Artificial Black Holes

) = 0, with g — det <7M„, in an effective curved spacetime with the metric g^v being entirely determined by the local speed of sound c(x) = — V ^ T T O ^ X ) ,

(2.3)

Luis Garay

39

and the background stationary velocity field v = -Vtf. m

(2.4)

Up to a conformal factor, this effective metric has the form 9w>

_(c2_v2) -V

-vT I

(2.5)

This class of metrics can possess event horizons. For instance, if an effective sink for atoms is generated at the center of a spherical trap (such as by an atom-laser out-coupling technique [17]), and if the radial potential profile is suitably arranged, we can produce densities p(r) and flow velocities v(x) = —u(r)x/r such that the quantity c2 — v 2 vanishes at a radius r = rh, being negative inside and positive outside. The sphere at radius r^ is a sonic event horizon completely analogous to those appearing in general relativistic black holes, in the sense that sonic perturbations cannot propagate through this surface in the outward direction [1, 2, 4]. This can be seen explicitly by writing the equation for the radial null geodesies of the metric
A>max{—,

\.

(2.6)

Otherwise they do not behave as sound waves since they lie outside the regime of validity of the hydrodynamic approximation. These short-wavelength modes must be described by the full Bogoliubov equations, which allow signals to propagate faster than the local sound speed, and thus permit escape

40

Acoustic black holes in dilute BECs

from sonic black holes. So, to identify a condensate state ^ s as a sonic black hole, there must exist modes with wavelengths larger than these lower limits, but also smaller than the black hole size. As it stands, this description is incomplete. The condensate flows continually inwards and therefore at r = 0 there must be a sink that takes atoms out of the condensate. Otherwise, the continuity equation V(pv) = 0, which must hold for stationary configurations, will be violated. From the physical point of view, such a sink can be accomplished by means of an out-coupler laser beam at the origin. Such out-couplers are the basic mechanisms for making trapped condensates into "atom lasers," and they have already been demonstrated experimentally by several groups. A tightly focused laser pulse changes the internal state of the atoms at a particular point in the trap, and can also be made to give them a large momentum impulse. This ejects them so rapidly through the always dilute condensate cloud that they do not significantly disturb it; effectively, they simply disappear. We have analyzed several specific systems which may be suitable theoretical models for future experiments, and have found that the qualitative behavior is analogous in all of them. We note that black holes which require atom sinks are both theoretically and experimentally more involved. Moreover, maintaining a steady supersonic flow into a sink may require either a very large condensate or some means of replenishment. We will therefore first discuss an alternative configuration which may be experimentally more accessible and whose description is particularly simple: a condensate in a very thin ring that effectively behaves as a periodic one-dimensional system (see figure 2.1). Under conditions that we will discuss, the supersonic region in a ring may be bounded by two horizons: a black hole horizon through which phonons cannot exit, and a 'white hole' horizon through which they cannot enter. Then we will analyze another simple one-dimensional model, of a long straight condensate with an atom sink at the center (see figure 2.5).

2.3

Black/white holes in a ring

In a sufficiently tight ring-shaped external potential of radius R, motion in radial (r) and axial (z) cylindrical coordinates is effectively frozen. We can then write the wave function as ty(z, r, 9, r) = f(z,r) $(#, r) and normalize $ to the number of atoms in the condensate J ^ d f l |$(#)| 2 = N, where, with the azimuthal coordinate 0, we have introduced the dimensionless time

41

Luis Garay

White hole horizon

Condensate cioud Figure 2.1: Ring-like configuration for an acoustic black hole: The tight ringshaped configuration, with both black and white horizons, and no singularity T -— (U/mR2)i. dimensional:

The Gross-Pitaevskii equation thus becomes effectively one/

i

it

\

(2.7) where U = 47roiVi?2

dz dr r | / ( 2 , r ) | 4

(2,8)

and Vext(0) is the dimensionless effective potential (in which we have already included the chemical potential) that results from the dimensional reduction. The stationary solution can then be written as <M#, r) = y/p(ff) e ' ^ 9 " ^ and the local dimensionless angular speed of sound as c(0) = sJQ p{B)/N. Periodic boundary conditions around the ring require the "winding number" w = (l/27r) J 0 * d9 v(8) to be an integer. The qualitative behavior of horizons in this system is well represented by the three-parameter family of speed-of-sound and flow-velocity fields [both of them related by the continuity equation de{pv) = 0] 2

c(8)

11

= ^-(l

+ bcos8),

v{9) =

2ir c(9)2

(2.9)

42

Acoustic black holes in dilute BECs

where b G [0,1]. Requiring that $ s (0, r) be a stationary solution to GrossPitaevskii equation then determines how the trapping potential must be modulated as a function of 9. All the properties of the condensate, including whether and where it has sonic horizons, and whether or not they are stable, are thus functions of U, b and w. For instance, if we require that the horizons be located at Oh = ±7r/2, which imposes the relation U = 2n w2 (1 — 62), then we must have c2 - v2 positive for 9 € (—7r/2,7r/2), zero at #/, = ±vr/2, and negative otherwise, provided that U < 27r w2. The further requirement that perturbations on wavelengths shorter than the inner and the outer regions are indeed phononic implies U » 27T, which in turn requires w » 1 and 1 ~> b » 1/w2. In fact, detailed analysis shows that w > 5 is sufficient. The mere existence of a black hole solution does not necessarily mean that it is physically realizable: it should also be stable over sufficiently long timescales. Since stability must be checked for perturbations on all wavelengths, the full Bogoliubov [11] spectrum must be determined. For large black holes within large, slowly varying condensates, this Bogoliubov problem may be solved using WKB methods that closely resemble those used for solving relativistic field theories in true black hole spacetimes [8]. A detailed adaptation of these methods to the Bogoliubov problem will be presented elsewhere [18]. The results are qualitatively similar to those we have found for black holes in finite traps with low winding number, where we have resorted to numerical methods because, in these cases, WKB techniques may fail for just those modes which threaten to be unstable. Our numerical approach for our three-parameter family of black/white holes in the ring-shaped condensate has been to write the Bogoliubov equations in discrete Fourier space, and then truncate the resulting infinitedimensional eigenvalue problem. Indeed, writing the wave function as $ = $s + tpeifd8vW,

(2.10)

decomposing the perturbation ip in discrete modes ip(0, T) = J2

e~iUT eine A,,„ uu>n(0) + eiu'T e~ine A^n < n ( 0 ) ,

(2.11)

and substituting into the Gross-Pitaevskii equation we obtain a set of algebraic equations for the modes uWi„ and uW]„. Eliminating Fourier components above a sufficiently high cutoff Q has negligible effect on possible instabilities, which can be shown to occur at relatively long wavelengths. We face then an eigenvalue problem for a [2(Q +

Luis Garay

43

1)] x [2(Q + 1)] matrix whose numerical solution, together with the normalization condition / d0 («„. ] n V,ti' - ^ • , n * W ) = <5nn' <W,

(2-12)

provides the allowed frequencies. Real negative eigenfrequencies for modes of positive norm are always present, which means that black hole configurations are energetically unstable, as expected. This feature is inherent in supersonic flow, since the speed of sound is also the Landau critical velocity. In a sufficiently cold and dilute condensate, however, the time scale for dissipation may in principle be made very long, and so these energetic instabilities need not be problematic [19]. More serious are dynamical instabilities, which occur for modes with complex eigenfrequencies. Since the Bogoliubov theory is based on a second quantized Hamiltonian that is Hermitian, there are certainly no complex energy eigenvalues; but the natural frequencies of normal modes can indeed be complex [in which case the usual rule, that energy eigenvalues are h(n +1/2) times the mode frequencies, simply breaks down]. A discussion of the quantum mechanics of dynamical instability is presented in section 2.5. For the time being, it suffices to note that complex eigenfrequencies are indeed genuine physical phenomena, and by no means a numerical artifact. For sufficiently high values of the cutoff {e.g., Q > 25 in our calculations), the complex eigenfrequencies obtained from the truncated eigenvalue problem become independent of the cutoff within the numerical error. The existence and rapidity of dynamical instabilities depend sensitively on (U,b,w). For instance, see figure 2.2 for a contour plot of the maximum of the absolute values of the imaginary part of all eigenfrequencies for w = 7, showing that the regions of instability are long, thin fingers in the (U, b) plane. It also shows the size of the largest absolute value of the instabilities for each point on the dashed curve. This figure illustrates the important fact that the size of the imaginary part, which gives the growth rate of the instability, increases quite rapidly with b (starting from zero) although it remains small as compared with the real part. The stability diagram of figure 2.2 suggests a strategy for creating a sonic black hole from an initial stable state. Within the upper subsonic region, the vertical axis 6 = 0 corresponds to a homogeneous persistent current in a ring, which can in principle be created using different techniques [20]. Gradually changing U and b, it is possible to move from such an initial state

44

Acoustic black holes in dilute BECs

0

0.1

0.2

0.3

0.4 b

Figure 2.2: Stability diagram for winding number w = 7. Solid dark-grey areas represent the regions of stability. Smaller plots at higher resolution confirm that the unstable 'fingers' are actually smooth and unbroken. Points on the dashed curve are states with horizons at &h — ±vr/2, so that the black/white hole Gils half the ring.

to a black/white hole state, along a path lying almost entirely within the stable region, and only passing briefly through instabilities where they are sufficiently small to cause no difficulty. Indeed, we have simulated this process of adiabatic creation of a sonic black/white hole by solving numerically (using the split operator method) the time-dependent Gross-Pitaevskii equation that provides the evolution of the condensate when the parameters of the trapping potential change so as to move the condensate state along various paths in parameter space. One of these paths is shown in figure 2.2 (light-grey solid line): we start with a current at w — 7, b = 0, and sufficiently high U [figure 2.4(a)]; we then

Luis Garay

45

0

0.1

0.2

0.3

0.4

0.5

b Figure 2.3: Stability diagram for black/white aiong the dashed line of figure 2.2.

holes of maximum

size, i.e.,

increase b adiabatically keeping U fixed until an appropriate value is reached [figure 2.4(b)]; finally, keeping b constant, we decrease U adiabatically (which can be physically implemented by decreasing the radius of the ring trap), until we meet the dashed contour for black holes of comfortable size [figure 2.4(c)]. Our simulations confirm that the small instabilities which briefly appear in the process of creation do not disrupt the adiabatic evolution. The final quantum state of the condensate, obtained by this procedure, indeed represents a stable black/white hole. We have further checked the stability of this final configuration by numerically solving the Gross-Pitaevskii equation for very long periods of time (as compared with any characteristic time scale of the condensate) and for fixed values of the trap parameters. This evolution reflects the fact that no imaginary frequencies are present, as predicted from the mode analysis, and that the final state is indeed stationary [figure 2.4(d)]. Once the black/white hole has been created, one could further change the parameters {U, b) so as to move between the unstable 'fingers' into a stable region of higher b (a deeper hole).

Acoustic black holes in dilute BECs

46

i-v(9)2/c(e)2

'<er

0.3

C«)

(b)

(o)

(d)

(e)

W

(g)

CO

rwn

CO

0

e/271

l

°

e/271

1

Figure 2.4: Simulation of creation of a stable black/white hole and subsequent evolution into an unstable region. Figures a-d are snapshots taken at the initial time (a), at an intermediate time, still within the subsonic region (b), when the black/white holes of maximum size is approached (c), and after a long time in that configuration (d). Then the parameters are changed along the dashed curve of figure 2.2 to enter an unstable region (e) and kept there (f-i). It can be observed that a perturbation grows at the black hole horizon and travels rightwards until it enters the white hole horizon.

4?

Luis Garay

Instead of navigating the stable region of parameter space, one could deliberately enter an unstable region [figure 2.4(e)-2.4(i)]. In this case, the black hole should disappear in an explosion of phonons, which may be easy to detect experimentally. Such an event might be related to the evaporation process suggested for real black holes, in the sense that pairs of quasiparticles are created near the horizon in both positive and negative energy modes. We will explain this point briefly in Sec. 2.5

2.4

Sink-generated black holes

We now present a simple model that exhibits the main qualitative features of more general situations and that can be studied analytically. Although in this model we study a condensate of infinite size, in more realistic models or experiments, it will suffice to take condensates which are sufficiently large, since the stability pattern is not significantly affected by the (large but finite) size of the condensate.

Outeoupler beam

•)

'Singularity'

Figure 2.5: SinJc configuration for an acoustic black hole: The tight cigarshaped conigmation, with two black hole horizons and a 'singularity' where condensate is out-coupled. Arrows indicate condensate Sow velocity, with longer arrows for faster Sow. Let us consider a tight cigar-shaped condensate of infinite size such that the motion in the (y, z) plane is effectively frozen. In appropriate dimensionless units, the effectively one-dimensional Gross-Pitaevskii equation thus

48

Acoustic black holes in dilute BECs

CCo-.

CO

-L

L

L

L+e

Figure 2.6: Profile for the speed of sound c(x) for the one-dimensional sinkgenerated black hole. becomes:

idT$=[--d2x

+ vext + u\\2).

(2.13)

In this equation, Vext is the dimensionless effective potential that results from the dimensional reduction, which already includes the chemical potential. In order to obtain a black hole configuration, let us choose the potential Vext so that it produces a profile for the speed of sound c(x) = \Jli p(x) of the form (figure 2.6) Co, \x\ < L c{x) = { co[l + (
(2.14)

with a > 1, and a flow velocity in the inward direction. The continuity equation then provides the flow velocity profile v(x) =

c(x)2

(2.15)

where VQ is the absolute value of the flow velocity in the inner region. As it stands this model fails to fulfill the continuity equation at x = 0. In order to take this into account, we will also introduce a sink of atoms at x = 0 that takes atoms out of the condensate (this can be physically implemented by means of a laser). From the mathematical point of view, it can be modeled by an additional term in the equation of the form — iE 5(x) which indeed

Luis Garay

49

induces loss of atoms at x = 0. boundary conditions of the form

Equivalently, it can be represented by

$(o + ,T)-$(rr,r) = o, $'(0+, r) - $'((T, r) = -2iE

$(0, r ) ,

(2.16)

which determine the flow velocity inside in terms of the characteristics of the out-coupler laser, namely, v0 = E. As a further simplifying assumption, we will assume that voe -C 1. Perturbations xp around this stationary state $ s = \fp^^ v^dx, such that $ = <3>s + %j) (note that for convenience we have chosen a different convention as compared with the ring in which $ = $ 6 +

_ I ^ + (V - ^ + Q v + c2 e » r v r,

(2.17)

Let us now expand the perturbation tp in modes

i> = Y, K * «<-,*(*)e_iWT + ^ v*(*)* e"*1 • Then, the modes uu^{x) equations

(2-18)

and vUtk{x) satisfy, in each region, the Bogoliubov

<*>«<•;,* = ~^K,k+

\

c 2

~ \ ) u^k + c2 e2if"v vuJt, c2

\v'U - (c2 - j ) vuJt - c2 c - * r • v .

(2.19)

The intermediate regions L < \x\ < L+e, provide the connection between the perturbation modes in the inner and outer regions. Once these connection formulas have been established, in the limit of small e, we will only need to study the inside and outside modes and their relation through such formulas. The case of an abrupt horizon, in which the background condensate velocity is steeply and linearly ramped within a very short interval, is obviously quite special; and it does not particularly resemble the horizon of a large black hole in Einsteinian gravity. But the connection formula that we derive for this case will qualitatively resemble those that are obtained, with considerably

Acoustic black holes in dilute BECs

50

more technical effort, for smoother horizons [18]. The results we will obtain for the global Bogoliubov spectrum of the condensate black hole will indeed be representative of more generic cases. The singular character of c"/c at \x\ = L, L + e can be mimicked by appropriate matching conditions at \x\ = L, L + e. Furthermore, the symmetry of the problem allows us to study only the region x > 0. These matching conditions for the perturbation ip, together with the form of the modes in the region L < x < L + e, provide the connection formulas between the "inside" (|x| < L) and "outside" (|a;| > L) modes (from now on we will drop the subindex u): uin,k{L) = -e u'0Ut>k(L) + - u 0Ut|fc (L), u

kk(L) = a u'out}lc(L),

and likewise for the modes ^in,outIn each of the regions (inside and outside), we can write uk{x) = uk e ^ - M H * - ^

Vk(x)

= Vk c i(*+M)(*-i).

( 2 .20)

Upon substitution of this expansion into the Bogoliubov equations (2.19), we obtain, for each region, the following set of algebraic equations: h~k uk + c2vk = 0,

c2 uk + h+ vk = 0,

(2.21)

where h^ = k2/2 + c2 ± (k\v\ + w). For these equations to have a solution, the determinant must vanish, thus providing the dispersion relation j + (c2 - i;2) k2 -2u\v\k-cj2

=0

(2.22)

which, for fixed w, is a fourth-order equation for k. For each of the four solutions Uk and Vk must be related by vk = hkuk

with

1 (k2 \ hk = — - I — + c2 - k\v\ - UJ 1 .

(2.23)

The constant coefficients uk can be regarded as normalization constants and will be set to unity. The possible solutions to the dispersion relation depend on whether w is real or complex. For complex frequencies, all four solutions are pure complex, two of them with positive imaginary part and two of

Luis Garay

51

them with negative one. For real frequencies, on the other hand, outside (c2 > v2), there are two real and two complex conjugate solutions. Of these two complex solutions, only one is allowed [the one with Im(A;) > 0] because the other grows exponentially. Inside (c2 < v2), for ui > w max there are two real and two complex conjugate solutions; for u < aj m a x there are four real solutions; the value w = w max is a bifurcation point. Since we are interested in the existence of dynamical instabilities, we will concentrate in the case in which u is complex. Then, as we have seen, the dispersion equation (2.22) has four complex solutions for k in each region. Inside, all four solutions k-mj, i = 1 . . .4 are in principle possible but outside those with Im(A;0Ut) < 0 will increase exponentially. Therefore (up to corrections coming from the finite size of the condensate which we ignore here) only modes associated with fc0ut,a> a = 1,2 such that Im(fc0Ut,Q) > 0 are allowed. Each mode uout,Q(a;) = e»(fcout,Q-«o/CT )(*-£) wjU match a linear combination u-inta(x) = Yli^ai Um,i{x) of modes u-mj(x) inside, and similarly for uout,a and uin,Q. After some straightforward calculations, it can be seen that these connecting coefficients Fai are given by the matrix Fai = ^ZjiM-1)^ Caj, where 1 M = "in,l "-jn,l

1

1

1

^in,2

*in,3

"-in,4 hjnA

"in,2 "-^,2 "in.S " i n ^

(2.24)

h'mA "in,4

1/ff - it kouUa a

Ua — (1/

CT

^out.a

~~ * e *'out,a)' l out,a

(2.25)

® "'out.a " , out,a

In these equations, kfni = km -x) allows us to study only the region x > 0, provided that we study the even and odd parity perturbations separately. For odd parity fluctuations [tp0(x,T) = —ip0(—x,T)}, the boundary conditions (2.16) become ipo(0, T) = 0 at all times r . This implies that the u and v components of ip

Acoustic black holes in dilute BECs

52

must separately satisfy the boundary condition. Since we can have any linear combination of the two solutions that decay outside the horizon, we therefore have a two-by-two matrix constraint. The condition that a non-zero solution exists is that ,

«in,l(0)

t

Uin,2(0)

= 0

(2.26)

[ t>in,l(0) Vini2(0) and therefore Y, Fu F2j (hinji - hin>j) e-<(*in.«+fctaJ)t

=

o.

(2.27)

For even fluctuations [rpe(x,T) = ipe(—x,T)], the boundary conditions (2.16) become %/}'e(0, T) + iv0 ipe(0, r) = 0, which implies that

5 3 F » F*i (^M -

h

^

k

^ *»J e-«*'-'+*'-j)1 = 0.

(2.28)

ij

For fixed L, U, VQ, and a, the quantities F, /ij n , and k\n that appear in equations (2.27) and (2.28) are only functions of w. Therefore the solutions to these equations are all the possible complex eigenfrequencies, which depend on the free parameters that determine the model, namely, the size 2L of the inner region, the speed of sound inside c0, the relative change of the speed of sound between the inner and the outer regionsCT,and the flow velocity inside v0 (related to the characteristics of the out-coupler laser). In practice, there are also other parameters of the condensate such as its size 2D (which has been made arbitrarily large) and the size of the intermediate regions e (which has been made arbitrarily small). Equations (2.27) and (2.28) can be solved numerically for different values of the parameters a, U, vo, and L. The numerical method employed is the following. The equations above have the form /(w; a,U, v0, L) = 0 , where / and w are in general both complex. We plot contours of constant absolute value of / in the complex u plane; where | / | approaches zero, we have an eigenfrequency. The distribution of complex solutions in the complex u plane depends on the size of the inner region L, for given a, U, and v0. Direct inspection of the numerical results shows that the number of instabilities increases by one when the black-hole size L is increased by -ir/k0 where ko — yjv% — c\. More explicitly, for L smaller than Tr/k0 — 6 (6 being much smaller than n/ko) there are no complex eigenfrequencies; for (L + 5)ko/n 6 [n, n-\-1] with n = 1,2 ...,

Luis Garay

53

we have n complex solutions except for L — (n+l/2)ir/k0, where we find n—1 complex solutions instead of n [i.e., there is one mode for which Im(w) = 0 within numerical resolution]. This can be easily interpreted qualitatively since the unstable modes are basically the bound states in the black hole, and the highest wave number k on the positive norm upper branch, for the barely bound state with w —> 0~, is exactly A;0. So the threshold is simply when the well becomes big enough to have a bound state; the small S displacement comes in because the horizon is not exactly a hard wall; and similarly for the extra bound state every n/k0. Thus stability can only be achieved for small sizes of the inner region, L < n/ko. As we discussed in Sec. II, the wavelength 2n/k of the perturbations must be smaller than this size, which implies k > 2n/L > 2ko. However, for these perturbations the hydrodynamic approximation, which requires k < 2fco, is not valid. Therefore there are no stable black hole configurations in a strict sense. The sizes of the imaginary parts of the complex solutions decrease as the size L of the interior of the black hole increases. Thus although a larger hole has more unstable modes, it is actually less unstable (and might even became quasi-stable in the sense that its instability time scale would be longer than the experimental duration).

2.5

Quasiparticle pair creation

In the language of second quantization, the perturbation field operator ip satisfies the linear equation ihj> = /i 0 (x) V + mc(x) 2 e ^ W $\

(2.29)

where ho(x) = - ^ - V 2 + Kxt + 2mc(x) 2 - /i. (2.30) 2m Taking into account that [^(x),r/>t(x')] = 6(x — x'), this equation can be written as ih ip — [ip, H], where the Bogoliubov Hamiltonian is H=

fdx. l ^ / i o W V + \mc{x)2

(e2i"(xW + e-^tytf)

(2.31)

The Hermiticity of the Bogoliubov linearized Hamiltonian implies that eigenmodes with complex frequencies appear always in dual pairs, whose frequen-

Acoustic black holes in dilute BECs

54

cies are complex conjugate. Indeed, expanding the perturbation field operator ip in modes V(x, t) = £

[ e - * ^ 4 ^ u„ tifc (x) + e ^ ' < , , «^ ifc (x)] ,

(2.32)

together with the normalization condition dx

/ '

{K'k,k uuk>,k' ~ vw'k,k vuk',k'j = Skk>,

(2.33)

we can write the linearized Hamiltonian as

H = hY^("k 4L.,fc Kk* + w*k AlhJt Aulk),

(2.34)

k

the only nonvanishing commutators among these operators being [A«k,k,Allnk,]

= 8kk,.

(2.35)

The asterisk on the subscript is important: the mode with frequency u*k is a different mode from the one with frequency ojk, and A'. k is not the Hermitian conjugate of AUk tk. It is therefore clear that none of these operators is actually a harmonic oscillator creation or annihilation operator in the usual sense. However, the linear combinations a-k

^(^k,k

+ Awl,k) ,

bk = ^ = « , t

+ Alj

(2.36)

and their Hermitian conjugates are true annihilation and creation operators, with the standard commutation relations, and in terms of these the Bogoliubov Hamiltonian becomes H = hJ2

[Re(wfc) (a\ak - b\bk) - lm{uk) ( a j $ + akbk)}

.

(2.37)

k

This interaction obviously leads to self-amplifying creation of positive and negative frequency pairs, which resembles the usual process of Hawking evaporation. That the Hamiltonian H is unbounded from below does not indicate anything unphysical about our model [21]: we have simply linearized about an unstable excited state of the nonlinear full Hamiltonian, which is bounded

Luis Garay

55

from below. Real negative frequencies u) indicate energetic instabilities, whereby the system will decay in the presence of dissipation. Complex frequencies, on the other hand, indicate dynamical instabilities. Classically, a dynamically unstable system will exponentially diverge from the initial stationary state if is perturbed, even without dissipation. Quantum mechanically, a dynamically unstable system has no normalizable stationary states, and it can be easily checked that this is indeed the case for the Bogoliubov Hamiltonian H above. If an initially stable system is driven into a state which is stationary but dynamically unstable at the classical (mean field) level, the initial state will have had finite Hilbert space norm, and hence under unitary evolution the final state will have the same norm. Thus it will not be a stationary state; one may say that quantum fluctuations will always trigger the dynamical instability. For a logarithmically long period of time, however, the linearized theory will still remain valid and, in this sense, our linearized description of quantum dynamical instabilities is sound.

2.6

Conclusions

We have seen that dilute Bose-Einstein condensates admit, under appropriate conditions, configurations that closely resemble gravitational black holes. We have analyzed in detail the case of a condensate in a ring trap, and proposed a realistic scheme for adiabatically creating stable sonic black/white holes and we have seen that there exist stable and unstable black hole configurations. We have also studied a model for a sink-generated sonic black hole in an infinite one-dimensional condensate. The dynamical instabilities can be interpreted as coming from quasiparticle pair creation, as in the well-known suggested mechanism for black hole evaporation. Generalizations to spherical or quasi-two-dimensional traps, with flows generated by laser-driven atom sinks, should also be possible, and should behave similarly. While our analysis has been limited to Bogoliubov theory, the further theoretical problems of back reaction and other corrections to simple mean field theory should be more tractable for condensates than for other systems analogous to black holes. We expect that experiments along the lines we have proposed, including both creation and evaporation of sonic black holes, can be performed with state-of-the-art or planned technology.

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Bibliography [1] W. G. Unruh, "Experimental black hole evaporation?", Phys. Rev. Lett., 46,(1981) 1351-1353. [2] W. G. Unruh, "Dumb holes and the effects of high frequencies on black hole evaporation", Phys. Rev. D 5 1 , 2827 (1995) [gr-qc/9409008]. [3] M. Visser, "Acoustic Propagation In Fluids: An Unexpected Example Of Lorentzian Geometry", gr-qc/9311028 (unpublished); S. Liberati, S. Sonego, and M. Visser, "Unexpectedly large surface gravities for acoustic horizons?", Class. Quantum Grav. 17, 2903 (2000) [gr-qc//0003105]. [4] M. Visser, "Hawking radiation without black hole entropy", Phys. Rev. Lett. 80, 3436 (1998) [gr-qc/9712016]; "Acoustic black holes: horizons, ergospheres, and Hawking radiation", Class. Quantum Grav. 15, 1767 (1998) [gr-qc/9712010]. [5] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973). [6] S. W. Hawking, "Black hole explosions?", Nature (London) 248, 30 (1974); "Particle creation by black holes", Commun. Math. Phys. 43, 199 (1975). [7] T. Jacobson, "Black hole evaporation and ultrashort distances", Phys. Rev. D 44 (1991) 1731. [8] S. Corley and T. Jacobson, "Black hole lasers", Phys. Rev. D59, 124011 (1999) [arXiv:hep-th/9806203]. 57

58

Acoustic black holes in dilute BECs S. Corley, "Computing the spectrum of black hole radiation in the presence of high frequency dispersion: An analytical approach", Phys. Rev. D 57 (1998) 6280 [arXiv:hep-th/9710075].

