can be seen as boundary conditions for the conformal factor in the resolution of the Hamiltonian constraint in a conformal decomposition, i.e. Eq. (11.1) [see however the end of Section 11.4 for other possibilities]. They express the same geometrical condition in terms of different sets of fields. The appropriate form to be used must be chosen according to the details of the problem we want to solve. Finally, we note that the boundary condition = 0 has been extensively studied in the literature (see [160] for a numerical perspective and [120,57] for an analytical one).
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11.2.2. Vanishing of the shear: ab = 0 From Eq. (10.116), the vanishing of the shear translates into 0 = (Lt q˜ab − 21 (Lt ln q) ˜ q˜ab ) + (2 D˜ a V˜b + 2 D˜ b V˜a − (2 D˜ c V c )q˜ab ) II: intrinsic geometry of St I: initial free data ˜ H˜ ab − 1 q˜ab H˜ ) . + (N−2 − b)( 2 III: “ extrinsic” geometry of St Defining, from Parts I and III in the previous equation, the symmetric traceless tensor ˜ H˜ ab − 1 q˜ab H˜ ) , Cab := − (Lt q˜ab − 21 (Lt ln q) ˜ q˜ab ) + (N−2 − b)( 2
(11.11)
(11.12)
we can write Eq. (11.11) as q˜bc 2 D˜ a V c + q˜ac 2 D˜ b V c − q˜ab 2 D˜ c V c = Cab .
(11.13)
a If we contract with 2 D˜ and use the Ricci equation (1.14) (properly contracted), we get a a q˜bc 2 D˜ 2 D˜ a V c + 2 R˜ db V d = 2 D˜ Cab .
(11.14)
Finally, defining C˜ ab := q˜ ca Cbc , we obtain the following elliptic equation for V a on St : 2 V ˜ a
b + 2 R˜ a b V b = 2 D˜ C˜ Cb a .
(11.15)
Once we have solved this equation on the sphere, we employ the solution as a Dirichlet boundary condition for the tangential part of the shift V. Therefore, the vanishing of the shear (vanishing of two independent functions) can be completely attained by an appropriate choice of the (two-dimensional) vector V. An important particular case occurs when we enforce the vanishing of Parts (I + II) and III separately. The vanishing of Part III is motivated in the literature in two different manners. On the one hand, this term cancels if we demand the coordinate radius of the horizon to remain fixed in a dynamical evolution [cf. Eq. (4.79)], something desirable from a numerical point of view. On the other hand, in order to make tractable the analytical study on the well-posedness of the initial data problem with quasi-equilibrium boundary conditions, results in the literature proceed by decoupling the momentum constraint from the Hamiltonian one. In particular, in this strategy, Part III must vanish on its own; if this is not the case, the presence of the conformal factor in the coefficient multiplying the extrinsic geometry part would couple the equation on with the equation on . Vanishing of Part (I + II) If Part III is zero, the vanishing of (I + II) is obtained by solving Eq. (11.15) with a traceless symmetric tensor Cab in Eq. (11.12) completely characterised by the traceless part of Lt q˜ab . A specially important case corresponds to the choice ˙˜ = 0 motivated by bulk quasi-equilibrium considerations [52,151]. In this case, the condition (I + II) = 0 reduces to q˜bc 2 D˜ a V c + q˜ac 2 D˜ b V c − q˜ab 2 D˜ c V c = 0
,
(11.16)
which states that V = − q() is a conformal Killing vector with respect to the metric q˜ , and hence to the (conformally related) metric q. These boundary conditions [generally (I + II) = 0] can be found in Refs. [54,99,58] (see also Ref. [66] for a previous related work). Remark 11.1. In the case of a stationary spacetime (M, g), it is natural to choose coordinates such that t coincides with the Killing vector associated with stationarity. Then Part I in Eq. (11.11) vanishes identically. Moreover if we ask H to be preserved by the spacetime symmetry, it must be transported to itself by t, which implies that t is tangent to H
H. Hence, from Eqs. (4.77) and (4.79), b = N . This results in the vanishing of Part III. Therefore for the choice of t
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as a Killing vector in a stationary spacetime, the NEH condition reduces to the vanishing of Part II. As shown above, this implies that V is a conformal Killing vector of (St , q). In the case where H is a black hole event horizon, this result is linked to a first step in the demonstration of Hawking’s strong rigidity theorem [88–90], which states that a H
stationary event horizon which is not static must be in addition axisymmetric. Indeed in the present case, l = t + V H
[Eq. (4.68) with b = N] and either V = 0 or V is a conformal symmetry of (St , q). Via Hawking’s theorem, this conformal symmetry is then extended to a full symmetry (i.e. axisymmetry) of (M, g). In particular l is then a Killing vector, hence the name rigidity: H’s null generators cannot move independently of the spacetime symmetries. Vanishing of Part III Two manners of imposing the vanishing of the Part III in Eq. (11.11) follow from the motivations presented after Eq. (11.15): (a) If we choose a coordinate system stationary with respect to the horizon (see Section 4.8), then Eq. (10.83) auto˜ Therefore, the Dirichlet condition for the radial part matically implies the vanishing of the coefficient (N −2 − b). of the shift b˜ = N−2
,
(11.17)
together with (I + II) = 0, guarantees the vanishing of the shear. This is the choice in [54,99]. (b) Even though the previous Dirichlet condition for the radial part of the shift b˜ is well motivated (since choosing a stationary coordinate system with respect to H is convenient from a numerical point of view), it presents the following problem for the solution of the constraints. If the value of b˜ is fixed on the boundary St , we loose control ˜ b˜ on St . In particular, this means via Eq. (10.90), that we cannot prescribe on the value of its radial derivative s˜ · D the sign of K(s, s). On the one hand, Eq. (11.7) then implies that we cannot guarantee St to be a future marginally trapped surface. On the other hand and perhaps more importantly, the positivity of the conformal factor cannot be guaranteed when solving the Hamiltonian constraint, since the sign of K(s, s) appearing in the “apparent horizon” boundary condition (11.8), must be controlled in order to apply a maximum principle to Eq. (11.1). This problem is discussed in Ref. [58]. The solution proposed there for guaranteeing the vanishing of Part III consists in choosing initial free data ˜ such that H˜ ab − 21 q˜ab H˜ = 0
,
(11.18)
is satisfied. This condition on the extrinsic curvature of the sphere St , i.e. on the shape of St inside t , is known as the umbilical condition. The boundary condition for the radial part of the shift is obtained in Ref. [58] by imposing K(s, s) = h1 ,
(11.19)
where h1 is a given function on St that can be considered as a free data on t . Using Eq. (10.90), this condition is expressed as a mixed condition on b˜ 2˜s k D˜ k b˜ − b˜ H˜ = 3N h1 − 2 D˜ k V k − 2V k D˜ s˜ s˜k − N K
.
(11.20)
Note the change of sign convention in the tangential part of the shift V with respect to [58], where in addition a maximal slicing K = 0 is assumed. Finally, we emphasize the fact that, in order to enforce the NEH structure, it is enough to impose the appropriate boundary conditions for the conformal factor and the tangential part of the shift: Eq. (11.8) for and Eq. (11.15) for V. Besides, a boundary condition for the normal part of the shift can be provided by making a choice relative to the coordinate system, Eq. (11.17), or by fixing K(s, s) on St through Eq. (11.20) [i.e. fixing (k) in the maximal slicing ˜ permit to fix case; cf. Eq. (11.7)]. In brief, the NEH structure together with an (additional) appropriate choice for b, boundary conditions for and . In other words, this first level in the isolated horizon hierarchy provides enough
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number of inner boundary conditions for addressing the resolution of the constraint equations, as exploited by Cook and Pfeiffer [54] (see also Ref. [7]). Incorporating Eq. (11.6) for the lapse in the construction of initial data (see Section 11.1.2) demands, in principle, some additional geometrical structure on H. This is considered in next section. 11.3. Boundary conditions on a WIH In Section 8 we showed how the addition of a WIH structure on a NEH permits to fix the foliation (St ) of the underlying null hypersurface H in an intrinsic manner. This determination of the foliation proceeded in two steps: firstly, by choosing a particular WIH class [l] (Section 8.4) and, secondly, by choosing a foliation (St ) compatible with that class (Section 8.5). This procedure in two steps is necessary when adopting an approach strictly intrinsic to the null surface, since in this case there is no privileged starting slice S0 in H. In brief, simply fixing l does not determines the foliation. This represents what we referred in the Introduction (Section 1) as an “up-down”approach. The situation changes completely when adopting a 3 + 1 point of view. A main feature in this case is the actual construction of the spacetime starting from an initial Cauchy slice 0 , which is then evolved by using Einstein equations. In this setting, in which 0 is given, fixing the lapse determines the foliation of M. Moreover, fixing the evolution vector l on H does fix the lapse on S0 and consequently the foliation, in contrast with the intrinsic approach to the geometry of H in the previous paragraph.30 This constitutes the “down-up” strategy mentioned in the Introduction. Even though we adopt here the “down-up” approach, the organization of this section rather follows the conceptual order dictated by the intrinsic geometry of H. Firstly we derive the implications of the choice of a WIH-compatible slicing. Then we apply the prescription in Section 8.4 for specifying a particular WIH and, finally, we revisit Section 8.5 and its determination of the foliation once the WIH class is chosen. As a result, we present different boundary values for the lapse associated with each step. 11.3.1. WIH-compatible slicing: = const. Evolution equation for the lapse Given an arbitrary but fixed WIH (H, [l]), demanding the slicing defined by N to be WIH-compatible (see Section 8.2) requires the non-affinity coefficient associated with the null vector l = N (n + s) ∈ [l] to be constant. Under the condition = const on the whole H, the null normal l actually builds a WIH structure31 or, in the language of Section 9.2, an (A, B)-horizon. If, motivated by the discussion in Section 8.6, we choose the representative l0 ∈ [l] with non-affinity coefficient given by the Kerr surface gravity, 0 = H (RH , JH ), then it follows directly from Eq. (10.9) H (RH , JH ) = HLl ln N + s i Di N − N K ij s i s j
.
(11.21)
This is an evolution equation for N on H (see [99]). As such, it can be employed to fix the values of N along the horizon H once the lapse has been freely chosen on a initial slice. This can be useful for fixing Dirichlet inner boundary conditions in the slow dynamical evolution of a quasi-equilibrium black hole (e.g. the evolution of a black hole during a late ringing down phase). On the contrary, in the context of the construction of initial data, Eq. (11.21) by itself does not prescribe a boundary condition for the lapse in Eq. (11.6). This is precisely due to the presence of the HLl ln N term, which cannot be expressed in terms of the data on an initial slice. This is in agreement with the fact that imposing the slicing to be WIH-compatible, through = H , does not determine the WIH class. A gauge choice has to be made to fix, up to a constant, l and therefore the family of slices (St ) (cf. footnote 31, Sections 8.4 and 8.5, as well as the rest of this section). Consequently, if we are indeed interested in using Eq. (11.21) for fixing a boundary condition for the lapse on the horizon, we are obliged to make a choice for the value of HLl N on S0 . In the quasi-equilibrium context, it is natural to demand the lapse not to evolve: HLl N = 0. More generally, writing HLl N = h2 , with h2 a function to be 30 As we have seen in Section 4.2 the converse is also true on H: fixing the foliation (S ) not only fixes the lapse (up to a “time” reparametrization) t but also the null normal l. This is in contrast with the bulk case, where (t ) fixes N but not the evolution vector t due to the shift ambiguity. The difference relies on the existence of a single null direction in H. 31 Note that the condition = const does not fix the WIH on H, since a transformation (8.7) changes the WIH without affecting the constancy of . In this section we are assuming a given [l]; in Section 11.3.2 the choice of a particular [l] is revisited.
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prescribed on S0 , Eq. (11.21) leads to the mixed boundary condition on S0 ! Eq. (11.21) ⇒ H (RH , JH ) = s i Di N − N K ij s i s j + h2 . +HLl N = h2
265
(11.22)
In this case, considered as a condition only on S0 , the corresponding l = N (n + s) is associated only with an infinitesimal (A)-horizon. Finally, another manner of looking at (11.21) consists in freely prescribing the values for N along the horizon H and consider Eq. (11.21) as a constraint on the rest of the fields, e.g. on the value of Kij s i s j (see Example 11.2 below). 11.3.2. Preferred WIH class: Ll (k) = 0 In Section 8.4 we prescribed a specific choice [l] of non-extremal WIH class among those that can be implemented on a generic NEH. This was achieved by imposing the derivative along l of the expansion associated with the ingoing null vector k, (k) , to vanish. In fact, such a condition could have been generalized to Ll (k) = h3 ,
(11.23)
with (l) = const, where the choice of h3 corresponds to the choice of gauge in the WIH structure [different choices for the function h3 fix distinct values for the function B in the transformation (8.18); the choice h3 = 0 corresponds to an (A, B, C)-horizon, again in the language of Section 9.2]. In a 3 + 1 formulation where a given starting slice S0 is specified and a WIH class is fixed, the choice of the only representative l ∈ [l] characterized by (l) = H (RH , JH ) determines the slicing on H. Therefore, a condition on the lapse must follow from Eq. (11.23). If we substitute expressions (10.15) and (10.20) in Eq. (8.16), we obtain indeed 2 D 2 D
N
− 2K s 2 D N
+(−2 D (q K s ) + q (K s )(K s ) −
12 2 R
+ 21 q R )N
(11.24)
+ H (R2H ,JH ) (D s − K s s + K) = N h3 . This equation can be used as a boundary condition for the lapse on a cross-section S0 of H. In this sense, it can be employed in combination with Eq. (11.21): Eq. (11.24) fixes the initial value of N on S0 whereas Eq. (11.21) dictates its “time” evolution. The freedom due to the presence of the HLl ln N term in Eq. (11.21) guarantees the compatibility between both equations. Cook [52] has proposed a condition32 very similar to Eq. (11.23) in order to fix the lapse on √ a initial slice S0 . Using √ the null normal normalized as in Eq. (4.14) together with its dual (4.26), i.e. lˆ := (n + s)/ 2 and kˆ := (n − s)/ 2, and imposing H
Llˆ (kˆ ) = 0
(11.25)
on S0 , leads to the condition on N proposed by Cook [52] and closely related to Eq. (11.24). Using Eqs. (4.13) and (4.27), together with the transformation law for (kˆ ) derived from Table 1, it follows H
Ll (k) = −(HLl ln N )(k) +H Llˆ (kˆ ) ,
(11.26)
where in addition (l) = const must be satisfied. Choosing as gauge condition in Eq. (11.23) h3 = −(HLl ln N )(k) , both conditions (11.23) and (11.25) are the same. Note also that in Eq. (11.25) we can always keep (l) constant as long as we let HLl ln N to be determined by Eq. (11.21). 11.3.3. Fixing the slicing: 2 D · = h. Dirichlet boundary condition for the lapse In Section 8.5 we concluded that, in the setting of an “up-down” approach, once a WIH class has been fixed on H the choice of the exact part exact of the Há´"iˇcek form determines the foliation (St ). We argued that, since its 32 See Refs. [54,134] for a discussion on the degeneracy occurring when using this boundary condition in conjunction with other quasi-equilibrium conditions.
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divergence-free part is fixed by relation (7.67), then the condition 2
D · = h4 ,
(11.27)
for some gauge choice of h4 , actually fixes the foliation. From the 3 + 1 perspective,33 this conclusion is a straight
forward consequence of Eq. (10.14). Indeed, contracting (10.14) with 2 D and inserting (11.27), we obtain 2 ln N
= 2 D (q K s ) + h4
.
(11.28)
" If we make now a gauge choice for h4 (satisfying St h4 2 = 0, e.g. h4 = 0 or the Pawlowski gauge as suggested in Section 8.5), we dispose of an elliptic equation that fixes N on St up to a constant value (or a function constant on St , if thinking in terms of H). This fixes the foliation (St ), understanding the latter as the ensemble of leaves in H [distinct values of the integration “constant” only entail different “speeds” to go through the slicing (St )]. In particular, the resulting lapse can be used as a Dirichlet boundary condition for the elliptic equation (11.6). 11.3.4. General remarks on the WIH boundary conditions As we can see, all boundary conditions derived at the WIH level, and aimed at being imposed on a initial sphere S0 , involve the choice of some function hi that cannot be fixed in the context of the initial data problem. This is the case of h2 =HLl N in Eq. (11.22), which shows that the WIH structure, with its “constant surface gravity” characterization, cannot be captured in terms of initial data. Regarding h3 in Eq. (11.24) and h4 in Eq. (11.28), they are directly related to the gauge ambiguity in the free data of a WIH (more precisely to the active and passive versions in Section 7.7.2, respectively). In sum, there exists an intrinsic ambiguity in the determination of the 2 + 1 slicing of a WIH. In consequence, we can conclude that the WIH on its own does not permit to fully determine the boundary conditions of the (extended) initial data problem and the prescription of an additional condition (a function on S0 ) is unavoidable.34 Therefore the approach in Refs. [54,7], where an effective boundary condition on S0 is chosen for the lapse, is fully justified from a geometrical point of view. Alternatively, we rather maintain here the geometrical expressions derived in this section, and encode their effective character (as boundary conditions on S0 ) through the free functions hi to be specified. Proceeding in this manner (i) the geometrical origin of the ambiguity is made explicit, and (ii) we can make use of the geometrical nature of the expressions to rearrange the correspondences between the different boundary conditions and the constrained parameter in the initial data problem, in such a way that the WIH effective condition is not necessarily related to the lapse. We illustrate this second point in the following section. 11.4. Other possibilities In the previous two sections we have translated the NEH and WIH geometrical characterizations into boundary conditions on the constrained parameters of the initial data problem, with special emphasis in the conformal thin sandwich approach. In particular, we have first interpreted Eq. (11.8) resulting from the vanishing of (l) as a boundary condition for the conformal factor . The vanishing of the shear has translated into the boundary condition (11.15) for the tangential part of the shift V, or simply into condition (11.16) if an additional condition on b˜ is enforced [either Eq. (11.17) or Eq. (11.20) with the umbilical condition (11.18)]. Finally, WIH conditions in Section 11.3 are mainly interpreted as boundary conditions for the lapse. However, a key feature of the present geometrical approach is the fact that each (non-linear) boundary condition is not necessarily associated with a single constrained parameter. It simply states a relation to be satisfied among different 3 + 1 fields. In case that such a correspondence between individual boundary conditions and individual constrained parameters is actually needed, the above-mentioned identification is a well motivated possibility, but it is not the only
33 We acknowledge B. Krishnan for the discussion in this section. 34 Note that this conclusion refers specifically to the initial data problem. A WIH structure contains notions which are essentially dynamical (“second derivatives” in time) and cannot be captured by the initial data . The WIH remains fully useful in the context of other problems. For instance, if we rather search an appropriate foliation for pursueing an a posteriori analysis of a full spacetime (not only a 3-slice) containing a NEH H (for instance the late time result of the numerical simulation of a collapse), in a first step we could disregard not WIH-compatible foliations. Then, after fixing a single spatial slice, the foliation on H can be completely determined by using Eq. (11.21).
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Table 2 Boundary conditions (b. c.) on S0 derived in Section 11, together with their geometrical content NEH b. c.
Non-eq. b. c. WIH b. c.
(l) = 0 =0
4˜s i D˜ i ln + D˜ i s˜ i + −2 Kij s˜ i s˜ j − 2 K = 0 2 ˜ V a + 2 R˜ a V b = 2 D˜ b C˜ a b
Eq. (11.15)
r = const Kij s i s j = h1
b˜ = N −2 2˜s k D˜ k b˜ − b˜ H˜ = 3N h1 − 2 D˜ k V k − 2V k D˜ s˜ s˜k − N K
Eq. (11.17) Eq. (11.20)
HL N = h l 2 HL l (k) = h3
H (RH , JH ) = s i Di N − N K ij s i s j + h2
2D · = h 4
b
2 D 2 D N − 2K s 2 D N + (−2 D (q K s )+
q (K s )(K s ) − 21 2 R + 21 q R )N + H (RH ,JH ) (D s − K s s + K) = N h3 2 2 ln N = 2 D (q K s ) + h 4
Eq. (11.8)
Eq. (11.22) Eq. (11.24)
Eq. (11.28)
one. In order to facilitate the choice of other possible combinations, we recapitulate the boundary conditions presented along Section 11 in the Table 2, where we make explicit the geometrical meaning of each of them. In particular, they are classified in NEH boundary conditions, in WIH-motivated conditions (since, as we have seen in the previous section, they do not actually construct a WIH) and a third set of boundary conditions, not necessarily related to a quasi-equilibrium regime, that can also be used in general dynamical settings.35 As an illustration of a possible alternative combination of boundary conditions, we provide a simple example (see Ref. [100]) which represents, at the same time, a non-trivial implementation of the isolated horizon boundary conditions beyond the analytical stationary examples provided in the rest of the article. Example 11.2. In the straightforward interpretation of Eqs. (11.22) and (11.28) in Section 11.3, they have been proposed as alternative boundary conditions for N, between which we must choose. However, we have pointed out that Eq. (11.22) can also be understood as fixing the value of Kij s i s j . In that case we can use it to determine the free ˜ Therefore a particular combination of boundary conditions in Table 2 function h1 in boundary condition (11.20) for b. is given by: vanishing expansion for , conformal Killing condition for V, condition (11.20) for b˜ with Kij s i s j fixed by Eq. (11.22) with h2 = 0 Kij s i s j =
1 i (s Di N − H (RH , JH )) , N
(11.29)
and, finally, condition (11.28) for the lapse. Fig. 17 shows the maximum and minimum values of Kij s i s j during the iteration of a numerical implementation of these boundary conditions [100], where we have chosen ˙˜ ij = K = 0, ˜ a flat metric, V = const · T (a symmetry on S0 ) and h4 = 0 in Eq. (11.28), together with an integration constant C = ln 0.2 for ln N. Since this implementation is performed in maximal slicing, in particular the constructed quasi-equilibrium horizon is a future marginally trapped surface. In brief, keeping boundary conditions in geometrical form we gain in flexibility for combining them in different manners. See Ref. [100] for other possibilities, in particular the enforcing of the vanishing of as a condition on the ˜ instead of a condition on . In conjunction with Eq. (11.15) for V, this means that the NEH normal part of the shift, b, condition = 0 can be completely fulfilled by an appropriate choice of the shift . 12. Conclusion In this article, we have developed an approach to null hypersurfaces based on the 3 + 1 formalism of general relativity, the main motivation being the application of the isolated horizon formalism to numerical relativity. Although the geometry of a null hypersurface H can be elegantly studied from a purely intrinsic point of view, i.e. without 35 The condition for fixing the slicing 2 D · = h could also be included in this category but, since we have mainly used the Há´"iˇcek 1-form 4 in the quasi-equilibrium context, we keep it as a WIH-motivated condition.
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0
Kijsisj
-0.05
-0.1
-0.15
-0.2
0
100
200 iteration step
300
400
Fig. 17. Values of max(Kij s i s j ) and min(Kij s i s j ) along the iteration of the simultaneous numerical implementation of Eqs. (11.22) and (11.28).
referring to objects defined outside H, the present 3 + 1 strategy proves to be useful, at least for two reasons. First of all, any 3 + 1 spacelike slicing (t ) provides a natural normalization of the null normal l to H, along with a projector onto H — fixing the ambiguities inherent to null hypersurfaces. Secondly, this permits to express explicitly H’s intrinsic quantities in terms of fields of direct interest for numerical relativity; like the extrinsic curvature K of the hypersurface t and its the 3-metric [or its conformal representation (, ˜ )]. In addition, we have adopted a fully 4-dimensional point of view, by introducing an auxiliary null foliation (Hu ) in a neighborhood of H. Not only this facilitates the link between the null geometry and the 3 + 1 description, but also reduces the actual computations to standard four-dimensional tensorial calculus (e.g. involving the spacetime connection ∇) and four-dimensional exterior calculus (e.g. the differential dl and dk of 1-forms associated with the null normal l and the ingoing null vector k, in connection with Frobenius theorem related to the submanifolds H and St = H ∩ t , or the decomposition of the curvature tensor following from Cartan’s structure equations). Thanks to the projector , we have performed a 4-dimensional extension of H’s second fundamental form , and of the Há´"iˇcek 1-form . Besides, we have introduced as a basic object the transversal deformation rate . By performing various projections of the Einstein equation, we have recovered, in addition to the null Raychaudhuri equation, the Damour–Navier–Stokes equation, and have derived an evolution equation for . Independently of the Einstein equation, the so-called tidal force equation (which involves the Weyl tensor) is recovered. All these equations constitute a set of evolution equations along the null generators of H. They hold for any null hypersurface. Following Há´"iˇcek and Ashtekar et al., we have then considered non-expanding null hypersurfaces (more specifically non-expanding horizons) as a first step in the modelization of a black hole horizon in quasi-equilibrium. At this stage, a ˆ on H compatible with the degenerate metric new geometrical structure enters into the scene, namely the connection ∇ q and induced by the spacetime connection ∇. This can be achieved thanks to the vanishing of the second fundamental ˆ then completely characterizes the geometry of a form for a non-expanding null hypersurface. The couple (q, ∇) non-expanding horizon. Once the null normal l is fixed by some 3 + 1 slicing (t ), this geometry is encoded in the fields (q, , , ) evaluated in a spatial cross-section St = H ∩ t . The change in time of these quantities is obtained by specializing the evolution equations to the case = 0. A number of possible constraints then follow for characterizing a horizon in quasi-equilibrium, beyond being simply non-expanding, leading to a hierarchy of structures on H. In particular, an intermediate notion of quasi-equilibrium is provided by the weakly isolated horizon structure introduced by Ashtekar et al. and defined by requiring (i) a time-independent and (ii) to be constant over H. This permits a quasi-local expression of physical parameters, like mass and angular momentum and provides constraints on the 3 + 1 slicing, even if it does not further constrain the geometry of H. On the contrary, the isolated horizon structure, that requires all fields (q, , , ) to be time-independent, represents the maximal degree of equilibrium imposed in a quasi-local manner. It really restricts H among all possible non-expanding horizons. It also provides tools for extracting information in the neighborhood of H. Thanks to explicit formulæ relating q, , , and to 3 + 1 fields, including the lapse function N and shift vector , we have then translated the isolated horizon hierarchy into inner boundary conditions onto an excised sphere in the
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spatial hypersurface t . This permits us to study the problem of initial data and spacetime evolution in a constrained scheme, making the links with existing results in numerical relativity. This connection with the 3 + 1 Cauchy problem illustrates the “down-up” strategy mentioned in the Introduction, i.e. the description of the null geometry on H by constructing it from initial data on a spacelike 2-surface. This provides an alternative point of view to the “up-down” picture, generally considered in the isolated horizon literature, and in which data on spatial slices are determined a posteriori from a given 3-geometry on H as a whole. In conclusion, the tools discussed in this article are aimed at providing a useful setting for studying black holes in realistic astrophysical scenarios involving regimes close to the steady state.
Acknowledgements We thank Víctor Aldaya, Lars Andersson, Marcus Ansorg, Abhay Ashtekar, Carlos Barceló, Silvano Bonazzola, Brandon Carter, Gregory B. Cook, Sergio Dain, Thibault Damour, John Friedman, Achamveedu Gopakumar, Philippe Grandclément, Mikolaj Korzynski, Badri Krishnan, José María Martín García, Jean-Pierre Lasota, Jurek Lewandowski, François Limousin, Richard Matzner, Guillermo Mena Marugán, Jérôme Novak, Tomasz Pawlowski, Harald Pfeiffer, José M. M. Senovilla, Koji Uryu, and Raül Vera for very instructive discussions and the referee for enlightening remarks. J.L.J. is supported by a Marie Curie Intra-European contract MEIF-CT-2003-500885 within the 6th European Community Framework Programme.
Appendix A. Flow of time: various Lie derivatives along l The choice (4.5) for l, as the tangent vector of H’s null generators associated with the parameter t, means that l can be considered as the “advance-in-time” vector associated with t. This is also manifest in the relation dt, l = 1 [Eq. (4.6)] or l = t + V + (N − b)s [Eq. (4.68)], which shows that l is equal to the coordinate time vector t plus some vector tangent to t [namely V + (N − b)s]. Therefore in order to describe the “time evolution” of the objects related to H, it is natural to introduce the Lie derivative along l. However, it turns out that various kinds of such Lie derivatives can be defined. First of all, there is the Lie derivative along the vector field l within the spacetime manifold M, which is denoted by Ll . But since l ∈ T(H), there is also the Lie derivative along the vector field l within the manifold H, which we denote by HLl . Finally, since l Lie drags the 2-surfaces St (cf. Section 4.2 and Fig. 9), one may define within the manifold St a Lie derivative “along l”, which we denote S Ll . We present here the precise definitions of these Lie derivatives and the relationships between them. A.1. Lie derivative along l within H: HLl Since l is a vector field on H, one may naturally construct the Lie derivative along l of any tensor field T on H. This results in another tensor field on H, which we denote by HLl T. Now, we may extend T into a tensor field on M thanks to the push-forward mapping ∗ for vectors [cf. Eq. (2.4)] and the operator ∗ for linear forms [cf. Eq. (4.58)]. It is then legitimate to ask for the relation between the Lie derivative Ll of this four-dimensional extension within the manifold M, and HLl T. Firstly we notice that both Lie derivatives coincide onto vectors:36 ∀v ∈ T(H),
HL
l v = Ll v
.
(A.1)
36 More precisely we should write Eq. (A.1) as HL v = L l ∗ l ∗ v, where ∗ is the push-forward operator associated with the embedding ∗ of H in M (cf. Section 2.1), but according to our four-dimensional point of view, we do not distinguish between v and ∗ v.
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Fig. 18. Geometrical construction showing that Ll v ∈ T(St ) for any vector v tangent to the 2-surface St : on St , a vector can be identified by a infinitesimal displacement between two points, p and q say. These points are transported onto the neighboring surface St+ t along the field lines of the vector field l (thin lines on the figure) by the diffeomorphism t associated with l: the displacement between p and t (p) is the vector t l. The couple of points ( t (p), t (q)) defines the vector t v(t) tangent to St+ t . The Lie derivative of v along l is then defined by the difference between the value of the vector field v at the point t (p), i.e. v(t + t), and the vector transported from St along l’s field lines, i.e. t v(t): Ll v(t + t) = lim t→0 [v(t + t) − t v(t)]/ t. Since both vectors v(t + t) and t v(t) are in T(St+ t ), it is then obvious that Ll v(t + t) ∈ T(St+ t ).
This follows from the very definition of the Lie derivative (cf. Fig. 18). Let us now consider an arbitrary 1-form on H: ∈ T∗ (H). Then invoking the Leibnitz rule on contractions and using the property (A.1) ∀ v ∈ T(H),
HLl , v = HLl , v − ,HLl v = Ll , v − , Ll v = Ll ∗ , v − ∗ , Ll v = Ll (∗ ), v .
(A.2)
We conclude that the 1-forms HLl and Ll (∗ ) coincide on T(H). Therefore their extensions to T(M) provided by the projector also coincide, and we can write ∀ ∈ T∗ (H),
∗ HLl = ∗ Ll (∗ ) .
(A.3)
By taking tensorial products, we above analysis can be extended straightforwardly to any field A of multilinear forms acting on Tp (H), so that we get ∀A ∈ T∗ (H)⊗n ,
∗ HLl A = ∗ Ll (∗ A)
.
(A.4)
A.2. Lie derivative along l within St : S Ll We have seen in Section 4.2 that l Lie drags the 2-surfaces St : St+ t is obtained from the neighboring surface St by an infinitesimal displacement tl of each point of St . As stressed by Damour [60], an immediate consequence of this is that the Lie derivative along l of any vector tangent to St is a vector which is also tangent to St : ∀v ∈ T(St ),
Ll v ∈ T(St ) .
(A.5)
This is obvious from the geometrical definition of a Lie derivative (see Fig. 18). It can also be established “blindly”: consider v ∈ T(St ); then l · v = k · v = 0, so that l · Ll v = l · (∇l v − ∇v l) = l · ∇l v − l · ∇v l = − v · (l) = 0 . =−v·∇l l
=0
(A.6)
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and k · Ll v = k · ∇l v − k · ∇v l = −v · ∇l k + l · ∇v k = dk(v, l) .
(A.7)
With expression (5.39) of the exterior derivative of k and the fact that l, v = 0 and l, l = 0, we get immediately k · Ll v = 0 .
(A.8)
Eqs. (A.6) and (A.8), by stating that Ll v is orthogonal to both l and k, show that Ll v is tangent to St [cf. Eq. (4.31)], and therefore establish (A.5). Property (A.5) means that, although l ∈ / T(St ), Ll can be viewed as an internal operator on the space T(St ) of vector fields tangent to St . We will denote it as S Ll to stress this feature and rewrite Eq. (A.5) as ∀v ∈ T(St ),
SL
lv
:= Ll v ∈ T(St )
.
(A.9)
The definition of S Ll can be extended to 1-forms on St by demanding that the Leibnitz rule holds for the contraction of a 1-form and a vector field: if ∈ T∗ (St ) is a 1-form on St , we define the Lie derivative S Ll of along l as the 1-form whose action on vectors is ∀v ∈ T(St ),
S Ll , v := Ll , v − , S Ll v .
(A.10)
Note that the right-hand side of this equation is well defined since S Ll v ∈ T(St ), so that we can apply the 1-form to it. We can extend the definition of the Lie derivative S Ll to bilinear forms on St , and more generally to multilinear forms, by means of Leibnitz rule: S
Ll (1 ⊗ 2 ) = S Ll 1 ⊗ 2 + 1 ⊗ S Ll 2 .
(A.11)
Taking into account the property (A.9), which also holds for any tensorial product of vectors, we finally conclude that the Lie derivative operator S Ll is defined for any tensor field on St : it is internal to St in the sense that it transforms a tensor field on St into another tensor field on St . This 2-dimensional operator has been introduced by Damour [60,61] and called by him the “convective derivative”. Now, any 1-form ∈ T∗ (St ) can also be seen as a 1-form on M thanks to the orthogonal projector q on T(St ) : it is the 1-form q ∗ defined by Eq. (5.11).37 Let us then investigate the relation between the “four-dimensional” Lie derivative Ll q ∗ and the “two-dimensional” one, S Ll . The first thing to notice is that the 1-forms Ll q ∗ and q ∗ S Ll coincide when restricted to T(St ). Indeed ∀ v ∈ T(St ),
Ll q ∗ , v = Ll q∗ , v − q∗ , Ll v = Ll , q (v) − , q (Ll v) = Ll , v − , Ll v = S Ll , v ,
(A.12)
where the third equality follows from property (A.5) and the fourth one from definition (A.10). Hence ∀ ∈ T∗ (St ),
S
Ll = (Ll q ∗ )|T(St ) .
(A.13)
Moreover, ∀ ∈ T∗ (St ),
Ll q ∗ , l = Ll q∗ , l − q∗ , Ll l =0
∗
Ll q , l = 0 .
=0
(A.14)
37 In this appendix, we make a distinction between ∈ T∗ (S ) and q
∗ ∈ T∗ (M), whereas in the remaining of the article we use the same t symbol to denote both applications, considering as the pull-back of q ∗ by the embedding of St in M.
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Since q∗ S Ll , l = 0 (for q (l) = 0), we can combine Eqs. (A.13) and (A.14) in ∀ ∈ T∗ (St ),
( q∗ S Ll )|T(H) = (Ll q ∗ )|T(H) .
(A.15)
But regarding the direction transverse to H one has ∀ ∈ T∗ (St ),
Ll q ∗ , k = Ll q∗ , k − q∗ , Ll k =0
=[l,k]
Ll q ∗ , k = , q ([k, l]) ,
(A.16)
where [k, l] denotes the commutator of vectors k and l. The right-hand side of Eq. (A.16) is in general different from zero. Indeed, a simple calculation using Eq. (5.39) shows that 1 N
, ∇l ln (A.17) [k, ] = k (∇ + ∇ ) − 2N 2 M so that q [k, ] = k q (∇ + ∇ ) .
(A.18)
For instance a sufficient condition for the right-hand side of Eq. (A.16) to vanish, and then Ll q ∗ to coincide with q ∗ S Ll , consists in demanding l to be a Killing vector of spacetime: ∇ + ∇ = 0. Another writing of Eq. (A.13) is ∀ ∈ T∗ (St ),
q ∗ S Ll = q ∗ Ll q ∗ ,
(A.19)
where each side of the equality is a 1-form on T(M) and the operators q ∗ added with respect to Eq. (A.13) effectively restrict the non-trivial action of these 1-forms to the subspace T(St ) of T(M). By taking tensorial products, the above analysis can be extended easily to any multilinear form A acting on T(St ). In particular Eq. (A.19) can be generalized to ∀A ∈ T∗ (St )⊗n ,
q ∗ S Ll A = q ∗ Ll q ∗ A
.
(A.20)
Note the similarity between this relation and Eq. (A.4) for HLl . Appendix B. Cartan’s structure equations Many studies about null hypersurfaces and isolated horizons make use of the Newman–Penrose framework, which is based on the complex null tetrad introduced in Section 4.6. An alternative approach is Cartan’s formalism which is based on a real tetrad and exterior calculus (see e.g. Chapter 14 of MTW [123] or Chapter V.B of Ref. [46] for an introduction). Cartan’s formalism is at least as powerful as the Newman–Penrose one, although it remains true that the latter is well adapted to null surfaces. B.1. Tetrad and connection 1-forms In the present context, it is natural to consider the following bases for, respectively, T(M) (vector fields) and T∗ (M) (1-forms) e = (l, k, e2 , e3 ) and
e = (−k, −l, e2 , e3 ) ,
(B.1)
where e2 and e3 are two vector fields tangent to the 2-surface St which constitute an orthonormal basis of T(St ) (with respect to the induced Riemannian metric q of St ) and e2 and e3 are the two 1-forms in T∗ (M) such that the basis (e ) of T∗ (M) is the dual of the basis (e ) of T(M), i.e. it satisfies e , e = ,
(B.2)
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where denotes the Kronecker symbol. The vector basis (e ) is usually called a tetrad, or moving frame or repère
mobile. Note that the ordering (e0 = l, e1 = k) and (e0 = −k, e1 = −l) has been chosen to ensure Eq. (B.2) for , ∈ {0, 1}, by virtue of the fact that l and k are null vectors and satisfy l · k = −1 [Eq. (4.28)]. Note that the tetrad (l, k, e2 , e3 ) is the same as that used to construct the complex Newman–Penrose null tetrad in Section 4.6. Thanks to properties (4.28) and (4.31), the metric tensor components with respect to the chosen tetrad are ⎛ ⎞ 0 −1 0 0 0 0 0⎟ ⎜ −1 g = g(e , e ) = ⎝ (B.3) ⎠ . 0 0 1 0 0 0 0 1
The connection 1-forms of the spacetime connection ∇ with respect to the tetrad (e ) are the sixteen 1-forms defined by ∇v e = , ve .
∀v ∈ T(M),
(B.4)
The expansions of the connection 1-forms on the basis (e ) of T∗ (M) define the connection coefficients38 of ∇ with respect to the tetrad (e ): = e
or
= e , ∇e e .
(B.5)
By direct computations using the formulas of Section 5, we get 0 0 = −1 1 = − N −2 ∇l l ,
(B.6)
1 0 = 0 1 = 0 ,
(B.7)
a 0 = 1 a = (a − ∇ea )l + ab eb ,
(B.8)
a 1 = 0 a = −a k − N −2 ∇ea l + ab eb ,
(B.9)
b a = −a b = −b a0 k − b a1 l + b ac ec ,
(B.10)
where is related to the lapse N and the metric factor M by = ln(MN ) [Eq. (4.17)] and we have introduced the abbreviation N 1 . (B.11) := ln 2 M a , ab and ab denote the components of respectively the Há´"iˇcek 1-form , the deformation rate and transversal deformation rate with respect to the basis (ea ) = (e2 , e3 ) of T∗ (St ): = a ea ,
= ab ea ⊗ eb
and = ab ea ⊗ eb .
(B.12)
Note that the above expressions are not restricted to T(St ) but do constitute four-dimensional writings of the 1-form and the bilinear forms and , since all these forms vanish on the vectors e0 = l and e1 = k [cf. Eqs. (5.14), (5.33), and (5.80)]. Note also that since the basis (ea ) is orthonormal, one has a b = ab and a b = ab . The symmetries (or antisymmetries) of the 1-forms when changing the indices and , as expressed in Eqs. (B.6)–(B.10), are due to the constancy of the components g of the metric tensor g in the basis e ⊗ e [cf. Eq. (B.3)]. Indeed this constancy, altogether with the metric compatibility relation dg = + (cf. e.g. Eq. (14.31b) of MTW [123]), implies = − , where := g . In particular 2 2 = 3 3 = 0. 38 Note that we are following MTW convention [123] for the ordering of the indices of the connection coefficients, which is the reverse of Hawking and Ellis’ one [90].
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B.2. Cartan’s first structure equation Cartan’s first structure equation states that the exterior derivative of each 1-form e is a 2-form which is expressible as a sum of exterior products involving the connection 1-forms:39 de = e ∧ .
(B.13)
These relations actually express the vanishing of the torsion of the spacetime connection ∇. For = 0, Eq. (B.13) results in de0 = e0 ∧ 0 0 + e1 ∧ 0 1 + ea ∧ 0 a , −dk = − k ∧ ( − N −2 ∇l l) + ea ∧ (−a k − N −2 ∇ea l + ab eb ) , dk = k ∧ − N −2 ∇l k ∧ l + a ea ∧k + N −2 ∇ea ea ∧ l − ab ea ∧ eb =0
=
= k ∧ ( − k) − k ∧ + N =N
−2
−2
(−∇l k + ∇ea e ) ∧ l a
(−∇k l − ∇l k + ∇ea ea ) ∧ l ,
dk = N −2 d ∧ l ,
(B.14)
where we have used the symmetry of ab , as well as the expression (5.35) of in terms of and k. Eq. (B.14) is nothing but the Frobenius relation (5.39). For = 1, Eq. (B.13) results in de1 = e0 ∧ 1 0 + e1 ∧ 1 1 + ea ∧ 1 a , −dl = − l ∧ (− + N −2 ∇l l) + ea ∧ [(a − ∇ea )l + ab eb ] , dl = ( − + ea ∇ea ) ∧ l − ab ea ∧ eb =(−k + ea ∇ea ) ∧ l =0
= (−∇l k − ∇k l + ∇ea e ) ∧ l , a
dl = d ∧ l ,
(B.15)
where we have used the symmetry of ab , as well as Eqs. (5.35) and (2.22). Again, we recover a previously derived Frobenius relation, namely Eq. (2.17). Finally, for = a = 2 or 3, Cartan’s first structure equation (B.13) results in dea = e0 ∧ a 0 + e1 ∧ a 1 + eb ∧ a b = − k ∧ [(a − ∇ea )l + ab eb ] − l ∧ (−a k − N −2 ∇ea l + ab eb ) + eb ∧ (−a b0 k − a b1 l + a bc ec ) dea = (2a − ∇ea )l ∧ k + (ab − a b0 )eb ∧ k + (ab − a b1 )eb ∧ l + a bc eb ∧ ec .
(B.16)
B.3. Cartan’s second structure equation Cartan’s second structure equation relates the exterior derivative of the connection 1-forms to the connection curvature: d = R − ∧ , 39 See Section 1.2.3 for our conventions regarding exterior calculus.
(B.17)
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where the R are the sixteen curvature 2-forms associated with the connection ∇ and the tetrad (e ). They are defined in terms of the spacetime Riemann curvature tensor (cf. Section 1.2.2) by ∀(u, v) ∈ T(M) × T(M),
R (u, v) := Riem(e , e , u, v) .
(B.18)
Note that due to the symmetry property (1.15) of the Riemann tensor, there are actually only 6, and not 16, independent curvature 2-forms. From Eq. (B.18), the curvature 2-forms can be expressed in terms of the components R of the Riemann tensor with respect to the bases (e ) and (e ) [cf. Eq. (1.13)] as R = R e ⊗ e = 21 R e ∧ e ,
(B.19)
where the second equality follows from the antisymmetry of the Riemann tensor with respect to its last two indices; it clearly exhibits that R is a 2-form. Conversely, one may express the Riemann tensor in terms of the curvature 2-forms as Riem = e ⊗ e ⊗ R .
(B.20)
For = = 0, Cartan’s second structure equation (B.17) results in d0 0 = R0 0 − 0 1 ∧ 1 0 − 0 a ∧ a 0 , d( − N −2 ∇l l) = R0 0 − (−a k − N−2 ∇ea l + ab eb ) ∧ [(a − ∇ea )l + ab eb ] ,
(B.21)
where, according to definition (B.18) and to the symmetry property (1.15) of the Riemann tensor, R0 0 = Riem(e0 , e0 , ., .) = Riem(−k, l, ., .) = Riem(l, k, ., .) .
(B.22)
Expanding Eq. (B.21) [cf. Eq. (1.22)] and using Eq. (B.15) leads to d = Riem(l, k, ., .) − b b a ea ∧ k + ac c b ea ∧ eb + {d(N −2 ∇l ) + N −2 ∇l d + a (a − ∇ea )k − [N −2 ∇eb b a + (b − ∇eb )b a ]ea } ∧ l . (B.23) For = 0 and = 1 (or = 1 and = 0), Cartan’s second structure equation (B.17) results in the trivial equation 0 = 0. For = 0 and = a it gives d0 a = R0 a − 0 0 ∧ 0 a − 0 1 ∧ 1 a − 0 b ∧ b a , d(−a k − N −2 ∇ea l + ab eb ) = R0 a − ( − N−2 ∇l l) ∧ (−a k − N−2 ∇ea l + ab eb )
+ (b k + N −2 ∇eb l − bc ec ) ∧ (−b a0 k − b a1 l + b ad ed ) . (B.24)
Expanding this expression and using Eqs. (B.14) and (B.15), as well as R0 a = −Riem(k, ea , ., .), leads to d(ab eb ) = − Riem(k, ea , ., .) + {N −2 (a d + ∇ea d + ∇ea ) + d(N −2 ∇ea ) + [N −2 (a ∇l + ∇eb b a0 ) − b b a1 ]k + [bc c a1 − N −2 (ab ∇l + c ab ∇ec )]eb } ∧ l + [da + a
+ (bc c a0 − c c ab )eb ] ∧ k − ab ∧ eb − bd d ac eb ∧ ec .
(B.25)
For = 1 and = 1, Cartan’s second structure equation yields the same result as for = 0 and = 0 (since 1 1 = −0 0 and R1 1 = −R0 0 ), namely Eq. (B.23). For = 1 and = a, it writes d1 a = R1 a − 1 0 ∧ 0 a − 1 1 ∧ 1 a − 1 b ∧ b a d[(a − ∇ea )l + ab eb ] = R1 a + ( − N−2 ∇l l) ∧ [(a − ∇ea )l + ab eb ] − [(b − ∇eb )l + bc ec ] ∧ (−b a0 k − b a1 l + b ad ed ) .
(B.26)
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Expanding this expression and using Eqs. (B.14) and (B.15), as well as R1 a = −Riem(l, ea , ., .), leads to d(ab eb ) = − Riem(l, ea , ., .) + {d(∇ea − a ) + (a − ∇ea )( − d ) + b a0 (∇eb − b )k + [N −2 ∇l ab + c ab (c − ∇ec ) + c a1 bc ]eb } ∧ l + bc c a0 eb ∧ k + ab ∧ eb − bd d ac eb ∧ ec .
(B.27)
As an application of this relation, we can express the Lie derivative of the second fundamental form along l restricted to the 2-plane T(St ), i.e. the quantity q ∗ Ll . Indeed, by means of expansion (B.12), let us write Ll as Ll = ea ⊗ Ll (ab eb ) + ab Ll ea ⊗ eb ⎤ ⎡
⎛
⎞
= ea ⊗ ⎣l · d(ab eb ) + d ab eb , l⎦ + ab ⎝l · dea + d ea , l⎠ ⊗ eb =0
=0
= ea ⊗ [l · d(ab eb )] + ab [(2a − ∇ea )l + (a c − a c0 )ec ] ⊗ eb ,
(B.28)
where we have used Cartan’s identity (1.26) to get the second line and Cartan’s first structure equation (B.16) to get the third one. Then Ll (ea , eb ) = d(ac ec )(l, eb ) + bc (c a − c a0 ) .
(B.29)
Applying the 2-form (B.27) to the couple of vectors (l, eb ) results in d(ac ec )(l, eb ) = −Riem(l, ea , l, eb ) + bc c a0 + ab .
(B.30)
Combining Eqs. (B.29) and (B.30) yields Ll (ea , eb ) = ab + ac c b − Riem(l, ea , l, eb ) .
(B.31)
Hence
− q ∗ Riem(l, ., l, .) , q ∗ Ll = + ·
(B.32)
i.e. we recover Eq. (6.27). For = a and = 0, Cartan’s second structure equation yields the same result as Eq. (B.27), since a 0 = 1 a and a R 0 = R1 a . For = a and = 1, it yields the same result as for = 0 and = a (since a 1 = 0 a and Ra 1 = R0 a ), namely Eq. (B.25). Finally, for = a and = b, Cartan’s second structure equation writes da b = Ra b − a 0 ∧ 0 b − a 1 ∧ 1 b − a c ∧ c b d(−a b0 k − a b1 l + a bc ec )
= a b − [(a − ∇ea )l + ac ec ] ∧ (−b k − N −2 ∇eb l + bd ed ) − (−a k − N −2 ∇ea l + ac ec ) ∧ [(b − ∇eb )l + bd ed ] − (−a c0 k − a c1 l + a cd ed ) ∧ (−c b0 k − c b1 l + c bf ef ) ,
(B.33)
which leads to d(a bc ec ) = Riem(ea , eb , ., .) + {a b0 N −2 d + da b1 + a b1 d + [(∇ea − a )b + (b − ∇eb )a − a c0 c b1 + a c1 c b0 ]k + [bc (a − ∇ea ) + N −2 (a c ∇eb − bc ∇ea ) − a c (b − ∇eb ) − a d1 d bc + a dc d b1 ]ec } ∧ l
+ [da b0 + (a c b − a bc − a d0 d bc + a dc d b0 )ec ] ∧ k + (a d bc − a c bd + a f d f bc )ec ∧ ed .
(B.34)
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B.4. Ricci tensor Having computed the tetrad components of the Riemann tensor via Cartan’s second structure equation, we can evaluate the tetrad components of the Ricci by means of definition (1.17) of the latter: R(e , e ) = Riem(e , e , e , e ) .
(B.35)
For = = 0, we get R(l, l) = Riem(e , l, e , l) = − Riem(k, l, l, l) − Riem(l, l, k, l) +Riem(ea , l, ea , l) =0
=0
= − Riem(l, ea , ea , l) ,
(B.36)
where we have used the symmetry property (1.15) of the Riemann tensor. Let us substitute Eq. (B.27) for Riem(l, ea , ., .). We notice that the long term {. . .} ∧ l vanishes when applied to (ea , l), as the term eb ∧ ec . Moreover since , l = [Eq. (5.5)], ab b a = [Eq. (5.65)] and ab b a0 = 0 (by symmetry of ab and antisymmetry of b a0 with respect to the indices a and b), we get R(l, l) = d(ab eb )(ea , l) + .
(B.37)
Let us express the exterior derivative of the 1-form ab eb by means of formula (1.25): d(ab eb )(ea , l) = ∇ea (ab eb ), l − ∇l (ab eb ), ea = − ab eb , ∇ea l − ∇l ab eb , ea + ab eb , ∇l ea = ab (eb , ∇l ea − eb , ∇ea l) − ∇l ab b a = ab (b a , l − b 0 , ea ) − ∇l = ab (b a0 − b a ) − ∇l = −ab ab − ∇l ,
(B.38)
where we have used the property ab b a0 = 0 noticed above. Inserting this relation into Eq. (B.37) yields R(l, l) = −∇l + − ab ab .
(B.39)
We thus recover the null Raychaudhuri equation (6.6). For = 0 and = a, Eq. (B.35) gives R(l, ea ) = Riem(l, k, l, ea ) − Riem(l, eb , eb , ea ) .
(B.40)
Substituting Eq. (B.23) for Riem(l, k, ., .) and Eq. (B.27) for Riem(l, eb , ., .), we get R(l, ea ) = d(l, ea ) + d(b c ec )(eb , ea ) + a + c bc b a ,
(B.41)
where we have used once again the symmetry property bc b ca = 0, as well as bc = b c . Thanks to Cartan’s identity (1.26), the first term in the right-hand side of Eq. (B.41) writes
d(l, ea ) = Ll − d , l, ea = Ll , ea − d, ea . =
(B.42)
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The second term in the right-hand side of Eq. (B.41) is expressed by means of formula (1.25): d(b c ec )(eb , ea ) = ∇eb (b c ec ), ea − ∇ea (b c ec ), eb = ∇eb b c ec , ea − b c ec , ∇eb ea − ∇ea b c ec , eb + b c ec , ∇ea eb = ∇eb b a − b c c ab − ∇ea b b + b c c ba =
=0
= db a , eb − c ab b c − d, ea .
(B.43)
Inserting expressions (B.42) and (B.43) into Eq. (B.41) leads to R(l, ea ) = Ll , ea + a − d, ea − d, ea + db a , eb + c bc b a − c ab b c .
(B.44)
with respect to the connection 2 D We recognize in the last term the component on ea of the covariant divergence of induced by ∇ in the 2-surface St :
a = db , eb + c b − c b . (2 D · ) a a c bc ab
(B.45)
Besides Ll , ea = Ll ( − k), ea = Ll − Ll k − Ll k, ea = Ll − l · dk, ea = Ll − N −2 l · (d ∧ l), ea = Ll , ea ,
(B.46)
so that Eq. (B.44) can be written as
ea , R(l, ea ) = Ll + − d( + ) + 2 D · ,
(B.47)
which is nothing but the Damour–Navier–Stokes equation under the form (6.13). Finally, let us consider the components of the Ricci tensor relative to T(St ), i.e. R(ea , eb ). From Eq. (B.35), we get R(ea , eb ) = −Riem(k, ea , l, eb ) − Riem(l, ea , k, eb ) + Riem(ec , ea , ec , eb ) .
(B.48)
The first term in the right-hand side can be expressed, thanks to Eq. (B.25) −Riem(k, ea , l, eb ) = d(ac ec )(l, eb ) − ∇eb a − a b − bc c a0 + c ab c + ab ,
(B.49)
whereas Eq. (B.27) yields −Riem(l, ea , k, eb ) = d(ac ec )(k, eb ) − ∇ea ∇eb + c ab ∇ec − ∇ea ∇eb + a ∇eb + b ∇ea + ∇eb a − c ab c − a b − N −2 ∇l ab − c a1 bc ,
(B.50)
and Eq. (B.34) gives Riem(ec , ea , ec , eb ) = d(c ad ed )(ec , eb ) − ac c b − ac c b + (k) ab + ab − c db d ac + c dc d ab .
(B.51)
Collecting together Eqs. (B.49)–(B.51) enables us to write Eq. (B.48) as R(ea , eb ) = d(ac ec )(l, eb ) − c a0 bc + ( + )ab + d(ac ec )(k, eb )
− c a1 bc + ((k) − N −2 ∇l )ab − ac c b − ac c b − ∇ea ∇eb + c ab ∇ec − ∇ea ∇eb + a ∇eb + b ∇ea − 2a b + d(c ad ed )(ec , eb ) − c db d ac + c dc d ab .
(B.52)
Let us first notice that the last three terms of this expression are nothing but the Ricci tensor 2 R of the connection 2 D associated with the induced metric in the 2-surface St , applied to the couple of vectors (ea , eb ). Indeed, using the
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moving frame (ea ) in St , the connection 1-forms 2 b a of 2 D are given by a formula similar to Eq. (B.5), except that the range of the summation index is now restricted to {2, 3}: 2
b a = b ac ec .
(B.53)
Expressing the curvature of 2 D by means of Cartan’s second structure equation, we get then 2
R(ea , eb ) = (d(2 c a ) − 2 c d ∧ 2 d a )(ec , eb ) = d(c ad ed )(ec , eb ) − c db d ac + c dc d ab .
(B.54)
Besides, we may express the term d(ac ec )(l, eb ) [resp. d(ac ec )(k, eb )] which appears in the right-hand side of Eq. (B.52) in terms of the Lie derivative Ll (resp. Lk ). Indeed a computation similar to that which led to Eq. (B.29) for Ll gives Ll (ea , eb ) = d(ac ec )(l, eb ) + bc (c a − c a0 ) .
(B.55)
Lk (ea , eb ) = d(ac ec )(k, eb ) + bc (c a − c a1 ) .
(B.56)
and
Inserting the above two equations, as well as Eq. (B.54), into Eq. (B.52) leads to R(ea , eb ) = Ll (ea , eb ) + Lk (ea , eb ) − 2ac c b − 2ac c b
+ ( + )ab + ((k) − N −2 ∇l )ab − 2a b − 2 D ea 2 D eb − 2 D ea 2 D eb + a 2 D eb + b 2 D ea + 2 R(ea , eb ) .
(B.57)
Appendix C. Physical parameters and Hamiltonian techniques In Section 8.6 we introduced quasi-local notions for the physical parameters associated with the horizon, when no matter is present on H. In that section we only presented the final results, since the actual derivations involved the use of Hamiltonian techniques not discussed in this article. In this appendix we aim at providing some intuition on the actual use of these tools. Instead of using a formal presentation (see [15,12,107]) we introduce the basic concepts by illustrating them with examples extracted from the isolated horizon literature, always in absence of matter. As indicated in Section 8.6, the physical parameters are identified with quantities conserved under certain transformations on the space of solutions of Einstein equation. More specifically, these envisaged solutions in contain a “fixed” WIH (H, [l]) as inner boundary. The transformations on relevant for the definition of the conserved quantities, are associated with the WIH-symmetries of this inner boundary. An appropriate characterization of this phase space , where each point is a Lorentzian manifold (M, g), must be therefore introduced. This is accomplished by setting a well-posed variational problem associated with spacetimes (M, g) containing a “given” WIH. A rigorous presentation of this whole subject requires a careful definition of the involved objects (domain of variation of the dynamical fields, variation of the fields at the boundaries of this domain, etc.). As mentioned above, we simply aim here at underlying the most relevant steps, through the use of (simplified) examples, referring the reader to the original references for the detailed formulations. As in the rest of the article, we restrict ourselves to the non-extremal case = 0. C.1. Well-posedness of the variational problem In our study of black hole spacetimes, we are interested in asymptotically flat solutions to Einstein equation with a WIH as inner boundary. In the variational formulation of spacetime dynamics, the presence of boundaries in the manifold generally demands the introduction of boundary terms in the action so as to compensate the variation of this action with respect to the dynamical fields. This is relevant for the differentiability of the action with respect to the dynamical fields as well as for guaranteeing that the derived equations of motion are actually the ones corresponding to the studied problem.
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Fig. 19. Domain of variation in M bounded by two Cauchy surfaces − and + , spatial infinity ∞ and the WIH (H, [l]). S− and S+ denote the cross-sections resulting from the intersections of the spatial slices − and + with the WIH.