[9] For a review, see, e.g., T. Jacobson, "Trans-Planckian redshifts and the substance of the space-time river", Prog. Theor. Phys. Suppl. 136, 1 (1999) [arXiv:hep-th/0001085]. 10] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 (1995); K. B. Davis et al, Phys. Rev. Lett. 75, 3969 (1995). 11] See, e.g., F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999). 12] V. M. H. Ruutu et al., Nature (London) 382, 334 (1996); T. A. Jacobson and G. E. Volovik, Phys. Rev. D 58, 4021 (1998); G. E. Volovik, Pis'ma Zh. Eksp. Teor. Fiz. 69, 662 (1999); J E T P Lett. 69, 705 (1999). 13] B. Reznik, "Origin of the thermal radiation in a solid-state analogue of a black hole", Phys. Rev. D 64 (2000) 044044 [gr-qc/9703076]. 14] U. Leonhardt and P. Piwnicki, Phys. Rev. Lett. 84, 822 (2000); Phys. Rev. A 60, 4301 (1999). 15] M. R. Matthews et al., Phys. Rev. Lett. 83, 2498 (1999); L. Denget et al. Nature (London) 398, 218 (1999); S. Burger et al., Phys. Rev. Lett. 83, 5198 (1999). 16] L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 85, 4643 (2000); Phys. Rev. A 63, 023611 (2001). 17] M. R. Andrews et al., Science 275, 637 (1997); I. Bloch, T. W. Hansen, and T. Esslinger, Phys. Rev. Lett. 82, 3008 (1999); E. W. Hagley et al., Science 283, 1706 (1999). 18] L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, (unpublished). 19] P. O. Fedichev and G. V. Shlyapnikov, Phys. Rev. A 60, R1779 (1999).

Luis Garay

59

[20] R. Dum, J. I. Cirac, M. Lewenstein, and P. Zoller, Phys. Rev. Lett. 80, 2972 (1998); J. Williams and M. Holland, Nature (London) 401, 568 (1999). [21] See G. Kang, Phys. Rev. D 55, 7563 (1997) for a concise pedagogical illustration, and references therein, especially S. A. Fulling, Aspects of Quantum Field Theory in Curved Spacetime (Cambridge University Press, Cambridge, 1989); B. Schroer and J. A. Swieca, Phys. Rev. D 2, 2938 (1970).

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Chapter 3 Slow light Ulf Leonhardt University of Saint Andrews North Haugh, Saint Andrews, KY16 9SS Scotland E-mail: [email protected]

Abstract: Laboratory-based optical analogues of astronomical objects such as black holes may rely on the creation of light with an extremely low or even vanishing group velocity (slow light). These brief notes represent a pedagogical attempt towards elucidating this extraordinary form of light. This chapter is intended as a primer, an introduction to the subject for non-experts, not as a detailed literature review.

3.1

Motivation

Creating a black hole with humble human resources is certainly a fantastic idea, yet perhaps not entirely lunatic. Recent experimental progress in quantum optics, in particular the generation of slow light [1, 2, 3] has opened a 61

Slow light

62

route towards the formation of optical event horizons for light or towards other less dramatic but equally interesting phenomena. Most of the ideas have been discussed in a series of papers [4] and review articles [5]. The underlying atomic and optical physics is perhaps less familiar to the audience of this book. Therefore, it might be worthwhile to develop some of the physics behind slow light from basic principles that are hopefully known to most readers.

3.2

Light-matter interaction

Consider an atomic medium capable of interacting with light, for example a cell filled with Rubidium vapor or one of the alkali Bose-Einstein condensates. The atoms that constitute the medium are electrically neutral but polarizable by light. Since atoms are usually much smaller than an optical wavelength, light experiences the atoms as dipoles (as the lowest order in the multipole expansion of an electrically neutral charge distribution [6]). In the dipole approximation, the energy density of the light-matter interaction is given by the negative scalar product —P • E of the matter polarization P (dipole-moment density) and the electric field E [6]. Therefore, the Lagrangian of light and atomic matter reads in SI units i f = £-^(E2 -c2B2)+P-E

+ ^A.

(3.1)

The first term describes the free electromagnetic field with c denoting the speed of light in vacuum. The last term of the Lagrangian characterizes the internal dynamics of the atoms. We represent the electric field E and the magnetic field B in terms of the vector potential A in Coulomb gauge [6] E = -dtA,

B = V x A,

V • A = 0.

(3.2)

Let us assume, for simplicity, that both the electromagnetic field and the medium are uniform in two spatial directions in Cartesian coordinates, but may vary in the third direction z. Furthermore, we consider light with fixed polarization so that we can concentrate on one component A of the vector potential A. The Lagrangian simplifies to if = |

((dtA)2 - c2{dzA)2)

-PdtA

+ J?A.

(3.3)

63

Ulf Leonhardt We obtain from the Euler-Lagrange equation the wave equation

(fl?-<*£)£ = - I t f P .

(3.4)

Atoms have a well-defined level structure such that light, oscillating 10 15 times per second, must match the atomic transition frequencies, because otherwise the effect of E in —P • E is rapidly washed out. Given a certain frequency range of light, the optical field thus interacts with a selected few of the atomic levels, which greatly simplifies matters. Let us describe the atoms quantum-mechanically, while regarding the electromagnetic field as classical. The simplest relevant atomic system involves just two levels, say the ground state \a) and the excited state \b). The Hamiltonian of the two-level atom is simply H

= huab\b)(b\-^f[A

A

= \a)(b\,

+ A^E,

(3.5) (3.6)

where u;0(, denotes the atomic transition frequency and Ka(, corresponds to the atomic dipole moment, a real number for transitions between bound states. In the Heisenberg picture the transition operator A oscillates with positive frequencies near wa(,. Therefore, A couples entirely to the negativefrequency component E^ of the electric field, while A* couples to the positive-frequency component E^+\ We arrive at the effective Hamiltonian H = t^ab\b){b\-^f(AE^

+ A^E^) .

(3.7)

To describe the quantum state of the atom, we employ a density matrix p that characterizes a statistical ensemble of pure states | ipa) with probabilities Pa,

P = J>a|a)(<M.

(3.8)

a

Probabilities are non-negative and sum up to unity. Consequently, the density matrix has non-negative eigenvalues and is normalized as tr(/5) = 1. In the Schrodinger picture, the density matrix evolves while the operators are invariant in time. According to Lindblad's theorem [7], the evolution of a normalized and non-negative density matrix is governed by the master equation [8]

^ = \ I* ^ - Z > {Al Arp-IMpA^^p

A\ A,) . (3.9)

Slow light

64

The Lindblad operators Ai describe dissipative processes occurring at the rates 7;, for example the spontaneous emission from the excited state to the ground state. As a result of the light-matter interaction, a medium of nA atoms per volume generates a matter polarization of P = nA ^ t r [p [A + i + ) } = nAnab Re (a \ p \ b>.

(3.10)

In this way the response of the atoms to the electric field modifies the propagation of light given by the wave equation (3.4). Consider a medium at rest in a regime of linear response. Here the medium integrates the local history of the electric field via the susceptibility X, +00

/

X{t

- t') E{?) dt'.

(3.11)

•00

Causality implies that P must not depend on the future of the E field, which restricts the integral (3.11) to the time interval (—00,0] by requiring X(t) = 0

for

t<0.

(3.12)

Consider the Fourier-transformed (spectral) susceptibility

* M = 2W

X{t)

^dt

=

27 J

xWe*"tdt-

( 3 - 13 >

Because x(t) is real, we get *(-") = * > ) .

(3-14)

Let us regard x(w) as a function of complex frequency CJ. When the imaginary part of CJ is positive, x(w) cannot have singularities, because here the Fourier integral contains a factor exp(—[Im u]t) that enforces convergence. Therefore, X(OJ) is analytic on the upper half plane. Causality thus implies analyticity [9]. For a non-dispersive medium x(w) is constant over the relevant frequency range and x M is reduced to a delta function, describing an instant response of the medium. In dispersive media, x ( w ) varies and the poles of X{OJ) on the lower half plane correspond to atomic resonances. At the real w axis the real part of x describes the dispersive properties of the medium, whereas the imaginary part accounts for dissipation. Given an analytic function x ( ^ ) on the upper half plane that decays sufficiently fast when OJ —¥ 00, the real and

Ulf Leonhardt

65

imaginary parts of x(w) at the real w axis are related to each other by Hilbert transformations [10] (Kramers-Kronig relations [9]). The imaginary part of X(u)) is thus uniquely determined by the real part and vice versa. Dispersion influences the group velocity of light pulses. Suppose that the medium properties do not vary significantly within the scale of an optical wavelength. In this case we can characterize completely the propagation of light pulses by the dispersion relation between the wave number k and the frequency u, k2 = ^[l

+ x(u,z)},

(3-15)

derived from the wave equation by Fourier transformation with respect to space and time. A light pulse propagates like a particle with Hamiltonian w and momentum k, subject to Hamilton's equations dfc

dui

dz

=

di "al'

dt

=

dui

,

9fc'

(3 16)

.

'

The group velocity vg is the velocity dz/dt of the fictitious light particle,

v

du

(dk\~l

' = dk = [*;)

=

c

^ZM'

r

n = v/rT

,„.,„..

*-

(3 17)

-

2n du Slow light [1] involves an extremely dispersive medium where dx/duj is large and, consequently, where the group velocity is very low (a few meters per second, or lower).

3.3

Ordinary media

Before we embark on discussing extremely dispersive media, let us consider an ordinary medium composed of two-level atoms at rest. Assume that the atoms are identical, with equal transition frequencies uab, and that they are affected by dissipative relaxation processes. The dissipation transfers excitations from the excited to the ground states, described by the Lindblad operator (3.6). Assume that the transition rate 7 dominates the time scale of the light-atom interaction. In this regime, relaxation forces the atomic dipoles to follow the fields. In the case of linear response we can decompose the electric field into Fourier components. To analyse the response, it is

66

Slow light

sufficient to study the reaction of one of the atoms to a single monochromatic wave with frequency LJ. We characterize the field-strength of the wave in terms of the Rabi frequency Q with ft e-*"* = ^ £ ( + ) . (3.18) ft In the absence of relaxation, an atom would oscillate between the ground and the excited state with frequency Q (Rabi flopping). On the other hand, relaxation leads to a stationary state. To describe the stationary regime we use an appropriate interaction picture. In an interaction picture, indicated by tildes over operators, the dynamics with respect to a partial Hamiltonian Ho is separated from the total evolution of the density matrix, p = UlpUQ,

U0 = exp (-±

H01\

.

(3.19)

To derive the evolution equation, we differentiate p with respect to t and apply the master equation (3.9). Suppose that the commutator between At and H0 is proportional to At, [i,,£0] =

fiwjj4j.

(3-20)

In this case U0AiU0 gives Aiexp(-iujit), and thus the dissipative part of the master equation (3.9) remains the same in the interaction picture. The Hamiltonian is transformed according to H = UlHU0-H0,

(3.21)

such that we obtain

ft

=

l [ p ' k ] ~ ^ 7 ' (4 f

A

IP-

2A

' PAI+P

M A) •

(3-22)

Returning to the two-level atom, we use the interaction picture with respect to Ha = hu\b)(b\ (3.23) that preserves the dissipative dynamics and leads to the time-independent Hamiltonian H = h{uiah-u)\b)(b\-\(AQr

+ A*si\ .

(3.24)

Ulf Leonhardt

67

The atom reaches a stationary state when the relaxation balances the optical transition, i [p, H] = 7 ( i t A p - 2A p i f + p i + i ) . (3.25) Given the normalization tr(/>) = 1, we can easily solve the linear equation (3.25) for the density-matrix components. Assuming linear response, we linearize the solution p in the Rabi frequency $7 and obtain

n* 2 ( w - CJab -

Q

ij)

(3.26)

0

2(w - u)ab + i'y)

The positive-frequency component of the matter polarization (3.10) is therefore P{+) = Y

K

«» < b I & P °l I a) = ^

Kab rab fi e"*"

(3.27)

where we have introduced Tab =

OJ

1 - uab + ij

1 Ua = -

(3.28)

W + CJab + 11

For monochromatic light, P ( + ' gives simply £QXE^+\ Considering the property (3.14) of the spectral susceptibility we obtain the Lorentzian

* - T S ( T - + " - >•

(3.29)

Transforming to a real-time susceptibility x W we see easily that x W responds within the relaxation time 7. In accordance with causality, the Fourier-transformed susceptibility x is analytic on the upper half plane. The single poles at ±ojab — iy correspond to the atomic two-level resonance. On the real frequency axis, \ IS peaked at ±u;0(, with the spectral line width 7. Figure 3.1 illustrates how the. real and the imaginary part of the spectral susceptibility depend on the frequency. The medium is most dispersive near the resonance frequency u)ab where, unfortunately, it is also most absorptive. Instead of slowing down light, the medium turns completely opaque.

Slow light

68

-1

Figure 3.1: Spectral susceptibility of light in an ordinary dielectric medium. The figure shows the real part (solid line) and the imaginary part (dashed line) of x(to) as a function of the detuning A — u — u>o in arbitrary units. The function is given by equations (3.28) and (3.29). The line width 7 was set to 1/3 in the units used. The imaginary part ofx(uj) has a peak at the atomic resonance frequency (for zero detuning). Outside the resonance the real part grows monotonically, corresponding, according to equation (3.17), to a positive group velocity. Near the resonance the dispersion reaches a maximum. At the resonance the real part of'x(w) decreases sharply (anomalous dispersion leading to a negative group velocity). However, the interesting spectral region of low or negative group velocity is totally overshadowed by absorption.

Ulf Leonhardt

3.4

69

Electromagnetically-Induced Transparency

Electromagnetically-Induced Transparency (EIT) [11] has served as a method to slow down light significantly [1] or, ultimately, to freeze light completely [2, 3]. Like other successful techniques, EIT is based on a simple idea [11]: A control beam of laser light couples the upper levels of an atom, and, in this way, the beam strongly modifies the optical properties of the atom. In particular, the coupling of the excited states affects the transition from the atomic ground state to one of the upper states, i.e., the ability of the atom to absorb probe photons with matching transition frequency. Destructive quantum interference between the paths of the transition process turns out to eliminate absorption at exact resonance [11]. A medium composed of such opticallymanipulated atoms is transparent in a spectral region where it would otherwise be completely opaque. In the vicinity of the transparency frequency WQ the medium is highly dispersive, i.e., the refractive index changes within a narrow frequency interval. In turn, probe light pulses with a carrier frequency at u)0 travel with a very low group velocity vg [12].

Probe

I

\

Control -^WWW

Figure 3.2: Three-level atom in a regime of Electromagnetically-Induced Transparency. The control beam couples the levels 2 and 3, which influences strongly the optical properties of the atom for a weaker probe beam tuned to the transition I«->3. Consider the three-level atom illustrated in figure 3.2. The atom is characterized by the energy-level differences Hu^ and hui23 with uj\2+^2z = <^i3 = WQ- Typically, the transition frequencies ui^ and w2z are in the optical range

Slow light

70

of the spectrum or in the near infrared (10 15 Hz), whereas the frequency w12 is much lower (10 9 Hz). The atom is subject to fast relaxation mechanisms (106Hz) that transport atomic excitations from the | 3) state down to 11) and from | 3 ) to | 2 ) , mainly caused by spontaneous emission. Hardly any excitations move from | 2 ) to | 1 ) , because the spontaneous emission rate is proportional to the cube of the frequency [13]. Here the relaxation may be dominated by other processes, for instance by spin-exchanging collisions. Without relaxation the dynamics of the atom is governed by the Hamiltonian

i^E^ 2

_1

0

(3.30)

H = 2 Ki3

£j(+) p

i

K

r(+)

The Hamiltonian represents the atomic level structure and describes the —PE interaction with light, considering here only the frequency components Ep and Ec that match approximately the level structure. The Ev and Ec fields are the probe and control light respectively. We describe relaxation phenomenologically by the flip processes i1

=

|l)(3|,

i2 = |2)(3|,

(3.31)

occurring at the rates 71 and 72, typically a few 106 Hz. Suppose that the three-level atom is illuminated with monochromatic control light at frequency UJC in exact resonance with the 2 «-> 3 transition, (3.32)

wc = w23 •

Consider a regime of linear response. In this case we can decompose the probe field into monochromatic waves, to describe completely the reaction of the atom. We characterized the two light fields involved by their Rabi frequencies Qc and Qp, defined as Clce

-iujct

«23

Qpe-iut

«13

h

E(+) p

(3.33)

The Rabi frequencies set the time scales of atomic transitions caused by the applied light fields. The control beam shall dominate all processes, l^d > l^pl, 7 i , 72-

(3.34)

Ulf Leonhardt

71

Mediated by relaxation, the atomic dipoles lose any initial oscillations they might have possessed and follow the optical fields. To find the stationary state, we utilize an interaction picture generated by 0 H0 = h

0 0

0

0 (3.35)

u> — u)c 0 0 a;

Due to commutation relations of the type (3.20) the dissipative part of the master equation is not changed in the interaction picture. The transformed Hamiltonian has become time-independent, " 0

±fi*

0

i

2 "p

0

H = -h

L 2 "P

UJ — 2 "

LJQ

c

(3.36)

2 "c

Ul —

LJQ

Similar to a two-level atom in a stationary state (3.25), the optical transitions should balance the relaxation processes, 6

[P, £] = £ 7i (A\ArP - 2AtpA\ + pAlAt)

(3.37)

i=i

We could solve exactly the linear equation (3.37) for the matrix elements of p with tr/5 = 1 (using computerized formula manipulation, for example), but without gaining much insight. Fortunately, since we are interested in the regime (3.34), we can find transparent approximations. Suppose first that also |fic|»|w-wd|. (3-38) We expand the solution of equation (3.37) to quadratic order in the small quantities (3.34) and (3.38), and get 1-

n; 2(w - wo) n

W

Ificl2

|fi P l 2 |fic| 2

fie 2(w-

-W0)

Ificl2

fiD

; (3.39)

72

Slow light

We proceed similarly to our analysis of the two-level atom and find, in the positive-frequency range, the spectral susceptibility In X = — (w - wb), Wo

(3.40)

given here in terms of the parameter

a=

nA 4 ,

tujp

T^^oW

(3 41)

-

The spectral susceptibility x depends linearly on the detuning u — UJQ and vanishes at the resonance frequency. Here the phase velocity of light is exactly the speed of light in vacuum, c, but the group velocity (3.17) is reduced by (1 + a),

*=Tk-

(3 42)

-

We call the a parameter (3.41) group index. The parameter is proportional to the ratio between the probe-light energy per photon, HCJ0, and the controllight energy per atom, e 0 \EC\2/TIA- Consequently, the less intense the control beam is the slower the probe light is, a paradoxical behavior. Taken to the extreme, the group velocity would vanish when the control beam is totally dimmed. However, in the stationary regime that we are considering, the control beam should dominate (3.34) and the detuning should be small compared with the modulus of the control's Rabi frequency. Apparently, the linear spectral slope (3.40) of the susceptibility ought to be limited. To find the limitation, we expand the exact stationary state of the master equation (3.22) to linear order in fip, in accordance with a regime of linear response. We obtain a spectral susceptibility (3.29) with u — u>o (w - u0)2 + i(w - u 0 )(7i + 72) - \ Pel2 '

(343)

One can easily verify that the poles of x are located on the lower half plane of the complex frequency w, in agreement with the causality of the medium's response. We expand formula (3.43) in powers of (ui — wo)/| S7C| and see that the medium becomes dissipative when the condition

\n 12

|w-o;o|«^£L 7i + 72

(3.44)

73

Ulf Leonhardt

is violated. For a large detuning we can ignore the \ |fi c | 2 term in the susceptibility (3.43). We obtain the simple Lorentzian (3.28) of an ordinary medium, with the combined line width 7 = 71 + 72- Outside the narrow transparency window of EIT, the absorption of the medium has even slightly increased, because the control beam couples the medium atoms to a second dissipative transition process. The maximally tolerable detuning for transparency is proportional to the group velocity, since vg is proportional to I Q c | 2 for sufficiently slow light. In practice the detuning is usually limited by eVgiMo/c with e in the order of a few 10~3. The transparency window concerns slow light in moving media, because of the Doppler effect. An atom with velocity u causes a Doppler detuning of u wo/c. If we fix the spectral range in the laboratory frame, the maximally tolerable velocity is evg. EIT is velocity-selective. Figure 3.3 illustrates the spectral susceptibility.

3.5

Dark-state dynamics

Suppose that a dominant and monochromatic control beam has, after relaxation, prepared the atom in the stationary state (3.39). How will the atom evolve when the control and probe strengths vary [14]? First we note that the state (3.39) is statistically pure [to quadratic order in the small quantities (3.34) and (3.38)] so that tr{/52} = 1.

(3.45)

When the purity condition (3.45) is satisfied the density matrix contains a single state vector [8] (3-46)

p=\rk)(^o\ with, in our case, |^o> = ^ ^ ( | l > - ^ | 2 > + ^ f ^ n

p

| 3 > )

.

(3.47)

The stationary state does not depend on the relaxation rates but only on the parameters of the Hamiltonian (3.36). Remarkably, even when the parameters vary, the state is protected from further relaxation, as long as the | 3 ) component is small, /»33 = < 3 | p | 3 > « l .

(3.48)

Slow light

74

<'

!!

;I:

ii> >

•

! '•

'

! | /

r

'

J

' ', hi

' l i l

i

*

Figure 3.3: Susceptibility of the probe light in a medium with Electromagnetically-Induced Transparency. The figure shows the real part (solid line) and the imaginary part (dashed line) of the spectral susceptibility x(w) as a function of the detuning A = u) — OJQ in units of the Rabi frequency O c of the control beam, given by equations (3.29) and (3.43). The line width 7i+72 was set to fic/3. The parameters used agree with the ones in figure 3.1. Comparing the two figures, we see that EIT radically alters the susceptibility in a spectral region around the probe resonance LJ0. Here the imaginary part of x(w) vanishes. The medium has become transparent where it would be completely opaque without the influence of the control beam. In the transparency window the real part of X(OJ) increases linearly with a steep slope, indicating that the medium is extremely dispersive. As a consequence of equation (3.17) the group velocity of the probe light is significantly reduced.

75

Ulf Leonhardt

Once the atom is in a pure state with sparsely populated top level, the purity (3.45) does not change during the evolution (3.9), d(tr{,32}) = 2tr{/5d/5} = 4 [ 7 l (l - pu) + 7 2 (1 - p22)] p33 d t .

(3.49)

The pure state so protected is called a dark state [15]. Although dark states are initially prepared due to the relaxation of the atomic dipoles, having so adapted to the light fields, they are no longer prone to dissipation. Suppose that the control and the probe strengths vary. How does a dark state follow the light? In the case (3.48) the state of the atom is dominated by its components in the subspace spanned by the two lower levels 11) and | 2). If we find a state | ip) that describes correctly the dynamics (3.9) in this subspace, the third component ( 3 | ^ ) must be correct as well, to leading order in P33. The lower ranks enslave the top level. Since the relaxation processes (3.31) do not operate within the lower subspace, we can ignore dissipation entirely, to find the dominant state of the atom. We represent both control and probe light in terms of variable Rabi frequencies, Clce

-iwct

h

npe-i»ot

c

=

?RE(+)

(3.50)

defined here with respect to the atomic transition frequencies LOC = w23 and Wo = W13. We write down the state vector

\i>) =

U0N{\l)

& > + % * >

(3.51)

with the abbreviations U0 Op Qc

1 0 0

0

0 0

iuut

e-

(3.53)

JO

2x

No N

(3.52)

-iujot

0

n i ~ = fi + ilMV N0 exp

/

\

f

7

1/2

(3.54) \Slp\2d8

\

|£2PP + |tt c | 2 y

(3.55)

Slow light

76

In a stationary regime under the conditions (3.34) and (3.38) the vector (3.51) agrees with the dark state (3.47). We see from the properties dtN

=-NN^dt^L,

dtN^L

=

N N

*

d t

^

(3.56)

that | ip) satisfies the differential equation ihdt\rl>) = H\i/>) +ihdt{3\i/>)\3).

(3.57)

Consequently, the vector (3.51) describes correctly the dynamics of the atom in the lower-level subspace. Therefore, the atom remains in the dark state (3.51), as long as the atom's evolution never leads to an overpopulation at the top level | 3). The initial relaxation-dominated regime has prepared the dark state, but later the atom follows dynamically without relaxation [14]. We calculate the matter polarization (3.10) generated by the dark states of the atoms that constitute the medium. The positive-frequency component of P i s P{+)

=

^3i<3|VXV<|i>

= »4^---*w + * c H' (3-58) with 9C = argfi c . Assume, for simplicity, that Qc is real. Otherwise we can easily incorporate the phase 6C of the control field in the phase of the electric field without affecting the wave equation (3.4), as long as 9C varies slowly compared with the optical frequency w0. We adopt the definition (3.41) of the group index a, and get i d t 2 P« « -N*a 2u0 (idt -UJO- i^\

4+)•

(3.59)

We approximate 2wb(*$ - w0) 4 + ) « {idt + uo){idt ~ ^o) 4 + ) = ~(d(2 + w2) EJ,+\

(3.60)

and obtain from the general wave equation (3.4) an equation that is valid for both the positive and the negative frequency component of the probe light, [d2t - c2dl + JVo4 (dtadt + aujl)]Ep

= 0.

(3.61)

Ulf Leonhardt

77

The dark-state dynamics may lead to a non-linear effect of the medium, described by the NQ factor in the wave equation (3.61). However, when the probe is significantly weaker than the control light, the medium responds linearly, (dt(l + a)dt - c2d2z + aui2)Ep = 0. (3.62) This wave equation governs the propagation of slow light in a regime of linear response and undisturbed dark-state dynamics.

3.6

Slow-light pulses

Consider a pulse of probe light in an EIT medium with variable group index (3.41). Suppose that the group velocity (3.42) does not vary much over the scale of an optical wavelength (0.5 x 10~6m) or an optical cycle (10~ 15 s). In this case we could apply the Hamiltonian theory (3.16) of a fictitious light particle to predict the position of the pulse peak. Because particle trajectories must not split, a slowly varying group index cannot cause reflection. Suppose that the pulse is traveling to the right. Then the pulse will continue to do so, and we can express the slow-light wave as Ev — £ exp(ikz — iuit) + £* exp(-ikz

+ iuit),

k = —,

(3.63)

assuming that the electric-field amplitude £ is slowly varying compared with the rapid optical oscillations. We approximate e x p H / c z + iuit) d2Ep+)

«

{-ui2 - 2iuidt) £ ,

exp(-ikz

«

— iui£,

«

(-k2+

exp{-ikz

+ iuit) dtE^ 2

+)

+ iu,t)d zEl

2ikdt) £ •

(3.64)

and get from the wave equation (3.62) -2iui(l

+ a) dt£

« =

((1 + a)u>2 + iuia - c2 k2 + 2ikc2dz - aui1) £ [2iuicdz + iuia + a(ui2 — ui^)) £

=

2iu(cdz£

+ ^£\

(3.65)

when the carrier frequency ui is equal to the transparency resonance UIQ- We thus obtain the propagation equation [14] (1 + a) dt£ + cdz£ + ^£ = 0.

(3.66)

Slow light

78

In order to understand the principal behavior of ordinary slow-light pulses, we consider two cases — a spatially varying yet time-independent group index and a spatially uniform yet time-dependent a. When the group index does not change in time, the propagation equation (3.66) has the simple solution

£(t,z)=£0(t-

[—),

(3.67)

in terms of the group velocity (3.42). At each space point z the pulse raises and falls in precisely the same way. However, because light is slowed down, the spatial shape of the pulse shrinks by a factor of vg/c compared with the pulse length in vacuum, for example by 10~ 7 for a group velocity of 30m/s. The intensity of the pulse is unaffected, despite the enormous pulse shortening, and the pulse energy has gone into the amplification of the control beam. Consider the other extreme, a spatially uniform EIT medium with adjustable group velocity. In this case, the solution of the propagation equation (3.66) is £{t, z) = £0(z-

I vg dt) y / V c .

(3.68)

The pulse envelope £ propagates again with the group velocity vg but the pulse length is not changed. However, the spectrum of the pulse around the carrier frequency w0 shrinks by a factor of vg/c. Additionally, the intensity drops by vg/c as well. The ratio between the control (3.41) and the pulse intensity (3.68) remains large,

Itor

V

l

K

\2

£

o /

even in the limit of a vanishing control field when vg vanishes, as long as UA is large (for a sufficiently dense medium). Therefore, the reaction of the probe field is remarkably consistent with the requirements for undisturbed dark-state dynamics [14]. One can freeze light without losing control [2, 3].

3.7

Effective field theory

After having studied two examples of pulse propagation in an EIT medium, we develop an effective field theory of slow light. We generalize the wave

Ulf Leonhardt

79

equation (3.62) to three-dimensional space and calibrate the electric field strength in appropriate units, / h\ Ep(t,x)=l-\

1/2

u)0
(3.70)

We introduce the Lagrangian #

= \{iX+ " ) ( W - c 2 (V V ) 2 - a o ; 0 V )

(3.71)

and see that the wave equation (3.62) is the resulting Euler-Lagrange equation. We have chosen the prefactor of the Lagrangian (3.71) such that Jz? agrees with the free-field Lagrangian (3.1) for zero a and frequencies around ijQ. Therefore we regard Jif as the effective Lagrangian of slow light. Let us use the Lagrangian (3.71) to calculate the energy balance of slow light. According to Noether's theorem [16] we obtain the energy density / = \ ((1 + a)(dt
(3.72)

and the energy flux (Poynting vector) S = -hc2{dtip){V
(3.73)

As a consequence of the wave equation (3.62) we obtain the energy balance dt/ + V - S = ^

V + o,0V).