Example C.1. This example shows how the WIH condition (8.2) guarantees the well-posedness of a first order action principle. Details can be found in [10,12,108]. An appropriate first order action for our problem can be written in terms of the cotetrad (eI ) and a real 1-form connection AIJ , where the capital Latin letters correspond to Lorentz indices which are raised and lowered with the Minkowski metric. We can write the action as # # I J ∧ F I J + I J ∧ A I J , (C.1) S(e, A) ∼ − M
∞
where I J = 21 I J KL eK ∧ eL , J FJI = dAJI + AK I ∧ AK ,
(C.2)
I J KL is the alternating symbol in four dimensions and ∞ denotes spatial infinity. In order to determine the equations of motion, the action is varied with respect to the fields eI and AIJ in a region M of M delimited by two Cauchy surfaces − and + (on which the variation of the fields, schematically denoted by (e, A), vanishes), by spatial infinity ∞ and the inner boundary (H, [l]) (see Fig. 19). In the variation of the action, the bulk term gives rise to a boundary term at infinity, but it is exactly cancelled by the variation of the boundary term at infinity in Eq. (C.1). The resulting variation can be expressed as # # 1
∧ 2 . (C.3)
S(e, A) = (Equations of motion) · (e, A) − 8G H The problem is well-posed and Einstein equations are recovered as an extremal value of this action ( S(e, A) = 0), if the boundary integral at H vanishes. This is the crucial point we want to make in this example: the WIH condition HL = 0, together with the NEH conditions and (e, A) = 0 on and , suffices to guarantee the vanishing of − + l this boundary term (see [10] for the details). C.1.1. Phase space and canonical transformations Once the variational problem is well-posed, we must determine the phase space . We firstly introduce some general concepts and notation (see, e.g. Refs. [1,8,80]). The phase space is constituted by a pair (, J), where is an (infinite-dimensional) manifold in which each point represents a solution to the equations of motion and J is a closed 2-form on known as the symplectic form: d J = 0, with d the exterior differential in . We can associate with each function F : → R, a Hamiltonian vector field F in T() (the space of vector fields on in the notation introduced in Section 1.2) as follows: i F J = d F ,
(C.4)
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where i F J := J( F , ·). Given two functions F and G with support on , their Poisson bracket is defined as {F, G} = J( F , G ) .
(C.5)
The pair (, J) is also known as a symplectic manifold (pre-symplectic if the kernel of J is non-trivial). Even though the phase spaces we are interested in are intrinsically infinite-dimensional, we shall skip all the subtleties related to infinite-dimensional symplectic spaces (see for instance [119]). An infinitesimal transformation on the space , generated by the vector field W , is a canonical transformation (also called symplecto-morphism) if it preserves the symplectic form J
L W J = 0 .
(C.6)
Using the closed character of J, this is locally equivalent to the existence of a Hamiltonian function HW , i.e.
W preserves (locally) the Poisson brackets
!
⇐⇒ ∃HW such that i W J = d HW ,
(C.7)
Applying this expression to a generic vector field in T(), yields J( W , ) = HW ,
(C.8)
where the notation HW := d HW ( ) is designed to mimic the intuitive physical notation. The evolution of a function G along the flow of the vector field W on (Hamilton equations), can be evaluated as
W G = {HW , G} .
(C.9)
In particular, due to the anti-symmetry of J, HW remains constant along the W trajectories. With these elements, the general strategy to associate physical parameters with the horizon will proceed via the following steps: (1) Construction of the appropriate phase space for our problem. (2) Extension of a given WIH-symmetry of (H, [l]) to an infinitesimal diffeomorphism W on each space-time M of , giving rise to a family of vector fields {W} . Definition of a canonical transformation W on out of the family {W} . (3) Identification of the physical parameter with the associated conserved quantity HW . We illustrate these steps by continuing Example C.1 (see again [15,12]). Example C.2. Phase space and canonical transformations. (1) Phase space. The phase space where we describe the dynamics defined by the action (C.1), can be parametrized by the pairs (eI , AIJ ) which satisfy Einstein equations and contain an inner boundary given by a “fixed” WIH (H, [l]). The so-called conserved current method [56] provides a standard manner to derive the relevant symplectic form from a given action. In our case this results in [12] # # 1 1 J( 1 , 2 ) = − ( 1 I J ∧ 2 AI J − 2 I J ∧ 1 AI J ) + ( 1 2 2 − 2 2 1 ) , (C.10) 16G 8G St where is a function on H such that HLl := (l) and 1 , 2 are arbitrary vector fields on T(). (2) Canonical transformations induced by spacetime transformations. On each point of , i.e. on each spacetime M represented by a pair (eI , AIJ ), we consider an infinitesimal spacetime diffeomorphism W(e, A) (we make explicit the dependence of this vector field on the particular spacetime) whose restriction to H is a WIH-symmetry. This family of spacetime vector fields {W(e, A)} permits us to define an
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Fig. 20. Illustration of the construction of a transformation W on from the family {W} of diffeomorphisms on spacetimes M. On each point of , a spacetime M(e, A), we consider a diffeomorphism W(e, A) whose restriction to H is a WIH-symmetry. The ensemble of these spacetime diffeomorphisms {W(e, A)} generates a transformation W on through Eq. (C.11).
infinitesimal transformation W on , which is defined in a point-wise manner (on each point M of ) by its action on the coordinates (eI , AIJ ) of ( W eI )|M := LW(e,A) eI ,
( W AIJ )|M := LW(e,A) AIJ ,
(C.11)
where right-hand side terms are evaluated on each spacetime M, associated respectively with the pair (eI , AIJ ). In sum, starting from a family of WIH-symmetries {W(e, A)|H } an infinitesimal transformation W has been induced in the phase space (see Fig. 20). The question now is to find out if such a transformation W is a canonical one. According to (C.7), one must contract the vector field W with the symplectic form (C.10) and check if the resulting 1-form on is (locally) exact. Following (C.8), this contraction is applied on an arbitrary vector field # −1 J( , W ) =
[q∗ W, 2 ] − q∗ W, 2 + (W) 2 8G St # 1 + AI J ∧ W, I J + W, AI J I J , (C.12) 16G ∞ where (W) = W − q∗ W, . (3) Conserved quantities and horizon physical parameters. The term at infinity is related to the standard ADM quantities (whenever W is a symmetry of the asymptotic metric at infinity). Consequently, it is associated with the exact variation of a function on , the corresponding ADM parameter. Likewise, in order to associate a conserved quantity with the horizon H itself, the integral on St must be written as the exact variation of a function on . We study this problem for the specific and physically relevant cases of the angular momentum and the energy.
C.2. Applications of examples (C.1) and (C.2) We offer some more details on the discussion developed in Sections 8.6.1 and 8.6.2. C.2.1. Angular momentum We restrict to its subspace of spacetimes containing a class II WIH. On the inner boundary H of each spacetime M, the same rigid azimuthal WIH-symmetry is fixed (in fact is an isometry of the cross-section St ). More specifically, we consider on every spacetime in a vector field which is a SO(2) axial isometry of the induced metric q on H with 2 affine length.
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This WIH-symmetry on H is then extended to a vector field on each spacetime M. Since we are interested in studying the angular momentum related to the horizon itself, this extension is enforced to vanish outside some compact neighborhood of the horizon. Evaluating expression (C.12) in this case, results in [12] # −1 2 J( , ) =
· . (C.13) 8G St The transformation induces directly a locally canonical transformation on . Making JH := H , the conserved quantity JH is identified as the angular momentum associated with the horizon and Eq. (8.32) follows. We also point out that this expression is conserved under the canonical transformation in , even if is not a WIH-symmetry. However, as mentioned in Section 8.6.1, in the absence of a symmetry the physical status of this expression is unclear. C.2.2. Mass As discussed in Section 8.6.2, the definition of the mass is related to the choice of an evolution vector t on each spacetime M, which plays now the role of the vector W in Example C.2. We fix expression (8.33) as the inner boundary condition for t. Regarding the outer boundary condition at infinity, we make t approach an observer t∞ inertial with respect to the flat metric. The first law of black hole mechanics (8.34) follows from imposing t to be a locally canonical transformation on . Expression (C.12) in this case simplifies to (t) t J( , t ) = EADM (C.14)
aH + (t) JH , − 8G " t corresponds to the ADM energy and aH = S 2 is the area of St . On each spacetime M in , (t) and where EADM (t) are constant. However, the actual values of these constants change from one spacetime to another: (t) and (t) are functions on . As a necessary condition for J(·, t ) to be an exact variation on so as to make t a canonical transformation via Eq. (C.8), the form J(·, t ) must be closed. Consequently, functions (t) and (t) depend on only through an explicit dependence on aH and JH , satisfying j(t) (aH , JH ) j(t) (aH , JH ) = 8G . jJH jaH
(C.15)
If t is indeed a canonical infinitesimal transformation, we can write the second term in the right-hand side of Eq. (C.14) t , and Eq. (8.34) follows. As mentioned in Section 8.6.2, this does not fix the functional forms as an exact variation EH t . Finally, these dependences of the physical parameters in a of (t) (aH , JH ), (t) (aH , JH ) and EH H and JH , which are the same for all spacetimes in , are fixed by making them to coincide with those of the Kerr family (a subspace of ), as explained in Section 8.6.2. Appendix D. Illustration with the event horizon of a Kerr black hole In Examples 2.5, 3.1, 4.6, 5.7 and 6.5, we have considered for simplicity a non-rotating static black hole (Schwarzschild spacetime). It is of course interesting to investigate rotating stationary black holes (Kerr spacetime) as well. In particular, the Há´"iˇcek 1-form which has been found to vanish for a Schwarzschild horizon [Eq. (5.105)] is no longer zero for a Kerr horizon. We discuss here the event horizon H of a Kerr black hole from the 3 + 1 decomposition introduced in Section 10, by considering a foliation (t ) based on Kerr coordinates. D.1. Kerr coordinates In standard textbooks, the Kerr solution is presented in Boyer–Lindquist coordinates (tBL , r, , BL ). However, being a generalization of Schwarzschild coordinates to the rotating case, these coordinates are singular on the event horizon H, as discussed in Example 2.5. We consider instead Kerr coordinates, which are regular on H. These are the coordinates in which Kerr originally exhibited his solution [104]; they are a generalization of Eddington–Finkelstein coordinates to the rotating case. Denoting them by (V , r, , ), they are such that the curves V = const, = const and
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= const are ingoing null geodesics (they form a so-called principal null congruence), as in the Eddington–Finkelstein case.40 As in the Schwarzschild case, we will use the coordinate t := V − r
(D.1)
instead of V [cf. Eq. (2.33)]. The coordinates (t, r, , ) are then simply a spheroidal version of the well-known Kerr–Schild coordinates (t, x, y, z): t is the same coordinate and (x, y, z) are related to (r, , ) by x = (r cos − a sin ) sin ,
y = (r sin + a cos ) sin ,
z = r cos ,
(D.2)
where a is the angular momentum parameter of the Kerr solution, i.e. the quotient of the total angular momentum J by the total mass m, a := J /m. The relation with the Boyer-Lindquist coordinates (tBL , r, , BL ) is as follows: dt = dtBL +
dr (r 2 + a 2 )/2mr − 1
and
d = dBL +
adr . r 2 − 2mr + a 2
The metric components with respect to the “3 + 1” Kerr coordinates (t, r, , ) are given by 2mr 4mr 4amr 2mr 2 dr 2 sin dt d + 1 + g dx dx = − 1 − 2 dt 2 + 2 dt dr − 2 2 2mr 2a 2 mr sin2 dr d + 2 d2 + r 2 + a 2 + sin2 d2 , − 2a sin2 1 + 2 2
(D.3)
(D.4)
with 2 := r 2 + a 2 cos2 .
(D.5)
The event horizon H is located at r = rH := m + m2 − a 2 .
(D.6)
Since rH does not depend upon nor , the Kerr coordinates are adapted to H, according to the terminology introduced in Section 4.8. Note that the metric components given by Eq. (D.4) are all regular at r = rH . On the contrary, most of them are singular at = 0, which, via Eq. (D.5), corresponds to r = 0 and = /2, and, via Eq. (D.2), to the ring x 2 + y 2 = a 2 in the plane z = 0. This is the ring singularity of Kerr spacetime. Note also that in the limit a → 0, then → r and the line element (D.4) reduces to the line element (2.34) in Eddington–Finkelstein coordinates. The metric (D.4) is clearly stationary and axisymmetric and the two vectors j j and 1 := (D.7) 0 := jt r,, j t,r, are two Killing vectors, 0 being associated with the stationarity and 1 with the axial symmetry of the Kerr spacetime. Remark D.1. The two Killing vectors 0 and 1 are identical to the “standard” two Killing vectors which are formed from the Boyer–Lindquist coordinates: j j and 1 = . (D.8) 0 = jtBL r,,BL jBL tBL ,r, This property follows easily from the transformation law (D.3) between the two sets of coordinates. Consequently, the metric coefficients gtt = 0 · 0 , gt = 0 · 1 and g = 1 · 1 , which can be read on Eq. (D.4) are the same than those for Boyer–Lindquist coordinates, as it can be checked by comparing with e.g. Eq. (33.2) in MTW [123]. 40 Note, however, a difference with Schwarzschild spacetime in Eddington–Finkelstein coordinates: in the rotating Kerr case, the hypersurfaces V = const are no longer null (see e.g. [69]).
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285
D.2. 3 + 1 quantities Let us consider the foliation of Kerr spacetime by the hypersurfaces t of constant Kerr time t. From the line element (D.4), we read the corresponding lapse function N=
2 + 2mr
,
(D.9)
the shift vector 2mr i = , 0, 0 2 + 2mr and the 3-metric ⎛
i =
and
2mr 2amr , 0, − 2 sin2 2
⎞ 2mr −a 1 + 2 sin2 ⎟ ⎜ ⎟ ⎜ 2 ⎟ , ij = ⎜ 0 0 ⎟ ⎜ ⎠ ⎝ 2mr A 2 −a 1 + 2 sin2 0 sin 2 ⎛ ⎞ a A 0 2 2 ⎜ ( + 2mr) ⎟ 2 ⎜ ⎟ ij −2 ⎟ , =⎜ 0 0 ⎜ ⎟ ⎝ ⎠ a 1 0 2 2 2 sin 1+
2mr 2
(D.10)
0
(D.11)
(D.12)
with A := (r 2 + a 2 )2 − (r 2 − 2mr + a 2 )a 2 sin2 = 2 (r 2 + a 2 ) + 2a 2 mr sin2 .
(D.13)
The unit timelike normal to t is deduced from the values of the lapse function and the shift vector via Eq. (3.24), which results in & ' 1 2mr n = 2 + 2mr, − , 0, 0 , (D.14) 2 + 2mr & n = −
2 + 2mr
' , 0, 0, 0
.
(D.15)
Finally the extrinsic curvature tensor of t is obtained from Eq. (3.42) with j ij /jt = 0: ⎛
2m(a 2 cos2 − r 2 )( 2 + mr) ⎜ 5 2 + 2mr ⎜ ⎜ ⎜ ⎜ sym. ⎜ Kij = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ sym. ⎝
2a 2 mr sin cos 3 2 + 2mr
2mr 2
2 + 2mr sym.
⎞ am(r 2 − a 2 cos2 ) sin2 2 + 2mr ⎟ 5 ⎟ ⎟ 3 3 ⎟ 2a mr sin cos ⎟ − ⎟ 3 2 + 2mr ⎟ . ⎟ ⎟ 2mr sin2 ⎟ × ⎟ 2 + 2mr ⎟ ⎠ 2 2 2 2 2 a m(a cos − r ) sin r+ 4
As a check of this formula, we may compare it with Eqs. (A2.33)–(A2.38) of Ref. [161].
(D.16)
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D.3. Unit normal to St and null normal to H The 2-surface St ⊂ t is defined by r = const = rH . Its outward unit normal s lying in t is obtained from si = (, 0, 0), with such that ij si sj = 1. We get ⎞ ⎛ * 2 + 2mr si = ⎝ , 0, 0⎠ , (D.17) A ⎞ ⎛ * * 2 + 2mr a A 1 ⎠ . si = ⎝ , 0, 2 + 2mr A
(D.18)
As a check, we verify that n and s given by Eqs. (D.14) and (D.18) coincide with the first two vectors of the orthonormal
) introduced by King et al. [105] [cf. their Eq. (2.4), noticing that their coordinate vectors are (j/jV )r,, = basis (E (j/jt)r,, and (j/jr)V ,, = (j/jr)t,, − (j/jt)r,, ]. We then get the null normal to H associated with the Kerr slicing, l, by inserting expressions (D.9), (D.14) and (D.18) into l = N(n + s) [Eq. (4.13)]: & √ ' A − 2mr a , 0, √ = 1, 2 . (D.19) + 2mr A H H The value on the horizon is obtained by noticing that A =(2mr H )2 ; we get =(1, 0, 0, a/(2mr H )), i.e., from Eq. (D.7), H
l = 0 + H 1
,
(D.20)
with41 H :=
a a = √ 2mr H 2m(m + m2 − a 2 )
.
(D.21)
Eq. (D.20) shows that on the horizon, the null normal l is a linear combination of the two Killing vectors 0 and 1 with constant coefficients (compare with the inner boundary (8.33) for the evolution vector t in Section 8.6.2). It is therefore a Killing vector itself. This implies H
Ll A = 0 ,
(D.22)
for any tensor field A which respects the stationarity and axisymmetry of the Kerr spacetime. Another phrasing of this is saying that H is a Killing horizon [36]. Comparing Eq. (D.20) with Eq. (4.80) (taking into account that t = 0 ), we get the surface velocity of H with respect to Kerr coordinates: V = H 1 = H
j j
.
(D.23)
t,r,
Hence the quantity H can be viewed as the angular velocity of H with respect to the coordinates (t, r, , ). The fact that H is a constant over H reflects the rigidity theorem of stationary black holes (see e.g. Theorem 4.2 of Ref. [37]; more generally, in the WIH setting of Section 8, the constancy of H guarantees t to be a WIH-symmetry on H). 41 The constant
H,
which constitutes an equivalent expression for H in Eq. (8.36), should not be confused with the Há´"iˇcek 1-form .
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287
D.4. 3 + 1 evaluation of the surface gravity We will need the orthogonal projector q on St . Its components with respect to the coordinates (r, , ) are given by the formula qji = ij − s i sj ; from Eqs. (D.17)–(D.18), we get (i = row index, j = column index) ⎞ ⎛ 0 0 0 ⎜ 0 1 0⎟ qji = ⎝ (D.24) ⎠ . a 2 − ( + 2mr) 0 1 A We will also need the 1-form K(s, .). From Eqs. (D.16) and (D.18), we get Krj s j =
m(r 2 − a 2 cos2 ) 2 2 2 √ [ a sin − 2( 2 + mr)(r 2 + a 2 )] , 4 ( 2 + 2mr) A
Kj s j =
2a 2 mr sin cos , √ ( 2 + 2mr) A
Kj s j =
am sin2 2 2 [r (3r + a 2 cos2 ) + a 2 (r 2 − a 2 cos2 )] . √ 4 A
(D.25)
Let us start by evaluating the non-affinity parameter from the 3 + 1 expression (10.10). The first part of this relation is computed from Eqs. (D.19) and (D.9): ∇ ln N =
m r 2 − a 2 cos2 √ A − 2mr . 2 2 ( 2 + 2mr)
(D.26)
H
Since A =(2mr H )2 , this implies H
∇ ln N = 0 ,
(D.27)
in agreement with Eq. (D.22). The second term in the right-hand side of Eq. (10.10) is computed from Eqs. (D.18) and (D.9) √ m (r 2 − a 2 cos2 ) A i s Di N = 2 , (D.28) ,2 + 2 + 2mr resulting in the following value on the horizon: H
s i Di N =
2 − a 2 cos2 ) 2m2 rH (rH
2H ( 2H + 2mr H )2
,
(D.29)
2 2 + a 2 cos2 = 2mr 2 with 2H := rH H − a sin . Finally from Eqs. (D.9), (D.25) and (D.18), we evaluate the last term which enters in formula (10.10), namely N K ij s i s j . The obtained expression is rather complicated; however combining its value on the horizon with the results (D.27) and (D.29) yields a very simple expression for the non-affinity parameter:
√ rH − m m2 − a 2 = = √ 2mr H 2m(m + m2 − a 2 )
.
(D.30)
Note that does not depend on , in agreement with the fact that H, endowed with the null normal l given by Eq. (D.20), is an isolated horizon [zeroth law of black hole mechanics, cf. Eq. (8.5)]. Actually we recover for the classical value of the surface gravity of a Kerr black hole (see e.g. Eq. (12.5.4) of Wald [167]).
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D.5. 3 + 1 evaluation of the Há´jiˇcek 1-form Let us now compute the Há´"iˇcek 1-form from the 3 + 1 formula (10.14). From Eqs. (D.9), (D.25) and (D.24), we get = −
2a 2 mr sin cos 2 + 2mr
= −
1 1 √ + 2 A
,
(D.31)
am sin2 2 2 [r (3r + a 2 cos2 ) + a 2 (r 2 − a 2 cos2 )] , √ 4 A
(D.32)
from which we deduce the following values on the horizon: H
= − H
=
a 2 sin cos , 2mr H − a 2 sin2
(D.33)
2 ) cos2 − r (r (2m2 − 3mr H + rH a H H + m) sin2 . 2 rH (rH + (2m − rH ) cos )2
(D.34)
As a check, let us recover the total angular momentum JH = am from the integral (8.32) which involves . The symmetry generator which appears in the integral is of course in the present case the Killing vector 1 = (j/j)t,r, , so that formula (8.32) results in (G = 1) # 1 JH = − 2 . (D.35) 8 St Let us express the integral in terms of the coordinates (, ) which span St : JH = −
1 8
# # 2 0
0
√ q d d ,
(D.36)
where q = det qab , with the 2-metric components qab read from Eq. (D.11): ⎛ 2 ⎞ 0 ⎠ . qab = ⎝ A 0 sin2 2 Hence
(D.37)
√ √ √ H q = A sin , so that q = 2mr H sin and the integral (D.36) becomes
JH = −
2mr H 8
# # 2 0
0
sin d d .
(D.38)
Substituting Eq. (D.34) for , we get am JH = − 4
# (2 − 1)( − 1) cos2 − (2 + 1) 0
( + (1 − ) cos2 )2
sin3 d ,
(D.39)
where we have set := rH /(2m). It turns out that the above integral is independent of and is simply equal to −4, hence JH = am
,
as it should be for a Kerr black hole.
(D.40)
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289
D.6. 3 + 1 evaluation of and In order to apply the formulæ derived in Section 10.3.2, let us first compute the second fundamental form H of the 2-surface St (as a hypersurface of t ). From the relation Hij = Dk sl q k j q l j [Eq. (10.30)] and expressions (D.17) and (D.24), we get the following values on the horizon: 2mr 2H
H
H =
(2mr H − a 2 sin2 )(4mr H − a 2 sin2 ) 2a 3 mr H sin3 cos
H
H = −
(2mr H − a 2 sin2 )3/2 (4mr H − a 2 sin2 )1/2
(D.41)
,
(D.42)
mr 3H sin2
H
H =
,
4(2mr H − a 2 sin2 )5/2 (4mr H − a 2 sin2 )1/2 % × 4m + 9mr H + 3rH − 4a 3
2
3
2
m − rH (rH + 3m) cos 2 + a 4 r2
( cos 4
.
(D.43)
H
We are then in position to evaluate H’s second fundamental form via Eq. (10.43): = N (H − q ∗ K). Using the value of K and q given by Eqs. (D.16) and (D.24), we get H
=0
.
(D.44)
Hence we recover the fact that the event horizon of a Kerr black hole is a non-expanding horizon. Regarding the transversal deformation rate , we use the formula = −1/(2N )(H − q ∗ K) [Eq. (10.44)]. Using expressions (D.41)–(D.43), (D.16), (D.24) and (D.9), we get H
= − H
=
2mr 2H
2mr H − a 2 sin2
,
2a 3 mr sin3 cos (2mr H − a 2 sin2 )2
(D.45)
,
% ( 2 ) sin2 a 2 m(a 2 cos2 − rH 2mr H sin2 rH + . = − 2mr H − a 2 sin2 (2mr H − a 2 sin2 )2 H
Contracting with q ab [obtained as the inverse of the matrix (D.37)], we get the transversal expansion scalar a2 m − 1 cos2 2m + 3rH − 1+ m rH H . (k) = − 2 2(2mr H − a sin2 )
(D.46)
(D.47)
(D.48)
As a check, one can easily verify that in the non-rotating limit (a = 0, rH = 2m), the values of , and (k) derived above reduce to that obtained in Example 5.7 for the Eddington–Finkelstein slicing of the Schwarzschild horizon [cf. Eqs. (5.104) and (5.105)].
Appendix E. Symbol summary The various metrics and associated connections (intrinsic geometries) used in this article are collected in Table 3, whereas the symbols used to describe the extrinsic geometries of the various submanifolds are collected in Table 4.
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E. Gourgoulhon, J.L. Jaramillo / Physics Reports 423 (2006) 159 – 294
Table 3 Metric tensors and associated connections used in this article Manifold
Metric
Signature
M H t t St St
g q ˜ q q˜
(−, +, +, +) (0, +, +) (+, +, +) (+, +, +) (+, +) (+, +)
HNEH
q
(0, +, +)
Compatible connection
∇
×
D ˜ D 2D 2D ˜ ˆ ∇
Ricci tensor
R ×
3R 3R ˜ 2R 2R ˜
Not used
The symbol ‘×’ in the second line means that there is not a unique connection on H compatible with q, for the latter is degenerate. The last line regards the particular case of a non-expanding horizon. Table 4 Extrinsic geometry of various submanifolds of M Submanifold of M
Projector onto it
Second fundamental form(s)/embedding manifold
Normal vector(s)
H t St St St St
q
q
q
q
/(M, g, l) K/(M, g) H/(t , ) ˜ /(t , ˜ ) H (, )/(M, g, l) ( q∗ K, H)/(M, g)
l n s s˜ ( l , k) ( n, s)
H NEH
0/(M, g)
l
Note that the projectors and q are orthogonal ones with respect to the ambient metric g, whereas is not the notation “/(M, g, l)” means that the second fundamental form depends not only on the ambient geometry (M, g), but also on the choice of the null normal l. The pairs (, ) and ( q∗ K, H) are required for the embedding of the 2-surfaces St in M because the correct object to describe such an two-dimensional embedding is not a single bilinear form but a type ( 21 ) tensor [see Eqs. (5.83) and (10.45)]. The last line regards the particular case of a non-expanding horizon.
References [1] R. Abraham, J. Marsden, Foundations of Mechanics, Addison Wesley, Reading, MA, 1978. [2] F. Acernese et al. (VIRGO Collaboration), Status of VIRGO, Class. Quantum Grav. 22 (2005) S869. [3] M. Alcubierre, The status of numerical relativity, in: Proceedings of the 17th International Conference on General Relativity and Gravitation (GR17), in press (preprint gr-qc/0412019). [4] M. Alcubierre, B. Bruegmann, Simple excision of a black hole in 3 + 1 numerical relativity, Phys. Rev. D 63 (2001) 104006. [5] M. Alcubierre, B. Bruegmann, D. Pollney, E. Seidel, R. Takahashi, Black hole excision for dynamic black holes, Phys. Rev. D 64 (2001) 061501. [6] TAMA Collaboration. Stable operation of a 300-m laser interferometer with sufficient sensitivity to detect gravitational-wave events within our galaxy, Phys. Rev. Lett. 86 (2001) 3950. [7] M. Ansorg, Double-domain spectral method for black hole excision data, Phys. Rev. D 72 (2005) 024018. [8] V. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1989. [9] R. Arnowitt, S. Deser, C.W. Misner, The dynamics of general relativity, in: Louis Witten (Ed.), Gravitation: An Introduction to Current Research, Wiley, New York, 1962, pp. 227–265, gr-qc/0405109 (Chapter 7). [10] A. Ashtekar, C. Beetle, S. Fairhurst, Isolated horizons: a generalization of black hole mechanics, Class. Quantum Grav. 16 (1999) L1. [11] A. Ashtekar, C. Beetle, O. Dreyer, S. Fairhurst, B. Krishnan, J. Lewandowski, J. Wi´sniewski, Generic isolated horizons and their applications, Phys. Rev. Lett. 85 (2000) 3564. [12] A. Ashtekar, C. Beetle, J. Lewandowski, Mechanics of rotating isolated horizons, Phys. Rev. D 64 (2001) 044016. [13] A. Ashtekar, C. Beetle, J. Lewandowski, Geometry of generic isolated horizons, Class. Quantum Grav. 19 (2002) 1195. [14] A. Ashtekar, J. Engle, T. Pawlowski, C. Van Den Broeck, Multipole moments of isolated horizons, Class. Quantum Grav. 21 (2004) 2549. [15] A. Ashtekar, S. Fairhurst, B. Krishnan, Isolated horizons: Hamiltonian evolution and the first law, Phys. Rev. D 62 (2000) 104025. [16] A. Ashtekar, B. Krishnan, Dynamical horizons: energy angular momentum fluxes and balance laws, Phys. Rev. Lett. 89 (2002) 261101. [17] A. Ashtekar, B. Krishnan, Dynamical horizons and their properties, Phys. Rev. D 68 (2003) 104030. [18] A. Ashtekar, B. Krishnan, Isolated and dynamical horizons and their applications, Living Rev. Relativity 7 (2004) 10 [Online article]: cited on 16 October 2005, http://www.livingreviews.org/lrr-2004-10.