(3-74)

Temporal changes in the control field, modifying the group index (3.41), do not conserve energy. In fact, the experiment [2] indicates that the control beam can amplify light stored in an EIT medium with zero group velocity. In the experiment [2], slow light enters the EIT sample and is then frozen inside by turning off the control field. Switching on the control releases the stored light. The pulse emerges with an intensity that depends on the control field and that may exceed the initial intensity, in agreement with equation (3.68). Clearly, this phenomenon is only possible if energy is indeed transferred from the control beam to the probe light.

Slow light

80

3.8

Moving media

An EIT medium slows down light, because the medium is extremely dispersive, reacting differently to the different frequency components of a pulse. The peak position of the pulse is the place where the frequency components interfere constructively. By slightly modifying the phase velocity of each component the medium influences strongly their interference, slowing down the pulse dramatically. The extreme spectral sensitivity of slow light can be also applied to observe optical phenomena in moving media, caused by the Doppler effect. A uniformly moving medium would not present an interesting case, though, because uniform motion just produces a global frequency shift. However, slow light is a superb tool in detecting non-uniform motion such as rotation [17]. To understand the principal effect of slow light in moving media, we modify the Lagrangian (3.71) to account for the Doppler effect. We assume that (3.71) is valid in frames co-moving with the medium and transform back to the laboratory frame. We note that {dt)2 is a Lorentz invariant and focus on the first-order Doppler effect in the a(dt(p)2 term, assuming the realistic case of non-relativistic medium velocities. We replace dt by dt + u • V and neglect the term quadratic in u. In this way we obtain the Lagrangian of slow light in a moving medium jSf = Jgfo - -^ S • u ,

S = -hc?{dtv){Vip)

(3.75)

in terms of the Lagrangian J£Q for the medium at rest (3.71). We see that the flow u couples to the Poynting vector S of slow light, similar to the Rontgen interaction of moving dipoles in electromagnetic fields [18]. We obtain from „£f the Euler-Lagrange equation (dt{l + a)dt - c 2 V 2 + aul + dtau • V + V • audt)

(3.76)

with the differential operators acting on everything to the right. For frequencies near the EIT resonance UIQ we represent the positive-frequency part tp^ of (p as
Ulf Leonhardt

81

with the effective Planck constant reduced by 2n c . A= —

(3.79)

Wo

The flow has a two-fold effect: On one hand, the velocity u appears similar to an effective vector potential, for example as the magnetic vector potential acting on an electron wave, and, on the other hand, the hydrodynamic pressure proportional to u2 acts similarly to an electric potential. A vortex flow will generate the optical equivalent of the Aharonov-Bohm effect, see reference [19] for details. A moving slow-light medium is also equivalent to an effective gravitational field [4]. Consider monochromatic light at exact resonance frequency LJ0. In this case, we can write the wave equation (3.76) in the form of a KleinGordon equation in a curved space-time,

d, {r

d„ip{+)) = 0,

(3.80)

with dv = (9 t /c,V)and

1

/"" = V=5 «T = au/c

au/c —1

(3.81)

Here g^v represents the effective space-time metric experienced by monochromatic slow light in a moving medium, to first order in u/c. We easily find the determinant g of the inverse of g**" from the relation

det/ = -g2/g = -g = - (l + a2 ^ V

(3.82)

The effective space-time line element ds 2 is, up to a conformal factor, ds 2 = c2 d*2 + 2a dt u • dx - dx 2 ,

(3.83)

resembling the line element of a moving coordinate system, ds 2 = (c2 - u2)dt2 + 2 d* u • dx - dx 2 ,

(3.84)

for example of a rotating system. The parameter a quantifies the degree to which the motion of the medium is transferred to the propagation of light in the medium, the degree of dragging. The group index is thus equivalent to

82

Slow light

Fresnel's dragging coefficient [17, 20]. For slow light a is very large indeed. Therefore, slow light is able to sense minute flow variations. Even subtle quantum flows imprint phase shifts onto slow light that are detectable using phase-contrast microscopy [21]. Sound waves in a fluid experience the flow as an effective space-time metric as well [22]. The acoustical line element is proportional to the element (3.84) of a moving system of coordinates, with two crucial differences: The flow is not subject to the rigidity of moving coordinate systems and, more importantly, in the acoustic metric the speed of light, c, is replaced by the speed of sound. A supersonic flow surpasses the sound barrier and can, under suitable circumstances, generate an artificial event horizon where the flow speed u reaches c. Here it is necessary that the g00 element of the metric vanishes. In the slow-light metric (3.83) the all-important term — u2dt2 is missing in 500, at least to the level of approximation we are considering here. We obtain a term proportional to — u2dt2 when we include effects of higherorder Doppler detuning. The critical velocity is then of the order of c/s/a. To observe the quantum effects of light generated by a horizon we would need to employ a steep profile of the flow speed. This causes a severe problem, because the Doppler detuning will exceed the transparency window of EIT. The Doppler effect plays a beneficial role in the sensitivity of slow light to motion, but it will also cause significant light absorption when one attempts to reach an artificial event horizon. The medium will certainly turn black, but not into a black hole. However, one could employ a spatially varying profile of the group index to create an interface that resembles an event horizon for slow light and that avoids this problem [23]. Slow light offers a variety of options for interesting experiments exploiting the analogues of light in media with effects in other areas of physics, and new ideas continue to emerge.

3.9

Summary

Light has been slowed down dramatically [1] or even stopped completely [2, 3]. To understand how this feat has been achieved, we studied the physics behind slow light, starting from basic first principles of the light-matter interaction. We first turned to ordinary optical media, so as to later contrast them with slow-light media based on Electromagnetically-Induced Transparency. We studied slow light in two regimes — in a stationary regime both dominated and limited by relaxation and the control light, and in a dynamic

Ulf Leonhardt

83

regime almost free from dissipation. Then we analyzed the typical behavior of slow-light pulses, before developing an effective field theory of slow light that we have subsequently generalized to moving media. It is certainly amazing how much a clever combination of atomic physics and optics can achieve, but it is also important to understand the principal limits of the techniques applied. These limits depend on the details of the physics behind the scene. We have tried to elucidate the details of slow light without going into too much detail, using models that are simple, but not too simple.

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Bibliography [1] L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999); M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, H. Welch, M. D. Lukin, Y. Rostovsev, E. S. Fry, and M. O. Scully, Phys. Rev. Lett. 82, 5229 (1999); D. Budiker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, ibid. 83, 1767 (1999); A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, Phys. Rev. Lett. 88, 023602 (2002). [2] Ch. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, Nature (London) 409, 490 (2001). [3] D. F. Philips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, Phys. Rev. Lett. 86, 783 (2001). [4] U. Leonhardt and P. Piwnicki, Phys. Rev. Lett. 84, 822 (2000); M Visser ibid. 85, 5252 (2000); U. Leonhardt and P. Piwnicki, ibid. 85, 5253 (2000); see also references [17, 19, 20, 23], and U. Leonhardt, Phys. Rev. A 62, 012111 (2000); J. Fiurasek, U. Leonhardt, and R. Parentani, Phys. Rev. A 65, 011802(R) (2002). [5] J. Anglin, Nature 406, 29 (2000); U. Leonhardt and P. Piwnicki, Contemp. Phys. 41, 301 (2000); Ph. Ball, Nature 411, 628 (2001). [6] J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975). [7] G. Lindblad, Commun. Math. Phys. 48, 119 (1976). 85

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[8] C. W. Gardiner, Quantum Noise (Springer, Berlin, 1991). [9] H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972). [10] M. J. Ablowitz and A. S. Fokas, Complex Variables (Cambridge University Press, Cambridge, 1997). [11] P. L. Knight, B. StoichefT, and D. Walls (eds.), Phil. Trans. R. Soc. Lond. A 355, 2215 (1997); S. E. Harris, Phys. Today 50(7), 36 (1997); M. O. Scully and M. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997) M. D. Lukin and A. Imamoglu, Nature 413, 273 (2001); A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, G. R. Welch, A. S. Zibrov, and M. O. Scully, Advances in Atomic, Molecular and Optical Physics 46, 191 (2001). 12] S. E. Harris, J. E. Field, and A. Kasapi, Phys. Rev. A 46, R29 (1992). 13] R. Loudon, The Quantum Theory of Light (Oxford University Press, Oxford, 2000). 14] M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett. 84, 5094 (2000). 15] E. Arimondo, Prog. Optics 35 257 (1996). 16] S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, 1999), Volume I. 17] U. Leonhardt and P. Piwnicki, Phys. Rev. A 62, 055801 (2000). 18] U. Leonhardt and P. Piwnicki, Phys. Rev. Lett. 82, 2426 (1999). 19] U. Leonhardt and P. Piwnicki, J. Mod. Optics 48, 977 (2001). 20] U. Leonhardt and P. Piwnicki, Phys. Rev. A 60, 4301 (1999). 21] E. Hecht, Optics (Addison-Wesley, Reading, 1989). 22] See the introductory chapter by M. Visser in this book. 23] U. Leonhardt, Nature 415, 406 (2002); U. Leonhardt, Phys. Rev. A65, 043818 (2002).

Chapter 4 Black hole and baby universe in a thin film of 3 He-A Ted Jacobson University of Maryland College Park Maryland USA E-mail: [email protected] Tatsuhiko Koike Keio University Hiyoshi, Kohoku Yokohama 223-8522 Japan E-mail: [email protected]

87

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Black hole and baby universe in

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He-A

Abstract: Condensed matter black hole analogues may provide guidance in grappling with difficult questions about the role of short distance physics in the Hawking effect. These questions bear on the very existence of Hawking radiation, the correlations it may or may not carry, the nature of black hole entropy, and the possible loss of information when a black hole evaporates. We describe a model of black hole formation and evaporation and the loss of information to a disconnected universe in a thin film of 3He-A, and we explain why the existence of Hawking radiation has not yet been demonstrated in this model.

4.1

Introduction and motivation

From the condensed matter point of view, black hole analogues are a curiosity. They certainly generate new questions, and they may lead to new insights into the interplay between bulk and microscopic physics. From the quantum gravity point of view, there is much more at stake. We are looking to condensed matter systems for the guidance they may provide in grappling with difficult questions about the role of short distance physics in the Hawking effect. These questions bear on the very existence of Hawking radiation, the correlations it may or may not carry, the nature of black hole entropy, and the possible loss of information when a black hole evaporates. Although the Hawking effect is a low-energy phenomenon for black holes much larger than the Planck mass, it cannot be deduced strictly within a low energy effective theory. The gravitational redshift from the event horizon is infinite, so the outgoing modes that carry the Hawking radiation emerge from the Planck regime. To derive the Hawking effect one needs only the assumption that near the horizon these modes emerge in their local ground state at scales much longer than the Planck length but still much shorter than the black hole radius [1]. This assumption is plausible, since the background is slowly varying in time and space compared to these scales. However, it is not derived. A condensed matter analogy has already provided some support for this picture. Unruh [2] introduced a sonic analogue, in which the black hole is modeled by a fluid flow with a supersonic "horizon", and the quantum

Ted Jacobson

and Tatsuhiko

Koike

89

field is replaced by the quantized perturbations of the flow. In a continuum treatment, Unruh argued that the horizon would radiate thermal phonons at the Hawking temperature hK,/2n, where K, which would be the surface gravity for a black hole, is here the gradient of the velocity field evaluated at the sonic horizon. The short-distance (atomic) physics of this analogue is fully understood in principle, hence it should be possible to understand the origin and state of the Hawking modes. As a first step inspired by this model, a number of studies have been carried out where the linearity of the quantum field equation is preserved but the short distance behavior is modified either by introducing nonlinear dispersion or a lattice cutoff, designed to mimic some aspects of the real atomic fluid. The consequences have been discussed in detail elsewhere (see e.g. [3] for a review). The main point is that, despite the exotic origin of the outgoing modes via "mode conversion" near the horizon, the short-distance physics does indeed deliver these modes in their local ground state near the horizon if they originate far from the horizon in their ground state. These models thus lend some (linear but nontrivial) support to the contention that Planck scale effects deliver the local vacuum at a black hole horizon. A controversial consequence of this simple picture, however, is that the Hawking effect produces a loss of information from the world outside the horizon. The reason is that the local vacuum condition at the horizon entails correlations between the field fluctuations inside and outside the horizon. The radiated Hawking quanta are thus correlated to "partners" that fall into the black hole. 1 The origin of these correlations is precisely the same as in the vacuum of flat spacetime, so it is difficult to see any reason to doubt this account. It is often doubted, however, since it implies that the process of formation and complete evaporation of a black hole entails nonunitary evolution in the Hilbert space restricted to the outside world, which is considered by many (not including the authors) to be a violation of quantum mechanics. It would be useful to have a down-to-earth condensed matter analogue in which the information loss question arises but the fundamental physics is understood. Related to the issue of information loss is the nature of black hole entropy. A spherical black hole of mass M emitting an energy dE = dMc2 : In addition to the information of correlations between Hawking quanta and their partners, any information that simply falls into the black hole from the outside appears to be lost.

90

Black hole and baby universe in

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He-A

in thermal radiation at the Hawking temperature T# = HC3/8TTGM loses an entropy dS = dE/TH = d(A/4lp), where A — 4irR2 is the area of the event horizon of radius Rs = 2GM/c2, and lp — (hG/c3)l/2 ~ 10 _ 3 3 cm is the Planck length. A black hole thus has one unit of entropy for every four units of Planck area. To understand the nature of the microscopic degrees of freedom counted by this entropy remains one of the outstanding problems of quantum black hole physics. To solve this problem will presumably require understanding the nature of the short-distance cutoff. Without a cutoff the entropy would seem to be infinite due to the quantum entanglement between field degrees of freedom on either side of the event horizon discussed in the previous paragraph. If a condensed matter horizon analogue produces Hawking radiation, then it would seem to also carry an entanglement entropy, so it may provide some guidance on the nature of black hole entropy. (There are important differences however, since in the condensed matter setting there is no relation between the energy of the system and the area of the horizon, the Einstein equation of course does not pertain to the evolution of the horizon area, and the area need not even change during the evolution of the system.) The basic question of the existence of the Hawking effect, as well as the issues of information loss and black hole entropy should be approachable in condensed matter analogues. The system we focus on here is particularly interesting in that it provides a model of the formation and evaporation of a black hole, with a disconnected part of the "universe", analogous to a so-called "baby universe", into which information can potentially be lost. It should be stated at the outset that at this stage we have only a model of the background spacetime geometry. A detailed analysis of issues pertaining to the Hawking effect remains to be carried out, and it is not yet clear that this system would produce analogue Hawking radiation. Another point worth stressing is that, for the purposes of obtaining guidance in quantum gravity issues, the experimental feasibility of actually setting up and observing a condensed matter analogue is not essential. Nevertheless, experimental observations are certainly one of the goals of the whole program, both to confirm the basic properties of the Hawking effect, and to gain insight into the short-distance physics at play. In the next section we introduce 3 He-A and discuss how this medium can be used to construct various black hole analogues. This is followed by a section focusing on the effective geometry and Hawking radiation in the moving thin-film domain-wall model. The last section describes the analogue of black hole formation and evaporation and the loss of information to a

Ted Jacobson and Tatsuhiko Koike

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disconnected universe in the that model. We would like this article to be accessible to researchers in both condensed matter and gravitational physics, hence we include more than the usual amount of introductory material. We use units with &B = 1.

4.2

Black hole analogues using 3 He

The Hawking effect is a quantum tunneling process that produces a gentle instability of the ground state due to the presence of unoccupied negative energy states in the ergoregion behind the horizon. The instability creates a flux of particles in a thermal state at the Hawking temperature. If a condensed matter system is to produce identifiable analogue Hawking radiation, therefore, the system should presumably be at least as cold as the Hawking temperature, 2 which is very low temperature for reasonable laboratory parameters. Moreover, there should be no other dissipation mechanisms that could swamp the Hawking effect. A natural place to start looking is therefore at superfluid systems at zero temperature. The case of superfluid 4 He was initially examined in [6], and further discussed in [7]. It was concluded that a sonic horizon cannot be established in superflow, because the flow is unstable to roton creation at the Landau velocity which is some four times smaller than the sound velocity.

4.2.1

3

He-A

Potentially more promising [8, 9,10,11] is the (anisotropic) A-phase of superfluid 3 He, which has a rich spectrum of massless quasiparticle excitations. In particular, there are fermionic quasiparticles — the "dressed" helium atoms — which have gapless excitations near the gap nodes at p = ±pp I on the anisotropic Fermi surface, and therefore can play the role of a massless relativistic field in a black hole analogue. The unit vector I is the direction of orbital angular momentum of the p-wave Cooper pairs and PF is the Fermi momentum. For the benefit of readers not familiar with 3 He-A we inject here a lightning sketch of the basics. (For complete introductions see [12, 13, 14].) 3 He 2 This condition may be avoided by observing instead a runaway quantum instability related to the Hawking effect that is expected to occur for bosonic fields when there is an inner horizon in addition to an outer horizon [4, 5].

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Black hole and baby universe in

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He-A

is a spin-1/2 fermion, and is described in a many-body fluid by a second quantized field operator ipA(x), where A is a two-component spinor index. The fluid has a phase transition at 2.7 mK to a superfluid state in which the order parameter (i(>A(x) ipB(y)} = / 4 B ( x — y) is non-zero. Below a pressure of about 33 bars and a temperature of order 1 mK the fluid is in the so-called B-phase. The A-phase is a very long-lived metastable phase that is coexistent with the B-phase and is stable in the region between 20 and 33 bars from 2.7 mK down to around 2 mK at the higher pressure. In the A-phase the order parameter, which can be thought of as the wave function of a Cooper pair, is a spatial p-wave and a spin triplet, and has the structure \L = l,mL = 1) |S = l,ms = 0). The coherence length £, which corresponds to the size of the Cooper pair wavefunction, is of order 500 A. The Fourier transform of fAB(x - y) evaluated near the Fermi momentum |p| = PF is proportional to the energy gap

A(p)=*<e,WBa±™i, PF

(4.1)

V2

where the three unit vectors 1, ei and e 2 form a right-handed orthonormal triad. The gap function (4.1) has nodes at p = ±pp I. Near these nodes the fermion quasiparticles can have arbitrarily low energies (above the Fermi energy), and they behave like massless relativistic particles. The velocity of these quasiparticles parallel to I in 3 He-A is the Fermi velocity vp ~ 55 m/s, while their velocity perpendicular to I is only c± — A/pF ~ 3 cm/s, where A ~ Tc ~ 1 mK is the energy gap.

4.2.2

Black hole candidates

Superflow It should be possible to set up an inhomogeneous superflow exceeding the slow speed c±_ in a direction normal to I, thus creating a horizon for the fermion quasiparticles. There is a catch, however, since the superflow is unstable when the speed relative to a container exceeds c± [9]. A possible way around this was suggested by Volovik [11], who considered a thin film of 3 He-A flowing on a substrate of superfluid 4 He, which insulates the 3 He from contact with the container. In such a film, the vector 1 is constrained to be perpendicular to the film. A radial flow on a torus, such that the flow velocity near the inner radius exceeds c±, would produce a horizon.

Ted Jacobson and Tatsuhiko Koike

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Theoretically this looks promising, however the Hawking temperature for a torus of size R is T = (h/2n)(dv/dr) ~ hc_i/R ~ (XF/R) mK, where XF is the Fermi wavelength, which is of the order of Angstroms. Thus, even for a micron sized torus, the Hawking temperature would be only ~ 1 0 - 7 K. Moving solitonic texture An alternative is to keep the superfluid at rest with respect to the container, but arrange for a texture in the order parameter to propagate in such a way as to create a horizon. For example, in [8] a moving "splay soliton" is considered. This is a planar texture in which the I vector rotates from +x to —i, along the x-direction perpendicular to the soliton plane. A quasiparticle moving in the i-direction thus goes at speed vp far from the soliton and at speed c±_ in the core of the soliton. If the soliton is moving at a speed greater than c±, the quasiparticles will not be able to keep up with it, so an effective horizon will appear. This example turns out to be rather interesting and complicated in the effective relativistic description. The null rays on the horizon have a transverse velocity, making it like that of a rotating black hole rather than a static black hole. In addition, since the I vector couples to the quasiparticles like an electromagnetic vector potential, its time and space dependence generates a strong "pseudo-electromagnetic" field outside the "black hole" which would produce quasiparticle pairs by analogy with Schwinger pair production [13]. (This latter process may be the same as what produces the so-called "orbital viscosity" [12] of a time-dependent texture.) The Hawking temperature also tends to be very low, and it seems likely that the Hawking effect would be masked by the pseudo-Schwinger pair production, though this has not been definitively analyzed. Thin film w i t h a moving domain wall In [10] a simpler system was studied, that of a thin film of 3 He-A, perhaps on a 4 He substrate, with a domain wall. The vector I, which is perpendicular to the film, has opposite sign on either side of the wall, and in the wall region the condensate is in a different superfluid phase. At the core of the wall the group velocity of the quasiparticles goes to zero, so if the wall itself is moving, a horizon will appear. For the rest of this article we focus on this example. The effective spacetime geometry of this system was first studied in [10]. Here we extend that analysis to the case of a wall that accelerates

Black hole and baby universe in 3He-A

94 and then comes to rest again.

4.3 4.3.1

Effective spacetime and Hawking effect from a moving domain wall texture Texture and spectrum

The order parameter for a domain wall texture in a thin film is described by a gap function of the same form as (4.1), with the unit vector ei replaced by something like §i = tanh(z/d) ei, where d ~ £ is the width of the wall. Here it is assumed that the film lies in the x-y plane with the wall along the y-axis. (See figure 4.1) The thickness of the film should be not much more than the coherence length £ in order for the domain wall to be stable.

domain wall

JJJJ/L L C XZEIIZE

XZEOZ3

Figure 4.1: Thin-Sim domain-wall texture. The quasiparticle spectrum in the A-phase takes the form E2(p) = vF (p - pp)2 + c\ (ei • p) 2 + c'i (e2 • p) 2 .

(4.2)

There is no excitation perpendicular to the thin film, hence we have px — pp, so p=

TJPF+PI+PI

= p F + —~(PI + pi) + " ' •

(4.3)

Using this expansion in (4.2) together with the replacement ei -4 ei we obtain the low energy spectrum for motion in the x-y plane in the domain wall texture: c2 (4.4) E2 = c{xf pi + c2± pi + -± f (p2x + p2,)2 +

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In the quartic term we have replaced ?; F /2A by £/h, to which it is roughly equal. The speed c(x), denned by c(x) = c±_ ta.nh(x/d),

(4.5)

goes to zero at the core of the domain wall. In the low energy limit the quartic term is negligible and the spectrum becomes that of a massless relativistic particle in two dimensions. The nonrelativistic quartic corrections become important at higher energy, when the wavelength is of order the coherence length £. Note that this is of order 500 A, much longer than the interatomic spacing. 3 The corrections produce "superluminal" group velocities at high momentum. If a quasiparticle is localized near the domain wall then these nonrelativistic corrections are important, since the width of the wall is of order £.

4.3.2

Spacetime of the stationary wall

The relativistic limit of (4.4) is that of a massless particle in a 2+1 dimensional spacetime with the line element ds 2 = -dt2 + c(x)-2dx2

+ cl2dy2.

(4.6)

The metric has translation invariance in the y-direction, so we will make a "dimensional reduction" to the 1+1 dimensional spacetime ds 2 = - d t 2 + c(x)- 2 da; 2 .

(4.7)

Clearly this spacetime is flat, since one can introduce a new spatial coordinate by da;* = dx/c(x) in terms of which the line element takes the manifestly flat form ds 2 = — dt2 -\-dx2. Note however that since c(x) goes to zero linearly as x —> 0 + , the coordinate a;* goes to — oo logarithmically as the domain wall is approached from the side of positive x. Therefore the film really corresponds to two complete copies of flat spacetime, joined "at infinity" at the wall. 3

In the analogy with quantum gravity, it would appear that the Planck scale should be identified with £ since this measures the "elasticity" of the background, and there is at present no analogue of the underlying atomic scale in fundamental theory except perhaps the string scale, which is usually taken to be longer than the Planck length, rather than shorter.

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4.3.3

3

He-A

Spacetime of the moving wall

Now suppose the domain wall texture is moving to the right at speed v < c± relative to the superfluid condensate. Then right moving quasiparticles sufficiently far from the wall will stay ahead of the wall, but those inside the point where c(x) = v will fail to stay ahead. There will be a black hole horizon where c(x) = v and a white hole horizon where c(x) — —v on the left hand side of the wall. In between the two horizons all low energy quasiparticles are purely left-moving relative to the wall texture (see figure 4.2). The closer v is to c± the farther apart the horizons lie. VW
c(x) = -v c(x) = 0

c(x) = v

Figure 4.2: Quasiparticles in a moving domain-wall

texture.

To determine the spacetime metric for the moving wall, we introduce coordinates xs and xw at rest with respect to the superfluid and the wall, respectively. These are related by the Galilean transformation xs = xw + vt. The dispersion relation is determined in the superfluid frame, so the line element (4.7) applies in the superfluid frame, however the argument of the function c(x) should be xw since this function describes the texture which is at rest in the wall frame. For the moving wall we thus have ds2

= = =

-dt2 + c(xw)~2dx2s 2

-dt +

2

c(xw)- {dxw 2

2

- ( 1 - v c(xw)~ )dt

2

(4.8) + vdtf

(4.9) 2

+ 2vc(xwy

2

2

dt dxw + c{xw)- dx w.

(4.10)

Perhaps surprisingly, this is no longer a flat spacetime. It has black and white hole horizons at c{xh) = ±^- The wall core at xw = 0 is now a spacelike line (since the coefficient of dt2 is positive there), and it lies at finite proper time along geodesies. The Ricci curvature scalar is given by R =

-Av2
c(xxy \c{xx)2

(4.11)

Ted Jacobson and Tatsuhiko Koike

97

This diverges like —(2v/xw)2 as xw = 0 is approached, so there is a curvature singularity at the core. The spacetime therefore looks rather like that of an eternal Schwarzschild black hole. The curvature at the horizon is given by ^horizon = -{2cjd)2

[l - {v/c±)4}.

(4.12)

Unlike the maximally extended Schwarzschild black hole, however, the spacetime of the moving wall is incomplete, in that geodesies can run off the edge of the coordinate system (t, xw) in a finite proper time or affine parameter. The location of the incomplete boundaries will be indicated below. Of course physical quasiparticles cannot escape, because this really is the entire physical spacetime. What happens is that as a quasiparticle heads in the direction of an incomplete boundary (either forward or backward in time), it is blueshifted into the part of the spectrum where the nonrelativistic corrections become important, at which point it propagates superluminally and the geodesies of the effective metric no longer determine its trajectory.

4.3.4

Hawking effect

Since the effective spacetime is that of a black hole, it is natural to suppose the horizon would radiate fermion quasiparticles at the Hawking temperature TH = (H/2TT)K, where K = dc/dx(x)l) is the surface gravity of the horizon. For the metric (4.10) with (4.5) we have explicitly TH(v) = TH(0) (1 - v'lc\),

r„(0) = ^ j .

(4.13)

Putting in the numbers we have TH{0) ~ 1 /iK. Equation (4.13) gives the Hawking temperature in the wall reference frame (the "Killing temperature" in the language of general relativity), which is related to the temperature in the asymptotic frame of the superfluid by a Doppler shift factor: TH^

= TH{0)(1

+ V/C±).