E. Gourgoulhon, J.L. Jaramillo / Physics Reports 423 (2006) 159 – 294
291
[19] L. Baiotti, I. Hawke, P.J. Montero, F. Löffler, L. Rezzolla, N. Stergioulas, J.A. Font, E. Seidel, Three-dimensional relativistic simulations of rotating neutron-star collapse to a Kerr black hole, Phys. Rev. D 71 (2005) 024035. [20] J.W. Bardeen, B. Carter, S.W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161. [21] C. Barrabès, P.A. Hogan, Lightlike signals in general relativity and cosmology, Phys. Rev. D 58 (1998) 044013. [22] R. Bartnik, J. Isenberg, The constraint equations, in: P.T. Chru´sciel, H. Friedrich (Eds.), The Einstein Equations and the Large Scale Behavior of Gravitational Fields, Birkhäuser, Basel, 2004, p. 1. [23] T.W. Baumgarte, G.B. Cook, M.A. Scheel, S.L. Shapiro, S.A. Teukolsky, General relativistic models of binary neutron stars in quasiequilibrium, Phys. Rev. D 57 (1998) 7299. [24] T.W. Baumgarte, S.L. Shapiro, Numerical integration of Einstein’s field equations, Phys. Rev. D 59 (1999) 024007. [25] T.W. Baumgarte, S.L. Shapiro, Numerical relativity and compact binaries, Phys. Rep. 376 (2003) 41. [26] M. Berger, B. Gostiaux, Géométrie différentielle: variétés, courbes et surfaces, Presses Universitaires de France, Paris, 1987. [27] F. Beyer, B. Krishnan, E. Schnetter, in preparation. [28] L. Blanchet, Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries, Living Rev. Relativity 5 (2002) 3 [Online article]: cited on 16 October 2005, http://www.livingreviews.org/lrr-2002-3. [29] S. Bonazzola, E. Gourgoulhon, P. Grandclément, J. Novak, Constrained scheme for the Einstein equations based on the Dirac gauge and spherical coordinates, Phys. Rev. D 70 (2004) 104007. [30] I. Booth, Metric-based Hamiltonians, null boundaries and isolated horizons, Class. Quantum Grav. 18 (2001) 4239. [31] I. Booth, Black hole boundaries, Canadian J. Phys., in press [preprint gr-qc/0508107]. [32] I. Booth, L. Brits, J. A. Gonzalez, C. Van Den Broeck, Marginally Trapped Tubes and Dynamical Horizons, preprint gr-qc/0506119. [33] I. Booth, S. Fairhurst, The first law for slowly evolving horizons, Phys. Rev. Lett. 92 (2004) 011102. [34] J.M. Bowen, J.W. York, Time-asymmetric initial data for black holes and black-hole collisions, Phys. Rev. D 21 (1980) 2047. [35] J.D. Brown, J.W. York, Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407. [36] B. Carter, Killing horizons and orthogonally transitive groups in space–time, J. Math. Phys. 10 (1969) 70. [37] B. Carter, General theory of stationary black hole states, in: C. DeWitt, B.S. DeWitt (Eds.), Black Holes—Les Houches 1972, Gordon & Breach Science Publishers, New York, 1973, p. 125. [38] B. Carter, Perfect fluid and magnetic field conservations laws in the theory of black hole accretion rings, in: C. Hazard, S. Mitton (Eds.), Active Galactic Nuclei, Cambridge University Press, Cambridge, 1979, p. 273. [39] B. Carter, The general theory of mechanical, electromagnetic and thermodynamic properties of black holes, in: S.W. Hawking, W. Israel (Eds.), General Relativity: An Einstein Centenary Survey, Cambridge University Press, Cambridge, 1979, p. 294. [40] B. Carter, Mathematical foundations of the theory of relativistic stellar and black hole configurations, in: B. Carter, J.B. Hartle (Eds.), Gravitation in Astrophysics, Plenum Press, New York, 1987. [41] B. Carter, Outer curvature and conformal geometry of an imbedding, J. Geom. Phys. 8 (1992) 53. [42] B. Carter, Basic brane theory, Class. Quantum Grav. 9 (1992) S19. [43] B. Carter, Extended tensorial curvature analysis for embeddings and foliations, Contemp. Math. 203 (1997) 207. [44] S.A. Caveny, M. Anderson, R.A. Matzner, Tracking black holes in numerical relativity, Phys. Rev. D 68 (2003) 104009. [45] S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford University Press, Oxford, 1992. [46] Y. Choquet-Bruhat, C. Dewitt-Morette, M. Dillard-Bleick, Analysis Manifolds and Physics, North-Holland, Amsterdam, 1977. [47] P.T. Chru´sciel, On the global structure of Robinson–Trautman space-times, Proc. Roy. Soc. London A 436 (1992) 299. [48] P.T. Chru´sciel, Black holes, in: J. Frauendiener, H. Friedrich (Eds.), Proceedings of the Tübingen Conference on Conformal Structure of Space-time, Lecture Notes in Physics, vol. 604, Springer, Berlin, 2002, p. 61. [49] G.B. Cook, Initial data for axisymmetric black-hole collisions, Phys. Rev. D 44 (1991) 2983. [50] G.B. Cook, Three-dimensional initial data for the collision of two black holes. II. Quasicircular orbits for equal-mass black holes, Phys. Rev. D 50 (1994) 5025. [51] G.B. Cook, Initial data for numerical relativity, Living Rev. Relativity 3 (2000) 5 [Online article]: cited on 16 October 2005, http://www.livingreviews.org/lrr-2000-5. [52] G.B. Cook, Corotating and irrotational binary black holes in quasicircular orbits, Phys. Rev. D 65 (2002) 084003. [53] G.B. Cook, M.W. Choptuik, M.R. Dubal, S. Klasky, R.A. Matzner, S.R. Oliveira, Three-dimensional initial data for the collision of two black holes, Phys. Rev. D 47 (1993) 1471. [54] G.B. Cook, H. Pfeiffer, Excision boundary conditions for black-hole initial data, Phys. Rev. D 70 (2004) 104016. [55] G.B. Cook, J.W. York, Apparent horizons for boosted or spinning black holes, Phys. Rev. D 41 (1990) 1077. [56] C. Crnkovi´c, E. Witten, Covariant description of canonical formalism in geometrical theories, in: S.W. Hawking, W. Israel (Eds.), Three Hundred Years of Gravitation, Cambridge University Press, Cambridge, 1987. [57] S. Dain, Trapped surfaces as boundaries for the constraint equations, Class. Quantum Grav. 21 (2004) 555. [58] S. Dain, J.L. Jaramillo, B. Krishnan, Existence of initial data containing isolated black holes, Phys. Rev. D 71 (2005) 064003. [59] T. Damour, Black-hole eddy currents, Phys. Rev. D 18 (1978) 3598. [60] T. Damour, Quelques propriétés mécaniques, électromagnétiques, thermodynamiques et quantiques des trous noirs, Thèse de doctorat d’État, Université Paris 6, 1979. [61] T. Damour, Surface effects in black hole physics, in: R. Ruffini (Ed.), Proceedings of the Second Marcel Grossmann Meeting on General Relativity, North-Holland, Amsterdam, 1982, p. 587. [62] M. Demia´nski, J.P. Lasota, Black holes in an expanding Universe, Nature Phys. Sci. 241 (1973) 53. [63] P. Diener, A new general purpose event horizon finder for 3D numerical spacetimes, Class. Quantum Grav. 20 (2003) 4901. [64] P. Diener, N. Jansen, A. Khokhlov, I. Novikov, Adaptive mesh refinement approach to the construction of initial data for black hole collisions, Class. Quantum Grav. 17 (2000) 435.
292 [65] [66] [67] [68]
E. Gourgoulhon, J.L. Jaramillo / Physics Reports 423 (2006) 159 – 294
O. Dreyer, B. Krishnan, D. Shoemaker, E. Schnetter, Introduction to isolated horizons in numerical relativity, Phys. Rev. D 67 (2003) 024018. D.M. Eardley, Black hole boundary conditions and coordinate conditions, Phys. Rev. D 57 (1998) 2299. R.J. Epp, Angular momentum and an invariant quasilocal energy in general relativity, Phys. Rev. D 62 (2000) 124018. A.E. Fischer, J.E. Marsden, The initial value problem and the dynamical formulation of general relativity, in: S. Hawking, W. Israel (Eds.), General Relativity: An Einstein Centenary Survey, Cambridge University Press, Cambridge, 1979. [69] S.J. Fletcher, A.W.C. Lun, The Kerr spacetime in generalized Bondi–Sachs coordinates, Class. Quantum Grav. 20 (2003) 4153. [70] Y. Fourès-Bruhat (Y. Choquet-Bruhat), Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Mathematica 88 (1952) 141. [71] J.L. Friedman, K. Uryu, M. Shibata, Thermodynamics of binary black holes and neutron stars, Phys. Rev. D 65 (2002) 064035. [72] G.J. Galloway, Maximum principles for null hypersurfaces and null splitting theorems, Ann. Henri Poincaré 16 (2000) 543. [73] G.J. Galloway, Null geometry and the Einstein equations, in: P.T. Chru´sciel, H. Friedrich (Eds.), The Einstein Equations and the Large Scale Behavior of Gravitational Fields, Birkhäuser, Basel, 2004, p. 379. [74] R. Geroch, J.B. Hartle, Distorted black holes, J. Math. Phys. 23 (1982) 680. [75] G. González, Status of LIGO data analysis, Class. Quantum Grav. 21 (2004) S1575. [76] E. Gourgoulhon, Generalized Damour–Navier–Stokes equation applied to trapping horizons, Phys. Rev. D 72 (2005) 104007. [77] E. Gourgoulhon, P. Grandclément, S. Bonazzola, Binary black holes in circular orbits. I. A global spacetime approach, Phys. Rev. D 65 (2002) 044020. [78] E. Gourgoulhon, P. Grandclément, K. Taniguchi, J.-A. Marck, S. Bonazzola, Quasiequilibrium sequences of synchronized and irrotational binary neutron stars in general relativity: method and tests, Phys. Rev. D 63 (2001) 064029. [79] P. Grandclément, E. Gourgoulhon, S. Bonazzola, Binary black holes in circular orbits. II. Numerical methods and first results, Phys. Rev. D 65 (2002) 044021. [80] V. Guillemin, S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1984. [81] C. Gundlach, Pseudospectral apparent horizon finders: an efficient new algorithm, Phys. Rev. D 57 (1998) 863. [82] P. Há´jiˇcek, Exact models of charged black holes. I. Geometry of totally geodesic null hypersurface, Commun. Math. Phys. 34 (1973) 37. [83] P. Há´"iˇcek, Can outside fields destroy black holes?, J. Math. Phys. 15 (1974) 1554. [84] P. Há´"iˇcek, Stationary electrovacuum spacetimes with bifurcate horizons, J. Math. Phys. 16 (1975) 518. [85] J.B. Hartle, Tidal friction in slowly rotating black holes, Phys. Rev. D 8 (1973) 1010. [86] J.B. Hartle, Tidal shapes and shifts on rotating black holes, Phys. Rev. D 9 (1974) 2749. [87] S.W. Hawking, Gravitational radiation from colliding black holes, Phys. Rev. Lett. 26 (1971) 1344. [88] S.W. Hawking, Black holes in general relativity, Commun. Math. Phys. 25 (1973) 152. [89] S.W. Hawking, The event horizon, in: C. DeWitt, B.S. DeWitt (Eds.), Black Holes, les astres occlus, Gordon and Breach, New York, 1973. [90] S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Spacetime, Cambridge University Press, Cambridge, 1973. [91] S.W. Hawking, J.B. Hartle, Energy and angular momentum flow into a black hole, Commun. Math. Phys. 27 (1972) 283. [92] S.W. Hawking, G.T. Horowitz, The gravitational Hamiltonian, action, entropy and surface terms, Class. Quantum Grav. 13 (1996) 1487. [93] S.A. Hayward, General laws of black hole dynamics, Phys. Rev. D 49 (1994) 6467. [94] S.A. Hayward, Spin-coefficient form of the new laws of black-hole dynamics, Class. Quantum Grav. 11 (1994) 3025. [95] S.A. Hayward, Energy and entropy conservation for dynamical black holes, Phys. Rev. D 70 (2004) 104027. [96] R.A. d’Inverno, 2 + 2 formalism and applications, in: J.-A. Marck, J.-P. Lasota (Eds.), Relativistic Gravitation and Gravitational Radiation, Cambridge University Press, Cambridge, 1997, p. 191. [97] R.A. d’Inverno, Combining Cauchy and characteristic codes in numerical relativity, in: J.-A. Marck, J.-P. Lasota (Eds.), Relativistic Gravitation and Gravitational Radiation, Cambridge University Press, Cambridge, 1997, p. 213. [98] R.A. d’Inverno, J. Smallwood, Covariant 2 + 2 formulation of the initial-value problem in general relativity, Phys. Rev. D 22 (1980) 1233. [99] J.L. Jaramillo, E. Gourgoulhon, G.A. Mena Marugán, Inner boundary conditions for black hole initial data derived from isolated horizons, Phys. Rev. D 70 (2004) 124036. [100] J.L. Jaramillo, F. Limousin, M. Ansorg, Numerical Implementation of Isolated Horizon Boundary Conditions, in preparation. [101] J. Jezierski, Geometry of null hypersurfaces, in: I. Racz, (Ed.), Relativity Today (Proceedings of the 7th Hungarian Relativity Workshop, 2003), Akadémiai Kiadó, Budapest (2004) [preprint gr-qc/0405108]. [102] J. Jezierski, J. Kijowski, E. Czuchry, Geometry of null-like surfaces in general relativity and its application to dynamics of gravitating matter, Rep. Math. Phys. 46 (2000) 399. [103] J. Jezierski, J. Kijowski, E. Czuchry, Dynamics of a self gravitating light-like matter shell: a gauge-invariant Lagrangian and Hamiltonian description, Phys. Rev. D 65 (2002) 064036. [104] R.P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11 (1963) 237. [105] A.R. King, J.P. Lasota, W. Kundt, Black hole and magnetic fields, Phys. Rev. D 12 (1975) 3037. [106] A. Komar, Covariant conservation laws in general relativity, Phys. Rev. 113 (1959) 934. [107] M. Korzynski, J. Lewandowski, T. Pawlowski, Mechanics of multidimensional isolated horizons, Class. Quantum Grav. 22 (2005) 2001. [108] B. Krishnan, Isolated horizons in numerical relativity. Ph.D. Thesis, Pennsylvania State University, 2002. [109] D.N. Kupeli, On null submanifolds in spacetimes, Geom. Dedicata 23 (1987) 33. [110] J. Lewandowski, Spacetimes admitting isolated horizons, Class. Quantum Grav. 17 (2000) L53–L59. [111] J. Lewandowski, T. Pawlowski, Quasi-local rotating black holes in higher dimension: geometry, Class. Quantum Grav. 22 (2005) 1573. [112] J. Libson, J. Mass, E. Seidel, W.-M. Suen, P. Walker, Event horizons in numerical relativity: methods and tests, Phys. Rev. D 53 (1996) 4335. [113] A. Lichnerowicz, L’intégration des équations de la gravitation relativiste et le problème des n corps, J. Math. Pures Appl. 23 (1944) 37 (reprinted in A. Lichnerowicz, Choix d’Wuvres mathématiques, Hermann, Paris, 1982).
E. Gourgoulhon, J.L. Jaramillo / Physics Reports 423 (2006) 159 – 294 [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163]
293
R. W Lindquist, Initial-value problem on Einstein–Rosen manifolds, J. Math. Phys. 4 (1963) 938. Proceedings of the 5th International LISA Symposium, held in Noordwijk (12–15 July 2004), Class. Quantum Grav. 22 (2005) S125–S537. C.-C.M. Liu, S.-T. Yau, Positivity of Quasilocal Mass, Phys. Rev. Lett. 90 (2003) 231102. P. Marronetti, G.J. Mathews, J.R. Wilson, Irrotational binary neutron stars in quasiequilibrium, Phys. Rev. D 60 (1999) 087301. M. Mars, J.M.M. Senovilla, Geometry of general hypersurfaces in spacetime: junction conditions, Class. Quantum Grav. 10 (1993) 1865. J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry, Springer, Berlin, 1999. D. Maxwell, Solutions of the Einstein constraint equations with apparent horizon boundary, Commun. Math. Phys. 253 (2004) 561. J.E. McClintock, R.A. Remillard, Black hole binaries, in: W.H.G. Lewin, M. van der Klis (Eds.), Compact Stellar X-ray Sources, Cambridge University Press, Cambridge, in press [preprint astro-ph/0306213]. C.W. Misner, The method of images in geometrostatics, Ann. Phys. (N.Y.) 24 (1963) 102. C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, Freeman, New York, 1973. M. Nakahara, Geometry, Topology and Physics, second ed., IOP Publishing Ltd., London, 2003. T. Nakamura, K. Oohara, Y. Kojima, General relativistic collapse to black holes and gravitational waves from black holes, Prog. Theor. Phys. 90 (Suppl.) (1987) 1. R. Narayan, Black holes in astrophysics, N. J. Phys. 7 (2005) 199. E. Newman, R. Penrose, An approach to gravitational radiation by a method of spin coefficients, J. Math. Phys. 3 (1962) 566 (errata in J. Math. Phys. 4 (1963) 998). P. Nurowski, D.C. Robinson, Intrinsic geometry of a null hypersurface, Class. Quantum Grav. 17 (2000) 4065–4084. N. Ó Murchadha, J.W. York, Initial-value problem of general relativity. I. General formulation and physical interpretation, Phys. Rev. D 10 (1974) 428. B. O’Neill, The Geometry of Kerr Black Holes, A.K. Peters, Wellesley, 1995. T. Padmanabhan, Gravity and the thermodynamics of horizons, Phys. Rep. 406 (2005) 49. R. Penrose, Gravitational collapse and spacetime singularities, Phys. Rev. Lett. 14 (1965) 57. H.P. Pfeiffer, Initial data for black hole evolutions, Ph.D. Thesis, Cornell University, 2003; gr-qc/0510016. H.P. Pfeiffer, The initial value problem in numerical relativity, in: Proceedings of the Miami Waves Conference 2004, in press [preprint gr-qc/0412002]. H.P. Pfeiffer, S.A. Teukolsky, G.B. Cook, Quasicircular orbits for spinning binary black holes, Phys. Rev. D 62 (2000) 104018. H.P. Pfeiffer, J.W. York, Extrinsic curvature and the Einstein constraints, Phys. Rev. D 67 (2003) 044022. H.P. Pfeiffer, J.W. York, Uniqueness and Nonuniqueness in the Einstein Constraints, Phys. Rev. Lett. 95 (2005) 091101. E. Poisson, A Relativist’s Toolkit, Cambridge University Press, Cambridge, 2004. E. Poisson, The gravitational self-force, in: Proceedings of the 17th International Conference on General Relativity and Gravitation (GR17), in press [preprint gr-qc/0410127]. R.H. Price, K.S. Thorne, Membrane viewpoint on black holes: properties and evolution of the stretched horizon, Phys. Rev. D 33 (1986) 915. M.J. Rees, Black holes in the real universe and their prospects as probes of relativistic gravity, in: G.W. Gibbons, E.P.S. Shellard, S.J. Rankin (Eds.), Future of Theoretical Physics and Cosmology, Cambridge University Press, Cambridge, 2003, p. 217. R.K. Sachs, Gravitational waves in general relativity: VIII. Waves in asymptotically flat space-time, Proc. Roy. Soc. London A 270 (1962) 103. G. Schäfer, A. Gopakumar, Minimal no-radiation approximation to Einstein’s field equations, Phys. Rev. D 69 (2004) 021501. E. Schnetter, Finding apparent horizons and other 2-surfaces of constant expansion, Class. Quantum Grav. 20 (2003) 4719. E. Schnetter, F. Herrmann, D. Pollney, Horizon pretracking, Phys. Rev. D 71 (2005) 044033. E. Seidel, W.-M. Suen, Towards a singularity-proof scheme in numerical relativity, Phys. Rev. Lett. 69 (1992) 1845. J.M.M. Senovilla, On the existence of horizons in spacetimes with vanishing curvature invariants, JHEP 0311 (2003) 046. J.M.M. Senovilla, Trapped submanifolds in Lorentzian geometry, in: Proceedings of the XIII Fall Workshop on Geometry and Physics (Murcia, Spain, 2004) [preprint math.DG/0412256]. M. Shibata, T. Nakamura, Evolution of three-dimensional gravitational waves: harmonic slicing case, Phys. Rev. D 52 (1995) 5428. M. Shibata, K. Uryu, J.L. Friedman, Deriving formulations for numerical computation of binary neutron stars in quasicircular orbits, Phys. Rev. D 70 (2004) 044044 (Erratum in Phys. Rev. D 70, 129901(E) (2004)). L. Smarr, J.W. York, Kinematical conditions in the construction of spacetime, Phys. Rev. D 17 (1978) 2529. J.R. Smith, et al., Commissioning, characterization and operation of the dual-recycled GEO 600, Class. Quantum Grav. 21 (2004) S1737. H. Stephani, General Relativity. An Introduction to the Theory of the Gravitational Field, Cambridge University Press, Cambridge, 1990. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact Solutions of Einstein’s Field Equations, Cambridge University Press, Cambridge, 2003. J. Stewart, Advanced General Relativity, Cambridge University Press, Cambridge, 1990. N. Straumann, General Relativity and Relativistic Astrophysics, Springer, Berlin, 1984. L.B. Szabados, Quasi-local energy–momentum and angular momentum in GR: a review article, Living Rev. Relativity 7 (2004) [Online article]: cited on 16 October 2004, http://www.livingreviews.org/lrr-2004-4. P. Szekeres, The gravitational compass, J. Math. Phys. 6 (1965) 1387. J. Thornburg, Coordinates and boundary conditions for the general relativistic initial data problem, Class. Quantum Grav. 4 (1987) 1119. J. Thornburg, Numerical relativity in black hole spacetimes, Ph.D. Thesis, University of British Columbia (1993); available at http://www.aei.mpg.de/jthorn/phd/html/phd.html J. Thornburg, Finding apparent horizons in numerical relativity, Phys. Rev. D 54 (1996) 4899. J. Thornburg, A fast apparent horizon finder for three-dimensional Cartesian grids in numerical relativity, Class. Quantum Grav. 21 (2004) 743.
294
E. Gourgoulhon, J.L. Jaramillo / Physics Reports 423 (2006) 159 – 294
[164] K.S. Thorne, Disk-accretion onto a black hole. II. Evolution of the hole, Astrophys. J. 191 (1974) 507. [165] K.S. Thorne, R.H. Price, D.A. MacDonald, Black Holes: the Membrane Paradigm, Yale University Press, New Haven, 1986. [166] K. Uryu, Y. Eriguchi, New numerical method for constructing quasiequilibrium sequences of irrotational binary neutron stars in general relativity, Phys. Rev. D 61 (2000) 124023. [167] R.M. Wald, General Relativity, University of Chicago Press, Chicago, 1984. [168] R.M. Wald, The thermodynamics of black holes, Living Rev. Relativity 4 (2001) 6 [Online article]: cited on 16 October 2005, http://www.livingreviews.org/lrr-2001-6. [169] J. Winicour, Characteristic evolution and matching, Living Rev. Relativity 4 (2001) 3 [Online article]: cited on 16 October 2005, http://www.livingreviews.org/lrr-2001-3. [170] J.W. York, Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial-value problem of general relativity, J. Math. Phys. 14 (1973) 456. [171] J.W. York, Kinematics and dynamics of general relativity, in: L.L. Smarr (Ed.), Sources of Gravitational Radiation, Cambridge University Press, Cambridge, 1979, p. 83. [172] J.W. York, Initial data for collisions of black holes and other gravitational miscellany, in: C.R. Evans, L.S. Finn, D.W. Hobill (Eds.), Frontiers in Numerical Relativity, Cambridge University Press, Cambridge, 1989, p. 89. [173] J.W. York, Conformal “thin-sandwich” data for the initial-value problem of general relativity, Phys. Rev. Lett. 82 (1999) 1350. [174] J.W. York, Velocities and momenta in an extended elliptic form of the initial value conditions, in: G. Ferrarese, T. Ruggeri (Eds.), Analysis, Manifolds and Geometric Structures in Physics (Elba, 2004), Pitagora Editrice (Bologna) [preprint gr-qc/0409102], to appear. [175] R.L. Znajek, Black hole electrodynamics and the Carter tetrad, Mon. Not. R. Astron. Soc. 179 (1977) 457. [176] R.L. Znajek, Charged current loops around Kerr holes, Mon. Not. R. Astron. Soc. 182 (1978) 639.
Physics Reports 423 (2006) 295 – 338 www.elsevier.com/locate/physrep
Polarized radiation in Mössbauer spectroscopy Krzysztof Szyma´nski∗ Institute of Experimental Physics, University of Białystok, Lipowa 41 Street, 15-424 Białystok, Poland Accepted 28 October 2005 editor: J. Eichler
Abstract Mössbauer polarimetry is a spectroscopic technique sensitive to the orientations of hyperfine fields. The technique is particularly effective with monochromatic, polarized radiation: the measured spectra do not contain many additional transitions present when nonmonochromatic radiation is used. The paper reviews recent achievements in the construction of sources of polarized monochromatic radiation. Recently, filter techniques were adopted for achieving circularly and linearly polarized radiation from commercially available radioactive isotopes. A synchrotron source with nano-eV energy width, suitable for Mössbauer measurements was constructed. Applications are reviewed, in particular determination of the direction of the hyperfine magnetic field and the orientation of the electric field gradient. Special attention is paid to cases when the distributions of the hyperfine fields and mixed interactions result in poorly resolved spectra. Recent achievements in methodology are described. An explicit form of the intensity tensor is derived, which allows the transition probabilities to be calculated omitting the diagonalization of the Hamiltonian. The concept of the velocity moments is introduced. It is shown that some averages of the whole Mössbauer spectra relate to the averages of hyperfine fields and possess tensor properties. © 2005 Published by Elsevier B.V. PACS: 61.18.Fs; 75.25.+z; 75.50; 75.50.Bb; 76.60.Cq; 76.60.Jx; 76.80.+y; 23.20−g; 23.20.Lv Keywords: The Mössbauer polarimetry; Nuclear magnetometry; The intensity tensor; Spin structure
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 2. The intensity tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 2.1. Concept of the intensity tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 2.2. The spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 2.3. Construction of the intensity tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 2.4. Explicit formulas for the intensity tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 3. Single-site Mössbauer spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 4. The velocity moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 4.1. Explicit formulas for the velocity moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 4.2. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 5. Randomly oriented electric field gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 6. Ambiguity problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 ∗ Tel.: +48 85 7457221; fax: +48 85 7457223.
E-mail address:
[email protected]. 0370-1573/$ - see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.physrep.2005.10.010
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7. Perturbation theory—dominating magnetic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 8. The sources of polarized resonant radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 8.1. Multiline polarized sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 8.2. Polarization by thermal agitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 8.3. Filter techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 8.4. Monochromatic, polarized synchrotron sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 9. Model of the resonant filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 9.1. Resonant filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 9.2. Comparison with real systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 9.2.1. A filter based on 57 FeBO3 for linear polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 9.2.2. Filter based on Fe3 Si for circular polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 9.2.3. Comment on the comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 9.3. Effect of quadrupole splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 10. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 10.1. Transmission of the polarized radiation through thick absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 10.2. Sign of the hyperfine magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 10.3. The magnetic texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 10.4. On the hyperfine magnetic field distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 10.5. Sign of the dominant electric field gradient component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 10.6. Tensor properties of the velocity moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 11. Selected experimental problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 11.1. On the measurement of the degree of polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 11.2. The drive and the permanent magnet system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 12. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 A.1. Useful constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 A.2. Useful identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 A.3. Transmission integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 A.4. Resonant filter (some details) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
1. Introduction The hyperfine structure of nuclear levels and the angular dependence of the nuclear transition probability bear important information about the nuclear spin interactions with its surrounding, the multipolarity of the transitions, and the strength and orientation of the hyperfine fields. The Hamiltonian of nuclear spin exposed to a magnetic field and an electric field gradient (EFG) was analysed at the end of the 1940s [1–4]. These works were stimulated mainly by the discovery of the nuclear hyperfine fields [5,6] and the observation of the nuclear magnetic resonance in solids [7,8]. The intensity of the emitted radiation and its dependence on the orientation of the hyperfine field is determined by conservation of the angular momentum of nucleus and photon system. Hyperfine interactions remove the degeneracy of the nuclear levels. The discovery of the Mössbauer effect resulted in the development of a new tool for investigation of the nuclear transitions. Because of the unusual energetic resolution of the Mössbauer effect, separate nuclear transitions, line intensities, and polarization of the interacting photons can be observed. During early applications of the Mössbauer technique, large effort was already being undertaken in order to construct a monochromatic, polarized source of resonant radiation [9–12]. A detailed review covering work up to 1981 was made by Gonser and Fischer in [13]. We report progress in the realization and application of the sources of monochromatic, polarized radiation. We focus on the important case of the 14.4 keV transition in 57 Fe [14–16]. One important field covered by Mössbauer spectroscopy concerns investigation of the properties of materials. Thus, our considerations are restricted mainly to standard Mössbauer absorbers, in which thickness effects are small. The cases of thick absorbers are reviewed in [13]. The standard Mössbauer technique deals with incoherent emission processes, and information about the phase of the scattered photons is not available. In contrast, the use of coherent nuclear scattering of synchrotron radiation offers access to the amplitudes and phases as well. Therefore, nuclear resonant scattering has become a powerful tool when new generation synchrotron sources permitted to achieve meV-resolution of X-ray monochromators. There is unquestioned
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expansion of the Mössbauer synchrotron techniques covering investigations of lattice dynamics, thin layers, highpressure studies, etc. For an extended review, see the special issue of the Hyperfine Interactions 123/124 (1999). Although in the nuclear resonant scattering technique fully polarized coherent radiation is used, the determination of hyperfine field distributions, particularly when mixed magnetic dipolar and electric quadrupolar interactions are present, presents a still nontrivial task. Hopefully, synchrotron radiation can also be used for carrying out Mössbauer experiments in energy domain with fully polarized radiation [16]. The hyperfine fields, important characteristics of matter, relate to the transition probabilities of nuclear resonances and their energies. The transition amplitudes are usually expressed in terms of spherical harmonics and matrix elements between eigenstates of the nuclear spin Hamiltonian [17,18]. In the case of dipole transitions, the concept of the socalled intensity tensor was introduced [19] and developed. Recently, the explicit form of the intensity tensor has been obtained without any calculation of the Hamiltonian eigenstates [20]. The line intensities for the mixed interactions can be expressed explicitly as functions of the hyperfine fields. This analytical approach allows one to solve [20] the ambiguity problem, to evaluate the line shape of the spectra of powdered absorbers in the presence of mixed interaction [21], and to explain the symmetry present in the Mössbauer spectra under inversion of the sign of the EFG [20]. Using the explicit form of the intensity tensor one can show that some averages over the whole Mössbauer spectrum of thin absorber posses tensor properties [22]. These averages are simply the velocity moments of the absorption line intensities. The paper is organized as follows: we start with a brief introduction of the intensity tensor formalism. Next, we show how to get the intensity tensor components for circularly polarized radiation as explicit functions of the hyperfine fields. The spectrum of a single site and the concept of the velocity moments of the spectrum is discussed in Sections 3 and 4. Since formulas are linear in intensities, the results can be used in the analysis of the distributions of hyperfine parameters. The applications of the Mössbauer polarimetry in resolving the structure of hyperfine fields when mixed interactions are present are discussed. The shape of the spectra of an absorber with randomly oriented principal axes of the EFG in a homogeneous external magnetic field, the solution of the ambiguity problem and a special case of small quadrupole interactions (with respect to the magnetic ones) are presented in Sections 5–7. In Section 8, we describe experimental efforts towards the construction of the sources of polarized resonant radiation. Special attention is paid to the monochromatic sources of polarized radiation obtained by the filter technique. In Section 9 a model of the monochromatic, polarized source obtained in the resonant filter technique is presented. Section 10 is dedicated to applications. We focus on experimental determination of orientations of the hyperfine magnetic fields (h.m.f.) and EFGs in the cases of mixed interactions and the presence of hyperfine fields distribution. Some methodological information concerning measurement of the degree of polarization and construction of permanent magnets can be found in Section 11. Physical constants used in the paper and some useful mathematical relations are given in the appendix.