(4.14)

The Hawking temperature in the frame of the superfluid is thus never less than Tf/(0). Although this is three orders of magnitude below the critical temperature, and extremely low in practical terms, it is nevertheless close to where the non-relativistic corrections become important (assuming d ~ £). While the black hole analogy looks compelling, it should be emphasized that the Hawking effect depends on behavior of the quantum field that may

98

Black hole and baby universe in

z

He-A

not be valid in this context. As discussed in the introduction, the required condition is that the high frequency outgoing modes near the horizon be in their quantum ground state. In this case these modes come from the singularity, since they propagate "superluminally". The propagation of these modes though the singularity may excite them. This has not yet been analyzed. 4 If they are excited, this may suppress the Hawking radiation by virtue of the Pauli principle. (Had the field been bosonic, the excitations would have produced extra, induced emission.) Another difference from the black hole case is that once a Hawking pair is produced, the negative energy partner rattles around trapped in the ergoregion between the two horizons. Once these available states fill up further Hawking radiation would be suppressed. (In the case of a black hole, by contrast, the negative energy partners fall into the singularity never to return.) For a discussion of aspects of the behavior of superfluids in the presence of a quasiparticle horizon see reference [15]. For the remainder of this article we assume that, in spite of Pauli-blocking effects, the moving domain wall produces at least some Hawking radiation, and we go on to study the analogue of the process of formation and evaporation of a black hole. According to (4.13) the Hawking temperature approaches a nonzero constant as v —> 0. Nevertheless it is clear that at v = 0 there can be no radiation since the wall is static and there is no horizon, hence there is no ergoregion with negative energy states to be filled. The radiation rate must therefore go to zero as v goes to zero. If it goes as a power of v then we have dE/dt = —bvn. The kinetic energy of the moving domain wall is proportional to v2 if, as seems plausible, the action is dominated by quadratic terms in the time and space derivatives. In this case E = [iv2/2 for some constant fj,. Integrating the energy loss, we find that it takes a finite time for the wall to come to rest if n < 2, but an infinite time if n > 2. Finally, a comment about entropy. It is tempting to try and define a thermodynamic entropy S for the moving domain wall, however it is by no means clear that this should be meaningful. In analogy to the black hole entropy, one might define S via dS = dE/TH,$f-, where E = fj,v2/2 as above and T//iS/ is the Hawking temperature in the superfluid frame, (4.14). This yields the formula S = (2ir[j,c±d/h) (v/c± — ln(l + v/c±)). This analogy seems flawed however, as the domain wall is not stationary in the superfluid frame so does not represent an "equilibrium" system. If we try to correct 4

However the related problem of quasiparticle tunneling across the stationary domain wall has been studied by Volovik in section 11.1 of [15].

Ted Jacobson and Tatsuhiko Koike

99

this by passing to the frame of the moving wall, we run into the problem that, as the wall slows down due to Hawking radiation, this comoving frame changes, unlike in the black hole case where the asymptotic rest frame of the black hole is fixed even as the black hole evaporates (or absorbs radiation).

4.4 4.4.1

Black hole formation and evaporation in the thin-film domain-wall model Carter-Penrose causal diagrams

To best exhibit the incompleteness of the black hole spacetime discussed in the previous section, as well as features of the case where the hole forms and then evaporates, it is helpful to use Carter-Penrose diagrams. We therefore pause here to explain what such a diagram is for the benefit of readers from the condensed matter side. The basic idea is to draw a picture representing the causal structure of a spacetime by showing light rays at 45°, with regions at infinite time or space brought into a finite location by a spacetime dependent conformal rescaling of the line element, ds 2 = Cl2 ds 2 , where Cl —> 0 at infinity. Since the causal structure is determined by the light cones, which are the solutions of ds 2 = 0, the causal structure of ds 2 is identical to that of ds 2 . (See for example [16].) As an example, consider 1+1 dimensional flat spacetime given by the line element ds 2 = — dt2 + dx2 — — dudv, where u — t — x and v = t + x. The conformal factor Cl(u, v) = (cosh u cosh v)"1 brings infinity into a finite location in the sense that the points at infinity for ds 2 lie at a finite proper distance for ds 2 . A diagram of the tilde spacetime is then just a diamond, figure 4.3 (a). The boundaries of the spacetime are at infinite time and/or space. Timelike geodesies (straight lines in this case) emerge from past timelike infinity i~ and terminate at future timelike infinity i+. Spacelike geodesies go from left spacelike infinity i°L to right spacelike infinity i°R, and null geodesies or light rays go from left or right past null infinity X~ ("scriminus") to right or left future null infinity J + ("scri-plus"). In four-dimensional flat spacetime the spherical symmetry can be used to reduce to a two dimensional diagram. In spherical coordinates the metric is ds 2 = — dt2 + dr2 + r2(d82 + sin2 0d2). If we now define u — t — r and v = t + r, the geometry of the t-r subspace for each set of polar angles is identical to the 1+1 dimensional case except that now only v > u is physical.

100

Black hole and baby universe in

(a)

r

(b)

Figure 4.3: Carter-Penrose

3

He-A

(c)

diagrams.

The spacetime is thus represented by figure 4.3(b), a diagram that is half of a diamond, with each point representing a 2-sphere except those on the vertical line on the left, which represents r = 0. This is shown as a dotted line. The spacetime of spherical matter that collapses to form a black hole looks like figure 4.3(c). The shaded region represents the collapsing matter. The dashed line represents the event horizon, and the thick-dashed line represents the curvature singularity inside the black hole. Note that the singularity is spacelike, and is the future terminus of any causal curve that goes beyond the horizon. It is unknown how or even if spacetime develops in any form to the future of the singularity, so a question mark is placed there. One common hypothesis is that a "baby universe" is born there, which is disconnected from the outside world. A controversial question is whether such a baby universe can harbor information unavailable to the outside world. The diagram for a black hole that forms by collapse and then evaporates

Ted Jacobson and Tatsuhiko Koike

101

away is shown in figure 4.3(d). After the black hole is gone, the origin of spherical coordinates appears shifted over in the diagram. No spacelike slice can enter the upper diamond (region F) and still be a Cauchy surface, since causal curves that cross the horizon into the black hole will never reach such a surface. This is the basis of the claim that only incomplete information is available to observers outside the horizon after the black hole is gone. Outside the matter the spacetime is the static Schwarzschild line element, which can be analytically extended to a complete spacetime, the diagram of which is given in figure 4.3 (e). This so-called "eternal" black hole spacetime is time-symmetric, with a white hole singularity in the past to match the black hole singularity in the future. It has two asymptotic spatial regions, connected through a "throat" at the center of the diagram.

4.4.2

Diagrams of static and uniformly moving walls

Static wall As explained in section 4.3.2 the spacetime of the static wall is just two complete copies of Minkowski spacetime. The causal diagram is figure 4.4. In strictly relativistic terms there is no connection at all between the two spacetimes, however the quasiparticles can travel superluminally and thus pass from one side of the wall to the other at a finite value of the time coordinate t. We indicate this physical connection by depicting the spacetimes as joined at the wall at spatial infinity.

Figure 4.4: Static wall.

Moving wall The causal structure of the spacetime of the moving wall (4.10) is shown in figure 4.5(a). It looks like what one would obtain by cutting the diagram

102

Black hole and baby universe in

3

He-A

for the eternal black hole (figure 4.3 (e)) along the white hole horizon, and sliding the lower half up so that the white hole singularity coincides with the black hole singularity. The result of this cut is to leave the spacetime incomplete along the cut, which corresponds to the pair of long-dashed lines in figure 4.5 (a). Geometrically it is not well-defined to extend the spacetime across the singularity, but physically in the thin film there is continuity in passing through the core of the domain wall. The other side of the wall thus plays the role of a baby universe. To clarify the relation between the conformal diagram and the physical spacetime we include in figure 4.5(a) lines of constant t, xs, and xw. Note that the incomplete boundaries are at t —> ±oo. They are the terminus of a null ray that runs parallel to the black hole horizon backward in time, or the white hole horizon forward in time. Such null rays asymptotically approach the horizon, blueshifting until superluminal terms in the dispersion relation become important, at which stage a quasiparticle would cross the horizon. Note also that the Newtonian time slices cross the singularity progressively from left to right.

4.4.3

Black hole creation and evaporation

Creation To construct an analogue of a black hole that forms by collapse we imagine that the wall velocity v is now a function of Newtonian time v(t) which is equal to zero before ii and thereafter smoothly approaches v. (We do not attempt at this stage to devise a mechanism for actually accelerating the wall in this manner.) The resulting spacetime should look like a portion of the static wall figure 4.4 below t\ and a portion of the moving wall figure 4.5 (a) above t\. This yields figure 4.5 (b). Note that the past incomplete boundary is now absent because the black hole forms at a finite time. Evaporation In the case of a real black hole the energy for the Hawking radiation comes from the mass of the hole. As the hole radiates it loses mass until it evaporates away completely, unless stabilized by conserved charges it cannot shed. In the case of the domain wall, the radiation occurs only when the wall is

moving, and it is possible that the back-reaction to the radiation causes

Ted Jacobson and Tatsuhiko

Koike

Figure 4.5: Causal diagrams of moving domain-wall

textures.

Black hole and baby universe in

104

3

He-A

the wall to slow down. On the other hand, as discussed in subsection 4.3.4, Pauli blocking may well terminate the Hawking process before the wall comes to rest. There is presumably another dissipation mechanism, such as pairbreaking due to the time-dependence of the moving texture, that eventually would stop the wall. In any case, for the purposes of creating a model of black hole evaporation, we can imagine simply that somehow or another the wall comes to rest at a time t2. The resulting spacetime should then again look like a portion of the static wall above t2, as shown in figure 4.5 (c). The causal structure for the wall that accelerates and then stops is similar but not entirely analogous to the spacetime of a black hole that evaporates. There is no region analogous to region F of figure 4.3(d), and in fact the spacetime is globally hyperbolic. The analogy is improved if we lean on the role of the Newtonian simultaneity to define what is accessible "at a given time". Thus, subsequent to t2, the spacetime consists again of two causally disconnected pieces analogous to F and the black hole interior of figure 4.3(d). To understand better what is going on it is helpful to use also a nonconformal diagram, in which the domain wall worldline is drawn vertically, figure 4.6. The singularity appears when the wall starts moving, and disappears when it comes to rest. During the motion the singularity is a window to the other side of the wall. Figure 4.6 shows the black and white hole horizons as dashed lines, as well as a quasiparticle worldline that crosses from right to left, and a Hawking pair that is created at the temporary horizon. Note that as it approaches the white hole horizon, the partner of the Hawking quantum is turned back toward the singularity since it is rightmoving with respect to the superfluid condensate. It "rattles" back and forth between the horizons until the wall stops moving.

4.4.4

Information loss

It seems clear from the previous discussion that quasiparticle information can fall across the horizon and be lost to the outside world. The information could come back carried by superluminal high energy quasiparticles, but it need not and there is no reason to suppose it does. The question of information loss by Hawking radiation is thornier. The partners remain in the ergoregion and fill the negative energy states. It seems that roughly half their information would wind up on the right side of the singularity to be available after the black hole "evaporates". Still, that leaves the other half that is lost.

Ted Jacobson and Tatsuhiko Koike

i

Figure 4.6: Temporary one-way window.

4.5

Conclusion

This is just the beginning of the story. Clearly a lot remains to be done to understand the nature of the Hawking effect in the setting of the thinfilm domain wall. Nevertheless, we hope that this analogue model will prove stimulating to researchers pondering the nature of Hawking radiation and information loss in quantum gravity on the one hand, and the physics of moving superfluid textures on the other.

Acknowledgment s: The research described here is based partly on joint work by G. Volovik and TJ. We would like to thank G. Volovik for helpful discussions, and R. Parentani for useful comments on a draft of this article. Most of this work was done while TK was visiting the Gravity Theory Group at University of Maryland, for whose hospitality he is very grateful. This research was supported in part by the Japan Society for the Promotion of Science and by the National Science Foundation under NSF grant PHY-9800967.

This page is intentionally left blank

Bibliography [1] T. Jacobson, "Black hole radiation in the presence of a short distance cutoff", Phys. Rev. D48, 728 (1993) [arXiv:hep-th/9303103]. [2] W.G. Unruh, "Experimental black-hole evaporation?", Phys. Rev. Lett. 46, 1351 (1981). [3] T. Jacobson, "Trans-Planckian redshifts and the substance of the space-time river", Prog. Theor. Phys. Suppl. 136, 1 (1999) [arXiv:hepth/0001085]. [4] S. Corley and T. Jacobson, "Black hole lasers", Phys. Rev. D 5 9 , 124011 (1999) [arXiv:hep-th/9806203]. [5] L.J. Garay, J. Anglin, J.I. Cirac, and P. Zoller, "Black holes in BoseEinstein condensates", Phys. Rev. Lett. 85, 4643 (2000) [arXiv:grqc/0002015]; "Sonic black holes in dilute Bose-Einstein condensates", Phys. Rev. A 63, 023611 (2001) [arXiv:gr-qc/0005131]. [6] T. Jacobson, "Black hole evaporation and ultrashort distances", Phys. Rev. D44, 1731 (1991). [7] T. Jacobson, "On the origin of the outgoing black hole modes", Phys. Rev. D 5 3 , 7082 (1996) [arXiv:hep-th/9601064]. [8] T.A. Jacobson and G.E. Volovik, "Event horizons and ergoregions in He-3", Phys. Rev. D58, 064021 (1998) [arXiv:cond-mat/9801308]. [9] N.B. Kopnin and G.E. Volovik, "Critical velocity and event horizon in pair-correlated systems with 'relativistic' fermionic quasiparticles", Pisma Zh. Eksp. Teor. Fiz. 67, 124 (1998) [arXiv:cond-mat/9712187]. 107

108

Black hole and baby universe in

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He-A

[10] T.A. Jacobson and G.E. Volovik, "Effective spacetime and Hawking radiation from moving domain wall in thin film of He-3-A", Pisma Zh. Eksp. Teor. Fiz. 68, 833 (1998) [JETP Lett. 69, 705 (1999)] [arXiv:grqc/9811014]. [11] G.E. Volovik, "Simulation of Painleve-Gullstrand black hole in thin 3HeA film", Pisma Zh. Eksp. Teor. Fiz. 69, 662 (1999) [JETP Lett. 69 705 (1999)] [arXiv:gr-qc/9901077]. [12] D. Vollhardt and P. Wolfle, The Superfluid Phases of Helium 3, (Taylor &; Francis, London, 1990). [13] G.E. Volovik, Exotic Properties of Superfluid Singapore, 1992).

3

He, (World Scientific,

[14] A. J. Leggett, "A theoretical description of the new phases of liquid 3He", Rev. Mod. Phys. 47, 331-414 (1975). [15] G.E. Volovik, "Superfluid analogies of cosmological phenomena", Phys. Rept. 351, 195 (2001) [arXiv:gr-qc/0005091]. [16] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Spacetime, (Cambridge University Press, 1973).

Chapter 5 Measurability of d u m b hole radiation? William Unruh CIAR Cosmology and Gravity Program University of British Columbia Vancouver Canada E-mail: [email protected]

Abstract: This chapter examines the possibilities of measuring the thermal radiation given off by dumb holes (the sonic analogues to black holes), using either superfluid 4He or "slow light" in order to illustrate the difficulties of designing an experimental realization of such analogues. (The latter case does not work at all.) While it may be possible to evade all of these difficulties, the challenge to do so is immense. 109

Measurability of dumb hole radiation?

110

5.1

Introduction

One of the most surprising conclusions in the study of quantum field theory in curved space-time was that black holes are not black, but have a temperature, proportional to the inverse of their mass. Unfortunately, not even solar mass black holes have ever been reliably found, and even if one were, the accompanying temperature (10~7 Kelvin for a solar mass black hole) would be so ridiculously low (10~7 of the surrounding cosmic background radiation) that there would be no chance of measuring it. In this context therefore, one's only hope is that small "primordial" black holes (currently of order 1015 gm) might exist as relics of the early universe, and that these black holes would just be nearing the end of their lives, to die in a spectacular explosion of ultrahigh-energy particles. No such sources have been found, and it is galling to simply have to wait for nature to deliver them to us. An alternative is to look for analogous situations in which one might expect such radiation to also occur. About 20 years ago [1], while teaching a course in fluid mechanics, I realized that there were similarities between the equations of motion obeyed by sound waves in a background flow, and the equations of motion of, say, a scalar field in the space-time near a black hole. Just as in the case of a scalar field around a black hole, the equations of motion of the linearized sound waves is a hyperbolic equation. The background time- and space-dependent fluid flow then acts like an effective metric for the field. Looking, in the first instance, at the irrotational perturbations of the fluid about the background flow, one finds a field equation for the velocity potential, <j>, defined via 5vt = 4>ti (5.1) where 5vi is the velocity perturbation of the fluid. The equation of motion becomes (dt - diV1) | (dt - v%) <j> - di (p («&«) drf) = 0. (5.2) (Here derivatives are assumed to act on everything to their right.) This is directly analogous with the equation of motion of a scalar field in a curved space-time background

4=^(vW^)=0,

(5.3)

where g = l/det(g A " / ) = det(g(/u/). We can identify the background fluid flow

William Unruh

111

with g^v via V~9 9°0 =

= p(£),

^

where c2 = dp/dp.

- p ( ^ ) ,

(5.4)

(5-5)

Since det( x /g p'"') = g we have 5 = /9 4 /c 2 , and to

=

-^(c2-^2),

(5.7)

9m

=

^(v1),

(5.8)

-jn.

(5.9)

9H =

This metric has a horizon where \v\ = c, and this surface plays exactly the same role as the horizon of a black hole does for scalar fields propagating in the curved space-time thereof. This sonic horizon also plays the same role as the black hole horizon does when one considers quantum fields in this background flow. Quantizing these irrotational sound waves (i.e., phonons) in this background classical flow, one finds, using exactly the same arguments that Hawking gave in his original article on black hole evaporation, that this horizon will not be silent, but rather will emit phonons with a thermal spectrum. The equivalent of the surface gravity in the case of a black hole here is given by the rate of change of the velocity of the background fluid at the horizon. The temperature is given by h

\

^2-KKBJ

d(|v|-c) ax

One of the key uses of this sonic analogue has been to test the dependence of the thermal emission on various theoretical assumptions. The first test was to ask whether the cutoff in the frequencies and wave-numbers given by the atomic nature of real fluids would be expected to alter the above prediction of thermal emission of sound waves. One might expect some alteration as Hawking's derivation linked the late-time thermal emission to vacuum fluctuations in the early state. (Before the black hole, or in our case, the dumb hole

112

Measurability of dumb hole radiation?

[dumb denoting "not being able to speak"], was formed.) Because of the exponential redshift suffered by the outgoing characteristic rays in the vicinity of the horizon, late-time emissions (at time t) at frequencies of Q = ksT/h are traced back to early-time frequencies larger by a factor of ent. These frequencies are of course absurd for times much larger than 1/Q. While the atomic physics of fluids is well understood (unlike quantum gravity), it is still impossible to calculate the quantum behaviour of 1025 atoms undergoing hypersonic flow. The realization by Jacobson [2] that one of the crucial effects of the atomic nature is to change the dispersion relation at high frequencies and wavenumbers for the sound waves allowed one to begin to understand the effects that the atomic nature of the fluid would have on the thermal emission. Much work has been done to examine the effects of a variety of possible alterations of the high frequency dispersion relations on the quantum emission from the hole. (See the chapter by Jacobson [3] in this volume for further references.) The conclusion of these investigations is that the altered dispersion relation essentially has no effect on the quantum emission of sound waves. One of the original investigations [4] is presented in figure 5.1 where the comparison between the outgoing radiation expected on the basis of the above thermal hypothesis and the directly calculated radiation in the presence of the altered dispersion relation is presented. The agreement of these curves, even at frequencies orders of magnitude larger than than the thermal frequency Q, demonstrates the insensitivity of the emitted radiation to alterations in the dispersion relation. Once one gets to frequencies of order the cutoff of course, significant alterations occur, but the growth of these deviations (with frequency) is slow. These theoretical investigations gave one hope that the Hawking phenomenon was robust, and did not (despite appearances from the derivation) depend on the physics in regimes in which the theory certainly failed. However, given the difficulty of finding small black holes to observe, another hope for the usefulness of such sonic analogues is in experimental tests of black hole evaporation. Can one observe the thermal emission from dumb holes, giving one faith that the assumptions going into the black hole calculations are correct? The difficulties with any such experimental realization are many. One must establish a trans-sonic flow. One must then have sufficient sensitivity in one's detection apparatus to detect the radiation given off. Furthermore one must ensure that there are not other sources of sound which would mask the effect under consideration. In the rest of this paper, I will outline these difficulties. (Some have already been treated by Jacobson [2].)

William Unruh

113

1

1

'

'

'

'

T

1

i

i

|

.

i

i

i

|

•

a - Pos. Freq. n - Neg. F r e q . A - Final

•

. -

A

*a

•

A

A

3

A H

*

•

. -

.

A A

a A H

' -

A

A

A

D A

a

4t A ^ A

% D

n i

,

,

,

,

A

* A *,

A"

1

i

CO

Figure 5.1: Comparison of positive and negative norm modes under the Thermal Emission assumption for a dumb hole.

5.2

Hypersonic flow

To establish a trans-sonic flow, about the only option is the use of a de Laval nozzle, a converging/diverging nozzle (figure 5.2). The fluid must be pushed into the nozzle at a rate sufficient to achieve sonic flow velocities at the throat of the nozzle (the only place where such sonic flow could occur). If the pressure on the downstream end is sufficiently low, the fluid will accelerate both going into the converging nozzle at subsonic speeds, and as it exits the nozzle in the diverging downstream end of the nozzle. We can derive the equations for flow in a de Laval nozzle by using the

114

Measurability of dumb hole radiation?

Phonons

Horizon (Sonic flow) de Laval Slot Nozzle Figure 5.2: A sketch of a slot type de Laval nozzle. With a sufficiently high pressure on the subsonic side and low pressure on the supersonic side, the fluid will go into hypersonic flow with the sonic surface at the narrowest part of the nozzle. Bernoulli equation -v2 + h{p) = Const,

(5.11)

where dp

« , ) - / !p *dp
(5.12)

Assuming steady flow, the mass conservation equation is p v A = Const,

(5.13)

where A is the area of the nozzle at the position under consideration. Assuming that all terms depend only on z (the distance along the axis of the nozzle), taking the derivative of both equations with respect to z, and using that the velocity of sound is c2 = p dh/dp we get c2 — v2 dv vc2 dz

l^dA Adz'

(5.14)

William Unruh

115

We have assumed that the velocity of sound is constant as a function of density, a reasonable assumption for liquid Helium which has [6] (dln(c)/dp « 1/30 atmosphere), to get

'»(;KHD>-'"(£)'

(515

»

with the transition from subsonic to supersonic occurring (if it does) at the narrowest part of the throat. Using the quadratic approximation for A at its narrowest portion, we have A = A> + pz2,

(5.16)

and c2-v2 = 2cz^.

(5.17) az

Therefore (5.18) The term 1/Li = ^/3/A0 is essentially the inverse distance over which the diameter of the nozzle increases by a factor of two in the vicinity of the throat. For the approximations which are used to derive the equations of the de Laval nozzle to be valid, this distance must be of the same order or greater than the diameter of the throat, which implies that the distance Li under consideration is of the same order as or larger than the diameter Lt of the nozzle. The biggest problem with this scenario is that the frequency, and thus the mean temperature of the phonons emitted due to the quantum instability (black hole evaporation analogue), is given by the velocity of sound over Li\ and the velocity of sound in general is very small (compared to that of light). The velocity of sound in superfluid helium is of the order of 250 m/s, a factor of approximately 10~5 of that of light. Thus, while a black hole with dimensions of the order of kilometers {i.e., a solar mass black hole) has a temperature of the order of 10~7 K, a dumb hole with typical dimensions of a centimeter would have the equivalent temperature of about 10~6 K. Even a micron sized nozzle would only have a temperature of about 3 • 1 0 - 4 K. That is, liquid helium is the only material in which one could hope to measure these effects.

116

Measurability of dumb hole radiation?

Let us estimate the energy radiated by one of these holes. Assuming a Boltzmann distribution of the energy radiated (an overestimate as a nonone-dimensional dumb hole — one where the curvature length at the center of the nozzle is of the same order as the diameter — will have a non-trivial albedo, especially at low frequencies), the power emitted from such a nozzle would be of the order of P = Cfsound T4

A0.

(5.19)

Here A0 is the area of the nozzle throat, T is the temperature of the dumb hole [^/(27rA)fl) dv/dz], and Osound -

12Q

(b.M)

c 2 h3

is the sonic Stefan-Boltzmann constant (for one rather than two polarizations), and c the velocity of sound in the material. Since T = hLt/c and A — L2, where Li is the longitudinal dimension of the nozzle, we have _2

P

~

7T2 ^

1.4

fc4

Kk%

3

120 c h

he2 2 • 10

h*Cc„ 44

n

B

2

(2TT) 4 Ll

r2

L\ 4

(roil

kg

(5.22)

L\

Thus the power emitted decreases rapidly for a given dimension of the dumb hole as the velocity of sound decreases (making attempts to produce situations with ever lower velocities of sound, in order to achieve trans-sonic flow more easily, counterproductive). For a one micron hole, with sound velocity of 250 m/s, and with Lt&Lt, we have 2 • 1 n~22 W

P « 2 • 10" 34 W m 2 L" 2 « i-P-. \ . (L/micron)' !

(5.23)

We could increase the emission rate by constructing many holes. For example, with 104 holes, produced say by micro-fabrication techniques, or a slit of length a centimeter, this could be increased to 1 0 - 1 8 W in phonon energy emitted. The frequency of the sound would be of order 0.5 GHz. A big problem would be the detection of this radiation. The best bolometric detectors for IR electromagnetic radiation have a sensitivity of about 10~16 W \fr [5], where r is the integration time. However, for electromagnetic bolometers one can make the fraction of the incident radiation absorbed

Measurability of dumb hole radiation?

118

2

jr

Wave Number

I

^ -i

2

Figure 5.3: Sketch of the dispersion curve for superfluid iHe. The local minimum located at 2A'1 is the so-called roton part of the dispersion curve.

wavenumber to negative frequencies. That is, the walls of the nozzle would have to be perfectly smooth on scales from microns to sub-atomic scales. This would rule out the creation of such nozzles by micro-lithography, as the walls of such nozzles would be expected to be very rough on such scales. The best that one could probably do would be to create a long slot nozzle, with the walls covered say by a sheet of mica with no dislocations or steps, or perhaps by the use of fire polished glass. While these could perhaps be sufficiently smooth on scales longer than atomic scales, one would expect the atomic constituents of even these walls to interact with the fluid He. However, more detailed calculations of phonon creation rates on such atomically smooth walls still need to be done. There is a further requirement on such a surface, and that is that the velocity of sound (in particular of surface waves) in the material of which the walls are made must be higher than the velocity of sound in the He. If the flow of the fluid past the walls is faster than the velocity of sound in the walls (or if the velocity of the fluid is higher than its own velocity of

William Unruh

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by the bolometer approximately unity, and by placing the bolometer into a vacuum, make the cooling time of the bolometer long. Coupling the sound waves in the liquid He to the bolometer is likely to produce a much lower absorption fraction. Furthermore, since the bolometer must be in sonic, and thus thermal contact with the ultra-low-temperature liquid (< 1 0 - 4 K), the cooling time is liable to be very short as well, making the sensitivity of a sonic bolometer much worse than for EM radiation.

5.3

Rot on creation

In the derivation of the thermal emission by a dumb hole, the assumption is that the only process which can lead to the creation of phonons in the convergent supersonic flow is due to the changing flow velocity near the horizon. This changing flow is smooth and leads to significant creation of phonons only for wavelengths which correspond to the scale of the changing flow. However a de Laval nozzle has walls, and one must worry about the interaction of the fluid with those walls. While the superfluid He at 10~4 K is not viscid, the interaction with the wall can result in the creation of phonons in the fluid. The dispersion relation of liquid 4 He is complex, and is sketched in figure 5.3. The local minimum in the frequency at a wavenumber of about 2 A - 1 is called the roton region. With increasing velocity of the fluid, the dispersion relation is tilted, so that ui = u — vk is the lab frame frequency. Once this frequency becomes negative, it is possible to create particles via a pair creation process with the equivalent negative frequency branch of the dispersion relation. Thus, when the flow velocity exceeds the Landau velocity (where u at the roton minimum goes negative), it is possible to create roton pairs. However in order to create such pairs, there must be a change in momentum of about twice the roton wavenumber. The smooth fluid flow cannot create these changes in momentum, but the walls of the nozzle could. The roton wavenumber, of about 2 A - 1 , corresponds to a wavelength of about 3 A. If the walls have variations on the scale of 1.5 A, those variations can create pairs once the velocity of the fluid exceeds the Landau velocity. But this scale is just the atomic scale, implying that the walls would have to be smooth on atomic dimensions. Furthermore, any variations in the walls on scales smaller than a micron would also be expected to create phonons once the fluid velocity takes portions of the dispersion curve with corresponding

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119

sound) an instability, causing exponential growth in sonic perturbations in both the wall and the fluid would be expected. The simplest derivation of this phenomenon is to assume that the fluid damps the motion of the walls (by the creation of sound waves in the fluid). The equation of motion of the walls is then of the form (where ex is the transverse wave velocity in the wall; not the speed of sound in the fluid) x + 2r] (X - vx') - c\x" = 0.

(5.24)

The dispersion relation is then w = —ir] ± yc2± k2 — r]2 + 2ir]vk

(5.25)

which gives, for large k u) = ±k-iT]{l±v/c_L).

(5.26)

This corresponds to exponential growth for modes running with the stream if v > C_L. That is, the sound waves running along the surface of the material in the direction of the flow are amplified. While this might not seem too bad, as these waves are flowing into the dumb hole, non-linearities in these exponentially growing waves and scattering from the end of the nozzle would be expected to create sound waves in the walls which propagated back into the subsonic flow region. Furthermore, the damping coefficient 77 would arise directly from the creation of sound waves within the fluid. So, even if the surface of the fluid is atomically smooth, one would expect such instabilities to cause the creation of significant sound within the fluid.