2. The intensity tensor 2.1. Concept of the intensity tensor In the 1960s, it was realized that the intensity of absorbed resonant radiation depends on the absorber orientation and reflects the symmetry of the hyperfine fields acting on the nucleus [23–25]. Then the formalism of the intensity tensor was introduced by Zimmermann [19], and developed in [13,26,27]. Since the intensity tensor formalism relates to the transition probabilities, it can be used for the thin absorbers or for the absorbers with well-separated absorption lines only. The theory of transmission of the polarized radiation through a resonantly absorbing medium of arbitrary thickness within the formalism of the density matrices was given in [13]. The intensity and the polarization state of the transmitted radiation through a medium with a known resonant absorption cross-section can be calculated for any thickness of the absorber. The density matrices of the nuclear transitions are constructed from the excited eigenstates, which has to be calculated numerically in the general case of mixed interactions. We start from a simple illustration of the concept of the intensity tensor in the case of circularly polarized radiation and an absorber with pure magnetic interaction. Nuclear spin I exposed to magnetic field splits into 2I + 1 sublevels. Transition probabilities ai between the excited Ie and the ground Ig states obey momentum and parity conservation
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m Ie =3/2
B
-3/2
ϑ
-1/2 +1/2
k
+3/2
velocity a3
Ig=1/2
a4
a6
+1/2
a1 a2
a5
-1/2
1 2 3 4 5 6 (a)
(b)
Fig. 1. (a) Zeeman splitting of the ground and excited nuclear levels in magnetic field B and (b) resulting Mössbauer absorption spectrum for circularly polarized radiation with the wavevector k.
rule. In the specific case of Ie = 23 , Ig = are proportional to [18]: a1 : a2 : a3 : a4 : a5 : a6 3 = 16 (1 ∓ cos ϑ)2 : 41 sin2 ϑ :
1 2
and M1 type of transition, transition probabilities ai in the Zeeman sextet
2 1 16 (1 ± cos ϑ)
:
2 1 16 (1 ∓ cos ϑ)
:
1 4
sin2 ϑ :
2 3 16 (1 ± cos ϑ)
,
(2.1)
where ϑ is an angle between the h.m.f. B and the photon wave vector k (Fig. 1). Upper and lower signs correspond to the two opposite circular polarizations. Each line intensity ai can be expressed as a certain tensor acting on the unit vector parallel to k. The tensor depends only on the hyperfine fields inside the absorber. Thus, the properties of the absorber can be separated from the properties of the photons in the spectra analysis. Indeed, let us consider the identities: (1 ± cos ϑ)2 = (1 ± m · )2 = Tr(1 − m ⊗ m) − · (1 − m ⊗ m) · ± 2m · , sin2 ϑ = 1 − (m · )2 = Tr(m ⊗ m) − · (m ⊗ m) · ,
(2.2)
where m and are unit vectors (m = B/B, = k/k), Tr is the trace of a matrix, 1 is the unit matrix and the ⊗ is the outer Cartesian product: (m ⊗ m)st = ms mt
(2.3)
with s, t = x, y, z. With the help of (2.2) each of the ai in (2.1) can be expressed as ai =
1 2
Tr Ii − 21 · Ii · − Gi · ,
(2.4)
where Ii is the symmetric part of the intensity tensor, while Gi is the pseudovector constructed from an antisymmetric part of the intensity tensor for ith absorption line (ith transition) of the single Fe site. For example I1 = 38 (1 − m ⊗ m),
G1 = ∓ 38 m .
(2.5)
All other components are listed in Table 1. The most important feature of the intensity tensor is that the spectrum of the thin absorber is linear in the intensity tensor components. Thus, in the case of any texture, one has to average the intensity tensor components over the texture function, then the line intensity for any orientation of the k vector with respect to the absorber frame can be obtained from the formula (2.4). To illustrate this, let us consider an absorber with magnetic texture (preferred h.m.f. orientation) P (), defined with respect to the Cartesian axes fixed to the absorber: P () d is the fraction of Fe atoms for which B is oriented between the angles and + d. See Fig. 2.
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Table 1 Components of the intensity tensor for pure magnetic interactions Line no. i
1
2
3
4
5
6
4 Tr I i ±8Ai 8Bi 8Ci
+3 −3 +3 −3
+2 0 0 +4
+1 +1 +1 −1
+1 −1 +1 −1
+2 0 0 +4
+3 +3 +3 −3
Ai , Bi and Ci coefficients are defined as follows: Gi = Ai · m, Ii = Bi · I + Ci m ⊗ m.
P (ϕ)
ϕ
Fig. 2. Example of two-dimensional angular distribution of h.m.f. and related texture function P ().
The function P () is normalized 2 P () d = d P () sin d = 1 . 0
(2.6)
0
By averaging the components of the intensity tensor given in Table 1 over a magnetic texture function P (), one gets the components of the intensity tensor of the textured absorber (in [26] called macroscopic intensity tensor), for example I1 = 38 (1 − m ⊗ m),
G1 = ∓ 38 m .
(2.7)
The brackets in (2.7) denote the angular averaging: for any function f (): f = f ()P () d .
(2.8)
The results (2.7), (2.4) and Table 1 show the possibility of experimental determination of the orientation of the average hyperfine field vector m. Indeed, from measured relative line intensities ai for a given orientation of the k (or ) vector of radiation (the probabilities ai are normalized: a1 + a2 + a3 + a4 + a5 + a6 = 1), we obtain m · = 43 (a1 − a6 ) .
(2.9)
Three experiments in which k vector is parallel to the x, y and z direction of the Cartesian system, respectively, yield the components mx , my and mz of the m vector, because of an obvious identity: ms = m · s, with s being the unit vector parallel to the s Cartesian axis (s = x, y, z). The line intensities ai determined for a textured absorber allow one to obtain another type of average: (m · )2 = a1 − a2 + a3 + a4 − a5 + a6 .
(2.10)
Three experiments with the aforementioned directions of k vector along x, y and z unit vectors yield m2x , m2y and m2z , respectively. In the case of an experiment with unpolarized radiation, the line intensities are correlated: a2 = a5 ,
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a1 = 3, a3 = 3, a4 = a6 ; thus relative intensity of lines 2 and 5 can be used for determination of the (m · )2 average. If one uses the common abbreviations a1 : a2 : a3 : a4 : a5 : a6 = 3 : z : 1 : 1 : z : 3, then (m · )2 =
4−z , 4+z
(2.11)
which is equivalent to (2.10). The averages m · and (m · )2 are characteristics of the angular distribution of h.m.f. in the absorber. Because of Schwartz’s inequality [28] we always have m · 2 (m · )2 .
(2.12)
If all B vectors have the same direction, (m · )2 = (m · )2 . Inequality (2.12) results in a relation between the line intensities [29]: |a3 − a4 | |a1 − a6 | 1 = 16 − z2 . a3 + a 4 a1 + a 6 4
(2.13)
Inequality (2.13) gives the upper limit of the asymmetry in line intensities 1, 6 (or 3, 4), typical for spectrum measured with circularly polarized radiation. The average m · is a component of the m vector in the direction. If we assume that, within a reasonable accuracy the atomic magnetic moment is proportional to the h.m.f., the m · · B gives element-selective information about the contribution of the element to the total magnetization. The average (m · )2 is a measure of the perpendicular to the component of the h.m.f.: (m · )2 = 0 means that the perpendicular component has achieved its maximum, while this component is zero when (m · )2 = 1. Preferred h.m.f. orientation, has been widely discussed in literature by use of a certain set of base functions, e.g. spherical harmonics Ylm [30,31]. Since only M1 dipolar transitions are measured in 57 Fe Mössbauer spectroscopy, unpolarized radiation delivers information on Y2m only, while other Y1m harmonics can be known when circularly polarized radiation is used. The knowledge of Y1m and Y2m components of the texture function P () is equivalent to the knowledge of averages m · and (m · )2 . Indeed, let us consider the contribution of the Y11 real spherical harmonic to P (), as an example: 3 P () sin cos d P ()Y11 d = − 4 3 3 = − P ()mx d = − m · x . (2.14) 4 4 We see that the contribution of the Y11 is proportional to m · with parallel to x. In a similar way one can show relation between contribution of Y1m , Y2m to P () and components ms , ms mt , where s, t = x, y, z [32]. In the presented introductory section we have seen how to use the intensity tensor formalism in attaining physically important information about angular distribution of the h.m.f. For purely magnetic interactions, the intensity tensor (Table 1) can be guessed from the angular dependences of the line intensities (2.1). A much more complicated situation takes place when the EFG is present as well. Moreover, Eqs. (2.9) and (2.10) usually cannot be directly applied to the measured spectra in the case of overlapping lines (the presence of hyperfine fields distribution) which usually prevent correct estimation of the intensities. These cases will be the subject of the next sections. We would like to demonstrate what kind of physical information about distributions of the hyperfine fields can be obtained when the polarization effects are employed in Mössbauer measurements. 2.2. The spin Hamiltonian Nuclear magnetic moment interact with externally applied magnetic field and with neighbouring electrons. For s and relativistic p1/2 electrons it is a contact interaction proportional to the unpaired electron spin density at the nucleus. Other electrons contribute to the magnetic dipolar interaction with the nucleus. The contact, the dipolar and the external magnetic field interaction can be regarded as the effective field B, acting on the nuclear magnetic moment.
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Nuclear electric quadrupolar moment interact with EFG. The EFG at the nuclear site arises from the local, low symmetry electron charge distribution in the atom and in the nearest neighbours. The EFG is a tensor of second rank; ij = −jEi /jxj are Cartesian components of the EFG tensor, where Ei is the component of the electric field acting on the nucleus. In the principal axes system (PAS) EFG is diagonal, and the axes can be chosen so, that |zz | |yy | |xx |. Because of Laplace equation zz + yy + xx = 0, diagonal elements are usually expressed by dominating component zz and so-called asymmetry parameter = (xx − xx )/zz . Diagonal elements of the EFG and orientation of the PAS system can be measured. Nuclear charge interact with electron density within the nucleus volume. This electrostatic interaction give rise to shifts of the nuclear levels. Experimentally observed shift (called the isomer shift) is proportional to the difference in the electron density at the nucleus site between absorber and emitter, and to the difference in nuclear radius between excited and ground states. Since nuclear properties are usually well known, interaction of the nucleus with electronic surrounding is used as an element selective probe for measuring local B, EFG and the isomer shift. We will focus on the determination of orientation of B and EFG. The Hamiltonian of the nuclear spin I in the hyperfine field B and the EFG written in the PAS of the EFG, see Fig. 3, is [33–35]: HI = −gI N I · B +
eQzz 2 3Iz − I2 + (I2+ + I2− ) , 4I (2I − 1) 2
(2.15)
where symbols are defined as follows: Q—nuclear quadrupole moment, gI —nuclear g-factor of the spin I , N —nuclear magneton, I± —I± = Ix ± iIy . The isomer shift was omitted in (2.15) for simplicity and will be introduced in Section 3. It is convenient to perform further calculations on the reduced form of the total 23 spin Hamiltonian H3/2 : H3/2 =
g3/2 N B H3/2 . 2
(2.16)
The H3/2 matrix elements in the |Ie , me basis of eigenfunctions of the z component of the I = operator are equal to [36]: ⎡
3/2
R − 3 cos ⎢ ⎢ √ ⎢ − 3 sin ei H3/2 = ⎢ ⎢ ⎢ R ⎢ √ ⎢ ⎢ 3 ⎣
1/2 √
− 3 sin ei −R − cos
−2 sin e−i
−2 sin e−i
−R + cos
R √ 3 where the symbols are defined as follows: 0
−1/2 R √ 3
√ − 3 sin ei
−3/2 0
⎤
⎥ ⎥ R ⎥ √ ⎥ ⎥ 3 ⎥ √ ⎥ i − 3 sin e ⎥ ⎥ ⎦ R + 3 cos
3 2
angular momentum
m 3/2 1/2 ,
(2.17)
−1/2 −3/2
R—the reduced dominant component of the EFG, R=eQzz /(2g3/2 N B). The R value is proportional to the commonly used ratio of the characteristic frequencies B = g3/2 N B/h¯ and E = eQzz /(12h) ¯ : R = 6E /B [16]. , —polar angles of the magnetic field B in the PAS system of the EFG, see Fig. 3. It is convenient to express the secular equation of the Hamiltonian (2.17) in a form which explicitly depends on the hyperfine interactions: the pseudovector B and the EFG : 4 + p 2 + q + r = 0 ,
(2.18)
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Fig. 3. Orientation of the h.m.f., m, and direction of the photon, , in the PAS of the EFG.
where coefficients (see also Appendix A.2): p = −10 −
4 3
Tr 2 ,
q = −16m · · m , r = 41 (p + 4)2 − 16m · 2 · m .
(2.19)
The reduced EFG in (2.19) = eQ/(2g3/2 N B); m and are unit vectors introduced already in Section 2.1. The secular equation (2.18) is independent of the coordination system used and the coefficients p, q and r depend explicitly on the hyperfine fields B and . Thus, all further results will be explicit functions of the hyperfine fields only. The roots of Eq. (2.18), , are proportional to the energies E of the excited levels: = 2E /(g3/2 N B). The explicit form of can be found in [36], while the numerical values of the physical constants used in the text are given in the Appendix A.1. 2.3. Construction of the intensity tensor We present briefly the construction of the intensity tensor for circularly polarized radiation following the ideas [26,27]. Let the |Ie , me and |Ig , mg be the eigenstates of the operator Iz for the excited Ie = 23 and the ground Ig = 21 spin, respectively. The excited and the ground eigenstates of the spin Hamiltonian (2.15) are linear combinations:
|e = me em |I , me and |g = mg gmg |Ig , mg , respectively. Let VM be the spherical component of the vector e e describing nuclear transition from the excited to the ground state:
∗ √ Ig L Ie
Ig −L+me VM = 2L + 1 . (2.20) em g (−1) e mg mg M −me me mg
The last expression in the parenthesis is Wigner’s 3j symbol. The star * is the complex conjugate. Any Cartesian vector V = i vi ei can be expressed in the spherical basis: V = i bi u∗i , where spherical basis vectors are defined as: u± = 2−1/2 (∓ex − iey ) and u0 = ez [37]. Spherical components of the intensity tensor for the transition are constructed in the following: ∗
ij I˜ = Vi Vj .
(2.21)
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The intensity tensor I˜ contains both a symmetric and an antisymmetric part, which will be treated separately. The symmetric part of the intensity tensor, I , is defined as ∗ I = (I˜ + I˜ )/2 .
(2.22)
The G vector is constructed from the antisymmetric part of the intensity tensor and its Cartesian components are given by jk Gi = −i ij k I˜ /2 ,
(2.23)
where ij k is the Levi–Civita symbol [28]. 2.4. Explicit formulas for the intensity tensor The intensity tensor (2.22), (2.23) depends on the eigenstates of the I = 23 spin Hamiltonian (2.17). Examples of eigenstates were given in [36,41]; however, their complicated form, probably not fully simplified, prevents practical use. In the special case of axially symmetric EFG ( = 0), eigenfunctions were derived in [16]. To our knowledge there have been no successful attempts of the use of eigenstates resulting in finding an explicit dependence of the intensity tensor on hyperfine fields in the case of mixed interactions. It has been shown in [20] that the tensor components (2.22), (2.23) and powers of the eigenvalues (2.18) form a set of independent linear equations. Solution of these equations results in explicit dependence of the intensity tensor on the hyperfine fields. The linear set can be obtained in the following. Any component of the intensity tensor, Eqs. (2.22) and (2.23), is a bilinear form of the excited and ground states. Thus, each component can be regarded as an operator constructed from ground states acting on excited states |e . For example, the trace of I can be written as Tr I = e |T |e ,
(2.24)
where T is a Hermitian operator, which in the |Ie , me basis is √ ⎡ 2 − 3sc e−i 0 −(3c √ − i1) ⎢ − 3sc e 1 −c2 −2sc e−i T = 1+ ⎣ 2 0 −2sc ei 2 4 √−s i 0 0 − 3sc e
⎤ 0 ⎥ √ 0 ⎦ . − 3sc e−i −(3s 2 − 1)
(2.25)
Index runs over two values: = +1 corresponding to the higher and = −1 to the lower energy in the ground state; s = sin /2, and c = cos /2. Consider an operator O acting on the eigenstates |e . For the eigenvalues E of the Hamiltonian H3/2 , and any power n we have the identity:
E n e |O|e = e |OHn3/2 |e = Tr OHn3/2 . (2.26)
Thus, evaluation of the left-hand sum requires summation of the diagonal elements of the operator OHn3/2 . In our example, using operator (4.2) as O, for n = 0, 1, 2 and 3 one gets
Tr I = 23 , (2.27)
Tr I = 25 ,
(2.28)
2 Tr I = − 41 (3p + q) ,
(2.29)
3 Tr I =
(2.30)
1 16 (−18q
+ (64 − 20p + p 2 − 4r)) .
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Eqs. (2.27)–(2.30) are all of the type x = y; the matrix contains only powers of the excited energies of the secular equation (2.18): ⎤ ⎡ ⎡ ⎤ 1 1 1 1 Tr I1 ⎢ 1 2 3 4 ⎥ ⎢ Tr I2 ⎥ ⎥ (2.31) =⎢ ⎣ 21 22 23 24 ⎦ , x = ⎣ Tr I3 ⎦ . Tr I4 3 3 3 3 1
2
3
4
The determinant of the matrix is known as the Vandermonde determinant and is equal to Det = ( 2 − 1 )( 1 − 3 )( 2 − 3 )( 1 − 4 )( 2 − 4 )( 4 − 3 ) . Because are eigenvalues of the traceless Hamiltonian (2.17),
= 0
(2.32)
(2.33)
and the inverse matrix can be reduced to ⎡ −r 21 + p 1 , , , ⎢ 1 w1 w1 w1 ⎢ 2 ⎢ −r 2 + p 2 ⎢ , , ⎢ w , w w −1 2 2 2 2 =⎢ ⎢ −r 23 + p 3 ⎢ , , , ⎢ w w w ⎢ 3 3 3 3 ⎣ 2 −r 3 + p 4 , , , 4 w4 w4 w4
⎤ 1 w1 ⎥ ⎥ 1 ⎥ ⎥ w2 ⎥ ⎥ , 1 ⎥ ⎥ ⎥ w3 ⎥ ⎦ 1 w4
(2.34)
where p = − 21 q
= − 13
2 , 3 ,
r = 1 2 3 4 , w = 4 3 + 2p + q .
(2.35)
The parameters p, q and r in (2.35) are the same as those in (2.19) because of Vieta’s formulas [38]. The solution of our exemplary linear problem (2.27)–(2.30) for the trace of the intensity tensor as well as for the other intensity tensor components (2.22), (2.23) is [20]: Tr I =
40 2 − 4q + (p + 4)(p + 16) − 4r 3 + , 8 16w
(2.36)
G = −
1 (162 + k2 + k1 1) · m , 8w
(2.37)
I =
1 (−16 s ⊗ s + 64 2 − 8(2 2 + p + 4 + 4 ) + k3 1) , 32w
where the following abbreviations are used: k1 = 10 2 + 3(p + 4) + (2 2 + p + 16) , k2 = 16 + 2 (2 2 + p + 16) ,
(2.38)
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k3 = 8 (2 2 + p + 4) + (32 2 − 4q + p 2 + 32p − 4r + 256) , s = 2 · m + ( + 3 )m . Let us discuss briefly the obtained results. The intensity tensor components (2.36)–(2.38) contain the eigenvalues
which can be expressed as functions of the p, q and r parameters [36] (solutions of the secular equation (2.18)). Thus, the derived (2.36)–(2.38) depend on the hyperfine fields only. One recognizes immediately from (2.37) that G is a pseudovector like m, while I is a symmetric second-order tensor, generally not diagonal in the principal axes of . The left-hand side of Eq. (2.27) was shown [39,27] to be equal to (2L + 1)/(2Ig + 1) which for L = 1, Ig = 21 reduces to 23 . An analogue of Eq. (2.28) was given previously in [39]. The sum of traces of (2.36) over ’s was shown to be equal to (2L + 1)/(2Ie + 1) [39,40,27] which for the specific case L = 1, Ie = 23 agrees with our result. More detailed discussion can be found in [20].
3. Single-site Mössbauer spectrum A Mössbauer spectrum for circularly polarized radiation in the case of mixed interaction can be easily calculated by use of the intensity tensor (2.36)–(2.38). The transition probability a for circularly polarized radiation, and for an arbitrary orientation of the and m vectors with respect to the principal axes of the EFG (see Fig. 3), can be expressed as [26,41]: a =
1 2
Tr I −
1 2
· I · − G · ,
(3.1)
where denotes one of the two circular polarization states of a photon. By setting = 0 one gets the results for an unpolarized beam. Absorption lines are located on the velocity scale at v (we recall that the and refer to the excited and the ground state, respectively): v =
B ( − 1/2 ) + , 2 3/2
(3.2)
where I = gI N c/E and I = 21 , 23 , see Appendix A.1 Parameter is the isomer shift. In the case of a randomly oriented absorber, (3.1) reduces to the simple form: a =
1 3
Tr I .
(3.3)
The explicit form of the intensity tensor allows one to explain some symmetry present in the simulated spectra [41]. Namely, if we change the sign of the dominant component of the EFG (R → −R) and the polarization of the beam (from one circular component to the opposite one, ( → −)), the spectrum will change its shape and become a mirror image of the previous one (v → −v), see Fig. 4. This symmetry, due to the time reversal symmetry of the problem, has been explained using the explicit form of the intensity tensor [20].
v
1
2
83
4 7 5
6
v
1
2
8
3
4 7
5
6
Fig. 4. Schematic absorption spectrum (left) and a spectrum corresponding to measurement with opposite polarization and an opposite sign of R illustrating R v symmetry (right). Thin vertical lines and arrows indicate the ground state splitting.
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4. The velocity moments 4.1. Explicit formulas for the velocity moments Some features of the hyperfine fields distribution can be obtained immediately from the features of the Mössbauer spectrum. For example, the centre of gravity (on the velocity scale) is approximately equal a mean isomer shift; the width of the poorly resolved spectrum of ferromagnet is roughly proportional to the average value of the h.m.f. distribution. We will show, using the concept of the velocity moments, that these approximate observations can be formulated strictly. The intensity tensor reflects an angular anisotropy of the resonantly absorbed radiation of well-defined transitions. Unfortunately, the formulas (2.36)–(2.38) can be applied to the case of spectra exhibiting well-separated lines only. We are going to show the tensor properties of the values related to the whole spectra, namely the velocity moments. The results will be particularly suitable for the case of poorly resolved spectra where transition probabilities and transition energies can hardly be determined. The nth velocity moment of the intensity tensor I component) can be defined as a sum of the nth power (or its n I . Using (2.27)–(2.30) and (3.2) we have the explicit of the velocity v multiplied by I , and is equal to v
moments of the trace:
Tr I = 3 , (4.1)
v Tr I = 3 ,
(4.2)
B 2 (621/2 − 201/2 3/2 − 3p23/2 ) + 32 ,
2 v Tr I =
1 8
3 v Tr I =
3 3 2 32 3/2 qB (21/2
− 33/2 ) − 63 + 3
(4.3) 2 v Tr I .
(4.4)
An analogue of Eq. (4.2) was given earlier by Karyagin [39]. A similar set of linear equations for the antisymmetric part of the intensity tensor can be obtained from Eqs. (2.37) and (3.2):
G = 0 , (4.5)
v G = 41 (1/2 − 53/2 )B ,
(4.6)
2 v G = 21 3/2 B(1/2 − 23/2 ) · B + 21 (1/2 − 53/2 )B ,
(4.7)
3 v G =
1 2 32 B (k4
· 1 − 1633/2 2 )B + 3
2 v G − 32
v G ,
(4.8)
where in (4.8): k4 = 231/2 − 3021/2 3/2 − (3p − 48)1/2 23/2 + (7p − 12)33/2 . Similarly, a set of linear equations for the symmetric part of the tensor can be obtained from Eqs. (2.38) and (3.2):
I = 1 , (4.9)
1 v I = · 1 − B3/2 , 2
(4.10)
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2 v I = k5 · 1 − B3/2 + 21 B 2 3/2 (1/2 − 33/2 )F1 , 3 v I = 3
+
2 v I + k6 · 1 +
2 1 3 16 B 3/2 (−61/2
2 3 3 3 2 2 B 3/2 + 4 B 3/2 (1/2
(4.11)
+ 121/2 3/2 − (4 − p)23/2 )
− 3/2 )F2 ,
(4.12)
where k5 = 18 B 2 (221/2 − 81/2 3/2 + (4 − p)23/2 ) + 2 , k6 =
1 3 2 32 B 3/2 q(31/2
− 43/2 ) − 23 ,
F1 = m ⊗ m , F2 = m ⊗ · m + · m ⊗ m . Left-hand sides of all Eqs. (4.1)–(4.12) are linear functions of the components of the intensity tensor, while the righthand sides are functions of the hyperfine fields. Using the above formulas, one can calculate velocity-weighted averages of the spectra—the quantities directly available in the experiment. The velocity moments have a dimension; so as to shorten notation, let us introduce the tensor U, proportional to the and to the EFG : U = 3/2 B =
eQc . 2E
(4.13)
The quadrupole splitting : eQzz c 2 2 = 3/2 BR 1 + 1+ = . 3 2E 3
(4.14)
In the case of pure quadrupole interactions, || measures the separation between the lines in the quadrupole doublet. The nth moment of the absorption spectrum is defined as n
v a n W = . (4.15)
a It follows from Eqs. (3.1), (3.2) and (4.1)–(4.12) that W1 = + 41 · U · − 41 (1/2 − 53/2 )B · ,
(4.16)
W2 = 41 (21/2 − 31/2 3/2 + 423/2 )B 2 − 41 3/2 (1/2 − 33/2 )(B · )2 + 2 + − 41 (1/2 W3 + W−3 2
− 43/2 ) · U · B − 21 (1/2
1 4
2 +
− 53/2 )B · ,
1 2
· U · (4.17)
= − 41 3/2 (31/2 − 53/2 )B · U · B − 43 3/2 (1/2 − 33/2 )(B · )2 + k7 · U · − 43 3/2 (1/2 − 3/2 )(B · )B · U · +
W−3 − W3 2
=
3 4
2 + 43 (21/2 − 31/2 3/2 + 423/2 )B 2 + 3 ,
1 3 16 (1/2
(4.18)
− 1521/2 3/2 + 391/2 23/2 − 4133/2 )B 2 B ·
+ 43 (1/2 − 53/2 )2 B · + 43 (1/2 − 43/2 )B · U · +
3 48 (31/2
− 73/2 )2 B · − 21 3/2 · U2 · B ,
where in the third line of Eq. (4.18) k7 =
2 1 16 (31/2
− 61/2 3/2 + 723/2 )B 2 +
3 48
2 +
3 4
2 .
(4.19)
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4.2. Discussion The velocity moments describe the property of the whole spectrum. To estimate the velocity moment properly one has to decompose the spectrum to the sum of Lorentzian lines with amplitudes proportional to a and perform summation, see (4.15). In the case of continuous hyperfine fields distribution (and a thin absorber) one has to integrate the deconvoluted with the Lorentzian shape [42] spectrum a (v): ∞ n v a (v) dv n ∞ . (4.20) W = −∞ −∞ a (v) dv Our experience is that the best method for receiving the deconvoluted spectrum a (v) is to use the Maximum Entropy Method [43,44]. The methodology is still under development [45]. The left-hand side of Eqs. (4.16)–(4.19) is estimated from the measured spectra via (4.15) and this information is used to deduce the hyperfine fields entered in the right-hand side of Eqs. (4.16)–(4.19). To estimate the velocity moment, one does not need to know which of the absorption line belongs to which transition because the summation in (4.15) covers over all possible transitions. The velocity moments are thus useful in the description of poorly resolved spectra, because one does not need to determine the hyperfine structure of the nuclear levels at all. Zimmerman [25] has shown how to measure the EFG tensor by detecting the anisotropy of the recoilless radiation of a given transition. Eqs. (4.16)–(4.19) can be considered as complementary results. Since the moments of the spectra are constructed from the hyperfine fields and the direction of , the dependence of the moments on the direction can be used for determination of the orientation of the hyperfine fields in the absorber. A particularly important value is the difference of the first moments (4.16) measured by two opposite circular polarizations, because the difference is proportional to the contribution of iron to the magnetization, as discussed in Section 2: B · =
2 (W−1 − W1 ) . 1/2 − 53/2
(4.21)
The brackets denote the averaging over orientations of the Mössbauer nuclei (2.8). The result (4.21) is very general, true for any texture or the presence of mixed interactions and distributions of hyperfine fields. The first moment is the quantity known in literature as the centre shift. An angular anisotropy of the first velocity moment (4.16), measured with an unpolarized beam, directly reflects the symmetry of the EFG, irrespective of the presence of h.m.f. In the special case of a single-crystal sample and an unpolarized beam: W01 = +
eQzz (3 cos2 − 1 + sin2 cos 2 ) , 16
(4.22)
where and are polar angles of the vector in the PAS of the EFG (see Fig. 3). In particular, the W01 moment can be used for determination of the sign of the dominant component of the EFG, zz . One important observation concerns measurements performed in the so-called texture free mode [46,47]. From Eq. (4.22) after averaging over the directions of U and B, we get the velocity moments for a texture free (subscript t–f) absorber: 1 Wt−f = , 2 Wt−f =
2 1 12 (31/2
(4.23) − 101/2 3/2 + 1523/2 )B 2 + 2 + 41 2 .
(4.24)
We see that the first velocity moment is a measure of the isomer shift when performed in a texture-free mode. The dependence of W1 and W2 on the orientation of with respect to the absorber, serves as a check for the random orientation of crystallites in powdered absorbers. From the (4.16) and (4.23) we see the possibility of experimental determination of the average · U · : 1 · U · = 2W−1 + 2W1 − 4Wt−f
for an absorber with any type of texture and the presence of mixed interactions.
(4.25)
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Obviously, the velocity moment treatment applied to the whole spectrum brings information averaged over all Mössbauer atoms. Therefore, the different Fe-site-selectivity is lost. The advantage of the velocity moment formalism is that the treatment is model free. It can be applied to absorbers possessing any magnetic or crystalline texture. A natural field for application of the velocity moments can be sought in various temperature studies (e.g. phase transitions). In the vicinity of magnetic transition temperature, when quadrupole interactions are present, the full Hamiltonian has to be used for reliable analysis. The velocity moment formalism offers data treatment which is easy and strict in the full range of the hyperfine parameters.
5. Randomly oriented electric field gradient A randomly oriented absorber with nonzero EFG exposed to a static external magnetic field is relatively easy to realize experimentally and commonly used in practice. Line intensities in the first-order perturbation treatment were given in [48] for the case of an axially symmetric EFG and a small magnetic field. Perturbation methods were widely used in calculations of the absorption line intensities [49], shapes [50–53] and moments of energy distribution [54–58] for large magnetic splitting. Blaes et al. [59] obtained an explicit expression for the line shape in the case of mixed interactions and an axially symmetric EFG. Using the explicit results for the intensity tensor [20], we have recently found its spectral density for 23 – 21 spin transitions in the case of circularly polarized radiation and randomly oriented EFG in a uniform magnetic field [21]. To obtain the energy distribution we have performed appropriate integration which results finally in functions of B, zz , , (R and in dimensionless units) and the g-factors of the excited and the ground states. Explicit results for the energy distribution of every component of the intensity tensor allow calculation of the spectrum for any orientation of the h.m.f. with respect to direction of the beam, see Fig. 5. The energy distribution shows singularities when one of the principal axes of the EFG coincides with the h.m.f. direction. A characteristic feature of the energy distribution of the powdered absorber is that the positions of the singular points do not depend either on the orientation of the magnetic field with respect to the , or on the deviation from the random orientation of the EFG in the absorber, see dotted vertical lines in Fig. 5. To the best of our knowledge this fact has never been exploited in the analysis of measured spectra. It offers a possibility for determining the B, zz , , parameters from the singular point positions in the spectra only [21].
6. Ambiguity problem Let us consider a single-crystal absorber in which iron occupies only one crystallographic site. The h.m.f. is oriented at unknown angles and with respect to the EFG axes, see Fig. 3. Our goal is to use the measured spectra to determine hyperfine fields and their orientations. It has been pointed out by Karyagin [39] that the knowledge of nuclear levels resulting from combined interactions is not sufficient for a unique determination of all hyperfine parameters. Indeed, from line positions one gets the hyperfine field B, the isomer shift, and the positions of excited levels. From the positions of excited levels, three parameters, p, q and r, can be obtained via Eq. (2.35). However, the number of unknowns zz , , and appearing in the Hamiltonian (2.17) is larger than the number of available data p, q and r. This is what is meant by talking about the “ambiguity problem”. Analysis of the ambiguity problem has been made in Refs. [60–63]. Ambiguity is not removed either by taking into account intensities for a single experiment with unpolarized radiation, or by changing the magnitude of the magnetic field [40]. However, it can be reduced by the use of polarized radiation [64]. It has been shown [20] that the measurements of the scalar product G · G , see Eq. (2.37), allow us to obtain one independent parameter more, namely (1 − 2 )R 3 , so together with p, q and r, the hyperfine parameters zz , , cos2 and cos 2 can be obtained. Some ambiguity still remains, because one cannot determine the orientation of the h.m.f. by knowing only cos2 and cos 2. However, further specification is possible by using circularly polarized radiation, sensitive for the sign of the h.m.f.
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(a)
(b)
arb. units
(c)
(d)
(e)
-6
0
6
2vEγ /(g3/2 µN Bc) Fig. 5. The energy distribution corresponding to the absorption spectrum for R = 2.5 and = 0.7, see text: (a) trace of the intensity tensor, (b) hyperfine magnetic field perpendicular to the observation direction, (c) hyperfine magnetic field parallel to the observation direction, (d) hyperfine magnetic field parallel to the observation direction; circularly polarized radiation, (e) hyperfine magnetic field antiparallel to the observation direction; circularly polarized radiation. The plots are not convoluted with Lorentzian shape.
7. Perturbation theory—dominating magnetic interactions Let us examine the case of dominating magnetic interaction (small R parameter). This is important because it is quite frequently encountered in practice. In addition, the results can be applied to the analysis of the spectra of absorbers with any type of texture, particularly those measured in an external magnetic field. In the experiments one observes easily the changes of line intensities, positions and widths which indicate the, usually partial, alignment of iron moments under an external magnetic field. The spectra of such partially ordered samples are difficult to interpret. An idealized situation of randomly oriented EFG and uniform magnetic field was presented in Section 5. However, even in the case of randomly oriented powder, when magnetic saturation is not reached, one needs to build models of magnetic moments arrangement for proper treatment of the spectra. Below, we discuss the possibility of estimating some averages of hyperfine fields from the measured line positions and intensities. Circular polarization effects are included in the treatment. The expressions presented are valid in the presence of: (i) crystalline texture, (ii) magnetic texture and (iii) a distribution of any hyperfine parameters. Perturbation methods have been widely applied in the analysis of the spectra for which quadrupole splitting was small as compared with the magnetic one. Line positions and line widths measured with unpolarized radiation have been analysed up to the second-order perturbation theory [56–58]. Series expansions obtained under assumption of the random distribution of the EFG and uniform hyperfine field were presented in [51,53–55]. The energies may be ordered so that 1 < 2 < 3 < 4 (see Fig. 6) and one can introduce the frequently used absorption line abbreviation [65,18,19,21] by integers 1, 2, . . . , 8, see Fig. 6. Lines 1–6 form a set known as a Zeeman sextet. Lines 7 and 8 correspond to the forbidden transitions in the case of pure magnetic interactions.
K. Szyma´nski / Physics Reports 423 (2006) 295 – 338
311
(E-Eγ)/Eγ .c γ 3/2B λ1/2 γ 3/2Bλ4/2 γ 3/2B λ3 /2
0
γ 3/2Bλ4 /2
∼ ∼
E/Eγ.c
c
γ1/2B/2 0 -γ1/2B/2 1 2 3 7 8 4 5 6 Fig. 6. Nuclear energy levels and the transitions for excited spin 23 and the ground 21 . Vertical axes correspond to the energy in Doppler velocity units. The energies are ordered so that 1 < 2 < 3 < 4 (the 3/2 B i /2 values are marked; because 3/2 < 0, 1 is on the top and 4 on the bottom).
Table 2 Indices and ascribed to the consecutive absorption lines for the case of 1 < 2 < 3 < 4 Line no.
1
2
3
4
5
6
7
8
4 +1
3 +1
2 +1
3 −1
2 −1
1 −1
1 +1
4 −1
Let us start our considerations from zero-order perturbation, e.g. pure magnetic interactions. This specific case is useful in magnetic texture investigations. Using explicit expressions for the intensity tensor of a single site [20] and setting R = 0, one gets the tensor components as functions of energy: Tr I = G = −
3 5 2 − 9 , + 8 8 ( 2 − 5) 5 2 − 9 + ( 2 + 3) 16 ( 2 − 5)
(7.1) ·m ,
6 + ( 2 + 9) 2 − 3 + 2
I = 2 ·1− m⊗m . 8( − 5) 8 ( 2 − 5)
(7.2)
(7.3)
Next for = −3, −1, 1 and 3 one obtains results for zero-order perturbation which have already been summarized in Tables 1 and 2. The zero-order results bring information about two types of averages, m · and (m · )2 , as discussed in Section 2. Using the results for the intensity tensor and the first-order expansion with respect to the small parameter R, contributions to the line intensities can be obtained, see Table 3, where the symbols are defined as follows: f1 = m · · − (m · · m)(m · ) ,
(7.4)
f2 = 2 · · + m · · m − 3(m · · m)(m · )2 ,
(7.5)
f3 = (m · · )(m · ) − (m · · m)(m · )2 .
(7.6)
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Table 3 Normalized transition probabilities in the first-order perturbation of the small parameter R (see text for explanation of the symbols) Line no. i
1
2
3
4
5
6
1 16
3 3 3 1 1 1
4 −4 0 −1 0 −2
1 1 −1 0 −1 1
1 1 1 0 1 −1
4 −4 0 −1 0 2
3 3 −3 1 −1 −1
(m · )2 /16 m · /8 f1 /4 f2 /32 f3 /8
The transition probability ai can be obtained by summing up the numbers from the column corresponding to the index i, multiplied by numbers 4 . 4 (m · )2 + · · · − 2 f . from column 1. For example, transition probability (i = 2) is equal to 16 16 8 3 Table 4 Position of the absorption lines vi (i = 1, . . . , 6) of the Zeeman sextet in the first-order perturbation in small parameter R (see text for explanation of the symbols) vi = (3/2 (0) − 1/2 )B/2 + (1) m · U · m/2 + + B · O(R 2 ), where tensor U is defined in Eq. (4.13) Line no. i
1
2
3
4
5
6
(0) (1)
+1 3 1
+1 1 −1
+1 −1 −1
−1 1 −1
−1 −1 −1
−1 −3 1
Parameters f1 , f2 and f3 are proportional to the small parameter R. The absorption line positions (3.2) expanded in small R are given in Table 4. From the transition probabilities, a1 , a2 , . . . , a6 , measured with polarized radiation for textured absorber, one can derive the parameters displayed in the first column of Table 3: m · = 2(a1 + a2 + a3 − a4 − a5 − a6 ) ,
(7.7)
(m · ) = −a1 − a2 + 7a3 + 7a4 − a5 − a6 ,
(7.8)
f1 = 2(a1 − 3a3 − 3a4 + a6 ) ,
(7.9)
2
f2 = 8(−a1 − 2a2 − 3a3 + 3a4 + 2a5 + a6 ) ,
(7.10)
f3 = 2(−a2 + a5 ) .
(7.11)
We recall that the probabilities ai are normalized: a1 + a2 + a3 + a4 + a5 + a6 = 1. Unlike the first two equations (7.7), (7.8), the left-hand sides of Eqs. (7.9)–(7.11) do not have a simple interpretation and contain terms depending on the absorber only (m · · m) as well as on the geometry of the experiment (m · , · · and m · · ), see (7.4)–(7.6). Other information is delivered by line positions, see Table 4 and Eq. (3.2): = 41 (v1 + v2 + v5 + v6 ) = 41 (v1 + v3 + v4 + v6 ) , B =
1 1/2
(v5 − v3 ) =
1 1/2
(v4 − v2 ) ,
m · U · m = 21 (v1 − v2 − v5 + v6 ) ,
(7.12) (7.13) (7.14)
Eqs. (7.12)–(7.14) valid to the O(R 2 ) were reported earlier in [36,54,51,57,58]. A remark must be added to the use of Eq. (7.14). The scalar m · U · m is proportional to the m · · m see Eq. (4.13), and we have the identity: m·U·m=
eQc m · · m = 3/2 Bm · · m . 2E
(7.15)
However, when the h.m.f. distribution is present, there is no general relation between m · U · m and m · · m. The same remark holds for · U · (Eq. (4.25)) and · · as well as for m · in (7.7) and B · in (4.21).
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One can easily get results for the second- and the higher-order terms in small parameter R using the explicit form of the intensity tensor (2.36)–(2.38) and the moments (4.16)–(4.19). However, the complete set of results is complicated and will not be discussed here.
8. The sources of polarized resonant radiation Polarized, resonance radiation can be obtained by separation of the nuclear transitions corresponding to the different hyperfine levels. A few methods have been reported. 8.1. Multiline polarized sources Interaction of nuclear spin with electronic surrounding cause lift of degeneracy of the nuclear levels. Radiation involved in the transition between sublevels have well-defined polarization. Some systems are of practical importance. One is a transversally magnetized 57 Co embedded in an -Fe foil [18,64,66–69]. Relatively weak field of a permanent magnet or an electromagnet is used for in-plane magnetic saturation of the foil. The source emits six lines of different photon energies. All emission lines are linearly polarized. Polarization plane of magnetic field vector of the photon is parallel to the applied external field for lines 2, 5 and perpendicular for lines 1, 3, 4, 6. Another system is a 57 Co in an -Fe foil magnetized perpendicularly to surface [18,70,71]. Large fields produced by superconducting coils are required for saturation of the foil. The source emits four lines of Zeeman sextet (1, 3, 4, 6) with circularly polarized radiation. Pair of lines 1, 4 have opposite polarization to the polarization of pair 3, 6. 57 Co isotope can be embedded into a single crystal having just one orientation of the local EFG and zero magnetic interaction. In such system two emission lines are present. For arbitrary orientation of the single crystal they are partially linearly polarized. In the specific case of axially symmetric EFG ( = 0) radiation from excited m = ± 23 states emitted perpendicularly to main principal axis of the PAS is linearly polarized [72,73]. Further details can be found in extended review [13]. 8.2. Polarization by thermal agitation Lowering the temperature of the magnetically split source to the mK range results in a nonequal population of energy levels [74–76]. A general discussion of the transition probability of the oriented nuclei from the point of view of group theory was presented in [77]. Transitions between the low-energy states are preferred and, in principle, a monochromatic polarized source can be obtained. However, the method prevents the use of intense sources because of the heating of the cryogenic system. So far, thermal agitation has been used rather as a tool in measuring temperature [78]. 8.3. Filter techniques One can filter out photons with one polarization from an unpolarized beam, keeping the filter and the source at resonance by a constant Doppler velocity, and using a second drive system for simultaneous variation of the Doppler velocity between the source and the absorber. The filter technique with a double drive spectrometer has been presented in [9,10]. An advanced drive system has been constructed especially for application to a polarized source. By choosing special materials for the source and the polarizer matrices, one can obtain resonance at zero Doppler velocity. Experiments with linearly polarized radiation and a single drive have been reported in [11,12,79]. One disadvantage of the reported method lies in the complicated technology of preparation of the 57 Co source in a CoO(I) matrix [11]. The authors of [11] also report the presence of an additional emission line in the source resulting from some admixture of the CoO(II) phase in the matrix, which complicated the interpretation of spectra. In certain experiments [12,79], single-crystal polarizer containing Fe2+ cations was used. However, it is difficult to prevent oxidation from Fe2+ to Fe3+ states, which changes the properties of the polarizer. One method of using 57 Co in a paramagnetic Pd matrix in a strong external magnetic field allows measurement of the Mössbauer spectrum without utilization of the Doppler motion [80]. It has been suggested [80] that 57 Co in the paramagnetic matrix and in an external field combined with a resonance filter can produce results equivalent to
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measurement with a circularly polarized monochromatic source. However, there are no published reports confirming experimental realization of this original idea. In 1995 the first experiments with 57 Co in a Cr matrix source and chemically stable polarizer based on an Fe3 Si alloy were performed [81]. The effect of linear polarization was very weak because of the small concentration of 57 Fe per unit area, but it confirmed the predicted good properties of Fe–Si for use in Mössbauer polarimetry [82]. A Fe3 Si-based polarizer enriched in 57 Fe was prepared. Special attention was paid to achieving a high degree of structural order in the Fe–Si alloy [83–85]. During our early research on structural ordering, the Fe atom at the so-called (A,C)-site of partially ordered Fe3 Si was identified and characterized [85]. We also worked out a method for the orientation of Fe–Si crystallites, which greatly enhanced magnetic ordering of the polarizer in the applied field of the permanent magnet. As a result of these operations, a monochromatic, circularly polarized Mössbauer source (MCPMS), with a degree of circular polarization 80%, has been achieved [14], see Fig. 7a,b. A transversally magnetized 57 FeBO3 or -57 Fe absorbers have been used as a filter for a monochromatic, linearly polarized source. A single drive and second-order Doppler shift was used for achieving resonance [15], see Fig. 7e,f. The authors performed a complete analysis of the transmission of the radiation through the resonantly absorbing medium and obtained detailed information about the intensity and polarization. Optimization allows the achievement of the linear polarization degree exceeding 90%. The advantage of the design is the possibility of using any type of commercially available single line source. The main technological advantage of the [14,15] constructions is use of the commercially available radioactive sources which greatly reduce the effort and costs of keeping the apparatus working. 8.4. Monochromatic, polarized synchrotron sources Recently a monochromatic, linearly polarized source of resonant radiation was constructed [16,86] on a synchrotron line. The synchrotron radiation was diffracted in Bragg geometry on single-crystal 57 FeBO3 . Strong suppression of the electronic scattering was achieved by use of pure nuclear diffraction from (3 3 3) crystal planes. Close to the Neel temperature the hyperfine structure makes it possible to obtain 100% linearly polarized and recoil free radiation, see Fig. 7g, h. Ideas for realizing Doppler modulation were presented [16]. A nonradioactive synchrotron source has brilliant perspectives because of its huge intensity, high collimation, and energetic dispersion comparable to the dispersion of the single line Mössbauer source. Moreover, a source of this type can be realized at bunch as well as at the continuous mode of the storage ring. The advantage of the device is its potential flexibility in polarimetric studies. One can adopt the idea of [87] and use the diamond quarterplate for easy changing of linear into circular polarization. Thus, two linear and two circular polarization states of the synchrotron beam would offer an effective method for nuclear polarimetry.
9. Model of the resonant filter 9.1. Resonant filter Let us start from the idealized, simple model of resonant filter. The source of unpolarized radiation is characterized by the Lamb–Mössbauer factor fs and emits an intensity N0 , composed of a fraction fs N0 /2 of resonant photons with polarization = +1, and fs N0 /2 with polarization = −1; (1 − fs )N0 photons belong to nonresonant radiation. Polarization = ±1 is one of two orthogonal photon polarizations; in this report, two circular or two linear polarizations will be considered. The resonant radiation has the energy distribution: p(E) =
1 . 2 (E − E )2 + (/2)2
(9.1)
The main absorption line of the filter (with polarization = +1) is centred at the emission line of the source. The energy distribution, p (E), of the fraction of photons with polarization which passes the polarizer is p (E) = p(E) · e−t (E) ,
(9.2)
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315
Fig. 7. Representative spectra of absorbers oriented in external magnetic field measured with monochromatic, polarized radiation. (a)–(d) circularly polarized radiation, radioactive source [14], (e), (f) linearly polarized radiation, radioactive source [15], (g), (h) linearly polarized radiation, synchrotron source [16]. Symbols ↑↑ and ↑↓ abbreviate parallel and antiparallel orientations, while and ⊥ abbreviate two perpendicular (orthogonal to each other) orientations of the net magnetization with respect to the k vector.
where (E) is the resonant cross-section for polarization : (E) =
a 2 /4 (E − E )2 + 2 /4
.
(9.3)
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0. 8
P
0. 6 0. 4 0. 2
20
40
60
80
100
t Fig. 8. Thickness dependence of the average polarization degree for the resonant cross-section of the filter given by Eqs. (9.3) (dotted line) and (9.8) (solid line).
Parameter t = n0 0 fa is the effective thickness, where n0 is the concentration per unit area of the Mössbauer atoms, 0 is the maximum resonant absorption cross-section, and fa is the recoilless fraction of the Mössbauer atom in the filter [88]. Relative weights a which are present in (9.3), a1 = 1 and a−1 = 0. The integrated intensity of the resonant radiation with polarization passing through the filter is +∞ p (E) dE , (9.4) N = 21 fs N0 e−m −∞
where m is the mass per unit surface and the electronic mass absorption coefficient. Polarization of transmitted radiation depends on the energy as a consequence of resonant cross-section (9.3). The average degree of polarization of the resonant radiation P is proportional to N−1 − N1 and in our idealized case P=
N−1 − N1 . N−1 + N1
Using the explicit form for the transmission integral [89] (see Appendix A.3) from Eq. (9.5) we get ⎧t small t , 1 − e−t/2 I0 (t/2) ⎨ 4 P= = 2 1 + e−t/2 I0 (t/2) ⎩ 1 − √ large t , t
(9.5)
(9.6)
where In is the Bessel function (see Appendix A.3). The average polarization (9.6) increases linearly with t for small t and slowly approaches to 1 when t tends to infinity, which is shown in Fig. 8. The linear increase of polarization P with t corresponds to the effective absorption of photons in the energy range E − , E + . In the range of effective thickness 0 < t < 1 the energy distribution of the beam, with polarization transmitted through the filter consists of a single peak. After exceeding t = 1, the distribution of transmitted photons, (9.2), has two maxima, see Fig. 9, at energies E± : E± = E0 ±
√ t −1 . 2
(9.7)
For thickness t larger than 1, the majority of transmitted photons with polarization have an energy belonging to the discussed maxima, and their absorption increases slowly with the increase of the filter thickness. This results in a slow approach to 1 of the polarization degree P. A similar double hump structure (as those shown in Fig. 9) appears when a white beam of synchrotron radiation travels through the thick absorber with a resonant scattering cross-section resulting in dynamical beats [90]. Eqs. (9.3) and (9.6), however, corresponds to the idealized situation. Only a fraction exp(−m) of photons passes through the filter with mass per unit surface m and electronic mass absorption coefficient . Thus, large values of t cause strong electronic absorption. To construct a filter which transmits reasonable intensity, one has to use material for which
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317
0.20 0.5
pζ(E) (Γ/2)
0.15
1.0
0.10
1.5
0.05 2.0 2.5
0.00
-4
-2
0
2
4
(E-Eγ ) / (Γ/2)
Fig. 9. Energy distribution of photons transmitted through a filter of various thickness with resonant cross-section (9.2). The thickness t is given below each solid line. The dotted line connecting the maxima of (9.3) reads: f (x) = (e(x 2 + 1))−1 , where x = (E − E )/(2).
resonant absorption is large in comparison with the electronic absorption. Detailed discussion of these conditions can be found in [15]. Moreover, there are two additional factors which make resonant filtering more complicated than that shown in Fig. 9, see solid line in Fig. 8. The first factor results from imperfect alignment of the hyperfine field in the filter or the presence of mixed interactions. Thus, there is a weak absorption line centred at the emission line of the source active for = −1 polarization: the relative weight in (9.3) a−1 = 0. An increase of thickness causes unwanted absorption of the photons with = −1 polarization. The second factor results in an unwanted, strong absorption line with polarization = −1 centred far from the main resonance. Only the tail of this line causes unwanted absorption. This line is present in all systems reported so far: in the case of Fe3 Si the 3rd line is used as a filter for circularly polarized radiation [14], the 4th line causes unwanted absorption; when the 3rd line is used as a filter for linearly polarized radiation [15], the 2nd line is unwanted; similarly, when one of the hyperfine levels split by EFG is used for receiving linearly polarized radiation [12], the second one causes unwanted absorption. To formulate a more reliable model one has to introduce the cross-section: (E) =
a1 2 /4 (E − E )2 + 2 /4
+
a2 2 /4 (E − E2 − E )2 + 2 /4
.
(9.8)
It is assumed that a1−1 >a11 (in the case of ideal filter a1−1 = 0), a2−1 is of the order of a11 and E2 ?. The total resonant cross-section is normalized, e.g. i ai = 1. This means that there may be some additional absorption lines with polarization , located far from the region between E and E− E2 . The absorption of these lines is neglected for simplicity. A typical example of the dependence P (t) with an absorption cross-section (9.8) is shown in Fig. 8. For small thickness, the polarization degree increases linearly with thickness because the main absorption line effectively filters out the photons with the energy located at E . Further increase of thickness becomes less effective, as discussed earlier. Simultaneous absorption of photons with polarization = −1 by the weak line a1−1 located at energy E becomes more effective. For large thickness, also the wing of the strong absorption line located at energy E2 starts to absorb photons with polarization = −1. Thus, at a certain thickness, the degree of polarization reaches a maximum. To evaluate the optimal thickness and the maximum polarization degree of material with intrinsic parameters E2 and ai , it is convenient to introduce parameters: u = a1−1 /a11 ,
x = ta 11 /2,
s = a2−1 /a11 ,
b = 2E2 / .
(9.9)
We assume a21 = 0, because the tail of the absorption line separated by large Doppler velocity negligibly disturbs the absorption of the main line with the amplitude a11 . Using (9.8) and (9.9) we get the expression for the polarization
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degree (9.5): P=
+∞
2 −∞ [1/(y +∞ 1/ −∞ [1/(y 2
1/
+ 1)] e(−2ux/y + 1)] e
2 +1)+(−2sx/(y−b)2 +1)
dy − e−x I0 (x)
(−2ux/y 2 +1)+(−2sx/(y−b)2 +1)
dy + e−x I0 (x)
.
(9.10)
The condition for a maximum of P (9.10) is given by jP /jx = 0. To our knowledge, there are no explicit results for the integral present in (9.10). The expression (9.10) with the condition jP /jx = 0 can be expanded into a power series of small parameters u and 1/b. It has been shown [91] that the optimal thickness xopt at which Pmax = P (xopt ) achieves its maximum, can be expressed as a function of the parameter : 2s = u+ 2 , (9.11) b resulting in the following series: xopt =
x0 x2 + ··· , + x1 + 2 s
1 p2 3 + ··· . Pmax = 1 − p0 + .p02 2 + p1 + 2 s Parameters xi and pi are smooth functions of the angle defined as 2s . = arctan ub2
(9.12) (9.13)
(9.14)
For the example 0.50 < x0 < 0.79, 1.7 < p0 < 1.9, see Appendix A.4. Parameters (9.11) and (9.14) are simply polar coordinates of the u1/2 and (2s/b2 )1/2 ( abbreviation should not be confused with those in Fig. 3). Returning to the initial parameters, for which the model of the filter was introduced, and neglecting the higher-order terms in (9.12) and (9.13), we have the explicit approximate results: a1−1 a2−1 a2−1 2 = + , tan = , a11 2a1−1 |E2 | 2a11 E22 topt ≈
4x0 E22 2a1−1 E22 + a2−1 2
,
Pmax ≈ 1 − p0 .
(9.15)
The parametrization (9.15) of the exact expression (9.10) with condition jP /jx=0 is valid when the intrinsic parameters of the filter are in a range which allows achievement of a high degree of polarization, see Fig. 10. Eq. (9.15) show explicitly how the two unwanted absorption lines influence the filter quality. The parameter controlling this process for the first one is the ratio u = a1−1 /a11 , while for the second one it is the relative separation (/E2 )2 . The two contributions can be quantitatively characterized by just one parameter (9.11). The optimal thickness of the filter is proportional to −2 , see Eq. (9.12). When the intrinsic quality of the material increases ( tending toward zero), the filter may be thicker while giving a degree of polarization smaller than 1 by a factor proportional to . 9.2. Comparison with real systems 9.2.1. A filter based on 57 FeBO3 for linear polarization The authors of [15] presented detailed measurements of the degree of polarization of radiation filtered by a singlecrystal 57 FeBO3 in an external magnetic field. To achieve resonance between the unpolarized source and the absorption line of the filter, the temperature variation of the hyperfine field was applied. The authors paid special attention to the minimization of the angular divergence of the beam and the alignment of the single crystal and the external field, so the influence of the unwanted component (with amplitude a1−1 ) was negligible. Thus, the value limiting the polarization degree was the influence of the tail. In Fig. 7 of [15] the results of the two experiments are presented, for which the
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thickness
319
polarization
0.20
2s / b
0.15
0.65
10
0.10
0.75
20 40 0.85
0.05
80 0.95
160 0.00 0.00
0.05
0.10
u
0.15
0.20 0.00
0.05
0.10
0.15
0.20
u
Fig. 10. Contours showing the constant values of (a) the optimal thickness and (b) the maximum degree of polarization [91]. Thick lines show the exact results (9.10) for s = 1.3 satisfying condition jP /jx = 0; dotted lines show approximate results including terms with the x0 and p0 coefficients; thin lines show approximate results including terms with the x0 –x2 , and p0 –p2 coefficients for Eqs. (9.12) and (9.13).
hyperfine fields were about 10 and 17 T. The parameter for these experiments was 0.13 and 0.074, respectively. This allows us to predict the optimal xopt = 28 and 91, respectively, which corresponds to topt = 112 and 364, respectively. The estimated polarization degree is 0.76 and 0.87, respectively. In the cited experiment the authors used a single crystal with t =290 at room temperature and obtained the polarization degree (estimated from the figure) 0.90 ± 0.08 and 0.92 ± 0.04, respectively. 9.2.2. Filter based on Fe3 Si for circular polarization There are two inequivalent sites of iron in a Fe3 Si crystal. 57 Fe in the site with smaller moment (the so-called A, C-site) has a hyperfine field of 20 T and a specific property of the accidental coincidence of its 3rd absorption line with the emission line of 57 Fe in the Cr matrix. Thus, the 3rd line is used for resonant filtering [14]. An unwanted absorption line with opposite polarization is located at a distance of 1.02 mm/s (the 4th line of the Zeeman sextet). The value of b = 21. The main problem in achieving a high polarization degree is the proper alignment of the hyperfine fields in the filter. The best alignment which has been achieved is monitored by suppression of the intensities of lines 2 and 5 to a level lower than 1%. This implies that the unwanted absorption line intensity at zero Doppler velocity corresponds to u = 0.01. The contribution from the tail of the unwanted line, entering into (9.11), is much smaller than the contribution from the nonperfect alignment. Both contributions lead to the parameter = 0.12 and we may estimate that the optimal xopt = 42. If we assume that all atoms from the A, C-site (a fraction of 23 ) are active in resonant filtering, we estimate that a11 = 41 · 23 and topt = 504. This value corresponds to approximately 31 mg/cm2 (assuming 90% enrichment). The polarization degree related to this value is 0.81. The polarizer used in [14] contains 34 mg/cm2 and the polarization degree is reported to be 0.80 ± 0.02. 9.2.3. Comment on the comparison The predicted polarization degree (9.13) corresponds to a filter optimized with respect to the thickness. The constructed devices were optimized not only to the polarization degree but also were designed for reliable intensity. Thus, the predicted maximal values seem to be too low in comparison with the values reported in [12] and [13]. The most plausible reason for this is the following one. In experimental determination of the polarization degree, one measures the
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amplitudes of the absorption lines (or the absorption area) in a rather narrow energy window (approximately 2). In the expression (9.10) all photons are taken into account, also those which are located far from zero Doppler velocity (few ). These photons contribute to a rather wide absorption shape, which is identified with the nonresonant background. Neglecting these photons, one causes an artificial enhancement of the experimentally determined P. 9.3. Effect of quadrupole splitting Let us discuss the influence of small quadrupole interactions on the degree of polarization. Such a case happens for an FeB polarizer, not fully analysed yet [92]. Assuming that h.m.f. acting on all the atoms of the polarizer is parallel to the photon direction, m · = 1 while directions of the PAS of the EFG tensor are randomly oriented (randomly oriented powder grains). Expanding the formula for the transition probability (5.1) in small parameter R up to the second order, and performing averaging over orientations of the EFG axes (see Appendix A.2), we may estimate parameter u for line no. 3 (or 4): 27 2 u= 1+ R2 . (9.16) 160 3 The h.m.f. of the main component FeB is about B = 12 T [93,94] at room temperature and the quadrupole splitting in the paramagnetic state in = 0.217 mm/s [95]. Using (4.14) and 3/2 from Table 7 we may estimate that R(1 + 2 /3)1/2 = 0.27 and u = 0.012. The distance between absorption line position is 0.611 mm/s an from (9.9) b = 12.6. Since s = 1 we see that u and 2s/b2 are of the same order, and = 0.15, thus the maximal polarization degree expected for an FeB filter is about 0.73. The reported polarization degree for a filter composed of 57 Fe2.85 Si1.15 and 57 FeB was 0.76 ± 0.01 [92].