5.4

Vorticity

I will not look into this difficulty except that the creation of vortex lines at the surface of the nozzle could be a dominant form of friction for the fluid along the walls of the nozzle. One would have to nucleate a short length of vortex string at the wall, but it could thereafter rapidly grow and penetrate into the fluid. This is roughly the analogue of turbulence of the superfluid. One would expect that any roughness in the surface of the nozzle would enhance any such nucleation process. The ultra-low temperatures needed in our system would imply that such a nucleation process should be a quantum tunneling process, as thermal activation should be highly suppressed by the low temperature. I will not attempt to estimate the nucleation probability here.

120

5.5

Measurability of dumb hole radiation?

Density changes

Since liquid He solidifies under high enough pressures (about 25 atmospheres = 2.5 • 106 Pascal), one must check that the pressure and density needed to drive the fluid through the supersonic nozzle are not too high. For the low-velocity part of the nozzle (i.e., on the subsonic side) and for A ^> A0, and thus large areas of the nozzle (i.e., much larger than the throat area) we have as the solution to the nozzle equation

h

(!) + h- to (s)

(5 27)

-

or p « A, v ^

(5.28)

where p0 is the density at the throat. So the density of the fluid needed to force the fluid through the nozzle supersonically is a factor of about 1.6 times the throat density (and similarly for the pressure). The required pressure is thus at least p = 0.6 po c2 « 5.6 • 106 Pascal « 56 atm, (5.29) which means that the liquid Helium solidifies before one can get it to a high enough pressure to force it through the nozzle. On the other hand, beyond the throat, the density drops roughly proportional to the inverse of the area. Thus the pressure rapidly drops below zero, and the liquid would begin to cavitate, creating large amounts of sound waves which could be transmitted back to the subsonic fluid through the walls of the nozzle. The overall conclusion is that the measurement of radiation of sound waves from the sonic horizon of a dumb hole is likely to present almost insurmountable difficulties. The creation of a dumb hole is difficult, requiring the acceleration of fluid flows to trans-sonic velocities. While searching for materials with low velocities would seem to be advantageous, this is counterproductive as the power radiated by a dumb hole of given dimension falls with the square of the sound velocity. Furthermore the generation of such trans-sonic flows creates severe problems in the interactions between the fluid and the walls of whatever nozzle is used to create those flows. (A free transsonic flow would be wonderful, but I cannot think of any way of creating it, barring the possession of a small black hole — in which case the fluid analogue would be irrelevant since one could study the black hole itself.)

William Unruh

5.6

121

Slow light

Recently it has been shown that the group velocity of light over a narrow frequency range can be reduced to a very low value, including zero [7]. This at first suggests that such "slow light" could be used to create a black hole analogue (sometimes called an optical black hole — which risks confusion with real black holes which are already optical) which could be used to test black hole evaporation at least over the narrow frequency range corresponding to the frequency region of slow light. Unfortunately, this hope is in vain. The critical issue is that what is important in the creation process of the thermal radiation from an optical black hole analogue is not the group velocity of the light, but rather the phase velocity. One must achieve a flow of the fluid in which the particle creation process is to take place such that the phase velocity, not the group velocity, with respect to an observer far from the system is zero. The reason is that the creation process is one in which one is creating particles by mixing positive and negative norm states of whatever field it is (sound or electromagnetic) in which one wishes to create the thermal radiation. But in order to get that mixing of positive and negative norm states, their frequencies must be equal (since the flow is stationary, and thus that frequency in the laboratory frame is conserved). In the fluid frame the frequencies of the positive and negative norm states are also positive and negative, as are the wavenumbers. Going from the fluid to the laboratory frame implies a change in frequency of W+iab = W+fluid -Vk+

(5.30)

where k is the wavenumber and the + refers to the positive-frequency component. For the negative-frequency component on the other hand we have the same relation but with fc_ = — k+ and w_flUid = -w+flUid- Thus, in order to get the particle production, we require that w_iab = w+iab or v = ——.

(5.31)

K+

This is the requirement that the velocity of the fluid must equal the phase velocity of the field, not the group velocity. Furthermore, in order to create the particles one must also have a way of changing the wavenumber of the field modes from k+ to — k+. In the black hole analogue this is provided by the rate of change of the velocity of the fluid at the horizon of the hole.

122

5.7

Measurability of dumb hole radiation?

Conclusion

While the sonic analogue, or other analogues, to black holes have been an exceedingly fruitful device for the theoretical understanding of the particle creation process near black holes, it would be exciting if they could also be used to carry out experimental investigations as well. However, the possibilities for such experimental investigations seems depressingly remote, and fraught with experimental problems. This chapter has shown some of the problems which arise in a particular realization of such dumb holes. While some are peculiar to the specific system being looked at {e.g., liquid 4 He, or slow light) most are generic. The low velocity of sound (or of any analogue to sound) means that even the smallest dumb holes conceivable (much smaller than micron size means that one is no longer operating with a temperature on the low-frequency linear portion of the dispersion relation) will have a temperature well below 1 K, and thus require a material which is liquid or gaseous at those temperature, a very limited number of materials. (Very dilute gases as in Bose-Einstein condensates have the problem that the interatomic spacing is such that the atomic cutoff in wavenumber and frequency occurs at very low values. Since the inter-atomic spacing is of the order of a micron, temperatures whose equivalent sonic wavelength is of that same order will not behave like analogues to black holes. The altered dispersion relation at thermal frequencies will make the emission decidedly non-thermal.) This implies ultra-low powers of thermal radiation emitted, and in a form which is very hard to measure or find transducers for. Some sort of nozzle is required to accelerate the fluid to supersonic velocities, and interaction of the fluid with the walls of the nozzle introduces potential sources of noise. Any deviation of the flow from the smooth ideal flow (e.g., by wall roughness) increases the possibility of introducing high-frequency instabilities in the fluid which will create particles once the fluid flow becomes sufficiently high velocity with respect to the walls, particles which will mask the sought for effect. The velocity of sound (or in particular of surface waves) in the nozzle walls can lead to instabilities once the fluid velocity exceeds that surface wave velocity, instabilities which again would be expected to created masking noise. Conduction of disturbances through the walls of the nozzle from the supersonic side (where instabilities in the fluid flow would be expected) can induce waves in the subsonic fluid. Finally, in the case of sound waves in liquid He, the pressures necessary to initiate the trans-sonic flow are also

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sufficient to solidify the superfluid 4 He or 3 He. All of these difficulties would be expected to exist no matter what the disturbance is that one is using as the analogue for light around a black hole, but they could be of lesser or greater importance depending on the nature of that disturbance. Finally, the most beguiling possibility, that of using slow light, seems to fail due to the requirement that the phase, not the group velocity of the light needs to be reduced to zero to trigger the Bogoliubov frequency mixing required to create particles, and thus the analogue of the black hole evaporation radiation. While it may be possible to evade all of these difficulties, the challenge to do so is immense.

Acknowledgments: I would like to thank the Canadian Institute for Advanced Research and the NSERC for support during this work. I would especially like to thank Walter Hardy, for teaching me about the properties of liquid He, Tom Tiedje for the suggestion of using mica to create a smooth surface and Mark Halpern for discussions about bolometers.

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Bibliography [1] W. G. Unruh, "Experimental black hole evaporation?", Phys. Rev. Lett., 46,(1981) 1351-1353. [2] T. Jacobson, "Black hole evaporation and ultrashort distances", Phys. Rev. D 44, 1731 (1991); [3] See the chapter by Ted Jacobson and Tatsuhiko Koike in this volume for additional references regarding quasi-particle black holes. [4] W. G. Unruh, "Dumb holes and the effects of high frequencies on black hole evaporation", Phys. Rev. D 5 1 , 2827 (1995) [gr-qc/9409008]. [5] Mark Halpern, Private Communication. Such bolometers are a key feature of the various balloon and satellite measurements of the Cosmic Microwave Background radiation. [6] For the properties of superfluid 4 He I have used: J. Wilks, The Properties of Liquid and Solid Helium, (Clarendon Press, Oxford, 1967). [7] See the chapter by Ulf Leonhardt in this volume for a review of slow light.

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Chapter 6 Effective gravity and q u a n t u m vacuum in superfluids Grigori Volovik Helsinki University of Technology Finland and Landau Institute for Theoretical Physics Moscow Russia E-mail: [email protected]

Abstract. Quantum liquids, in which an effective Lorentzian metric (and thus some kind of gravity) gradually arise in the low-energy corner, are systems where the problems related to the quantum vacuum can be investigated in detail. In particular, they provide the possible solution of the cosmological constant problem: Why is the observed vacuum energy at least 120 orders of magnitude smaller than the naive estimate derived from the relativistic quantum field theory? The almost complete cancellation of the cosmological constant does not require any fine tuning and comes from the fundamental "trans-Planckian" physics of quantum liquids. The remaining vacuum energy is generated by the perturbation of quantum vacuum caused by matter 127

128

Effective gravity and quantum vacuum in superftuids

(quasiparticles), curvature, and other possible sources, naturally leading to a smooth component — the quintessence. This provides the possible solution of another cosmological constant problem: Why is the present cosmological constant on the order of the present matter density of the Universe? We discuss here some properties of the quantum vacuum in quantum liquids: the vacuum energy under different conditions; excitations above the vacuum state and the effective acoustic metric for them provided by the motion of the vacuum; Casimir effect, etc.

6.1

Introduction

Quantum liquids, such as 3 He and 4 He, represent systems of strongly interacting and strongly correlated atoms, 3 He and 4 He atoms correspondingly. Even in their ground states, such liquids are rather complicated objects, whose many body physics requires extensive numerical simulations. However, when the energy scale is reduced below about 1 K, we can no longer resolve the motion of isolated atoms in the liquid. The smaller the energy the better is the liquid described in terms of its collective modes and a dilute gas of particle-like excitations — quasiparticles. This is the Landau picture of the low-energy degrees of freedom in quantum Bose and Fermi liquids. The dynamics of collective modes and quasiparticles is described in terms of what we now call "the effective theory". In superfluid 4 He this effective theory, which incorporates the collective motions of the ground state — the quantum vacuum — and the dynamics of quasiparticles in the background of the moving vacuum, is known as two-fluid hydrodynamics [1]. Such an effective theory does not depend on details of microscopic (atomic) structure of the quantum liquid. The type of effective theory is determined by the symmetry and topology of the ground state, and the role of the microscopic physics is only to choose between different universality classes on the basis of the minimum energy consideration. Once the universality class is determined, the low-energy properties of the condensed matter system are completely described by the effective theory, and the information on the microscopic (trans-Planckian) physics is lost [2]. In some condensed matter systems the universality class produces an

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effective theory which is strongly reminiscent of relativistic quantum field theory. For example, the collective fermionic and bosonic modes in superfluid 3 He-A reproduce chiral fermions, gauge fields, and even (in many respects) the gravitational field [3]. This allows us to use quantum liquids for the investigation of the properties related to the quantum vacuum in relativistic quantum field theories, including the theory of gravitation. The main advantage of quantum liquids is that in principle we know their vacuum structure at any relevant scale, including the interatomic distance, which plays the part of one of the Planck length scales in the hierarchy of scales. Thus the quantum liquids can provide possible routes from our present low-energy corner of the effective theory to the "microscopic" physics at Planckian and trans-Planckian energies. One of the possible routes is related to the conserved number of atoms N in the quantum liquid. The quantum vacuum of a quantum liquid is constructed from discrete elements, the bare atoms. The interaction and zeropoint motion of these atoms compete and provide an equilibrium ground state of the ensemble of atoms; and this state can exist even in the absence of external pressure. The relevant energy and the pressure in this equilibrium ground state are exactly zero in the absence of interaction with the environment. Translating this to the language of general relativity, one deduces that the cosmological constant in the effective theory of gravity in the quantum liquid is exactly zero without any fine tuning. The equilibrium quantum vacuum is not gravitating. This route shows a possible solution of the cosmological constant problem: Why does estimating the vacuum energy using relativistic quantum field theory give a value which is 120 orders of magnitude higher than its upper experimental limit? In quantum liquids there is a similar discrepancy between the exact zero result for the vacuum energy and the naive estimation within the effective theory. We shall also discuss here how different perturbations of the vacuum in quantum liquids lead to a small nonzero energy of the quantum vacuum. In the language of general relativity, in each epoch the vacuum energy density must be either of order of the matter density of the Universe, or of its curvature, or of the energy density of the smooth component — the quintessence. Here we mostly discuss the Bose ensemble of atoms: a weakly interacting Bose gas, which experiences the phenomenon of Bose condensation, and a real Bose liquid — superfluid 4 He. The consideration of the Bose gas allows us to use the microscopic theory to derive the ground state energy of the

130

Effective gravity and quantum vacuum in superBuids

quantum system of interacting atoms and the excitations above the vacuum state — quasiparticles. We also discuss the main differences between the bare atoms, which comprise the vacuum state, and the quasiparticles, which serve as elementary particles in the effective quantum field theory. Another consequence of the discrete number of elements comprising the vacuum state is related to the Casimir effect. The discreteness of the vacuum — the finite-TV effect — leads to mesoscopic Casimir forces, which cannot be derived within the effective theory. For these purposes we consider the Fermi ensembles of atoms: Fermi gas and Fermi liquid.

6.2 6.2.1

Einstein gravity and cosmological constant problem Einstein action

Einstein's geometrical theory of gravity consists of two main elements [4]. (1) Gravity is related to a curvature of space-time in which particles move along geodesic curves in the absence of non-gravitational forces. The geometry of space-time is described by the metric g^ which is the dynamical field of gravity. The action for matter in the presence of a gravitational field 5M, which simultaneously describes the coupling between gravity and all other fields (the matter fields), is obtained from the special relativity action for the matter fields by replacing everywhere the flat Minkowski metric by the dynamical metric g^, and the partial derivative by the g-covariant derivative. This follows from the principle that the equations of motion do not depend on the choice of coordinate system (the so-called principle of general covariance). This also means that motion in the non-inertial frame can be described in the same manner as motion in some gravitational field — this is the equivalence principle. Another consequence of the equivalence principle is that the space-time geometry is the same for all particles: gravity is universal. (2) The dynamics of the gravitational field is determined by adding the action functional SQ for g^, which describes propagation and self-interaction of the gravitational field: S = SG + 5 M . (6.1) The general covariance principle requires that SQ is the functional of the

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curvature. In the original Einstein theory only the first order curvature term is retained:

l&rGJ

(6.2)

d 4 £\/—ff H i

where G is the gravitational Newton constant, and ~R is the Ricci scalar curvature. The Einstein-Hilbert action is thus

5=

~i6b/dW=^+5M

(6.3)

Variation of this action over the metric field g^„ gives the Einstein equations: SS

-sM^-^H

= 0,

(6.4)

where T™v is the energy-momentum of the matter fields. Bianchi identities lead to the "covariant" conservation law for matter T% = 0 , o r a , {T^V^g) = \V=9 Tam dv9a& ,

(6.5)

But actually this "covariant" conservation takes place in virtue of the field equation for "matter" irrespective of the dynamics of the gravitational field. 6.2.2

Vacuum energy and cosmological term

In particle physics, field quantization allows a zero-point energy, the constant energy when all fields are in their ground states. In the absence of gravity, only the difference between zero points can be measured. For example in the Casimir effect, while the absolute value is unmeasurable by non-gravitational means, Einstein's equations react to T ^ and thus to the value of vacuum energy itself. If the vacuum energy is taken seriously, the energy-momentum tensor of the vacuum must be added to the Einstein equations. The corresponding action is given by the so-called cosmological term, which was introduced by Einstein in 1917 [5]: 5A = -PA j d^y/^j , T£ = -^jjjrv

= P^»

(6.6)

Effective gravity and quantum vacuum in superfiuids

132

The energy-momentum tensor of the vacuum shows that the quantity PA\f-~9 is the vacuum energy density, and the equation of state of the vacuum is = -PA ,

PA

(6-7)

where PA^/--g is the partial pressure of the vacuum. Einstein's equations are modified in the following way: 1

'R»»-l-n9liV\=Ti

8TTGV ^

6.2.3

+ T™.

(6.8)

2

Cosmological c o n s t a n t p r o b l e m

The most severe problem in the marriage of gravity and quantum theory is "why is the vacuum not gravitating?" [6]. The vacuum energy density can be easily estimated: the positive contribution comes from the zero-point energy of the bosonic fields and the negative — from the occupied negative energy levels in the Dirac sea. Since the largest contribution comes from the high momenta, where the energy spectrum of particles is massless, E = cp, the energy density of the vacuum is PA\fZZ9

= 77 [ ^bosons ^2

V

\

^Cp

- ^fermions X I CP

P

P

(6"9) /

where V is the volume of the system; bosons is the number of bosonic species and fermions is the number of fermionic species. The vacuum energy is divergent and the natural cut-off is provided by gravity itself. The cut-off Planck energy is determined by Newton's constant:

= /^V /2

^Planck = ( - £ - )

>

(6-10)

It is on the order of 1019 GeV. If there is no symmetry between the fermions and bosons (supersymmetry) the Planck energy scale cut-off provides the following estimation for the vacuum energy density: 1_ ^Planck . (6-11) c with the sign of the vacuum energy being determined by the fermionic and bosonic content of the quantum field theory. Here we considered the flat space with Minkowski metric g^v = d i a g ( - l , c ~ 2 , c ~ 2 , c ~ 2 ) . PAV1^

~ ± - ^3P l a n c k = ±X^9

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133

The "cosmological constant problem" is a huge disparity between the naively expected value in equation (6.11) and the range of actual values. The experimental observations show that p^ is less than or on the order of 10 _120 .Ep lanck [7]. In case of supersymmetry, the cut-off is somewhat less, being determined by the scale at which supersymmetry is violated, but the huge disparity persists. This disparity demonstrates that the vacuum energy in equation (6.9) is not gravitating. This is in apparent contradiction with the general principle of equivalence, according to which the inertial and gravitating masses must coincide. This indicate that the theoretical criteria for setting the absolute zero point of energy are unclear and probably require physics beyond general relativity. To clarify this issue we can consider such quantum systems where the elements of the gravitation are at least partially reproduced, but where the structure of the quantum vacuum is known. Quantum liquids are the right systems.

6.2.4

Sakharov induced gravity

Why is the Planck energy in equation (6.10) the natural cutoff in quantum field theory? This is based on the important observation made by Sakharov that the second element of the Einstein's theory can follow from the first one due to the quantum fluctuations of the relativistic matter field [8]. He showed that vacuum fluctuations of the matter field induce the curvature term in the action for g^. One can even argue that the whole Einstein-Hilbert action is induced by vacuum polarization, and thus gravity is not the fundamental force but is determined by the properties of the quantum vacuum. The magnitude of the induced Newton's constant is determined by the value of the cut-off: G~l ~ hE^utoS/c5. Thus, in Sakharov's gravity induced by quantum fluctuations, the causal connection between the gravity and the cut-off is reversed: the physical high-energy cut-off determines the gravitational constant. The Cutoff dependence of the inverse gravitational constant explains why gravity is so small compared to the other forces, whose running coupling "constants" have only a mild logarithmic dependence on EcutosThe same cut-off must be applied for the estimation of the cosmological constant, which thus must be of order of Cutoff- But * m s i s m severe contradiction with experiment. This shows that, though the effective theory is appropriate for the calculation of the Einstein curvature term, it is not applicable for the calculation of the vacuum energy: trans-Planckian physics

134

Effective gravity and quantum vacuum in superQuids

must be evoked for that. The Sakharov theory does not explain the first element of the Einstein's theory: it does not show how the metric field g^ appears. This can be given only by the fundamental theory of quantum vacuum, such as string theory where the gravity appears as a low-energy mode. The quantum liquid examples also show that the metric field can naturally and in some cases even appear as an emergent low-energy collective mode of the quantum vacuum.

6.2.5

Effective gravity in quantum liquids

The first element of the Einstein theory of gravity (that the motion of quasiparticles is governed by the effective curved space-time) arises in many condensed matter systems in the low-energy limit. An example is the motion of acoustic phonons in a distorted crystal lattice, or in the background flow field of a superfluid condensate. This motion is described by the effective acoustic metric [9, 10, 11, 12]. For this "relativistic matter field" (acoustic phonons with dispersion relation E = cp, where c is the speed of sound, simulate relativistic particles) the analogue of the equivalence principle is fulfilled. As a result, the covariant conservation law in equation (6.5) does hold for the acoustic mode if g^ is replaced by the acoustic metric. The second element of the Einstein's gravity is not easily reproduced in condensed matter. In general, the dynamics of acoustic metric g^ does not obey the equivalence principle in spite of the Sakharov mechanism of induced gravity. In existing quantum liquids, the Einstein-Hilbert action induced by the quantum fluctuations of the "relativistic matter field" is much smaller than the non-covariant action induced by the "non-relativistic" high-energy component of the quantum vacuum, which is overwhelming in these liquids. Of course, one can find some very special cases where the Einstein-Hilbert action for the effective metric is dominating, but this is not a rule. Nevertheless, in spite of the incomplete analogy with the Einstein theory, effective gravity in quantum liquids can be useful for investigation of the cosmological constant problem.

Grigori Volovik

6.3 6.3.1

135

Microscopic 'Theory of Everything' in quant u m liquids Microscopic and effective theories

There are two ways to study quantum liquids: (i) The fully microscopic treatment. It can be realized completely (a) by numerical simulations of the many body problem; (b) analytically for some special models; (3) perturbatively for some special ranges of the material parameters, for example, in the limit of weak interaction between the particles. (ii) Phenomenological approach in terms of effective theories. The hierarchy of the effective theories correspond to the low-frequency long-wave-length dynamics of quantum liquids in different ranges of frequency. Examples of effective theories: Landau theory of Fermi liquid; Landau-Khalatnikov two-fluid hydrodynamics of superfluid 4 He [1]; theory of elasticity in solids; Landau-Lifshitz theory of ferro- and antiferromagnetism; London theory of superconductivity; Leggett theory of spin dynamics in superfluid phases of 3 He; effective quantum electrodynamics arising in superfluid 3 He-A; etc. The latter example indicates, that the existing Standard Model of electroweak, and strong interactions, and the Einstein gravity too, are the phenomenological effective theories of high-energy physics, which describe its low-energy edge, while the microscopic theory of the quantum vacuum is absent.

6.3.2

Theory of Everything for quantum liquids

The microscopic "Theory of Everything" for quantum liquids — "a set of equations capable of describing all phenomena that have been observed" [2] in these quantum systems — is extremely simple. On the "fundamental" level appropriate for quantum liquids and solids, i.e., for all practical purposes, the 4 He or 3 He atoms of these quantum systems can be considered as structureless: the 4 He atoms are the structureless bosons and the 3 He atoms are the structureless fermions with spin 1/2. The Theory of Everything for a collection of a macroscopic number N of interacting 4 He or 3 He atoms is contained in the non-relativistic many-body Hamiltonian

136

Effective gravity and quantum vacuum in superBuids

acting on the many-body wave function ^ ( r i , T2, •-., r^,..., r,-,...). Here m is the bare mass of the atom; C/(r; — r,) is the pair interaction of the bare atoms i and j . When written in the second quantized form it becomes the Hamiltonian of the quantum field theory

U-\xM

=

fdx^{x)

- 2 ^ - "

i>(x)

+\fdxdyU(x-y)tl>1(x)il>1(y)tl>(y)1>(x).

(6.13)

In 4 He, the bosonic quantum field ip(x) is the annihilation operator of the 4 He atoms. In 3 He, ip(x) is the fermionic field and the spin indices must be added. Here Af = J dx ip^(x)ip(x) is the operator of particle number (number of atoms); fj, is the chemical potential — the Lagrange multiplier which is introduced to take into account the conservation of the number of atoms.

6.3.3

Importance of discrete particle number in microscopic theory

This is the main difference from the relativistic quantum field theory, where the number of any particles is not restricted: particles and antiparticles can be created from the quantum vacuum. As for the number of particles in the quantum vacuum itself, this quantity is simply not determined by current knowledge. At the moment we do not know the structure of the quantum vacuum and its particle content. Moreover, it is not clear whether it is possible to describe the vacuum in terms of some discrete elements (bare particles) whose number is conserved. On the contrary, in quantum liquids the analogue of the quantum vacuum — the ground state of the quantum liquid — has a known number of atoms. If N is big, this difference between the two quantum field theories disappears. Nevertheless, the mere fact that there is a conservation law for the number of particles comprising the vacuum leads to a definite conclusion regarding the value of the relevant vacuum energy. Also, as we shall see below in section 6.7, the discreteness of the quantum vacuum can be revealed in the mesoscopic Casimir effect.

Grigori Volovik

6.3.4

137

Enhancement of symmetry in the low energy corner: Appearance of effective theory

The Hamiltonian (6.13) has very restricted number of symmetries: It is invariant under translations and 5 0 ( 3 ) rotations in 3D space; there is a global U(l) group originating from the conservation of the number of atoms: % is invariant under gauge rotation ip(x) —> e"*ip(x) with constant a; in 3 He in addition, if the relatively weak spin-orbit coupling is neglected, % is also invariant under separate rotations of spins, SO(3)s- At low temperature the phase transition to the superfluid or to the quantum crystal state occurs where some of these symmetries are broken spontaneously. For example, in the 3 He-A state all of these symmetries, except for the translational symmetry, are broken. However, when the temperature and energy decrease further, the symmetry becomes gradually enhanced in agreement with the anti-grand-unification scenario [13, 14]. At low energy, the quantum liquid or solid is well described in terms of a dilute system of quasiparticles. These are bosons (phonons) in 4 He and fermions and bosons in 3 He, which move in the background of the effective gauge and/or gravity fields simulated by the dynamics of the collective modes. In particular, phonons propagating in the inhomogeneous liquid are described by the effective Lagrangian ^effective = V^H ff"" ^ Q 8vtt

,

(6.14)

where g**" is the effective acoustic metric provided by inhomogeneity and flow of the liquid [9, 10, 12]. These quasiparticles serve as the elementary particles of the low-energy effective quantum field theory. They represent the analogue of matter. The type of the effective quantum field theory — the theory of interacting fermionic and bosonic quantum fields — depends on the universality class of the fermionic condensed matter (see review [3]). The superfluid 3 He-A, for example, belongs to the same universality class as the Standard Model. The effective quantum field theory describing the low energy phenomena in 3 He-A, contains chiral "relativistic" fermions. The collective bosonic modes interact with these "elementary particles" as gauge fields and gravity. All these fields arise together as emergent quantities, together with their associated Lorentz and gauge invariances, and with elements of the general covariance from the fermionic Theory of Everything in equation (6.13).

138

Effective gravity and quantum vacuum in superHuids

These emergent phenomena do not depend much on the details of the Theory of Everything [2], in our case, on the details of the pair potential U(x — y). Of course, the latter determines the universality class in which the system enters at low energy. But once the universality class is established, the physics remains robust to deformations of the pair potential. The details of U(x — y) influence only the "fundamental" parameters of the effective theory ("speed of light", "Planck" energy cut-off, etc.) but not the general structure of the theory. Within the effective theory the "fundamental" parameters are considered to be phenomenological.

6.4

Weakly interacting Bose gas

The quantum liquids are strongly correlated and strongly interacting systems. That is why, though it is possible to derive the effective theory from first principles in equation (6.13), if one has enough computer time and memory, this is a rather difficult task. It is instructive, however, to consider the microscopic theory for some special model potentials U(x — y). This allow us to solve the problem completely or perturbatively. In case of the Bose liquids the proper model is the Bogoliubov weakly interacting Bose gas, which is in the same universality class as a real superfluid 4 He. Such a model is very useful, since it simultaneously covers the low-energy edge of the effective theory, and the high-energy "trans-Planckian" physics.

6.4.1

Model Hamiltonian

Here we follow mostly the book by Khalatnikov [1]. In the model of weakly interacting Bose gas the pair potential in equation (6.13) is weak. As a result, most of the particles at T = 0 are in the Bose condensate, i.e., in the state with the momentum p = 0. The Bose condensate is characterized by the nonzero vacuum expectation value (vev) of the particle annihilation operator at p = 0:
(aj > = o ) = 0 ^ e - i * .

(6.15)

Here A'o is the particle number in the Bose condensate, and $ is the phase of the condensate. The vacuum is degenerate over global U(l) rotation of the phase. Further we consider the particular vacuum state with $ = 0.