10. Applications 10.1. Transmission of the polarized radiation through thick absorbers Transition probabilities are proportional to the observed absorption line intensities in the thin absorber. In thicker absorbers saturation effects are present which can be described by the transmission integral [88]. This treatment is valid for unpolarized radiation. In the case of an absorber with oriented hyperfine fields, each absorption line exhibits characteristic polarization (say, linear ). An unpolarized beam can be considered as a set of photons with and and polarization, and only states pass without absorption. In the thick absorber, all photons with states are absorbed while those with states are transmitted. Thus, a thick unpolarized resonant absorber absorbs more intensity than a polarized one [96,97]. A particularly interesting situation occurs when the energies of two absorption lines with opposite polarizations are close to each other. In that case a thick absorber produces an unusual shape for the absorption spectrum: maximum absorption occurs at the energy between the transition energies of the two lines. This behaviour has been observed in the spinel MgFe2 O4 [98]. Qualitatively, the effect is easy to understand: at energy between the two transition energies both polarizations of photons are absorbed, giving almost 100% absorption. The scattering of gamma radiation is characterized by a complex refraction index. A real part of the refraction index is related to the dispersion. As in optics, the change of the polarization state (from linear to circular and vice versa) and the Faraday rotation of the polarization plane is possible. A Faraday rotation of about 19◦ was observed when linearly polarized gamma radiation passed through an MgFe2 O4 spinel [99]. A complete description of the transmission of gamma radiation through resonantly absorbing media was presented in a review [13] within the formalism of the density matrix. 10.2. Sign of the hyperfine magnetic field A h.m.f. is a pseudovector and it has the property of a sense (or sign) which changes under time inversion. Experimental determination of the sign of the h.m.f. is important because the h.m.f. is related to the electronic magnetic moment.
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321
Fig. 11. Amorphous Fe0.66 Er 0.19 B0.15 measured in an axial magnetic field of Hext 8 × 105 A/m using a monochromatic circularly polarized Mössbauer radiation [104]. Direction of the Fe, Er moments and the net magnetization are shown schematically by the arrows (net magnetization parallel to k). The used polarization was the same as the one for the experiment shown in Fig. 7a,b.
In his pioneering work with -Fe absorber in an external magnetic field [100], S. S. Hanna demonstrated that the h.m.f. is oriented antiparallel with respect to the net magnetization. Experiments with a multiline source split by an external magnetic field of a few teslas were performed to determine the sign of the h.m.f. of iron in various matrices [70]. Another way to determine the sign of the h.m.f. is to use the sign of the Faraday rotation in a resonantly absorbing medium in an external magnetic field [101]. The monochromatic circularly polarized Mössbauer source (MCPMS) was constructed so as to be able to perform measurements in a similar way to those carried out on standard spectrometers. Neither single crystals nor absorbers enriched in 57 Fe are needed. We have demonstrated that, using -Fe powder as an absorber, one gets a spectrum with an unusually strong asymmetry between the 1st and 6th line intensities, clearly distinguishing transitions with a momentum change m = +1 and m = −1. The observed asymmetry determines unambiguously the sign of the h.m.f. [14]. On the example of HoFe2 powder, we have demonstrated the antiparallel orientation of Fe magnetic moments with respect to the direction of net magnetization [14], see Fig. 7g,h. The MCPMS technique was applied for measuring the direction of the dominant, external magnetic field-induced contribution to the h.m.f. in Pd2 TiAl:57 Fe. The alloy has rather unusual properties: it behaves like a ferromagnet with very low magnetization and a relatively high Curie temperature [102]. The induced h.m.f. direction is parallel to the external field and originates from conduction electron polarization [103]. An important class of magnetic systems consists of two subsystems ordered antiparallel. MCPMS technique is particularly suitable for measurements of details of magnetic ordering of the subsystems. In Fig. 11, the spectra of the amorphous Fe–Er–B alloy are presented. It is clear without any data treatment that at low temperatures, Fe magnetic moments are oriented antiparallel to the net magnetization while above the so-called compensation point (Tcomp ≈ 150 K) the Fe subsystem is oriented parallel to the net magnetization. The details of the arrangement of magnetic moments are discussed in [104]. 10.3. The magnetic texture Important information about the angular averages of Fe magnetic moments can be inferred from the relative line intensities in the Zeeman sextet, measured in a standard experiment [105–107,67]. The intensities of lines nos. 2 and 5 for pure magnetic interactions are proportional to the average square of the sine of the angle between the photon and magnetic moments direction (see Section 2). Orientation of magnetic moments in the sample plane can be investigated by linearly polarizer radiation. As a source of multiline polarized radiation usually a transversely magnetized 57 Co embedded in -Fe matrix was used. As an
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K. Szyma´nski / Physics Reports 423 (2006) 295 – 338 2θ
(a)
1-ε
(b) ε /2
ε /2
Fig. 12. Spin structures which can be distinguished in an MCPMS experiment. Arrows lengths and directions indicate fractions and directions of magnetic moments, respectively.
example we quote investigations of the tensile stress applied to the amorphous Fe40 Ni40 P14 B6 ribbon along so-called ribbon direction. It was demonstrated that magnetic moments are fully aligned in the direction of the stress [108]. In similar type of experiment, magnetic field was applied in the Fe80 B20 amorphous ribbon plane. Nonperfect alignment was detected and quantitative analysis allows determination of average square of the projection of Fe magnetic moment on the direction of applied magnetic field [67]. Recently, a multiline linearly polarized source was used in investigation of an Fe–Cr multilayer system by detecting the conversion electrons [109]. We remind that in experiments with linearly polarized radiation only the (m·)2 type of averages can be determined. MCPMS offers determination of the average cosine between photon and magnetic moment directions. The average cosine, m · , brings important information concerning sample magnetization, particularly in the region of incomplete magnetic saturation. As discussed in Section 2, the magnetic moment of iron multiplied by average m · is simply the contribution to magnetization of the Fe atom. This is valid, irrespective of the mechanism which causes magnetic moment misalignments: the remaining domain structure, structural defects, spin canting, or local anisotropy.An example of the measurement of Fe contribution to total magnetization in UFe4 Al8 is given in [110]. Below, we would like to illustrate the use of information about the averages m · and (m · )2 in a simple example. In standard Mössbauer spectroscopy, one measures the symmetric part of the intensity tensor only, and only the (m · )2 average can be estimated. Let us consider two apparently different cases of 3-dimensional spin alignment, shown schematically in Fig. 12. In the first one, the spins exhibit some canting angle , ( abbreviation should not be confused with those in Fig. 3) while in the second one, the dominant fraction (1 − ) of the spins are parallel while fraction is oriented perpendicularly to the vector. In fact the arrangement shown in Fig. 12 was considered in [98,97] because the measurements performed on -57 Fe foil absorber in the external magnetic field indicate the presence of about 2% Fe magnetic moments with a clearly noncollinear arrangement. The symmetric part of the intensity tensor of line no. 1 or 6, for example, in its PAS is a a = Iyy = Ixx b b Ixx = Iyy =
3 8 3 8
− −
2 3 16 sin , b 3 Izz 16 ,
a Izz =
= 38 ,
3 8
sin2 , (10.1)
where indices a, b correspond to the situations in Fig. 12a and b, respectively. We see that if = sin2 , then Ia = Ib and a standard experiment cannot detect any difference between the spectra corresponding to the two structures. Unlike the symmetric, the antisymmetric part of the intensity tensor G (see Tables 1 and 2) is different for these two structures: Gaz = − 38 cos ,
Gbz = − 38 (1 − ) .
(10.2)
Therefore, if =sin2 , apparently |Ga | < |Gb |. Thus, for the MCPMS experiment one should observe a larger asymmetry in the spectra (for the difference between a1 and a6 or between a3 and a4 , see (2.9)) in the case of spin canting rather than in the case of perpendicular orientation of moments. These two spin structures, looking identical in the standard experiment, can be distinguished in an experiment with circularly polarized radiation. However, for the specific case considered, experimental detection of the difference would be difficult since great precision is required. For example, in the case of = 10◦ , |Ga | and |Gb | would differ by less than 1.6%.
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10.4. On the hyperfine magnetic field distribution In the more than 40 years since the development of the Mössbauer technique, advanced methods of data treatment have appeared. Particular activity has taken place in the domain of analysis of hyperfine field distributions. The Mössbauer spectrum can be considered as a function of velocity, S(v). For any h.m.f. distribution p(B): S(v) = Z(v, B)p(B) dB , (10.3) where Z(v, B) is a known function representing the Zeeman sextet. The thickness effects, angular beam divergence, apparatus broadening, and drive nonlinearity were neglected in (10.3). Eq. (10.3) is a Fredholm equation of the first class and usually one can get a solution for (10.3), which is model independent (see detailed discussion in [111]). If two or more kinds of hyperfine interactions are present and exhibit distributions, then a continuous class of solutions for Eq. (10.3) exists [112]. Nevertheless, one can obtain physically sound information about the distribution in some cases. The first case takes place when a functional dependence between different kinds of hyperfine interactions exists. For example, it is quite common to assume a linear correlation between isomer shift and h.m.f. [113–116]. Such an assumption results in reduction of the problem to the (10.3) type of equation. The second case corresponds to the well-separated peaks in the Zeeman sextet. Parameters of the hyperfine field distribution can be obtained uniquely when the average and the variance of the h.m.f. belong to the so-called validity diagram for h.m.f. calculation presented in [53]. The measurements with MCPMS bring additional information as compared with the measurements with unpolarized radiation: one is able to find which transitions correspond to m = ±1 and which to m = 0. This information allows one to reach valid conclusions about the shape of the magnetic h.m.f. in the case of more overlapping spectra than given by a standard validity diagram [53]. We have demonstrated that in the case of the overlapping spectra of an Fe–Cr–Al alloy, standard analysis leads to an ambiguous answer as far as the shape of the h.m.f. distribution is concerned. Our experiment with MCPMS has removed the ambiguity and brought additional information about magnetic moments alignment [117]. Another system studied with MCPMS technique was bcc disordered Fe–Cr–Mn alloy. Ternary Cr-rich Cr–Fe–Mn alloys with bcc structure exhibit complex magnetic behaviour [118,119]. At a certain composition range a coexistence of ferro- and antiferromagnetic order has been found. We have identified two Fe subsystems with different orientations under external magnetic field [120]. In contrast, in the Fe–Ni invar alloys, two Fe subsystems with different magnetic moments exhibit the same kind of angular arrangement [121]. As discussed in Section 2, MCPMS offers measurements of the m · and (m · )2 averages. Let us abbreviate m · ≡ c1 and (m · )2 ≡ c2 . In the case of the presence of distribution p(B) (10.3), one can in principle measure the averages c1 and c2 as functions of B : c1 (B) and c2 (B). The condition (2.12) must be fulfilled: (c1 (B))2 c2 (B) .
(10.4)
To reduce ambiguity it is advisable to measure the same absorber with a different polarization state of the photon beam. The algorithm for simultaneous evaluation of the spectra under constraints (10.4) was described in [122]. The method was applied to a nanocrystalline Fe48 Al52 system. The example of the p(B) and c1 (B) distribution is shown in Fig. 13. The measurements were performed at T = 13 K and in external magnetic field 1.1 T. One sees clearly that for h.m.f. smaller than 5 T orientation of the h.m.f. is on average antiparallel to the net magnetization. One should stress that such unusual distributions would be hardly possible to obtain if a simultaneous evaluation of the spectra measured with different polarization states of photons had not been used, which greatly reduces ambiguity. The MCPMS and extended X-ray absorption fine structure (EXAFS) measurements were combined for the determination of chemical environments responsible for the noncollinear magnetic order in the nanocrystalline Fe–Al [123]. 10.5. Sign of the dominant electric field gradient component The MCPMS technique can be used to determine the sign of the dominant component of the EFG tensor. The method is similar to the standard technique in which polycrystalline dia- or paramagnets are exposed to the external
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Fig. 13. An example of the distributions of h.m.f. obtained from the experiment (up) and the h.m.f. dependence on the average cosine between the h.m.f. vector and magnetization (down), measured at T = 13 K and Bext = 1.1 T [123]. Vertical bars split the p(Bhf ) distribution into sectors numbered by k = . . . 9, 8, 7 . . . , where k is the number of Al atoms in the two first coordination shells.
Table 5 Isomer shift and functions of the moments of the spectra measured with MCPMS at different fields, see text for details Sample SNP Pd2 TiAl:57 Fe Pd2 TiAl:57 Fe Pd:57 Fe Pt 3 Fe Fe2 Ti Fe2 Ti FeTi1.008 FeTi0.996 FeSi Mn3 P -FeSi2 -FeSi2.04 -FeSi2.35 UFe4 Al8
Ref.
[103] [103] [124] [125]
[41]
[110]
B (T)
IS (mm/s)
1 )/2 (mm/s) (W1 + W−
x (T)
y (T)
1.1(1) 1.1(1) 0.085(7) 1.4(1) 1.4(1) 1.4(1) 1.1(1) 1.1(1) 1.1(1) 1.4(1) 1.4(1) 1.1(1) 1.4(1) 1.4(1) 1.1(1)
−0.220(4) 0.130(2) 0.130(2) 0.179(3) 0.314(3) −0.277(2) −0.277(2) −0.153(2) −0.162(2) 0.278(5) 0.270(2) 0.087(2) 0.082(1) 0.079(1) 0.156(3)
−0.215(3) 0.136(1) 0.144(1) 0.186(3) 0.32(1) −0.273(2) −0.270(2) −0.1465(25) −0.1510(14) 0.273(3) 0.270(1) 0.089(2) 0.088(2) 0.090(2) 0.154(1)
0.93(11) 0.78(24) 0.2(2) 0.87(8) 1.236(85) 1.33(16) 1.26(9) 1.08(15) 1.13(10) 1.41(11) 1.22(8) 0.98(6) 1.37(4) 1.39(8) 0.844(95)
0.90(3) 0.85(1) 0.087(8) 0.97(4) 1.10(15) 0.45(3) 0.14(2) 0.75(3) 0.775(15) 1.276(41) 1.15(2) 0.81(2) 1.40(2) 1.42(3) 0.67(1)
SNP denotes sodium nitroprusside—the standard used in Mössbauer spectroscopy. Details of sample preparation and characterization can be found in references shown in the 2nd column.
magnetic field [59]. In the MCPMS experiment with two polarizations of a photon beam, the spectra are better resolved and contain more information [41]; thus, determination of the sign of the dominant EFG component is easier than in standard methods which use unpolarized radiation. 10.6. Tensor properties of the velocity moments In order to check some of the relations introduced in Section 4, we have analysed the spectra already measured as well as some unpublished ones. The substances used in the experiment are listed in Table 5. The samples were chosen so as to exhibit different values of isomer shift and quadrupole splitting (including the sign), and to be in a paramagnetic state. The last property ensures that, after applying the external magnetic field, the hyperfine fields will be collinear with the applied one. The magnetic field sources will be characterized in Section 11.2,
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see also the 3rd column of Table 5. The Mössbauer absorbers were prepared from powders to have random angular distribution of the crystallites. The average isomer shift relative to the -Fe absorber is given in the 4th column of Table 5. Every spectrum was deconvoluted using a Lorentzian line shape, with FWHM determined in a calibration experiment, see the solid lines in Fig. 14. Next, the moments (4.16) and (4.17) were estimated. Because of the random orientation of crystallites in the absorbers, the average of the second term in Eq. (4.16) vanishes, and the sum of the first moments is proportional to the isomer shift: =
W1 + W−1 2
.
(10.5)
This relation is examined in Fig. 15 in which the isomer shift of every absorber and the sum of the first moments are compared. A straight line with slope equal to 1, crossing the axes origin, is shown for comparison. The sums of the first moments are equal, within the error bar, to the isomer shift. This also shows that the degree of polarization of leftand right-handed polarized beams is the same. To check Eq. (4.17) consider two spectra measured in a standard experiment: one in an applied field parallel to the photon direction; the second, in a zero field. The difference between the second moments (4.17) is equal to W02 (B) − W02 (0) = 41 (21/2 − 41/2 3/2 + 723/2 )B 2 .
(10.6)
We see that the easily measured left-hand side of Eq. (4.17) is proportional to the square of the field, irrespective of the strength of the quadrupole interaction. The field obtained from Eq. (10.6) can be compared with the field determined from the first moment, as described by Eq. (4.16): the difference of the first moments measured for two photon helicities is proportional to the product B · . Let us define two parameters x and y as follows: x = x0 sign(B · ) W02 (B) − W02 (0) , (10.7) y = y0
W−1 − W1 P
,
(10.8)
where P is the degree of polarization of the beam. Constants x0 and y0 are determined by nuclear g-factors and for are equal to
57 Fe
x0 =
y0 =
2 21/2 − 41/2 3/2 + 723/2
= 7.1311(11)
2 sT = 4.3639(8) . 1/2 − 53/2 mm
sT , mm
(10.9)
(10.10)
Both parameters, x and y, under the assumption that B is parallel to the , should be equal to the intensity of h.m.f. The experimentally determined x and y values are listed in Table 5 and displayed in Fig. 16. The straight line in this figure has a slope equal to 1 and crosses the axes origin. All but two experimental points show linear correlation and consistency with the formalism presented in Section 4. The presented examples also show that model independent moments of the spectra can be used to determine the average values of hyperfine fields. The two exceptional points, for Fe2 Ti alloy, are discussed below. In the case of the intermetallic Fe2 Ti alloy, the difference of the first moments, y, is unexpectedly small. Two points correspond to two independent measurements carried out with two different magnets and different polarized sources. As can be seen from Table 5, the intermetallic FeTi alloys were also measured and exhibited no anomaly. Thus, we conclude that the deviations observed for Fe2 Ti must be due to the microscopic properties of this alloy. The alloy is antiferromagnetic at temperatures T < TN = 280 K [126]. It was concluded in [127] that charge fluctuations between electronic levels with different symmetry (t2g and eg ) are responsible for invar-type behaviour. Thus, similarly as in the typical invar alloys [128], the existence of ferromagnetic spin fluctuations in the paramagnetic range can be expected. In the ferromagnetic cluster the hyperfine field is antiparallel to the atomic magnetic moment of Fe. Its contribution to the first velocity moment is opposite to the contribution of paramagnetic Fe atoms in the external magnetic field. Thus, the presence of spin fluctuations in the paramagnetic phase results in the decrease of the y-value (10.8).
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C0471
F0142
Fe2Ti
4%
5%
Mn3P
F0144
relative transmission
relative transmission
F0062
B=1.4T
6%
G0155
3%
5%
B=1.4 T
↑↑ 1.5%
G0201
↑↑
G0156
3%
↑↓
↑↓
G0203
1.5%
-2
-1
0
1
2
-2
-1
velocity [mm/s]
0
1
2
velocity [mm/s]
A0640
A0440
Pt3Fe
6%
3%
SNP
F0118
relative transmission
3%
G0151
↑↑
E0424
relative transmission
B=1.4T
6%
B=1.1 T
3%
E0416
↑↑
1%
3%
G0152
↑↓
E0417
1%
-2
-1
0 velocity [mm/s]
1
2
-2
↑↓
-1
0
1
2
velocity [mm/s]
Fig. 14. Samples of the Mössbauer spectra measured in a standard experiment (top), in the field parallel to the beam and with MCPMS (two spectra at the bottom).
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327
(W1ζ+W1-ζ)/2 [mm/s]
0.2
0.0
-0.2
-0.2
0.0
0.2
IS [mm/s] Fig. 15. Correlation between the sum of the first moments of the spectra, measured by MCPMS, and the isomer shift.
1.5
y [T]
1.0
0.5
0.0 0.0
0.5
1.0
1.5
x [T] Fig. 16. Correlation between the first and second moments of the spectra measured by MCPMS. Two large points correspond to Fe2 Ti alloy, see text for detailed explanation.
In the two examples presented so far, the absorbers were made from powders and we have assumed that the crystallites were randomly distributed. Linear correlations of the experimental data shown in Figs. 15 and 16 confirmed our assumption. In the third example we should like to demonstrate a possibility for measuring the term · U · . Before presenting the results of the experiments, let us summarize briefly the literature data on FeB, which we used for our demonstration. The FeB intermetallic compound has an orthorhombic B27-type crystal structure [129]. Iron atoms occupy a single site and form linear chains along the c-axis. The magnetic structure was shown to be a canted ferromagnet with spins forming an angle of about 20◦ with respect to the b-axis, in the ab-plane [130]. The Curie temperature Tc = 594 K
K. Szyma´nski / Physics Reports 423 (2006) 295 – 338
relative transmission [arb.units]
328
0
C0653
30
C0659
54.7
C0666
60
C0665
90
C0655
-3
-2
-1
0
1
2
3
velocity [mm/s] Fig. 17. Examples of the spectra of absorbers prepared from in-field oriented powder of FeB. Measurements were performed in a zero applied field. The angles between the directions of photon and the field orienting the powder are shown.
[131]. In the paramagnetic state the quadrupole splitting is 0.217(8) mm/s, while in the FM state it is 0.092(7) mm/s, which indicates that the magnetic field in the FM state is inclined with respect to the zz -axis of the EFG by 38◦ (2) [95]. The Mössbauer spectrum of the FeB compound consists of a six line pattern [93,94] and the small addition of the second Zeeman component, which is interpreted as originating from the substitution of the neighbouring iron by an extra boron atom [94]. The FeB compound was considered for use as a resonant polarizer [92]. During our work, we prepared polarizers from crystallites oriented in an external magnetic field. The powder was mixed with epoxy glue of appropriate viscosity and exposed to an external field of about 1.2 T. The sample was in a magnetic field until the hardening of the glue. Then we measured absorbers under different angles between the direction of the beam and the direction of the preferred orientation. The Mössbauer measurements were performed at room temperature and in a zero applied magnetic field. Examples of the spectra are shown in Fig. 17. The spectra consist of two Zeeman components, whose parameters agree with those reported in literature [94]. In Fig. 17 the variation of line intensities with the angle resulting from the preferred orientation of the crystallites is shown. The angular dependence on W01 is shown in Fig. 18. Since W01 is a tensor component of the second order, the angular change of the average W01 should be quadratic in cos . The experimental results were fitted by W01 =0.257(3) mm/s+0.056(4) mm/s cos2 , see the solid line in Fig. 18, confirming the validity of our approach. Assuming that the low field component seen in the spectra in the paramagnetic phase has the same splitting as the dominant component, the latter being equal to 0.217 mm/s [95], we get the maximal possible change of W01 equal to 0.081 mm/s < W01max − W01min < 0.094 mm/s, where the two limits depend on . The change observed is smaller. This can be expected because of the nonideal orientation of crystallites. We see that when the direction of a photon is parallel to the direction of the field applied during orientation, the W01 attains its maximum value, indicating that zz is positive. This result agrees with the sign of zz obtained from measurement of the line positions [95].
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329
0.32
W10 [mm/s]
0.30
0.28
0.26
0.0
0.1
0.2
0.3
0.4
0.5
θ/π Fig. 18. Angular dependence of the W01 moment for FeB in-field oriented absorbers.
relative transmission [arb.units]
296 K 250 K 200 K 150 K 100 K 70 K 50 K 12 K -4
-2 0 2 velocity [mm/s]
4
Fig. 19. Spectra of the y = 6 sample (Cr 79 Fe20 Mn1 ) measured in an applied field and with circularly polarized radiation (the same polarization as in Fig. 7a,b).
Finally, we should like to illustrate the use of velocity moments for analysis of the angular distribution of the magnetic moments in a Cr–Fe–Mn system, exhibiting a poorly resolved spectra. Two series, Cr 75+x Fe16+x Mn9−2x and Cr 75+y Fe16−y Mn9 , called here x- and y-series, respectively, have been prepared. The x-series is characterized by constant electron density while the electron concentration within the y-series is maximally varied [132]. Samples with x = 4 and y = −6 exhibit ferromagnetic properties. Samples with x = −4 and y = +6 are characterized by low iron concentrations, and there are strong experimental indications that they are ordered antiferromagnetically. MCPMS measurements were performed at temperatures between 12 and 296 K in external magnetic field B = 1.1 T parallel to the direction of the gamma rays [133], see example in Fig. 19.
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0.1
10 x=+4
0.0
(W 1↑↑ -W1↑↓)/2 [mm/s]
Bhf [Tesla]
8
x=-4 y=+6
6
4
2
0
x=+4
-0.1
x=-4
-0.2
y=+6 -0.3
-0.4
0
100
200
T [ K]
(a)
-0.5
300
(b)
0
100
200
300
T [ K]
1 )/2 [133], uncorrected Fig. 20. (a)Average value of h.m.f. intensity from in-field experiments and (b) experimentally determined difference (W1 −W− for the polarization degree.
For the antiferromagnetic-like alloys (x =−4 and y =+6), the evolution of the h.m.f. distribution in wide temperature range is observed. Measurements with polarized radiation show that at higher temperatures the average z component of the h.m.f. corresponds to the applied external magnetic field (Fig. 20b), while the average h.m.f. (Fig. 20a) definitely exceeds the applied field. This indicates that at high temperatures, in samples with low Fe concentration, the iron moments form a structure which can hardly be modified by application of an external magnetic field. However, Fe magnetic moments can be partly aligned at low temperatures, with apparently antiparallel orientation of the average z component of the h.m.f. An experiment with polarized radiation performed on ferromagnetic alloys (x = +4) shows that projection of the average h.m.f. is antiparallel to the direction of magnetization up to T = 150 K, Fig. 20b. Thus, the h.m.f. is likely to be of core origin. A nearly linear decrease of average h.m.f (Fig. 20a) with increasing temperature, accompanied by an almost linear increase of the difference of first moments (Fig. 20b), both for in-field experiments at T < 150 K, indicates that the ratio of these values remains approximately constant within wide (compared to 0–Tc ) temperature range. This indicates that angular distribution of Fe magnetic moments does not change dramatically with temperature. Summing up, it has been demonstrated that the concept of the velocity moments is useful in the investigation of the geometry of the h.m.f. in the case of poorly resolved spectra. 11. Selected experimental problems 11.1. On the measurement of the degree of polarization The degree of polarization can be measured using an absorber with magnetic moments oriented in an external magnetic field [14,32]. For pure magnetic interaction in the absorber, the degree of polarization, P, is equal to P =C
|1 − a1 /a6 | |1 − a4 /a3 | =C , 1 + a1 /a6 1 + a4 /a3
(11.1)
where ai (i =1, . . . , 6) are intensities of the absorption lines corrected for the thickness effect. The parameter C depends on the magnetic texture of the absorber: C=
1 + (m · )2 , 2m ·
(11.2)
where brackets denote averaging over the orientations of the hyperfine field in the absorber. The averages in (11.2) are usually unknown; to reduce the ambiguity one tends to align magnetic moments by an external magnetic field as much as possible. Measuring the Zeeman sextet with line intensities 3: z: 1: 1: z: 3 in a standard experiment, we get the
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331
average of (m · )2 given by Eq. (2.10). Next, the Schwarz’s inequality (2.12) allows one to estimate the lower limit of the degree of polarization: P√
4 16 − z2
|1 − a1 /a6 | |1 − a4 /a3 | 4 =√ . 2 1 + a1 /a6 16 − z 1 + a4 /a3
(11.3)
Let us discuss the influence of the assumption concerning imperfect spin alignment on the factor C, used in determining the degree of polarization. Consider three examples: (i) if all magnetic moments are inclined by the same angle with respect to the vector (Fig. 12a), the factor C in Eq. (11.1) is equal to C=√
4 16 − z2
,
(11.4)
(ii) in a more realistic case, the magnetic moments are distributed homogeneously within a cone, and the factor C can be expressed as C=
4+z+
√
16 , 3(4 + z)(12 − 5z)
(11.5)
(iii) in the third example the dominant fraction of the magnetic moments is oriented parallel to the magnetic field, while the remaining fraction of moments is oriented perpendicularly to the field (Fig. 12b), resulting in: C=
4 . 4−z
(11.6)
In the experiment in which the degree of polarization was measured, the typical value of z was in the range 0.04–0.06 [14,92]. The difference between the results of (11.4), (11.5) and (11.6) for z = 0.05 is smaller than 1.5%. However, in the case of Fe powder in the external magnetic field of 1 T, which was typical for our experiments, assumption (iii) is rather unphysical. The difference in factors C given by Eq. (11.4) and (11.5) is much smaller than the experimental precision. Thus, we conclude that either formula, (11.4) or (11.5) can safely be used for the determination of the degree of polarization. The discussion performed so far concerns the situation where a six line pattern can be measured. However, Mössbauer experiments are performed also in a velocity range in which not all the peaks of the -Fe standard are measured. In this case the degree of polarization can be estimated from measurements of the first velocity moments for standard paramagnetic absorbers in an external magnetic field. It follows from (4.16), for which experimental tested is presented in Section 10.6, that if the hyperfine field B is parallel to the photon direction, the measurements of W1 and W−1 give P=
y0 |W1 − W−1 | B
,
(11.7)
where the constant y0 for 57 Fe was already given in Eq. (10.10). 11.2. The drive and the permanent magnet system The active transition in the resonant filter used in [14] was one of the inner lines of the Zeeman sextet. The efficiency of the filtering may be increased by the use of the outer absorption lines, since they are three times more intense. However, there are strong experimental restrictions caused by a large velocity at which such resonance can be achieved—in the case of -Fe this velocity is equal to 5.3 mm/s. There is, though, an experimental trick [134] which makes it possible to use a resonance filter and a source driven by a standard triangular signal. The trick consists in using resonance at the velocity v1 during the first half of the period of the transducer, while during the second half using the resonance at velocity v6 , which is nearly equal to −v1 . The construction of a waveform generator dedicated to this kind of work has been presented in [134]. Design of the signal used in the programmable generator and test measurements have also been presented [134].
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z
ρ
2r
Fig. 21. Permanent magnet and the absorber in an axial, external magnetic field, perpendicular to the sample plane. The arrows indicate direction of permanent magnetization.
Another important part of the construction of the MCPMS is formed by compact, permanent magnets allowing for orientation of the magnetic moments of the polarizer and the sample, if needed. We have shown [92] the construction of a transducer in which two simultaneous experiments with a polarized beam can be performed. The magnet system is used both for producing the field in which the drive coil is moving, and for aligning magnetic moments of the polarizer. At the same time the field acting on the radioactive source should be as small as possible in order to minimize line broadening of the source. Quantitative estimation of this broadening shows that it does not exceed 0.05 mm/s [92]. Below, we would like to consider the effect of the inhomogeneity of the field acting on the sample or polarizer in our permanent magnet system. The problem has axial symmetry, see Fig. 21. A cylinder with a hole of radius r, made of hard magnetic material, is magnetized and an inhomogeneous field acts on the sample located in the centre. The sample has the shape of a disc with radius r and geometrical thickness h. Experimentally it is easy to measure Bz (z) along the z-axis. Approximating the field by quadratic terms only, we have: Bz = B0 +
z 2 j2 B z , 2 jz2
(11.8)
where the magnetic field B0 and j2 Bz /jz2 are taken at the axis origin. The scalar potential in the vicinity of the axes origin in cylindrical coordinates , z, is (z, ) =
j2 Bz z (32 − 2z2 ) 2 − B0 z . 12 jz
(11.9)
The potential fulfills the equation = 0. Finally, the field B = −G, acting on the sample (at −h/2 < z < h/2 and < r), where is given by (11.9), exhibits a distribution with an average value and standard deviation equal to: 1 j2 B 2 2 1 2 r − h , (11.10) |B| = B0 − 8 jz2 3 2 r 2 h 2 j2 B z 1 2 j2 Bz B = √ r 2 1 + . (11.11) jz 48B02 jz2 8 3 From the symmetry of the problem it follows that B at z = 0 is parallel to the z-axis. Averaging m2z over the sample volume leads to: 2 h 2 r 2 j2 B z 2 mz = 1 − . (11.12) 96B02 jz2
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333
Table 6 Characteristics of the magnets used in the MCPMS experiments r (mm) 5.0(1) 6.0(1) 6.0(1)
B0 (T) 1.300(2) 0.918(1) 0.0941(1)
j2 Bz /jz2 (T/mm2 ) −0.045(1) −0.036(2) + 0.0020(1)
B (T) 1.441(7) 1.08(1) 0.0851(6)
B (T)
1 − m2z
Ref.
0.115(5) 0.132(9) 0.0073(4)
6.9 × 10−3
[41,92] [14,103,117,110] this work
5.8 × 10−4 1.7 × 10−4
Variation of the field along the magnet axis was measured by Hall probe and Lake Shore 450-type gaussometer.
The characteristics of the magnets used in our experiments are given in Table 6. As a geometrical sample thickness we take h = 1 mm. We see that m2z is very close to 1, so the angular misalignment of the field in the vicinity of the sample volume is much smaller than the m2z values of magnetic moments observed in experiments. Thus, the moment misalignment is caused by a lack of saturation and not by the inhomogeneity of the field.
12. Summary Recent progress in methodology of the Mössbauer polarimetry is presented. The explicit form of the intensity tensor for circularly polarized radiation allows calculation of the transition probabilities omitting the diagonalization problem. The line intensities can be expressed explicitly as functions of the hyperfine fields. With the help of the explicit form of the intensity tensor an analytic form of the energy distribution for randomly oriented EFG and uniform h.m.f. was derived and a solution to the ambiguity problem is presented. The concept of velocity moments has been introduced and tested. This treatment is a generalization of the intensity tensor methods applied so far to the absorbers in a paramagnetic state. The velocity moments can be applied to poorly resolved spectra. We report the progress made in realization of the monochromatic polarized source of resonant radiation.A synchrotron source allowing measurements in energy mode was announced. So far only linear polarization is available. A quarter plate could be applied to obtain a circular polarization state. Thus, a strong synchrotron source with the option of linear or circular polarization would be a powerful tool in modern nuclear polarimetry. The radioactive sources of monochromatic, polarized radiation obtained by resonant filtering are rather cheap and simple constructions. They are equipped with commercially available 57 Co sources, and can be easily implemented in laboratories. Application of Mössbauer polarimetric methods in investigation of the orientations of hyperfine fields are reviewed. Polarimetric methods provide measurements with different polarization states of the photon. Emerging computational power and the possibility of simultaneous analysis of results reduces ambiguity and allows for access to the distribution of hyperfine fields and their orientations.
Acknowledgements I am deeply grateful to Professor Ludwik Dobrzy´nski who built a group for nuclear techniques in solid-state physics in Białystok and invited me to build a Mössbauer lab. I thank the professor for giving us, the members of his group, freedom in action, constructive criticism and warm help in any trouble. Special gratitude for his careful reading of the manuscript and useful comments.
Appendix A A.1. Useful constants Frequently used physical constants are listed in Table 7 .
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Table 7 Frequently used physical constants Quantity
Symbol
Numerical value
Unit
Ref.
m s−1
[135] [135] [135] [135] [135] [135] [136] [136] [136] [136] [137]
Speed of light in vacuum Nuclear magneton Elementary charge Planck constant Avogadro constant Atomic mass unit Nuclear g-factor of the 57 Fe, I = 21 1/2 = g1/2 N c/E Nuclearg-factor of the 57 Fe, I = 23 3/2 = g3/2 N c/E Nuclear quadrupole moment, 57 Fe, I = 23 eQc/(4I3/2 (2I3/2 − 1)E ) 57 Fe resonant absorption cross-section 57 Fe source line width W = 2 Energy of the 23 – 21 57 Fe nuclear transition
c
W E
299 792 458 5.050 783 43(43) × 10−27 1.602 176 53(14) × 10−19 6.626 0693(11) × 10−34 6.022 1415(10) × 1023 1.660 538 86(28) × 10−27 0.181 21(2) 0.118 821(12) −0.103 544(26) −6.789 7(17) × 10−2 1.6 × 10−29 2.8 × 10−23 2.56(3) × 10−22 0.1940(3) 14.412 497(3)
Wavelength for the 23 – 21 57 Fe nuclear
8.602 547 4(16) × 10−11
m
[138]
m57Fe ER
5.693 539 87(15) × 10−2 1.958 330 6(9) 2.119 5.5847 × 10−2 6.37 5.91 × 10−25
kg mol−1 meV % kg mol−1 m2 kg−1 m2
[139]
transition, = hc/E Atomic mass of 57 Fe isotope Recoil energy ER = E2 /(2m57Fe c2 ) 57 Fe abundance in natural Fe Atomic mass of natural Fe Fe mass attenuation coefficient for E Total electronic absorption cross-section per Fe atom for E , el = mFe /NA
N
e h NA u g1/2
1/2
g3/2
3/2 Q
0
mFe
el
J T−1 C Js mol−1 kg
mm s−1 T−1 mm s−1 T−1 m2 mm s−1 V−1 m2 m2 mm s−1 keV
[136] [136] [138]
[140] [141]
A.2. Useful identities is a dimensionless tensor proportional to the EFG, defined after Eq. (2.19). In the PAS of the EFG: ⎡ −1 + R⎣ = 0 2 0
⎤ 0 0 −1 − 0 ⎦ . 0 2
(A.1)
Coefficients p, q and r, of the secular equation (2.18) can be expressed explicitly by polar angles of m in the PAS of the EFG: 2 p = −10 − 2R 2 1 + , 3 q = −8R(3 cos2 − 1 + sin2 cos 2) , 2 2 r = 9 + 2R 2 (6 sin2 − 5 + 4 sin2 cos 2 + 2 cos 2) + R 4 1 + , 3
(A.2)
K. Szyma´nski / Physics Reports 423 (2006) 295 – 338
Invariants obtained from powers of the EFG tensor expressed by p, q and r coefficients: q , m··m=− 16 (p + 4)2 − 4r , m · 2 · m = 64 16R(8R 2 + 3p + 30) + 3q(p + 10) m · 3 · m = , 128 2 3(4r − (p + 4) )(p + 10) − 4qR(8R 2 + 3p + 30) . m · 4 · m = 512 Angular averages of (A.3), when m is randomly oriented with respect to the PAS of the EFG:
335
(A.3)
m · · m = 0 , m · 2 · m =
2 2 1 15 (3 + )R
,
m · 3 · m = 41 (1 − 2 )R 3 , m · 4 · m =
2 2 4 1 24 (3 + ) R
.
(A.4)
(m · · m)2 = 16 (3 + 2 )R 2 .
(A.5)
Another useful average:
The trace of the powers of the EFG tensor: (1 + )n + (1 − )n Tr n = 1 + . Rn (−2)n
(A.6)
Particularly Tr = 0 , Tr 2 = − 43 (p + 10) = 21 (3 + 2 )R 2 , Tr 3 = 43 (1 − 2 )R 3 , Tr 4 =
9 32 (p
+ 10)2 = 18 (3 + 2 )2 R 4 .
(A.7)
A.3. Transmission integral Transmission integral proportional to the dip of the transmission line [89]: 1 +∞ t 1 [−t/(z2 +1)] I (t) = 1 − e−t/2 , ·e dz = 1 − I0 −∞ z2 + 12 2
(A.8)
where In is the Bessel function [28], for x real and n integer defined as a series: In (x) = (x/2)n
∞
(x 2 /4)k . k!(n + k)!
(A.9)
k=0
For small t I (t) = 21 t −
3 2 16 t
+
5 3 96 t
−
35 4 3072 t
+ ··· .
For large t Eq. (A.8) becomes a series: 9 75 1 1 + + + · · · . 1+ I (t) = 1 − √ 4t 32t 2 128t 3 t
(A.10)
(A.11)
336
K. Szyma´nski / Physics Reports 423 (2006) 295 – 338
A.4. Resonant filter (some details) Explicit form of the coefficients xi () and pi () entering into (9.12) and (9.13) are the following [91]. x0 () ≡ x0 =
z , cos2
x1 ≡ x1 () =
1 , 2(1 + 4x02 − 4z2 )
x2 () =
x02 x2 sin2 [12 ex0 cos2 + 4I0 (z) sin2 + (ez (−15 + 13x0 + (3 − 16x0 ) cos 2) I0 (z) + 3x0 cos 4 − 8x02 sin2 )] .
(A.12)
Parameter was defined in (9.14). Parameter z ≡ z() is given implicitly by I1 (z) 1 1 + = , 2z I0 (z) cos2
(A.13)
where In is the Bessel function (A.9). 2 ex0 p0 = , x0 I0 (z) p1 = −
ex0 (4 e2x0 + I02 (z)) , 4 23 x03 I03 (z)
p2 =
x0 ex0 sin2 [2(I0 (z) + ex0 ) 2 4I02 (z) + 2(−I0 (z) + 2 ex0 + ez − 2 ez x0 ) cos 2 + ez (−10 + 3x0 + x0 cos 4)] .
References [1] P.R. Weiss, Phys. Rev. 73 (1948) 470; P.R. Weiss, Phys. Rev. 74 (1948) 1478 [2] R.V. Pound, Phys. Rev. 79 (1950) 685. [3] C. Kikuchi, Phys. Rev. 77 (1950) L746. [4] E.F. Carr, C. Kikuchi, Phys. Rev. 78 (1950) L470. [5] R.P. Penrose, Nature 163 (1949) 922. [6] B. Bleaney, D.J.E. Ingram, Nature 164 (1949) 116. [7] R.V. Pound, Phys. Rev. 74 (1948) 1247. [8] R.V. Pound, Proc. Phys. Soc. 61 (1948) 576. [9] S. Shtrikman, Solid State Commun. 5 (1967) 701. [10] S. Shtrikman, S. Somekh, Rev. Sci. Instr. 40 (1969) 1151. [11] J.P. Stampfel, P.A. Flinn, Mössb. Effect Meth. 6 (1971) 95. [12] F. Varret, P. Imbert, G. Jehanno, R. Saint-James, Phys. Stat. Sol. (a) 27 (1975) K99. [13] U. Gonser, H. Fischer, in: U. Gonser (Ed.), Mössbauer Spectroscopy II, Springer, Berlin, 1981, p. 99. [14] K. Szyma´nski, L. Dobrzy´nski, B. Prus, M.J. Cooper, Nucl. Instr. and Meth. B 119 (1996) 438. [15] J. Jäschke, H.D. Rüter, E. Gerdau, G.V. Smirnov, W. Sturhahn, J. Pollmann, Nucl. Instr. and Meth. B 155 (1999) 189. [16] G.V. Smirnov, U. van Bürck, A.I. Chumakov, A.Q.R. Baron, R. Rüffer, Phys. Rev. B 55 (1997) 5811. [17] H.H.F. Wegener, F.E. Obenshain, Z. Phys. 163 (1961) 17. [18] H. Frauenfelder, D.E. Nagle, R.D. Taylor, D.R.F. Cochran, W.M. Visscher, Phys. Rev. 126 (1962) 1065. [19] R. Zimmermann, Chem. Phys. Lett. 34 (1975) 416. [20] K. Szyma´nski, J. Phys.: Condens. Matter 12 (2000) 7495. [21] K. Szyma´nski, Nucl. Instr. and Meth. B 171 (2000) 515. [22] K. Szyma´nski, Habilitation, Warsaw University, unpublished, 2000.
(A.14)
K. Szyma´nski / Physics Reports 423 (2006) 295 – 338 [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82]
337
P. Zory, Phys. Rev. A 140 (1965) 1401. R. Ingalls, K. Ono, L. Chandler, Phys. Rev. 172 (1968) 295. R. Zimmermann, Nucl. Instr. and Meth. 128 (1975) 537. R. Zimmermann, in: B.V. Thosar, P.K. Iyengar (Eds.), Advances in Mössbauer Spectroscopy Applications to Physics, Chemistry and Biology, Elsevier, Amsterdam, 1983, p. 273. H. Spiering, in: G.J. Long (Ed.), Mössbauer Spectroscopy Applied to Inorganic Chemistry, vol. 1, Plenum Press, New York, 1984, p. 77. I.A. Stegun, in: M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, USA, 1972. K. Szyma´nski, D. Satuła, L. Dobrzy´nski, J. Phys.: Condens. Matter 11 (1999) 881. H.-D. Pfannes, H. Fischer, Appl. Phys. 13 (1977) 317. H.-D. Pfannes, R.M. Paniago, Hyperfine Interactions 71 (1992) 1499. K. Szyma´nski, Nucl. Instr. and Meth. B 134 (1998) 405. L.C. Brown, P. Parker, Phys. Rev. 100 (1955) 1764. P.M. Parker, J. Chem. Phys. 24 (1956) 1096. E. Matthias, W. Schneider, R.M. Steffen, Phys. Rev. 125 (1962) 261. L. Häggström, Report UUIP-851, Uppsala, 1974. A.R. Edmonds, in: R. Hofstadter (Ed.), Angular momentum in Quantum Mechanics, Princeton, 1974. http://mathworld.wolfram.com/VietasFormulas.html S.W. Karyagin, Fizika Tverdogo Tela 8 (1966) 493. J. van Dongen Torman, R. Jagannathan, J.M. Trooster, Hyperfine Interactions 1 (1975) 135. K. Szyma´nski, Nucl. Instr. and Meth. B 168 (2000) 125. T.M. Lin, R.S. Preston, Mössb. Effect Meth. 9 (1974) 25. D.S. Sivia, Data Analysis: A Bayesian Tutorial, Clarendon Press, Oxford, 1996. R.A. Brand, G. Le Caër, Nucl. Instr. and Meth. 34 (1988) 272. L. Dobrzy´nski, K. Szyma´nski, D. Satuła, Nukleonika 49 (Suppl. 3) (2004) S89. J.M. Greneche, F. Varret, J. Phys. Lett. 43 (1982) L-233. J.M. Greneche, F. Varret, J. Phys. C: Solid State Phys. 15 (1982) 5333. R.L. Collins, J. Chem. Phys. 42 (1965) 1072. E. Kreber, U. Gonser, Nucl. Instr. and Meth. 121 (1974) 17. G.N. Belozerskii, Yu.P. Khimich, V.N. Gitsovich, Phys. Stat. Sol. (b) 79 (1977) K125. D.I.C. Pearson, J.M. Williams, Nucl. Instr. and Meth. 164 (1979) 363. G. Le Caër, J.M. Dubois, J. Phys. E: Sci. Instr. 12 (1979) 1083. G. Le Caër, J.M. Dubois, H. Fischer, I.U. Gonser, H.G. Wagner, Nucl. Instr. and Meth. B 5 (1984) 25. R.A. Brand, Nucl. Instr. and Meth. B 28 (1987) 398. R.A. Brand, Nucl. Instr. and Meth. B 28 (1987) 417. M. Eibschütz, M.E. Lines, Phys. Rev. B 25 (1982) 4256. M. Eibschütz, M.E. Lines, Phys. Rev. B 26 (1982) 2288. M. Eibschütz, M.E. Lines, H.S. Chen, M. Tsuyoshi, J. Phys. F: Met. Phys. 14 (1984) 505. N. Blaes, H. Fischer, U. Gonser, Nucl. Instr. and Meth. B 9 (1985) 201. P.G.L. Williams, G.M. Bancroft, Mössb. Effect Meth. 7 (1971) 39. L.J. D¸abrowski, J. Piekoszewski, J. Suwalski, Nucl. Instr. and Meth. 91 (1971) 93. A. Ito, M. Tanaka, S. Morimoto, T. Tokoro, Nat. Sci. Rep., Ochanomizu Univ. 27 (1976) 149. L.J. D¸abrowski, J. Piekoszewski, J. Suwalski, Nucl. Instr. and Meth. 103 (1972) 545. D. Barb, D. Tarina, A. Ito, S. Morimoto, J. Phys. C: Solid State Phys. 14 (1981) 497. R.H. Nussbaum, R.M. Housley, Nucl. Phys. 68 (1965) 145. U. Gonser, R.W. Grant, Appl. Phys. Lett. 9 (1966) 18. M. Ghafari, R. Gomez Escoto, Hyperfine Interactions 110 (1997) 51. M.T. Hirvonen, T. Katila, K. Riski, J. Tanttu, Phys. Rev. B 24 (1981) 11. P. Helistö, M.T. Hirvonen, T. Katila, K. Riski, Hyperfine Interactions 12 (1982) 27. N. Blum, L. Grodzins, Phys. Rev. 136 (1964) 133. H. Ullrich, J. Hesse, J. Magn. Magn. Mater. 45 (1984) 315. R.M. Housley, Nucl. Instr. and Meth. 62 (1968) 321. U. Gonser, H. Sakai, W. Keune, J. Phys. (Paris) Coll. (Suppl.) 12 (1976) C6-709. J.G. Dash, R.D. Taylor, D.E. Nagle, P.P. Craig, W.M. Visscher, Phys. Rev. 122 (1961) 1116. G.J. Ehnholm, T.E. Katila, O.V. Lounasmaa, P. Reivari, Phys. Lett. A 25 (1967) 758. A.Ya. Parshin, V.P.J. Peshkov, J. Low Temp. Phys. 15 (1974) 417. J.M. Daniels, J.M. Catignola, Can. J. Phys. 40 (1962) 1342. G.M. Kalvius, T. Katila, O.V. Lounasmaa, Mössb. Effect Meth. 5 (1970) 231. M. Henry, F. Varret, Phys. Stat. Sol. (a) 44 (1977) 601. N. Sakai, H. Sekizawa, Hyperfine Interactions 29 (1986) 1521. K. Szyma´nski, Proceedings of the Conference of ICAME 95, vol. 50, 1995, p. 891. K. Szyma´nski, Hyperfine Interactions 97/98 (1996) 573.
338 [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103]
K. Szyma´nski / Physics Reports 423 (2006) 295 – 338
K. Szyma´nski, L. Dobrzy´nski, J. Waliszewski, A. Wi´sniewski, Acta Phys. Pol. A 59 (1989) 411. K. Szyma´nski, L. Dobrzy´nski, R. Burzy´nska, Hyperfine Interactions 59 (1990) 477. K. Szyma´nski, L. Dobrzy´nski, D. Satuła, Hyperfine Interactions 97/98 (1996) 76. G.V. Smirnov, Hyperfine Interactions 125 (2000) 91. C. L’abbé, J. Meersschaut, W. Sturhahn, J.S. Jiang, T.S. Toellner, E.E. Alp, S.D. Bader, Phys. Rev. Lett. 93 (2004) 037201. S. Margulies, J.R. Ehrman, Nucl. Instr. and Meth. 12 (1961) 131. R. Mössbauer, W.H. Wiedemann, Z. Phys. 159 (1960) 33. G.V. Smirnov, Hyperfine Interactions 123/124 (1999) 31. K. Szyma´nski, Nucl. Instr and Meth. (2005), in press, doi:10.1016/j.nimb.2005.09.007. K. Szyma´nski, L. Dobrzy´nski, B. Prus, Yu.N. Mal’tsev, B.I. Rogozev, M.Y. Silin, Hyperfine Interactions C 3 (1998) 265. T. Shinjo, F. Itoh, H. Takaki, Y. Nakamura, N. Shikazono, J. Phys. Soc. Japan 19 (1964) 1252. L. Takacs, M.C. Cadeville, I. Vincze, J. Phys. F: Met. Phys. 5 (1975) 800. J.B. Jeffries, N. Hershkowitz, Phys. Lett. A 30 (1969) 187. J.M. Williams, J.S. Brooks, Nucl. Instr. and Meth. 128 (1975) 363. U. Gonser, H. Fischer, Hyperfine Interactions 72 (1992) 31. U. Gonser, F. Aubertin, S. Stenger, H. Fischer, G. Smirnov, G. Klingelhöfer, Hyperfine Interactions 67 (1991) 701. R.M. Housley, U. Gonser, Phys. Rev. 171 (1968) 480. S.S. Hanna, J. Heberle, G.J. Perlow, R.S. Preston, D.H. Vincent, Phys. Rev. Lett. 4 (1960) 513. U. Gonser, R.M. Housley, Phys. Lett. A 26 (1968) 157. K.-U. Neumann, J. Crangle, J.G. Smith, N.K. Zayer, K.R.A. Ziebeck, J. Magn. Magn. Mater. 137 (1994) 264. K. Szyma´nski, M. Biernacka, L. Dobrzy´nski, K-U. Neumann, K. Perzy´nska, D. Satuła, P. Zaleski, K.R.A. Ziebeck, J. Magn. Magn. Mater. 195 (1999) 651. [104] K. Szyma´nski, B. Kalska, D. Satuła, L. Dobrzy´nski, A. Broddefalk, R. Wäppling, P. Nordblad, J. Magn. Magn. Mater. 251 (2002) 271. [105] H.D. Pfannes, H. Fischer, Appl. Phys. 13 (1977) 317. [106] J. Lauer, W. Keune, Phys. Rev. Lett. 48 (1982) 1850. [107] M. Rots, R. Coussement, J. Claes, L. Hermans, Hyperfine Interactions 11 (1981) 185. [108] H. Fischer, U. Gonser, R.S. Preston, H.G. Wagner, J. Magn. Magn. Mater. 9 (1978) 336. [109] F. Tanczikó, L. Deák, D.L. Nagy, L. Bottyán, Nucl. Instr. and Meth. B 226 (2004) 461. [110] K. Szyma´nski, K. Re´cko, L. Dobrzy´nski, D. Satuła, J. Phys.: Condens. Matter 11 (1999) 6451. [111] G. Le Caër, R.A. Brand, Hyperfine Interactions 71 (1992) 1507. [112] G. Le Caër, R.A. Brand, J. Phys.: Condens. Matter 10 (1998) 10715. [113] I. Turek, J. Magn. Magn. Mater. 98 (1991) 119. [114] I. Vincze, F. van der Woude, J.M. Friedt, Phys. Rev. B 33 (1991) 5050. [115] R.J. Pollard, Z.S. Wronski, A.H. Morrish, Phys. Rev. B 29 (1984) 4864. [116] C.L. Chien, H.S. Chen, J. Phys. (Paris) Coll. 40 (1979) C2/118. [117] K. Szyma´nski, D. Satuła, L. Dobrzy´nski, J. Phys.: Condens. Matter 11 (1999) 881. [118] W.M. Xu, W. Steiner, M. Reissner, A. Pösinger, M. Acet, W. Pepperhoff, J. Magn. Magn. Mater. 104–107 (1992) 2023. [119] M. Acet, T. Schneider, H. Zähres, W. Stamm, E.F. Wassermann, W. Pepperhoff, Physica B 161 (1989) 63. [120] K. Szyma´nski, D. Satuła, L. Dobrzy´nski, K. Perzy´nska, M. Biernacka, P. Zaleski, J. Magn. Magn. Mater. 263 (2001) 56. [121] D. Satuła, K. Szyma´nski, L. Dobrzy´nski, K. Re´cko, J. Waliszewski, Nukleonika 40 (Suppl.) (2003) S71. [122] K. Szyma´nski, D. Satuła, L. Dobrzy´nski, Hyperfine Interactions 156/157 (2004) 21. [123] K. Szyma´nski, D. Satuła, L. Dobrzy´nski, E. Voronina, E.P. Yelsukov, T. Miyanaga, Phys. Rev. B 72 (2005) 104409-1–104409-12. [124] K. Szyma´nski, L. Dobrzy´nski, A. Andrejczuk, R. Chrenowicz, A. Vantomme, S. Degroote, G. Langouche, J. Phys.: Condens. Matter 8 (1996) 5317. [125] K. Szyma´nski, L. Dobrzy´nski, J. Magn. Magn. Mater. 140–144 (1995) 107. [126] J. Pelloth, R.A. Brand, W. Keune, Proceedings of the Conference of ICAME 95, vol. 50, 1995, p. 315. [127] E.F. Wassermann, B. Rellinghaus, Th. Roessel, J. Kästner, W. Pepperhoff, Eur. Phys. J. B 5 (1998) 361. [128] B. Rellinghaus, J. Kästner, T. Schneider, E.F. Wassermann, P. Mohn, Phys. Rev. B 51 (1995) 2983. [129] W.B. Pearson (Ed.), in: Handbook of Lattice Spacing and Structure of Metals, vol. 2, Pergamon, Oxford, 1967. [130] H. Bunzel, E. Kreber, U. Gonser, J. Phys. (Paris) Coll. (Suppl.) 35 (1974) C6-609. [131] N. Lundquist, H.P. Meyer, Ark. Fiz. 20 (1961) 463. [132] D. Satuła, K. Szyma´nski, J. Waliszewski, L. Dobrzy´nski, B. Prus, Mol. Phys. Rep. 30 (2000) 151. [133] K. Szyma´nski, D. Satuła, L. Dobrzy´nski, M. Biernacka, K. Perzynska, P. Zaleski, Hyperfine Interactions 141/142 (2002) 231. [134] K. Szyma´nski, L. Dobrzy´nski, Nucl. Instr. and Meth. B 173 (2001) 383. [135] P.J. Mohr, B.N. Taylor, Rev. Mod. Phys. 77 (2002) 1. [136] in: J.W. Robinson (Ed.), Handbook of Spectroscopy, CRC Press, Boca Raton, FL, 1981. [137] P. Dufek, P. Blaha, K. Schwartz, Phys. Rev. Lett. 75 (1995) 3545. [138] Yu.V. Shvyd’ko, M. Lerche, J. Jäschke, M. Lucht, E. Gerdau, M. Gerken, H.D. Rüter, H.-C. Wille, P. Becker, E.E. Alp, W. Sturhahn, J. Sutter, T.S. Toellner, Phys. Rev. Lett. 85 (2000) 495. [139] G. Audi, A.H. Wapsta, Nucl. Phys. A 595 (1995) 409 Also: http://physics.nist.gov/Comp [2002, April 22]. [140] R.R. Kinsey, Trans. Am. Nucl. Soc. 75 (1996) 120. [141] J.H. Hubbell, S.M. Seltzer, Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients, 1997. Available: http://physics.nist.gov/xaamdi.
Physics Reports 423 (2006) 339
Contents of Volume 423 A. Lue The phenomenology of Dvali–Gabadadze–Porrati cosmologies
1
Y. Oh, K. Nakayama, T.-S.H. Lee Pentaquark H1 ð1540Þ production in cN ! KKN reactions
49
M. Gran˜a Flux compactifications in string theory: A comprehensive review
91
E. Gourgoulhon, J.L. Jaramillo A 311 perspective on null hypersurfaces and isolated horizons
159
K. Szyman´ski Polarized radiation in Mo¨ssbauer spectroscopy
295
Contents of volume
339