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139

If there is no interaction between the particles (an ideal Bose gas), all the particles at T = 0 are in the Bose condensate, N0 = N. Small interactions induce a small fraction of particles which are not in the condensate. These particles have small momenta p. As a result, only the zero-momentum Fourier component of the pair potential is relevant, and equation (6.13) has the form: H-nN +

S P^O

= -nN0 + ^

(6.16)

( | ^ ~ ^) aPflp + ~W S ( 2 a p a P + 2 a -P a -P + a P a -P + a P a -p) • ^ ' P^o (6.17)

Here N0 = aja0 = a0al = ojaj = a0a0 is the particle number in the Bosecondensate (we neglected quantum fluctuations of the operator a0 and consider ao as a c-number); U is the matrix element of the pair interaction for zero momenta p of particles. Minimization of the main part of the energy in equation (6.16) over No gives UNQ/V = \i and one obtains:

H-vM=-^V + Y,np;

(6.18)

p^O

Hp =

6.4.2

5 Qm + W ( apflp + a -P a - p )

+

2 (apa_p +

apa p

" ) ' (6 ' 19)

Pseudo-rotation — Bogoliubov transformation

At each p the Hamiltonian can be diagonalized using the following consideration. The operators A = ^(apap + a-pffl-p + l) , A + iC2 = a P a l p , £i - i£ 2 = a p a_ p (6.20) form the group of pseudo-rotations, SU(1,1) (the group which conserves the form x\ + x\ — x\), with the commutation relations: [£3, A] - t£j , [£2, £3] = iCi , [£1, £2] = -iC3 ,

(6.21)

In terms of the pseudomomentum the Hamiltonian in equation (6.19) has the form

w

+

£ + £

+

>=(£ *) ' " '-H^ ")-

(622)

-

Effective gravity and quantum vacuum in superfiuids

140

In case of the nonzero phase <£ of the Bose condensate one has Hp=

( | ^ + M ) £ 3 + MCOs(2$)£ 1 + / x s i n ( 2 $ ) £ 2 - i ( ' | - + ^

. (6.23)

The diagonalization of this Hamiltonian is achieved by first rotating by an angle 2$ around the z axis, and then by performing a Lorentz transformation — a pseudo-rotation around the y axis: £ 3 = £3 c o s h x + A sinhx , C\ = £1 c o s h x + A sinhx y , tanhx = —5

•

(6.24) This corresponds to a Bogoliubov transformation and gives the following diagonal Hamiltonian:

Hp

" =

(L+»)+ti(L+»)

2 Urn

-»2

(6 25)

-

± £ ( p ) (o)pap + aL p a_ p ) + \ {E{$) " ( | ^ + /*) ) . ( 6 26)

where a p is the operator of annihilation of quasiparticles, whose energy spectrum E(p) is

6.4.3

Vacuum and quasiparticles

The total Hamiltonian now represents the ground state — the vacuum — and the system of quasiparticles

H-nAf=(H-

nAf)^ + Y,

£(PK«P

( 6 - 28 )

The lower the energy the more dilute is the system of quasiparticles and thus the weaker is the interaction between them. This description in terms of the vacuum state and a dilute system of quasiparticle is generic for condensed matter systems, and is valid even if the interaction of the initial bare particles is strong. The phenomenological effective theory in terms of vacuum state

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141

and quasiparticles was developed by Landau both for Bose and Fermi liquids. Quasiparticles (not the bare particles) play the role of elementary particles in such effective quantum field theories. In the weakly interacting Bose-gas in equation (6.27), the spectrum of quasiparticles at low energy (i.e., at p
6.4.4

Particles and quasiparticles

It is necessary to distinguish between the bare particles and quasiparticles. Particles are the elementary objects of the system on a microscopic "transPlanckian" level, these are the atoms of the underlying liquid ( 3 He or 4 He atoms). The many-body system of the interacting atoms form the quantum vacuum — the ground state. The non-dissipative collective motion of the superfluid vacuum with zero entropy is determined by the conservation laws experienced by the atoms and by their quantum coherence in the superfluid state. Quasiparticles are the particle-like excitations above this vacuum state, they serve as elementary particles in the effective theory. The bosonic excitations in superfluid 4 He and fermionic and bosonic excitations in superfluid 3 He represent the matter in our analogy. In superfluids they form the viscous normal component responsible for the thermal and kinetic lowenergy properties of superfluids. Fermionic quasiparticles in 3 He-A are chiral fermions, which are the counterpart of the leptons and quarks in the Standard Model [3].

142

Effective gravity and quantum vacuum in superfluids

6.4.5

Galilean transformation for particles and quasiparticles

The quantum liquids considered here are essentially nonrelativistic: under laboratory conditions their velocity is much less than the speed of light. That is why they obey with great precision the Galilean transformation law. Under the Galilean transformation to the coordinate system moving with the velocity u, the superfluid velocity — the velocity of the quantum vacuum — transforms as v s —> v s + u. As for the transformational properties of bare particles (atoms) and quasiparticles, it appears that they are essentially different. Let us start with bare particles. If p and E(p) are the momentum and energy of the bare particle (atom with mass m) measured in the system moving with velocity u, then from Galilean invariance it follows that its momentum and energy measured by the observer at rest are correspondingly p = p + m u , E(p) = E{p + mu) = E{p) + p • u + - m u 2 .

(6.29)

This transformation law contains the mass m of the bare atom. However, as far as the quasiparticles are concerned, one expects that characteristics of the microscopic world (such as the bare mass m) will not enter the transformation law for quasiparticles. This is because quasiparticles in effective low-energy theory have no information on the trans-Planckian world of the bare atoms comprising the vacuum state. All the information on the quantum vacuum, which the low-energy quasiparticle has, is encoded in the effective metric g^. Since the mass m must drop out from the transformation law for quasiparticles, we expect that the momentum of quasiparticle is invariant under the Galilean transformation: p —• p , while the quasiparticle energy is simply Doppler shifted: E(p) —> E(p) + p • u. Such a transformation law allows us to write the energy of quasiparticle in the moving superfluid vacuum. If p and E(p) are the quasiparticle momentum and energy measured in the coordinate system where the superfluid vacuum is at rest (i.e., v s = 0, we call this frame the superfluid comoving frame), then its momentum and energy in the laboratory frame are p = p , E(p) = E(p) + p-vs.

(6.30)

The difference in the transformation properties of bare particles and quasiparticles comes from their different status. While the momentum and

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143

energy of bare particles are determined in "empty" space-time, the momentum and energy of quasiparticles are counted from that of the quantum vacuum. This difference can be easily visualized if one considers the spectrum of quasiparticles in the weakly interacting Bose gas in equation (6.27) in the limit of large momentum p » mc, when the energy spectrum of quasiparticles approaches that of particles, E -> p2/2m. In this limit the difference between particles and quasiparticles disappears, and at first glance one may expect that quasiparticle should obey the same transformation property under Galilean transformation as a bare isolated particle. To add more confusion let us consider an ideal Bose gas of noninteracting bare particles, where quasiparticles have exactly the same spectrum as particles. Why are the transformation properties so different for them? The ground state of an ideal Bose gas has zero energy and zero momentum in the reference frame where the Bose condensate is at rest (the superfluid comoving reference frame). In the laboratory frame the condensate momentum and energy are correspondingly (^)vac = ^ ™ v s , (^)vac = ^

(6.31) •

(6-32)

The state with one quasiparticle is the state in which N — 1 particles have zero momenta, p = 0, while one particle has nonzero momentum p ^ 0. In the comoving reference frame the momentum and energy of such state with one quasiparticle are correspondingly {V)vac+lqp = p and {'H)v3lC+lqp = E(p) = P2/2rn. In the laboratory frame the momentum and energy of the system are obtained by Galilean transformation (^)vac + i 9P = (N-

l ) m v , + (p + mv s ) = (V)vac + p ,

W«+i f f = (N ~ ! ) ^ + (P + 2 7 S ) 2

=

< W >~ + E{P)

(6.33) +P

• Vs (6.34)

Since the energy and the momentum of quasiparticles are counted from that of the quantum vacuum, the transformation properties of quasiparticles are different from the Galilean transformation law. The part of the Galilean transformation which contains the mass of the atom is absorbed by the Bosecondensate, which represents the quantum vacuum.

144

6.4.6

Effective gravity and quantum vacuum in superBuids

Effective metric from Galilean transformation

The right-hand sides of equations (6.33) and (6.34) show that the energy spectrum of quasiparticles in the moving superfiuid vacuum is given by equation (6.30). Such spectrum can be written in terms of the effective acoustic metric: (E - p • v s ) 2 = c V , or g^p^ =0 . (6.35) where the metric has the form: g00 = - 1 , g0i = -vt , tfi = cH* - u&" ,

fl» = ~ ( x - 5"J ' Soi = - ^ f , 9ij = jkj V ^ - c- 3 .

(6.36)

,

(6.37) (6.38)

Equation (6.35) does not determine the conformal factor. The derivation of the acoustic metric with the correct conformal factor can be found in references [10, 11, 12].

6.4.7

Broken Galilean invariance

The modified transformation law for quasiparticles is a consequence of the fact that the mere presence of a gas or liquid with nonzero number TV of atoms breaks Galilean invariance. While for the total system, quantum vacuum 4quasiparticles, Galilean invariance is a true symmetry, it is not applicable to the subsystem of quasiparticles if it is considered independently of the quantum vacuum. This is a general feature of broken symmetry: the vacuum breaks Galilean invariance. This means that in the Bose gas, and in the superfiuid 4 He, two symmetries are broken: the global U(l) symmetry and the Galilean invariance.

6.4.8

Momentum versus pseudomomentum

On the other hand, due to the presence of quantum vacuum, there are two different types of translation invariance at T = 0 (see detailed discussion in reference [12]): (i) Invariance under the translation of the quantum vacuum with respect to the empty space; (ii) Invariance under translation of quasiparticle with respect to the quantum vacuum.

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145

The operation (i) leaves the action invariant provided that empty space is homogeneous. The conserved quantity, which comes from translation invariance with respect to empty space is the momentum. The operation (ii) is a symmetry operation if the quantum vacuum is homogeneous. This symmetry gives rise to the pseudomomentum. Accordingly bare particles in empty space are characterized by the momentum, while quasiparticles — excitations of the quantum vacuum — are characterized by pseudomomentum. That is why the transformation properties for the momentum of particles in equation (6.29) and the pseudomomentum of quasiparticles in equation (6.30) are different. Galilean invariance is the symmetry of the underlying microscopic physics of atoms in empty space. It is broken and fails to work for quasiparticles. Instead, it produces the transformation law in equation (6.30), in which the microscopic quantity — the mass m of bare particles — drops out. This is an example of how the memory of the microscopic physics is erased in the lowenergy corner. Furthermore, when the low-energy corner is approached and the effective field theory emerges, these modified transformations gradually become part of the more general coordinate transformations appropriate for the Einstein theory of gravity.

6.4.9

Vacuum energy of weakly interacting Bose gas

The vacuum energy of the Bose gas as a function of the chemical potential H is

< w - ^ = -£v- + i £ ( * ( p > - | ^ + ^ ) -

<«•)

p

The last term in round brackets is added to take into account the perturbative correction to the matrix element U [1]. If the total number of particles is fixed, the corresponding vacuum energy is a function of N:

Wvac = EUN)

= \N™?

+ \ £

(E<J>)

- ^

- mc2 + ^ )

. (6.40)

Inspection of the vacuum energy shows that it does contain the zero-point energy of the phonon field, \ J2pE(p)This divergent term is balanced by three counterterms in equation (6.40). They come from microscopic physics

146

Effective gravity and quantum vacuum in superfluids

(they explicitly contain a microscopic parameter — the mass m of atom). This regularization, which naturally arises in the microscopic physics, is absolutely unclear within the effective theory. After the regularization, the contribution of the zero-point energy of the phonon field in equation (6.40) becomes

i£*(P> - i E „ p , - I E ( £ w - ^ ) p reg

p 8

p

157T

Nmc2

0m

(,«, '

3 3

„, 2

v

c

, „ „„,

,

(6.42)

where n = N/V is the particle density in the vacuum. Thus the total vacuum energy is E^{N)

= e(n) V 1 . . 2( Vmc n+

= 2

=

F

(^

(6.43) tCAA. (6 44)

3 3

16 m c \

{ T^^r)

n 2 +

15^

-

m 3 / 2 C / 5 / 2 n 5 / 2

)

(6 45)

-

In the weakly interacting Bose gas the contribution of the phonon zero-point motion (the second terms in equations (6.44) and (6.45)) is much smaller than the leading contribution to the vacuum energy, which comes from the interaction (the first terms in equations (6.44) and (6.45)). The small parameter, which regulates the perturbation theory in the above procedure is mca/h «C 1 (where a is the interatomic distance: a ~ n - 1 / / 3 ), or mU/h2a
6.4.10 Planck energy scales Microscopic physics also shows that there are two energy parameters, which play the role of the Planck energy scale: he ^Planck l = mc

,

-Epianck 2 = — •

(6.46)

a The Planck mass, which corresponds to the first Planck scale .Epianck 1 > 1S the mass of Bose particles m, that comprise the vacuum. The second Planck scale •Epianck 2 reflects the discreteness of the vacuum: the microscopic parameter,

147

Grigori Volovik

which enters this scale, is the mean distance between the particles in the vacuum. The second energy scale corresponds to the Debye temperature in solids. In a given case of weakly interacting particles one has Epianck 1 *C .Epianck 2, i.e., the distance between the particles in the vacuum is so small, that the quantum effects are stronger than interaction. This is the limit of strong correlations and weak interactions. Below the first Planck scale E
^~2^~9

Planck!,

(6-47)

where g = —1/c6 is the determinant of the acoustic metric in equation (6.38). This contribution to the vacuum energy has the same structure as the cosmological term in equation (6.11). However, the leading term in the vacuum energy, equation (6.40), is higher and is determined by both Planck scales: \Vs/=-g E P l a n c k

6.4.11

2

£Planck ! .

(6.48)

Vacuum pressure and cosmological constant

The relevant vacuum energy of the grand ensemble of particles is the thermodynamic potential at fixed chemical potential: (H — nAf)^. It is related to the pressure of the liquid as (see the prove of this thermodynamic equation below, equation (6.55)) P = -~CH-»Af)vac

.

(6.49)

Such relation between pressure and energy is similar to that in equation (6.7) for the equation of state of the relativistic quantum vacuum, which is described by the cosmological constant. This vacuum energy for the weakly interacting Bose gas is given by (H - HM)^

= ±Vy/=j

(-£Planck

2 £ P .a„ck

1+ ^^Pianck i)

•

(6-50)

148

Effective gravity and quantum vacuum in superfluids

Two terms in equation (6.50) represent two contributions to the vacuum pressure in the weakly interacting Bose gas. The zero-point energy of the phonon field, the second term in equation (6.50), which coincides with equation (6.42), does lead to the negative vacuum pressure as is expected from the effective theory. However, the magnitude of this negative pressure is smaller than the positive pressure coming from the microscopic "trans-Planckian" degrees of freedom (the first term in equation (6.50) which is provided by the repulsive interaction of atoms). Thus the weakly interacting Bose-gas can exist only under positive external pressure.

6.5 6.5.1

Quantum liquid Real liquid 4 He

In the real liquid 4 He the interaction between the particles (atoms) is not small. It is a strongly correlated and strongly interacting system, where the two Planck scales are of the same order, mc2 ~ he/a. This means that the interaction energy and the energy of zero-point motion of atoms are of the same order. This is not a coincidence but reflects the stability of the liquid state. Each of the two energies depends on the particle density n. One can find the value of n at which the two contributions to the vacuum pressure compensate each other. This means that the system can be in equilibrium even at zero external pressure, P = 0, i.e., the quantum liquid can exist as a completely autonomous isolated system without any interaction with the environment. This is what we must expect from the quantum vacuum in cosmology, since there is no external environment for the vacuum. Consider the case of a collection of a large but finite number N of 4 He atoms at T = 0. The atoms do not fly away as happens for gases, but are held together to form a droplet of liquid with a finite mean particle density n. This density n is fixed by the attractive interatomic interaction and repulsive zero-point oscillations of atoms, only a part of this zero-point motion being described in terms of the zero-point energy of phonon mode. The only macroscopic quantity which characterizes the homogeneous stationary liquid at T = 0 is the mean particle density n. The vacuum energy density is a function of n ^ ) = ^>vac ,

(6.51)

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149

and this function determines the equation of state of the liquid. The relevant vacuum energy density — the density of the thermodynamic potential of grand ensemble e(n) = e(n) -

Mn

= i (H - nM)^

.

(6.52)

Since the particle number N = nV is conserved, e(n) is the right quantity which must be minimized to obtain the equilibrium state of the liquid at T = 0 (the equilibrium vacuum). The chemical potential JJ, plays the role of the Lagrange multiplier responsible for the conservation of bare atoms. Thus an equilibrium number of particles no(n) is obtained from equation:

Here we discuss only the spatially homogeneous ground state, i.e., with spatially homogeneous n, since we know that the ground state of helium at T = 0 is homogeneous: it is a uniform liquid, not a crystal. From the definition of pressure, d(Ve(N/V))

p=

-

. , de £(n) + n

lc _,, (6 54)

~^' dv and from equation (6.53) for the density n in equilibrium, one deduces that in equilibrium the vacuum energy density e and the vacuum pressure P are related by Cvac eq — ~~Pvac

•

(6.55)

The thermodynamic relation between the energy and pressure in the ground state of the quantum liquid P = —e, is the same as obtained for the vacuum energy and pressure from the Einstein cosmological term. This is because the cosmological term also does not contain derivatives. Close to the equilibrium state one can expand the vacuum energy in terms of deviations of the particle density from its equilibrium value. Since the linear term disappears due to the stability of the superfluid vacuum, one has e(n) = e{n) - fin = -P^

+ -—r^(n

~ n*(v)?

•

(6-56)

150

6.5.2

Effective gravity and quantum vacuum in superfiuids

Gas-like versus liquid-like vacuum

It is important that the vacuum of real 4 He is not gas-like but liquid-like, i.e., it can be in equilibrium at T = 0 without interaction with the environment. Such property of the collection of atoms at T = 0 is determined by the sign of the chemical potential. If it is counted from the energy of an isolated 4 He atom, fi is positive in a weakly interacting Bose gas, but is negative in a real 4 He where n ~ - 7 K [15]. Due to the negative [i the isolated atoms are collected together forming the liquid droplet which is self-sustained without any interaction with the outside world. If the droplet is big enough, so that the surface tension can be neglected compared to the volume effects, the pressure in the liquid is absent, P\ac = 0, and thus the vacuum energy density e is zero in equilibrium: ^vacuum of self—sustaining system

==

U •

[0.0 t J

This condition cannot be fulfilled for gas-like states for which \i is positive and thus they cannot exist without an external pressure.

6.5.3

Model liquid state

It is instructive to discuss some model energy density e(n) describing a stable isolated liquid at T = 0. Such a model must satisfy the following condition: (i) e(n) must be attractive (negative) at small n and repulsive (positive) at large n to provide an equilibrium density of liquid at intermediate n; (ii) The chemical potential must be negative to prevent evaporation; (iii) The liquid must be locally stable, i.e., the eigen-frequencies of collective modes must be real. All these conditions can be satisfied if we modify equation (6.45) in the following way. Let us change the sign of the first term describing interaction, and leave the second term coming from vacuum fluctuations intact assuming that it is valid even at high density of particles. Due to the attractive interaction at low density, the Bose gas collapses forming the liquid state. Of course, this is rather artificial construction, but it qualitatively describes the liquid state. So we come to the following model €(n)

=-\an2+

\tln5l2

,

(6.58)

though, in addition to a and /?, one can use also the exponents of n as fitting parameters. An equilibrium particle density in terms of the chemical

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151

potential is obtained from the minimization of the relevant vacuum energy e = e — fin over n: — = H -* an

- an0 + /3nJ

= (j.

(6.59)

The equation of state of such a liquid is P(no) = - (e(no) - M%) = -\ar%

+\ ^

2

(6-60)

This equation of state allows the existence of an isolated liquid droplet, for which the external pressure is zero, P = 0. The equilibrium density, chemical potential and speed of sound in the isolated liquid are c \ 5a

2

"o(P = 0 ) = ^ - J M(P

= 0) = - i n 0 a , D

,

(6.61) (6.62)

(6 63)

•^(^L-C-SL-h"--*!"!- -

This liquid state is stable: the chemical potential \x is negative preventing evaporation, while c2 is positive, i.e., the compressibility is negative, which indicates the local stability of the liquid. Equation (6.60) shows that the quantum zero-point energy produces a positive contribution to the vacuum pressure, instead of the negative pressure expected from the effective theory and from equation (6.50) for the weakly interacting Bose gas. Let us now recall that in this model we changed the sign of the interaction term, compared to that in the weakly interacting Bose gas. As a result both terms in equation (6.50) have changed sign. The equilibrium state of the liquid is obtained due to the competition of two effects: attractive interaction of bare atoms (corresponding to the negative vacuum pressure in equation (6.60)) and their zero-point motion which leads to repulsion (corresponding to a positive vacuum pressure in equation (6.60)). These effects are balanced in equilibrium, that is why the two "Planck" scales in equation (6.46) become of the same order of magnitude.

152

Effective gravity and quantum vacuum in superQuids

6.5.4

Quantum liquid from the Theory of Everything

The parameters of liquid 4 He at P = 0 have been calculated in the exact microscopic theory, where the many-body wave function of 4 He atoms has been constructed using the "Theory of Everything" in equation (6.13) with a realistic pair potential [15]. For P = 0 one has n 0 ~ 2 • 1022 cm" 3 , n = ^ ^ n0

~ - 7 K , c ~ 2.5 • 104 cm/sec ,

(6.64)

mc2 ~ 30 K , hcnl/3 ~ 7 K ,

(6.65)

I = 0 . (6.66) These derived parameters are in good agreement with their experimental values.

6.6

Vacuum energy and cosmological constant

6.6.1

Nullification of the "cosmological constant" in quantum liquid

If there is no interaction with the environment, the external pressure P is zero, and thus in equilibrium the vacuum energy density e — \m — — P in equations (6.49) and (6.55) is also zero. The energy density e is the quantity which is relevant for the effective theory: just this energy density enters the effective action for the soft variables, including the effective gravity field, which must be minimized to obtain the stationary states of the vacuum and matter fields. Thus e is the proper counterpart of the vacuum energy density, which is responsible for the cosmological term in the Einstein gravity. Nullification of both the vacuum energy density and the pressure in the quantum liquid means that PA = — p^ — 0, i.e., the effective cosmological constant in the liquid is identically zero. Such nullification of the cosmological constant occurs without any fine-tuning or supersymmetry. Note that supersymmetry — the symmetry between fermions and bosons — is simply impossible in 4 He, since there are no fermionic fields in the Bose liquid. The same nullification occurs in Fermi liquids, in superfluid phases of 3 He, since these are also quantum liquids with negative chemical potential [3]. Some elements of supersymmetry can be found in the effective theory of superfluid 3 He [16, 3], but this is certainly not enough to produce the nullification.

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153

Applying this to the quantum vacuum, the mere assumption that the "cosmological liquid" — the vacuum of the quantum field theory — belongs to the class of states which can exist in equilibrium without external forces, leads to the nullification of the vacuum energy in equilibrium at T = 0. Whether this scenario of nullification of the cosmological constant can be applied to the cosmological fluid (the physical vacuum) is a question under discussion (see discussion in reference [17], where the inflaton field is considered as the analogue of the variable n in a quantum liquid).

6.6.2

Role of the zero point energy: Bosonic and Fermionic fields

The advantage of the quantum liquid is that we know both the effective theory and the fundamental Theory of Everything in equation (6.13). That is why we can compare the two approaches. The microscopic wave function used for microscopic calculations contains, in principle, all the information on the system, including the quantum fluctuations of the low-energy phonon degrees of freedom, which are considered in the effective theory in equation (6.67). That is why the separate treatment of the contribution to the vacuum energy of the low-energy degrees of freedom described by the effective theory makes no sense: this leads at best to double counting. The effective theory of quantum Bose liquids contains phonons as elementary bosonic quasiparticles but no fermions. That is why the analogue of equation (6.9) for the vacuum energy produced by the zero-point motion of "elementary particles" is PA=

2V

5 Z °P ~ ^ P l a n c k = V-9 phonons

^Planck •

(6"67)

Here g is the determinant of the acoustic metric in equation (6.38). The "Planck" energy cut-off can be chosen either as the Debye temperature, -^Debye = ^c/o = hcn^3 in equation (6.65) with a being the interatomic distance, which plays the role of the Planck length; or as mc2 which has the same order of magnitude. The disadvantages of such a naive calculation of the vacuum energy within the effective field theory are: (i) The result depends on the cut-off procedure; (ii) The result depends on the choice of the zero from which the energy is counted: a shift of the zero level leads to a shift in the vacuum energy.

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Effective gravity and quantum vacuum in supeiQuids

In the microscopic theory these disadvantages are cured: (i) The cut-off is not required; (ii) The relevant energy density, I = e — fin, does not depend on the choice of zero level: the shift of the energy / d 3 r e is exactly compensated by the shift of the chemical potential fi. At P — 0 the microscopic results for both vacuum energies characterizing the quantum liquid are: e(n 0 ) = 0, e(n 0 ) = fin0 < 0. Both energies are in severe disagreement with the naive estimation in equation (6.67) obtained within the effective theory: p^ in equation (6.67) is nonzero in contradiction with e(n 0 ) = 0; comparing it with e(n 0 ) one finds that p\ is about the same order of magnitude, but it has an opposite sign. This is an important lesson from condensed matter physics. It shows that the use of the zero-point fluctuations of bosonic or fermionic modes in equation (6.9) in the trans-Planckian effective theory is absolutely irrelevant for the calculations of the vacuum energy density. Whatever are the lowenergy modes, fermionic or bosonic, for equilibrium vacuum they are exactly cancelled by the trans-Planckian degrees of freedom, which are not accessible within the effective theory.

6.6.3

Why is equilibrium vacuum not gravitating?

We discussed the condensed matter view to the problem, why the vacuum energy is so small, and found that the answer comes from "fundamental trans-Planckian physics". In the effective theory of the low energy degrees of freedom, the vacuum energy density of a quantum liquid is of order EpUnck, with the corresponding "Planck" energy appropriate for this effective theory. However, from the exact "Theory of Everything" of the quantum liquid, i.e., from the microscopic physics, it follows that the "trans-Planckian" degrees of freedom exactly cancel the relevant vacuum energy without fine tuning. The vacuum energy density is exactly zero, if the following conditions are fulfilled: (i) there are no external forces acting on the liquid; (ii) there are no quasiparticles (matter) in the liquid; (iii) no curvature or inhomogeneity in the liquid; and (iv) no boundaries which give rise to the Casimir effect. Each of these factors perturbs the vacuum state and induces a nonzero value of the vacuum energy density on the order of the energy density of the perturbation, as we shall discuss below. Let us, however, mention, that the actual problem for cosmology is not why the vacuum energy is zero (or very small when it is perturbed), but why the vacuum is not (or almost not) gravitating. These two problems are

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not necessarily related since in the effective theory the equivalence principle is not fundamental physical law, and thus does not necessarily hold when applied to the vacuum energy. That is why one cannot exclude the situation that the vacuum energy is huge, but it is not gravitating. Condensed matter also provides an example of such situation. The weakly interacting Bose gas discussed above is just the proper object: This gas-like substance can exist only at positive external pressure, and thus it has negative energy density. The translation to the relativistic language gives a huge vacuum energy on the order of the Planck energy scale (see equation (6.50)). Nevertheless, the effective theory remains the same as for the quantum liquid, and thus even in this situation the equilibrium vacuum, which exists under an external pressure, is not gravitating. Only the small deviations from equilibrium state are gravitating. Exactly this situation was discussed in reference [17].

In condensed matter physics effective gravity appears as an emergent phenomenon in the low energy corner. The gravitational field is not fundamental, but is one of the low energy collective modes of the quantum vacuum. This dynamical mode provides the effective metric (the acoustic metric in 4 He and weakly interacting Bose gas) for the low-energy quasiparticles which serve as an analogue of matter. This emergent gravity does not exist on the microscopic (trans-Planckian) level and appears only in the low energy limit together with the "relativistic" quasiparticles and the acoustics itself. The bare atoms, which live in the "trans-Planckian" world and form the vacuum state there, do not experience the "gravitational" attraction experienced by the low-energy quasiparticles, since the effective gravity simply does not exist at the microscopic scale (we neglect here the real gravitational attraction of the atoms, which is extremely small in quantum liquids). That is why the vacuum energy cannot serve as a source of the effective gravity field: the pure completely homogeneous equilibrium vacuum is not gravitating.

On the other hand, the long-wave-length perturbations of the vacuum are within the sphere of influence of the low-energy effective theory; such perturbations can be the source of the effective gravitational field. Deviations of the vacuum from its equilibrium state, induced by different sources discussed below, are gravitating.

156

6.6.4

Effective gravity and quantum vacuum in superBuids

Why is the vacuum energy unaffected by phase transitions?

It is commonly believed that the vacuum of the Universe underwent one or several broken symmetry phase transitions. Since each of the transitions is accompanied by a substantial change in the vacuum energy, it is not clear why the vacuum energy is (almost) zero after the last phase transition. In other words, why does the true vacuum have zero energy, while the energies of all other false vacua are enormously big? What happens in quantum liquids? According to the conventional wisdom, the phase transition, say, to the broken symmetry vacuum state, is accompanied by the change of the vacuum energy, which must decrease in a phase transition. This is what usually follows from the Ginzburg-Landau description of phase transitions. However, let us compare the energy densities of the false and the true vacuum states. Let us assume that the phase transition is of first order, and the false vacuum is separated from the true vacuum by a large energy barrier, so that it can exist as a (meta)stable state. Since the false vacuum is stable, equation (6.57) can also be applied to the false vacuum, and one obtains an apparently paradoxical result: in the absence of external forces the energy density of the false vacuum must be the same as the energy density of the true vacuum, i.e., the relevant energy density e must be zero for both vacua. Thus the first order phase transition occurs without a change in the vacuum energy. To add more confusion, note that equation (6.57) can be applied even to the unstable vacuum which corresponds to a saddle point of the energy functional, if such a vacuum state can live long enough. Thus the vacuum energy density does not change in the second order phase transition either. There is no true paradox, however: After the phase transition to a new state has occurred, the chemical potential fj, will be automatically adjusted to preserve the zero external pressure and thus the zero energy e of the vacuum. Thus the relevant vacuum energy is zero before and after the transition, which means that T = 0 phase transitions do not disturb the zero value of the cosmological constant. Thus the scenario of the nullification of the vacuum energy suggested by the quantum liquids survives even if the phase transition occurs in the vacuum. The first order phase transition between superfluid phases 3 He-A and 3 He-B at T = 0 and P = 0 provides an example [3].

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157

Why is the cosmological constant nonzero?

We now come to another problem in cosmology: Why is the vacuum energy density presently of the same order of magnitude as the energy density of matter pM, as is indicated by recent astronomical observations [7]? While the relation between pu and ps. seems to depend on the details of transPlanckian physics, an order of magnitude estimation can be readily obtained. In equilibrium and without matter the vacuum energy is zero. However, the perturbations of the vacuum caused by matter and/or by the inhomogeneity of the metric tensor lead to imbalance. As a result the deviations of the vacuum energy from zero must be on the of order of the perturbations. Let us consider how this happens in quantum liquids for different types of perturbations. That is, how is the vacuum energy (which is zero at T = 0 and in complete equilibrium in the absence of external forces) influenced by different factors? How can this lead to a small but nonzero value of the cosmological constant?

6.6.6

Vacuum energy from finite temperature

A typical example derived from quantum liquids is the vacuum energy produced by temperature. Let us consider for example the superfluid 4He in equilibrium at finite T without external forces. If T
= -PA = PM = \pM = V = P ^ T * ,

(6.68)

where g = — c~6 is again the determinant of the acoustic metric. In this example the vacuum energy density p\ is positive and always on the order of the energy density of matter. This indicates that the cosmological constant is not actually a constant but is adjusted to the energy density of matter and/or to the other perturbations of the vacuum discussed below.

158

6.6.7

Effective gravity and quantum vacuum in superBuids

Vacuum energy from the Casimir effect

Another example of the induced nonzero vacuum energy density is provided by the boundaries of the system. Let us consider a finite droplet of 4 He with radius R. If this droplet is freely suspended then at T = 0 the vacuum pressure P A must compensate the pressure caused by the surface tension due to the curvature of the surface. For a spherical droplet one obtains the negative vacuum energy density: 2cr

-^Debye _

,

„3

tic

where a is the surface tension. This is an analogue of the Casimir effect, in which the boundaries of the system produce a nonzero vacuum pressure. The strong cubic dependence of the vacuum pressure on the "Planck" energy •Epianck = -Ebebye reflects the trans-Planckian origin of the surface tension a ~ E'oebye/a2 ~ hc/a3: it is the energy (per unit area) related to the distortion of atoms in the surface layer of the size of the interatomic distance a. Such terms in the Casimir energy, of order Epl&nck/R, have been considered in reference [18]. In reference [19] such a vacuum energy, with R being the size of the horizon, has been connected to the energy of the Higgs condensate in the electroweak phase transition. This form of the Casimir energy — the surface energy 4TTR2<J normalized to the volume of the droplet — can also serve as an analogue of the quintessence in cosmology [20]. Its equation of state is Pa = — (2/3)pa: ^R2a " = ^ W

P

=

3a -R '

P

°

=

2a 2 --R=-3P°-

(6 70)

-

The equilibrium condition within the droplet can be written as P = P\+Pa = 0. In this case the quintessence is related to the wall — the boundary of the droplet. In cosmology the quintessence with the same equation of state, (Pa) = — (2/3) (/9CT), is represented by a wall wrapped around the Universe or by a tangled network of cosmic domain walls [21]. The surface tension of the cosmic walls can be much smaller than the Planck scale.

6.6.8

Vacuum energy induced by texture

Nonzero vacuum energy density, with a weaker dependence on .Epianck, is induced by the inhomogeneity of the vacuum. Let us discuss the vacuum

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energy density induced by texture in a quantum liquid. We consider here the twist soliton in 3 He-A, since such texture is related to the Riemann curvature in general relativity [3]. Within the soliton the field of the 3 He-A order parameter — the unit vector 1 — has a form l(z) = x cos (j){z) + y sin (f>(z). The energy of the system in the presence of the soliton consists of the vacuum energy p\{4>) and the gradient energy: P = P\{) + Pw»A ,

PK{4>)

=

PA(

= 0) + -Tsin2(t>,

p s r a d = K(dz)2 , (6.71)

where £p is the so-called dipole length [22]. Here we denoted the energy I by p to make the connection with general relativity, and omitted s/—g assuming that c = 1. The solitonic solution of the sine-Gordon equation, tan (0/2) = ez^D, gives the following spatial dependence of vacuum and gradient energies: PA{Z) -P\( = 0)= Pgrad(z) = 75

v2/ Ic \ '

(6-72)

££,cosh'(z/£ D ) Let us consider for simplicity the 1+1 case. Then the equilibrium state of the whole quantum liquid with the texture can be discussed in terms of the partial pressure of the vacuum, P\ = — PA, and that of the inhomogeneity, ^grad = Pgrad- The latter equation of state describes the so called stiff matter in cosmology. In equilibrium the external pressure is zero and thus the positive pressure of the texture (stiff matter) must be compensated by the negative pressure of the vacuum: P = PA(z) + Pg rad (z) = 0 .

(6.73)

This equilibrium condition produces another relation between the vacuum and the gradient energy densities PA{Z) = -PA{Z)

= Pgrad(^) = PZrad(z) .

(6.74)

Comparing this equation (6.74) with equation (6.72) one finds that in equilibrium pA(4> = 0) = 0 , (6.75) i.e.,. as before, the main vacuum energy density — the energy density of the bulk liquid far from the soliton — is exactly zero if the isolated liquid is in

160

Effective gravity and quantum vacuum in superBuids

equilibrium. Within the soliton the vacuum is perturbed, and the vacuum energy is induced being on the order of the energy of the perturbation. In this case p\{z) is equal to the gradient energy density of the texture. The induced vacuum energy density in equation (6.72) is inversely proportional to the square of the size of the region where the field is concentrated: M ^

~ V ^ Planck ( f )

2

-

(6-76)

In the case of the soliton R ~ ^D. Similar behavior for the vacuum energy density in the interior region of the Schwarzschild black hole, with R being the Schwarzschild radius, was discussed in reference [23]. In cosmology, the vacuum energy density obeying the equation (6.76) with R proportional to the Robertson-Walker scale factor has been suggested in reference [24], and with R being the size of the horizon, R = R^, in reference [19]. Following the reasoning of reference [19], one can state that the vacuum energy density related to the phase transition is determined by equation (6.76) with R = Rn(t) at the cosmological time t when this transition (or crossover) occurred. Applying this to, say, the cosmological electroweak transition, where the energy density of the Higgs condensate is of order of Te4w, one obtains the relation Te2w = .Epianck^c/Rn(t = tevl). It also follows that the entropy within the horizon volume at any given cosmological temperature T is Sn ~ Epiinci./T3 for the radiation-dominated Universe.

6.6.9

Vacuum energy due to Riemann curvature

The vacuum energy ~ R~2, with R proportional to the Robertson-Walker scale factor, comes also from the Riemann curvature in general relativity. It appears that the gradient energy of a twisted 1-texture is equivalent to the Einstein curvature term in the action for the effective gravitational field in 3 He-A [3]:

" 1 6 ^ Jd3r^n

= * / d M ( l • (V x l))2 .

(6.77)

Here 1Z is the Riemann curvature calculated using the effective metric experienced by fermionic quasiparticles in 3 He-A ds 2 = -dt2 + cj 2 (i x dr) 2 + c^ 2 (l • dr) 2 .

(6.78)

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The order parameter vector 1 plays the role of the Kasner axis; cy and Cj_ correspond to the speed of "light" propagating along the direction of 1 and in transverse direction; c\\ 3> cj_. The analogy between the textural (gradient) energy in 3 He-A and the curvature in general relativity allows us to interpret the result of the previous section, equation (6.74), in terms of the vacuum energy induced by the curvature of space. It appears that in cosmology this effect can be described within general relativity. We must consider the stationary cosmological model, since the time dependent vacuum energy is certainly beyond the Einstein theory. The stationary Universe was obtained by Einstein in his work where he first introduced the cosmological term [5]. It is the closed Universe with positive curvature and with matter, where the effect of the curvature is compensated by the cosmological term, which is adjusted in such a way, that the Universe remains static. This is just the correct and probably unique example, of how the vacuum energy is induced by curvature and matter within general relativity. Let us recall this solution. In the static state of the Universe two equilibrium conditions must be fulfilled: P = PM + PA + Pn = 0,

P = PM + PA + Pn = 0.

(6.79)

The first equation in (6.79) reflects the gravitational equilibrium, which requires that the total mass density must be zero: P = PM + PA + PK = ® (actually "gravineutrality" corresponds to a combination of the two equations in (6.79), p + 3P — 0, since p + 3 P serves as the source of the gravitational field in the Newtonian limit). This gravineutrality is analogous to the electro-neutrality in condensed matter. The second equation in (6.79) is equivalent to the requirement that for the "isolated" Universe the external pressure must be zero: P = PM + PA + Pn = 0. In addition to matter density pM and vacuum energy density pA, the energy density pn stored in the spatial curvature is added:

P

* = " l i b = -8^H* ' PR = ~\Pn '

(6 80)

-

Here R is the cosmic scale factor in the Friedmann-Robertson-Walker metric ds2 = - d i 2 + i ?

2

( - ^ ^ + rWWsin

2

0d/)

,

(6.81)

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Effective gravity and quantum vacuum in superfluids

the parameter k = (—1,0,4-1) for an open, flat, or closed Universe respectively; and we again removed the factor \f—g from the definition of the energy densities. For the cold Universe with PM = 0, the equations (6.79) give Ph

= l-PM = -l-pn

= -

^

and for the hot Universe with the equation of state 1

, PM

(6.82) =

(1/3)PM,

3A;

Since the energy of matter is positive, the static Universe is possible only for positive curvature, k = + 1 , i.e., for the closed Universe. This is the unique solution, which describes an equilibrium static state of the Universe, where the vacuum energy is induced by matter and curvature. And this solution is obtained within the effective theory of general relativity without invoking trans-Planckian physics, and thus does not depend on details of the trans-Planckian physics.

6.6.10

Necessity of Planck physics for time-dependent cosmology

The condensed matter analogue of gravity provides a natural explanation of why the cosmological constant is zero with great accuracy, when compared with the result based on naive estimation of the vacuum energy within the effective theory. It also shows how a small effective cosmological constant of relative order 10" 120 naturally arises as the response to different perturbations. We considered the time-independent perturbations, where minimum energy considerations and the equilibrium condition provided the solution of the problem. For the time-dependent situation, such as an expansion of the Universe, the calculation of the vacuum response is not as simple even in quantum liquids. One must solve self-consistently the coupled dynamical equations for the motion of the vacuum and matter fields. In the case of general relativity this requires the equation of motion for the vacuum energy p\, but this is certainly beyond the effective theory since the time dependence of pis, violates Bianchi identities. Probably some extension of general relativity,

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towards the scalar-tensor theory of gravity such as discussed in reference [25], will be more relevant for that. On the other hand the connection to Planck physics can help to solve the other cosmological problems. For example, there is the flatness problem: To arrive at the Universe we see today, the Universe must have begun extremely flat, which means that parameter A; in the Friedmann-Robertson-Walker metric must be zero. In quantum liquids the general Friedmann-RobertsonWalker metric in equation (6.81) describes a spatially homogeneous spacetime as viewed by the low-energy quasiparticles within the effective theory. However, for the external or high-energy observer the quantum liquid is not homogeneous if k ^ 0. The same probably happens in gravity: If general relativity is an effective theory, then invariance under the coordinate transformations exists only at low energy. For the "Planck" observer the Friedmann-Robertson-Walker metric in equation (6.81) is viewed as space dependent if k ^ 0. That is why the condition, that the Universe must be spatially homogeneous not only on the level of the effective theory but also on the fundamental level, requires that k = 0. Thus, if general relativity is the effective theory, the truly homogeneous Universe must be spatially flat.

6.7 6.7.1

Effects of discrete number N of particles in the vacuum Casimir effect in quantum liquids

Until now we used the conservation law for the particle number TV, the number of bare atoms in the quantum vacuum, to derive the nullification of the vacuum energy in the grand ensemble of particles. Now we consider another possible consequence of the discrete nature of the quantum vacuum in quantum liquids. This is related to the Casimir effect. The attractive force between two parallel metallic plates in vacuum induced by the quantum fluctuations of the electromagnetic field has been predicted by Casimir in 1948 [26]. The calculation of the vacuum pressure is based on the regularization schemes, which allows to separate the effect of the low-energy modes of the vacuum from the huge diverging contribution of the high-energy degrees of freedom. There are different regularization schemes: Riemann's zeta function regularization; introduction of the exponential cutoff; dimensional regularization, etc. People are happy when dif-

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Effective gravity and quantum vacuum in superQuids

ferent regularization schemes give the same results. But this is not always so (see e.g. [27, 18, 28], and in particular the divergences occurring for spherical geometry in odd spatial dimension are not cancelled [29, 30]). This raises some criticism against the regularization methods [31] or even some doubts concerning the existence and the magnitude of the Casimir effect. The same type of Casimir effect arises in condensed matter, due to thermal (see review paper [32]) or/and quantum fluctuations. When considering the analogue of the Casimir effect in condensed matter, the following correspondence must be taken into account, as we discussed above. The ground state of quantum liquid corresponds to the vacuum of quantum field theory. The low-energy bosonic and fermionic excitations above the vacuum — quasiparticles — correspond to elementary particles forming matter. The low energy modes with linear spectrum E = cp can be described by a relativistictype effective theory. The analogue of the Planck energy scale i?pianck is determined either by the mass m of the atom of the liquid, -Z?pianck = mc2, or by the Debye energy, E'pianck = he/a (see equation (6.46)). The traditional Casimir effects deals with low energy massless modes. The typical massless modes in quantum liquid are sound waves. The acoustic field is described by the effective theory in equation (6.14) and corresponds to a massless scalar field. The walls provide the boundary conditions for the sound wave modes, usually these are Neumann boundary conditions. Because of the quantum hydrodynamic fluctuations there must be a Casimir force between two parallel plates immersed in the quantum liquid. Within the effective theory, the Casimir force is given by the same equation as the Casimir force acting between the conducting walls due to quantum electromagnetic fluctuations. The only modifications are: (i) the speed of light must be substituted by the speed of sound c; (ii) the factor 1/2 must be added, since we have the scalar field of the longitudinal sound wave instead of two polarizations of light. If d is the distance between the plates and A is their area, then the d-dependent contribution to the ground state energy of the quantum liquid at T = 0 which follows from the effective theory must be _ c

~

hen2 A ~ lUOd3

(6.84)

Such microscopic quantities of the quantum liquid as the mass of the atom m and interatomic space a do not enter explicitly in equation (6.84): the traditional Casimir force is completely determined by the "fundamental" parameter c of the effective scalar field theory.

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165

Finite-size versus finite-iV effect

However, we shall show that equation (6.84) is not always true. We shall give here an example, where the effective theory is not able to predict the Casimir force, since the microscopic high-energy degrees of freedom become important. In other words "trans-Planckian physics" shows up and the "Planck" energy scale explicitly enters the result. In this situation the Planck scale is physical and cannot be removed by any regularization. Equation (6.84) gives a finite-size contribution to the energy of quantum liquid. It is inversely proportional to the linear dimensions of the system, Ec oc l/R for a sphere of radius R. However, for us it is important to consider not only the finite-size effect, but also the finite-N effect, EQ OC TV-1/3, where N is the number of atoms in the liquid in the slab. As distinct from R, the quantity TV* is discrete. Since the main contribution to the vacuum energy is oc R3 oc JV, the relative correction of order TV-4/3 means that the Casimir force is the mesoscopic effect. We shall show that in quantum liquids, the essentially larger mesoscopic effects, of the relative order TV-1, can be more pronounced. This is a finite-TV effect, which reflects the discreteness of the vacuum and cannot be described by the effective theory dealing with the continuous medium, even if the theory includes the real boundary conditions with the frequency dependence of dielectric permeability. We shall start with the simplest quantum vacuum — the ideal onedimensional Fermi gas — where the mesoscopic Casimir forces can be calculated exactly without invoking any regularization procedure.

6.7.3

Vacuum energy from microscopic theory

We consider a system of N bare particles, each of them being a one-dimensional massless fermion. The continuous energy spectrum is E(p) = cp, with c playing the role of speed of light. We assume that these fermions are either "spinless" (this means that they all have the same direction of spin and thus the spin degrees of freedom can be neglected) or 1+1 Dirac fermions. If the fermions are not interacting the microscopic theory is extremely simple: in the vacuum state fermions simply occupy all energy levels below the chemical potential /j.. In the continuous limit, the total number of particles TV, and the total energy of the system in the one-dimensional "cavity" of size d, are

166

Effective gravity and quantum vacuum in superfluids

expressed in terms of the Fermi momentum pp = fi/c in the following way

"-•"'• » - £ • £ - £ •

(6 85)

E = c(„)« , t ( „ ) = £

(6-86)

-

^ c p = g | = f *»> -

Here e(n) is the vacuum energy density as a function of the particle density. The relation between the particle density and chemical potential [i = nhcn = ppc also follows from minimization of the relevant vacuum energy: d(e(n) — fj,n)/dn — 0. In the vacuum state the relevant vacuum energy density and the pressure of the Fermi gas are e = e(n) — p,n = ——hen2 , P = — e = —hen2 . dt

(6.87)

dt

Fermi gas can exist only at positive external pressure provided by the walls.

6.7.4

Vacuum energy in effective theory

As distinct from the microscopic theory, which deals with bare particles, the effective theory deals with quasiparticles — fermions living at the level of the chemical potential \i = epp- There are 4 different quasiparticles: (i) quasiparticles and quasi-holes living in the vicinity of the Fermi point at p = +PF have spectrum Eqp(p+) = \E(p) — /j,\ = c\p+\, where p — pz — pp\ (ii) quasiparticles and quasi-holes living in the vicinity of the other Fermi point at p = —pF have the spectrum Eqp(p~) = \E(p) — fi\ = c\p-\, where p_ = p + pF. In the effective theory the energy of the system is the energy of the Dirac vacuum

£ = -£c|p+|-5>|p_|. p+

(6.88)

v-

This energy is divergent and requires an explicit cut-off. With the proper cut-off provided by the Fermi-momentum, ppianck ~ PF, the negative vacuum energy density e(n) in equation (6.87) can be reproduced. This is a rather rare situation when the effective theory gives the correct sign of the vacuum energy.

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167

Vacuum energy as a function of discrete N

Now let us discuss the Casimir effect — the change of the vacuum pressure caused by the finite size effects in the vacuum. We must take into account the discreteness of the spectrum of bare particles or quasiparticles (depending on which theory we use, microscopic or effective) in the slab. Let us start with the microscopic description in terms of bare particles (atoms). We can use two different boundary conditions for particles, which give two kinds of discrete spectrum: »-. Ek

, 7ic7r = k— ,

*-{'+*)¥•

(6.89)

<"°>

Equation (6.89) corresponds to the spinless fermions with Dirichlet boundary conditions at the walls, while equation (6.90) describes the energy levels of the 1+1 Dirac fermions with no particle current through the wall; the latter case with the generalization to d + 1 fermions has been discussed in [33]. The vacuum is again represented by the ground state of the collection of N noninteracting particles. We know the structure of the system completely and thus the vacuum energy in the slab is well defined: it is the energy of N fermions in ID box of size d

E(N,d) = J2Ek = ^N(N + l) , for Ek = k^f ,

(6.91)

k=\

E(N,d) = Y,Ek = ?Z-tf , for Ek=(k + ±)!f. 6.7.6

(6.92)

Leakage of vacuum through the wall

To calculate the Casimir force acting on the wall, we must introduce vacuum on both sides of the wall. Thus let us consider three walls: at z = 0, z = di < d and z = d. Then we have two slabs with sizes d\ and di — d — d\, and we can find the force acting on the wall separating the two slabs, i.e., on the wall at z = d\. We assume the same boundary conditions at all the walls. But we must allow the exchange the particles between the slabs, otherwise the main force acting on the wall between the slabs will be determined simply by the

168

Effective gravity and quantum vacuum in superfiuids

difference in bulk pressure in the two slabs. This can be done by introducing, say, very small holes (tunnel junctions) in the wall, which do not violate the boundary conditions and thus do not disturb the particle energy levels, but still allow particle exchange between the two vacua. This situation can be compared with the traditional Casimir effect. The force between the conducting plates arises because the electromagnetic fluctuations of the vacuum in the slab are modified due to boundary conditions imposed on electric and magnetic fields. In reality these boundary conditions are applicable only in the low-frequency limit, while the wall is transparent for the high-frequency electromagnetic modes, as well as for the other degrees of freedom of real vacuum (fermionic and bosonic), that can easily penetrate through the conducting wall. In the traditional approach it is assumed that those degrees of freedom, which produce the divergent terms in the vacuum energy, must be cancelled by the proper regularization scheme. That is why, though the dispersion of dielectric permeability does weaken the real Casimir force, nevertheless in the limit of large distances, d\ 3> c/u0, where wo is the characteristic frequency at which the dispersion becomes important, the Casimir force does not depend on how easily the high-energy vacuum "leaks" through the conducting wall. We consider here just the opposite limit, when (almost) all the bare particles are totally reflected. This corresponds to the case when the penetration of the high-energy modes of the vacuum through the conducting wall is highly suppressed, and thus one must certainly have the traditional Casimir force. Nevertheless, we shall show that due to the mesoscopic finite-./V effects the contribution of the diverging terms to the Casimir effect becomes dominating. They produce highly oscillating vacuum pressure in quantum liquids. The amplitude of the mesoscopic fluctuations of the vacuum pressure in this limit exceeds by a factor ppianckd/h the value of the conventional Casimir pressure. Continuous effective low-energy theories are not applicable for their description

6.7.7

Mesoscopic Casimir force in Id Fermi gas

The total vacuum energy in two slabs for spinless and Dirac fermions is correspondingly

EiN,diM =

^m±»+M^,

(MS)

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169

„,„ , , , E{N,dud2)

=

thCTT /TV2 JV22\ _ 2 \ -d\L + -d22. ,

(6.94)

where A^ and N2 are the particle numbers in each of the two slabs: N1 + N2 = N , dx + d2 = d

(6.95)

Since particles can transfer between the slabs, the global vacuum state in this geometry is obtained by minimization over the discrete particle number Ni at fixed total number N of particles. If the mesoscopic 1/N corrections are ignored, one obtains JVi « (di/d)N and N2 « (d2/d)N; the two vacua have the same pressure, and thus there is no force acting on the wall between the two vacua. However, JVt and N2 are integer valued, and this leads to mesoscopic fluctuations of the Casimir force. The global vacuum with given values of Ni and N2 is realized only within a certain range of parameter d\. If d\ increases, it reaches some threshold value above which the energy of the vacuum with the particle numbers N\ +1 and N2 — l has lower energy and it becomes the global vacuum. The same happens if d\ decreases and reaches some threshold value below which the vacuum with the particle numbers N\ — \ and N2 + 1 becomes the global vacuum. The force acting on the wall in the state (N\, N2) is obtained by variation of E(Ni, N2, dx,d — dx) over d\ at fixed JVi and N2: F(NuN2,dud2)

= _ d W f f •*.*) ddi

+

m N M d dd2

2

)

When d\ increases and reaches the threshold, where E{N\,N2,di,d2) = E(N\ + 1,N2 — 1, d\, d2), one particle must cross the wall from the right to the left. At this critical value, the force acting on the wall changes abruptly (we do not discuss here an interesting physics arising just at the critical values of ai, where degeneracy occurs between the states {Ni,N2) and (JVi + 1, N2 — 1); at these positions of the wall (or membrane) the particle numbers Nx and A^2 are undetermined and are actually fractional due to quantum tunneling between the slabs [34]). Using for example the spectrum in equation (6.94) one obtains for the jump of the Casimir force: F{N1±l,N2*l)-F(N1,N2)

=

h

^{±22d2^1

-

± ^ . did2

+

±2

fd2~l) (6.97)

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Effective gravity and quantum vacuum in supertiuids

The same result for the amplitude of the mesoscopic fluctuations is obtained if one uses the spectrum in equation (6.93). In the limit di
(6.98)

This is smaller, by a factor l/Ni — (nh/dipF) = (nh/dippianck), than the vacuum energy density in equation (6.86). On the other hand, it is larger (by the factor pFd\ = ppianckdi) than the traditional Casimir pressure, which in one-dimensional case is Pc ~ hc/df. The divergent term which linearly depends on the Planck momentum cutoff ppianck as in equation (6.98) has been revealed in many different calculations (see e.g. [30]), and attempts have been made to invent a regularization scheme which would cancel the divergent contribution.

6.7.8

Mesoscopic Casimir pressure in quantum liquids

Equation (6.98) for the amplitude of the mesoscopic fluctuations of the vacuum pressure can be immediately generalized for a d-dimensional space: if Vi is the volume of the internal region separated by almost impenetrable walls from the outside vacuum, then the amplitude of the mesoscopic vacuum pressure must be of order |Pmeso| ~ %

^

•

(6.99)

The mesoscopic random pressure comes from the discrete nature of the underlying quantum liquid, which represents the quantum vacuum. The integer value of the number of atoms in the liquid leads to mesoscopic fluctuations of the pressure: when the volume V\ of the vessel changes continuously, the equilibrium number TVx of particles changes in step-wise manner. This results in abrupt changes of pressure at some critical values of the volume: Pmeso ~ P{NX ± 1) - P(JVx) = ±^~

= ±

^

= ±^»f* ,

(6.100)

where again c is the speed of sound, which plays the role of the speed of light. The mesoscopic pressure is determined by microscopic "trans-Planckian" physics, and thus such microscopic quantities as the mass m of the atom, the "Planck mass", enters this force.

Grigori Volovik

171

For the spherical shell of radius R immersed in the quantum liquid the mesoscopic pressure is mc

(fie \

Pmeso ~ ± - ^ " = i V ^ P l a n c k ( ^ J

6.7.9

•

(6.101)

Mesoscopic vacuum pressure versus conventional Casimir effect

Let us compare the mesoscopic vacuum pressure in equation (6.101) with the traditional Casimir pressure obtained within the effective theory for the same spherical shell geometry. In the effective theory (such as electromagnetic theory in the case of the original Casimir effect, and the low-frequency quantum hydrodynamics in quantum liquids) the Casimir pressure comes from the bosonic and fermionic low-energy modes of the system (electromagnetic modes in the original Casimir effect or quanta of sound waves in quantum liquids). In superfiuids, in addition to phonons other low-energy sound-like collective are possible, such as spin waves. These collective modes with linear ("relativistic") spectrum in quantum liquids play the role of the relativistic massless scalar field. They obey typically Neumann boundary conditions, corresponding to the (almost) vanishing mass or spin current through the wall (almost, because the vacua inside and outside the shell must be connected). If we believe in the traditional regularization schemes which cancel out the divergent terms, the effective theory gives the Casimir pressure for the spherical shell as

PC

=

_ < S dV

=

* ^ ( | ) \

( 6 ,„ 2 )

8TT

where K = —0.4439 for the Neumann boundary conditions; K = 0.005639 for the Dirichlet boundary conditions [30]; and c is the speed of sound or of spin waves. The traditional Casimir pressure is completely determined by the effective low-energy theory, it does not depend on the microscopic structure of the liquid: only the "speed of light" c enters this force. The same pressure will be obtained in case of the pair correlated fermionic superfiuids, if the fermionic quasiparticles are gapped and their contribution to the Casimir pressure is exponentially small compared to the contribution of the collective massless bosonic modes.

172

Effective gravity and quantum vacuum in superfluids

However, at least in our case, the result obtained within the effective theory is not correct: the real Casimir pressure is given by equation (6.101): (i) It essentially depends on the Planck cut-off parameter, i.e., it cannot be determined by the effective theory; (ii) it is much bigger, by factor p P l a n c k i?//i, than the traditional Casimir pressure in equation (6.102); and (iii) it is highly oscillating. The regularization of these oscillations by, say, averaging over many measurements; by noise; or due to quantum or thermal fluctuations of the shell; etc., depend on the concrete physical conditions of the experiment. This shows that in some cases the Casimir vacuum pressure is not within the responsibility of the effective theory, and the microscopic (trans-Planckian) physics must be evoked. If two systems have the same low-energy behavior and are described by the same effective theory, this does not mean that they necessarily experience the same Casimir effect. The result depends on many factors, such as the discrete nature of the quantum vacuum, and the ability of the vacuum to penetrate through the boundaries. It is not excluded that even the traditional Casimir effect which comes from the vacuum fluctuations of the electromagnetic field is renormalized by the high-energy degrees of freedom Of course, the extreme limit which we consider, is not applicable to the original (electromagnetic) Casimir effect, since the situation in the electromagnetic Casimir effect is just opposite. The overwhelming part of the fermionic and bosonic vacuum easily penetrates the conducting wall, and thus the mesoscopic fluctuations are small. But are they negligibly small? In any case, this example shows that the cut-off problem is not mathematical, but physical, and that Planck physics dictates the proper regularization scheme or the proper choice of the cut-off parameters.

6.8

Conclusion

We have discussed the problems related to the properties of the quantum vacuum in general relativity using the known properties of the quantum vacuum in quantum liquids, where some elements of the Einstein gravity arise in the low-energy corner. We have found that in both systems there are similar problems, which arise if the effective theory is exploited. In both systems the naive estimate of the vacuum energy density within the effective theory gives PA ~ ^pianck with the corresponding "Planck" energy appropriate for each of the two systems. However, as distinct from the general relativity, in quantum

Grigori Volovik

173

liquids the fundamental physics, "The Theory of Everything", is known, and it shows that the "trans-Planckian" degrees of freedom exactly cancel this divergent contribution to the vacuum energy. The relevant vacuum energy is zero without fine tuning, if the vacuum is stable (or metastable), isolated and homogeneous. Quantum liquids also demonstrate how a small vacuum energy is generated if the vacuum is disturbed. In particular, thermal quasiparticles — which represent the matter in general relativity — induce a vacuum energy of the order of the energy of the matter. This example shows a possible answer to the question: Why is the present cosmological constant of the order of the present matter density in our Universe? It follows that in each epoch the vacuum energy density must be of order of either the matter density of the Universe, or of its curvature, or of the energy density of the smooth component — the quintessence. However, a complete understanding of the dynamics of the vacuum energy in the time-dependent regime of the expanding Universe cannot be achieved within general relativity and requires an extension of this effective theory. In principle, one can construct the artificial quantum liquid, in which all the elements of general relativity are reproduced in the low energy corner. The effective metric g^v acting on "relativistic" quasiparticles arises as one of the low-energy collective variables of the quantum vacuum, while the Sakharov mechanism leads to the Einstein curvature and cosmological terms in the action for this dynamical variable. In this liquid the low energy phenomena will obey the Einstein equations (6.8), with probably one exception: the dynamics of the cosmological "constant" will be included. It would be extremely interesting to realize this programme, and thus to find out the possible extension of general relativity, which takes into account the properties of the quantum vacuum. The most important property of the quantum vacuum in quantum liquids is that this vacuum consists of discrete elements — bare atoms. The interaction and zero-point oscillations of these elements lead to the formation of the equilibrium vacuum, and in this equilibrium vacuum state the cosmological constant is identically zero. Thus the discreteness of the quantum vacuum can be the possible source of the (almost complete) nullification of the cosmological constant in our present Universe. If so, one can try to exploit the other possible consequences of the discrete nature of the quantum vacuum, such as the mesoscopic Casimir effect discussed in section 6.7. Analogy with the quantum vacuum in quantum liquids allows us to dis-

174

Effective gravity and quantum vacuum in superBuids

cuss other problems related to the quantum vacuum in general relativity: the flatness problem; the problem of a big entropy in the present Universe; the horizon problem, etc..

Acknowledgments: This work was supported in part by the Russian Foundation for Fundamental Research and by the European Science Foundation.

Bibliography [1] I.M. Khalatnikov: An Introduction to the Theory of Superfluidity, Benjamin, New York, 1965. [2] R.B. Laughlin and D. Pines, The Theory of Everything, Proc. Natl. Acad. Sc. USA 97, 28-31 (2000). [3] G.E. Volovik, Superfluid analogies of cosmological phenomena, Physics Reports 351 (2001) 195-348 [gr-qc/0005091]. [4] T. Damour, Gravitation and experiment, gr-qc/9711061. [5] A. Einstein, Kosmologische Betrachtungen zur allgemeinen Relativitatstheorie, Sitzungsber. Konigl. Preuss. Akad. Wiss., 1, 142-152 (1917). [6] S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. 6 1 , 1-23 (1989); S. Weinberg, The cosmological constant problems, astro-ph/0005265. [7] Supernova Cosmology Project (S. Perlmutter et al.), Measurements of fi and A from 42 high redshift supernovae, Astrophys. J. 517, 565-586 (1999). [8] A. D. Sakharov: Vacuum quantum fluctuations in curved space and the theory of gravitation, Dokl. Akad. Nauk 177, 70-71 (1967) [Sov. Phys. Dokl. 12, 1040-41 (1968)] [9] W. G. Unruh, Experimental black-hole evaporation?, Phys. Rev. Lett. 46, 1351-1354 (1981); Sonic analogue of black holes and the effects of high frequencies on black hole evaporation, Phys. Rev. D 51, 2827-2838 (1995). 175

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Effective gravity and quantum vacuum in superfluids

[10] M. Visser, Acoustic black holes: horizons, ergospheres, and Hawking radiation, Class. Quantum Grav. 15, 1767-1791 (1998) [11] M. Stone, Iordanskii force and the gravitational Aharonov-Bohm effect for a moving vortex, Phys. Rev. B 6 1 , 11780 — 11786 (2000). [12] M. Stone, Acoustic energy and momentum in a moving medium, Phys. Rev. B62, 1341- 1350 (2000). [13] C D . Frogatt and H.B. Nielsen, Origin of Symmetry, World Scientific, Singapore - New Jersey - London - Hong Kong, 1991. [14] S. Chadha, and H.B. Nielsen, Lorentz invariance as a low-energy phenomenon, Nucl. Phys. B217, 125-144 (1983). [15] C.W. Woo, Microscopic calculations for condensed phases of helium, in The Physics of Liquid and Solid Helium, Part I, eds. K.H. Bennemann and J.B. Ketterson, John Wiley k Sons, New York, 1976. [16] Y. Nambu, Fermion-boson relations in the BCS-type theories, Physica D15, 147-151 (1985). [17] R. Brout, Who is the Inflaton? gr-qc/0103097. [18] F. Ravndal, ph/0009208.

Problems with the Casimir vacuum

energy,

hep-

[19] J. D. Bjorken, Standard model parameters and the cosmological constant, hep-ph/0103349. [20] R.R. Caldwell, R. Dave, and P.J. Steinhardt, Cosmological imprint of an energy component with general equation of state, Phys. Rev. Lett. 80, 1582-1585 (1998). [21] M.S. Turner and M. White, CDM models with a smooth component, Phys. Rev. D56, R4439 — R4443 (1997). [22] D. Vollhardt, and P. Wolfle, The superfluid phases of helium 3, Taylor and Francis, London - New York - Philadelphia, 1990. [23] G. Chapline, E. Hohlfeld, R.B. Laughlin, and D.I. Santiago, Quantum phase transitions and the breakdown of classical general relativity, grqc/0012094.

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[24] G. Chapline, The vacuum energy in a condensate model for spacetime, Mod. Phys. Lett. A14, 2169-2178 (1999). [25] B. Boisseau, G. Esposito-Farese, D. Polarski, A.A. Starobinsky, Reconstruction of a scalar-tensor theory of gravity in an accelerating universe, Phys. Rev. Lett. 85, 2236-2239 (2000) [26] H.B.G. Casimir, Proc. K. Ned. Wet. 51 793 (1948). [27] G. Esposito, A. Yu. Kamenshchik, K. Kirsten, Casimir energy in noncovariant gauges, Int. J. Mod. Phys. A14, 281 (1999); Casimir energy in the axial gauge, Phys. Rev. D62, 085027 (2000). [28] H. Falomir, K. Kirsten, K. Rebora, Divergences in the Casimir energy for a medium with realistic ultraviolet behavior, hep-th/0103050. [29] K.A. Milton, Dimensional and dynamical aspects of the Casimir effect: Understanding the reality and significance of vacuum energy, hepth/0009173. [30] G. Cognola, E. Elizalde, K. Kirsten, Casimir energies for spherically symmetric cavities, hep-th/9906228. [31] C.R. Hagen, Cutoff dependence and Lorentz invariance of the Casimir effect, quant-ph/0102135; Casimir energy for spherical boundaries, Phys. Rev. D 6 1 , 065005 (2000). [32] M. Kardar and R. Golestanian, The "friction" of vacuum, and other fluctuation-induced forces, Rev. Mod. Phys. 7 1 , 1233-1245 (1999). [33] R.D.M. De Paola, R.B. Rodrigues, N.F. Svaiter, Casimir energy of massless fermions in the slab-bag, Mod. Phys. Lett. A14, 2353-2362 (1999). [34] A.F. Andreev, Superfluidity, superconductivity, and magnetism in mesoscopics, Uspekhi Fizicheskikh Nauk 168, 655-664 (1998).

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Chapter 7 Emergent relativity and t h e physics of black hole horizons George Chapline Physics and Advanced Technology Division Lawrence Livermore National Laboratory Livermore, California 94550 E-mail: [email protected] Robert B . Laughlin Department of Physics Stanford University Stanford, California 94305 E-mail: [email protected] David I. Santiago Department of Physics Stanford University Stanford, California 94305 E-mail: [email protected] 179

180

Emergent relativity and the physics of black hole horizons

Abstract: Classical general relativity predicts that nothing physical happens as one crosses a black hole event horizon. We propose that what is actually happening at a black hole horizon is a quantum phase transition of the vacuum of spacetime analogous to the liquid-vapor critical point of a Bose superfluid. The horizon paradoxes in these systems are the consequence of an incorrect extrapolation of a classical or long-wavelength description beyond the point where it is valid. The underlying quantum mechanics is well-defined and free of paradoxes even at the horizon, i.e., critical point. Classical general relativity holds on either side of the infinite-redshift surface yet fails at the surface itself through divergence of a characteristic quantum mechanical coherence length £. We predict distinctive physical effects right at the event horizon that can be observed from the exterior of the black hole.

7.1

Introduction

One of the most disturbing and interesting problems in physics is the incompatibility between quantum mechanics and classical general relativity. The incompatibility arises because quantum mechanics cannot be defined without a universal time. In general relativity there exist spacetimes which evolve into black hole spacetimes. These are characterized by surfaces (event horizons or infinite-redshift surfaces) at which time as measured by a stationary exterior observer stands still. Although the pathological nature of such a surface is generally dismissed as a coordinate system artifact, infinite-redshift surfaces are in fact incompatible with ordinary quantum mechanics. One of the classical signatures of a gravitational field is that clocks run at different rates at different points of space. As pointed out by Landau [1] one can use this fact to map out the gravitational field provided one can synchronize clocks. Unfortunately physical synchronization of clocks — which is equivalent to choosing a coordinate system where the off-diagonal components g0i of the metric tensor vanish — requires two-way radio or light-wave communications. The synchronized proper time is

dr=

9oo~Yl9oi/9ii dt .

(7.1)

George Chapline/Robert

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181

Evidently the synchronized proper time is well denned as long as the quantity inside the square root remains positive. As it happens, this quantity vanishes at the event horizon of either a Schwarzschild or a Kerr black hole. The incompatibility between classical general relativity and quantum mechanics, i.e., the non-universality of time in Einstein gravity, is fundamental and deep. Black hole spacetimes with their inherent lack of universal time are predicted to exist by the Einstein field equations as the endpoint of the evolution of certain stars. The Einstein field equations, however, are little more than a logical consequence of the principles of relativity and equivalence. Thus the development of a black hole event horizon actually follows from the relativity principle, and cannot be avoided unless relativity fails in some way. Notice that spacetime curvature is finite at the horizon. The physical meaning of this is that a free-falling observer sees no local pathology at the event horizon and passes through in finite proper time. Therefore the Einstein field equations are stable across the horizon, but quantum mechanics is not: the lack of a universal time prevents the usual many-body Schrodinger equation ih—

=H *

(7.2)

from being well-defined. This equation is the foundation of quantum field theory and statistical mechanics, i.e., of our microscopic understanding of the whole natural world outside gravity. Without universal time the Hamiltonian % cannot be consistently defined, and without a Hamiltonian one is not guaranteed the existence of thermodynamics, the unitarity of the S-matrix, the absence of an information paradox, or Feynman rules. Most of the other conceptual difficulties of quantum gravity such as the lack of a lower bound to the Einstein action and the need to sum over wormhole configurations in path integrals [2] are equivalent to this problem. Our proposal to resolve the incompatibility between general relativity and quantum mechanics is to start with the many-body Schrodinger equation (7.2) and weaken the relativity principle, although in a most subtle fashion: The black hole event horizon is a phase transition of the vacuum of spacetime [3]. This requires that the relativity principle itself be emergent, i.e., a low-energy collective phenomenon of the matter that makes up the vacuum, rather than fundamental [4, 5]. This idea is actually implicit in the way we think about elementary particles. The standard model is a de-facto aether theory, in that it ascribes to the vacuum spectroscopic properties similar to

182

Emergent relativity and the physics of black hole horizons

those of ground states of ordinary matter — polarizability, strong mixing of the fermionic and bosonic degrees of freedom, and so forth — and then simply postulates the gravitationally-relevant energy density to be zero. String theory is more explicitly an aether theory, in that it allows the gravitational degrees of freedom to mix arbitrarily with others. All theories of the vacuum that purport to describe it accurately are renormalizable, this being one of its experimental properties. However, renormalizability is also a property of low-energy effective field theories of phases of matter. It follows from the emergence of universal spectroscopic properties as the energy scale is lowered and formally describes the independence of these low-energy properties from the model-dependent high-energy ones. Similarly, in particle physics the renormalizability of effective field theories of the vacuum guarantees that high-energy properties are completely decoupled from low-energy ones and thus have no consequence for existing experiment. Thus emergent relativity is consistent with the facts as we know them. To illustrate these ideas let us consider a much simpler problem, the liquid-vapor critical point of a Bose superfluid. The properties of 4 He demonstrate that both zero-temperature phase transitions of bosonic matter and the liquid phase exist [6]. 4 He is a solid at pressures above 25 atmospheres and zero temperature. As the pressure is dropped below 25 bar it melts into a liquid with unmeasurably small quantum vapor pressure — meaning that it puddles at the bottom of a container larger than itself and will not evaporate at zero temperature. The vapor phase of bosonic matter also exists in nature in the newly-discovered "Bose-Einstein condensates" — a name that is somewhat misleading as these systems exhibit a finite sound speed [7]. Moreover, they are metastable excited states rather than ground states, and are more aptly called supersaturated quantum vapors. Their behavior is fully consistent with Bogoliubov's original description of superfluid broken symmetry in 4 He, which was based on weak repulsive potentials and was actually a description of the quantum gas [8]. The nature of the zero-temperature liquid-vapor transition in these systems is, however, controversial. A realistic variational study [9] of Lennard-Jones fluids found that the critical point could not be reached by varying pressure. However there is no general principle leading to that conclusion. In fact, recent studies based on different model assumptions [10] find behavior more consistent with the existence of such a transition. We will proceed on the assumption that both models were solved correctly, but the existence of the transition is model-dependent. In the Bose superfluids the emergent principle is quantum hydrodynamics

George Chapline/Robert

Laughlin/David

Santiago

183

rather than relativity, so the details are different from real quantum gravity, but the physical idea is the same. In the Bose superfluid, the approach to the critical surface is signaled by the vanishing of the speed of sound, while in a black hole the approach to the horizon is signaled by the vanishing of the time dilation factor defined in equation (7.1). The critical surface acts as an infinite-redshift surface (or horizon) for sound. The principles of hydrodynamics can, and do, fail at such a continuous phase transition: the classical description of the "vacuum" on either side of the horizon fails on length scales smaller than a coherence length £. This length, which is purely quantum-mechanical in nature, is proportional to the inverse speed of sound and thus diverges at the critical point, i.e., the horizon. The classical equations remain exactly valid up to the horizon only in context of a special, unphysical order of limits. If a distance to the critical surface or horizon is chosen first, there exists a frequency for which hydrodynamics holds up to that distance. If the frequency is chosen first on the other hand, as it would be in a more realistic experiment, hydrodynamics holds up to a short distance above the horizon, then fails. In this way the classical description, be it hydrodynamics or relativity, can be true arbitrarily close to the horizon yet fail there.

7.2

Horizons in Bose fluids

As an introduction to the Bose superfluid model, let us recall the functional form of the Van der Waals equation of state for a classical fluid: (V - b) ( P + ^ ) = NkBT.

(7.3)

A quantum realization of this is the phenomenological equation of state

(V2 -b)(p + £)=c

(7.4)

is shown in figure 7.1. The Active temperature c is a parameter in the manybody Hamiltonian. The equation of state (7.4) is generated in the mean-field approximation from the field theory

c = r (ih^+M) v - ^ i v v f - wavi2), with |^| 2 interpreted as the density p = N/V and

(7-5)

Emergent relativity and the physics of black hole horizons

184

P/Pc

1

Figure 7.1: Phenomenological equation of state defined by equation (7.4) for various values of the parameter c near the critical value. The dotted lines indicate the Maxwell loops.

U=

In 2VbV"*\V-Vb)

3V4"

(7.6)

At zero temperature this system exhibits the phenomenon of Bose condensation — i.e., acquires a superfluid order parameter ip with low-energy dynamics described by the extremal condition


(7.7)

This is the Gross-Pitaevskii equation [11]. The particle density and current density, defined by h (7.8) W ) , 2Mi satisfy hydrodynamic conservation of both particle number and momentum [12] \tP\2 = p,

and

|

pv

= ^T7T(V*VV -

+v-w

= o,

(7.9)

George Chapline/Robert

M

l^

Laughlin/David

+ V Mv2 +

{\

»-^

Santiago

V2lnp+i(Vlnp)2

185

0.

(7.10)

The quiescent state of the fluid is described by the uniform solution tp0 satisfying U'(\A\2)=V

= (PV + E0)/N.

(7.11)

The particle density is fixed by suitably adjusting the chemical potential \i. Small perturbations to this solution, i> = ip0 + 5ipR + i5iph

(7.12)

satisfy both AWR)

dt

2

* V = -^17 2M W i )

(7-13)

and ^

^

= -^7V2(^R) + —(^R),

(7-14)

to linear order. This yields the dispersion relation for compressional sound:

K = \/(favz)2+(i7inr) , \2M

with

vs^j—.

(7.15)

This also identifies £ = h/Mvs as the length scale for the failure of hydrodynamics. This same scale appears in the Bogoliubov solution [8]. There is one and only one point in the diagram where the bulk modulus B — —V(dP/dV) is zero, namely the critical point. Consider the thought experiment illustrated in figure 7.2 in which a tall tank on the surface of the earth is filled with a quantum superfluid characterized by a critical equation of state (c is chosen so that the curve in figure 7.1 passes through the critical point). The pressure increases toward the bottom of the tank due to gravity and at some critical depth reaches, and then surpasses, the critical pressure. Sound waves are refracted toward this surface just as light is refracted toward a black hole horizon and for the same

186

Emergent relativity and the physics of black hole horizons

tz

' / /

Vapor

'7/.

Critical Surface

Liquid

Figure 7.2: Illustration of thought experiment in which pressure increases toward the bottom of a tank of quantum Ruid. Sound emitted from a transducer on the side of the tank is refracted downward toward the critical surface where the sound speed collapses to zero. reason, namely that the propagation speed measured by a clock at infinity vanishes there. For the specific equation of state defined by equation (7.4) with c = 8a/27b, working near the critical surface at z = 0 we have 1 Pc =

73&'

Pr.=

276 2 '

v, =

(7.16)

MPc

with P_

- 1 ^ 1 2 ( ^ - 1 ^ 1 - ^ ,

Pr

v0

9* 12t;2

1/3

(7.17)

Small density fluctuations p —> p + dp then propagate according to d2(SP) v-k v(M] = dt2 ' This is qualitatively the same as the scalar wave equation 2

(7.18)

(7.19)

George Chapline/Robert

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187

which one obtains from [13]

hv^9

»*5F)= 0

<7 20)

'

using the gravitational metric ds 2 = g^ da^dx" = dx2 + dy2 + dz2 - v2dt2.

(7.21)

It is clear from equation (7.18) that, just as for the scalar field moving in the gravitational field of a black hole, the evolution of a sound wave moving in the Bose fluid seems to stop where the speed of sound vanishes, i.e., at the sound horizon.

7.3

Horizons in quantum magnets

While the occurrence of the quantum liquid-vapor transition in the Bose superfluid is controversial, there are other phase transitions in the same universality class, notably the spin-flop transition of a quantum ferromagnet, for which there is no controversy. These provide a sound precedent for creating horizon-like critical surfaces in ordinary matter and have the important advantage of being experimentally accessible. Let us briefly review the quantum mechanics of this transition for the anisotropic Heisenberg model. The Hamiltonian is

n

= -J <jk> E {SJS*+e[5^+5?5*]}

(J

> 0) -

(7 22)

-

where < jk > denotes the set of nearest-neighbor pairs on a 3 dimensional cubic lattice. The lattice spacing is d. When e < 1 the system is in its Ising phase, which has two degenerate ground states | t t ••• 1> a n d | 14- ••• 4>When e > 1 the order parameter "flops" down into the x-y plane but can point in any direction in this plane. Let us first consider the Ising phase, the simpler of the two. The lowenergy excitations out of the | t t • • • 1> ground state are given exactly by Sq\ t t ••• t>) where

188

Emergent relativity and the physics of black hole horizons

Sq ='*Texp(iq-rj)

Sj

and

S + = 5 > x p H < f • TV) S+,

(7.23)

with Sf = SJ ± iSj. The energy of these spin waves is Eq = J {3 - e [cos(qxd) + cos(qyd) + cos(^d)]} .

(7.24)

This dispersion relation is plotted for various values of e in figure 7.3. Note the energy gap at q = 0, which is characteristic of the phase. If we force the eigenvalue of Sz = J^. SL which commutes with the Hamiltonian, to be 0 then the system phase-separates into f and J, regions. These may be thought of as the fog of pure vapor and pure liquid that characterizes the 2-phase region of the liquid-vapor phase diagram. The z component of total spin in the magnet is thus roughly analogous to pressure. It must be tuned to its critical value, 0, for the transition in e to be continuous. Let us next solve the xy phase for e > 1 approximately using the Bogoliubov method. Rotating the system in spin space by n/2 around the y axis we obtain

s3S'k + -s?s*k + s»js»

<jk>

(7.25)

Then taking the "bare" vacuum to be IV'o > = | t t ••• t>> a n d the corresponding "bare" spin wave annihilation operators to be aq = S+/V~N, so that

[aq,al,)~6qq>,

[aq, a„>] ~ 0 ,

(7.26)

in the context of |V'o>, we obtain U ~

3N — eJ + ^

\£qalaq + &q(aW-q + «-?«?)

(7.27)

—— [cos(qxd) + cos(qyd) + cos(qzd)] \ ,

(7.28)

where = J < 3e

A , = J —— [cos(qxd) + cos(qyd) + cos(qzd)}.

(7.29)

George Chapline/'Robert

EJ\J\

Laughlin/David

Santiago

189

-

0.5 qxb/n Figure 7.3: Left: Spin wave dispersion relation for t = 0 , 0 . 2 , . . . , 1 . The energy gap in this case is characteristic of the Ising phase. Right: Spin wave dispersion relation for e = 1,1.2,...,2. The linearity at small q is characteristic of the xy phase. The canonical transformation aq = uq bq + vq b_q with

uq — cosh(0 g ),

vq = sinh(0,),

and

tanh(20 g ) = -

2A„

(7.30)

then gives

«-^+E

E„q [I b% " vq + l \ q

\e0

(7.31)

where Ec =

S^\- 4A2

(7.32)

The ground state

l^) = I l e x p ( | 7 a J a ^ ) l^>«

(7 33)

-

is annihilated by every bq. The dispersion relation Eq, the spin wave excitations created by b\, is also plotted for various values of e in figure 7.3. It is linear at small q with velocity

Jd J 3e(e-l)

(7.34)

190

Emergent relativity and the physics of black hole horizons q - q1 - q 2

q2

q

qi

Figure 7.4: Lowest-order scattering processes in the critical region. The process on the left causes mass renormalization and decay at zero temperature. The one on the right causes critical opalescence. Thus the critical point of this model is the isotropic Heisenberg ferromagnet. Both the spin wave velocity of the xy phase and the energy gap of the Ising phase converge to zero at e = 1, and the spin wave dispersion relations of the two phases limit to ^critical

=

jU

_ L o s ( 9 x d)

+

cog^d)

+

cos(fcd)] } .

(7.35)

At small q this reduces to Eq ~ J(qd)2/2, the dispersion relation of a nonrelativistic boson. In the limit that e is only slightly larger than 1, the dispersion relation is linear at arbitrarily small q but quickly rolls over at q £ ~ 1, where

? to the critical expression. transition.

7.4

-VS

(736)

-

The coherence length £ diverges at the phase

Quantum criticality

In quantum mechanical systems the paradoxes associated with horizons or infinite-redshift surfaces are spurious. They are the consequence of an incorrect extrapolation of a long wavelength (hydrodynamic, spin wave, relativistic) description beyond the point where it is valid. In contrast to the case of classical general relativity, the paradoxes of sound propagation in a Bose fluid and xy spin wave propagation in a magnet near the critical surface have a simple quantum-mechanical resolution: Sound and spin waves

George Chapline/Robert

Laughlin/David

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191

\R\*

LOTQ

Figure 7.5: Reflectivity as a function of UJTQ, where TQ is a measure of how fast the speed of sound is collapsing as the critical surface is approached. cease to make sense near the horizon because the long-wavelength description fails on length scales smaller than the correlation length £ and time scales shorter than £/vs. At the horizon both diverge to infinity. A sound or spin wave quantum with fixed frequency u propagating toward the horizon reaches the point at which u> > vs/£ in finite time and decays there into the soft excitations of the critical point. These are dense, so most of the energy thermalizes. These phenomena are accounted for by the underlying quantum Hamiltonian. The long-wavelength description fails, but quantum mechanics does not. Notice that even if hydrodynamics fails at the horizon for the Bose superfluid, the quantum description that follows from the Lagrangian of equation (7.5) after canonical quantization is perfectly well defined even at the horizon. Let us consider the properties of the horizon in a little more detail. At the critical point the Lagrangian is effectively [3]

£eff = r ( i f t | - t^jrp ~ ^ i V V f - ^ ( I V I - V>o)4-

(7.37)

The fourth-order interaction term will lead to a finite lifetime for nonrelativistic bosons proportional to their energy squared [3]:

h _ J _ /AT

(7.38) 2 T ~ 3^2 Vfi;)' . (*)' <M* The important decay and scattering processes are shown in figure 7.4. The elementary excitations become more and more sharply defined as the energy

192

Emergent relativity and the physics of black hole horizons

dSldu'

0.05

Figure 7.6: Differential cross-section given by equation (7.39) for inelastic scattering of sound from a critical surface as a function of scattered frequency u)' for values of 6 ranging from 0 to 7r/2. The maximum value of UJ'/U for sound reflected normally (8 = 0) is 1/9. is lowered, so that in the low-energy limit one obtains the ideal noninteracting Bose gas. The mass of these bosons is larger than M because it is strongly renormalized by the interaction [3]. A phonon impinging obliquely on the surface at zero temperature is coherently reflected or transmitted depending on its energy. The reflection coefficient is shown in figure 7.5. The resonances become narrower and narrower with increasing thickness of the interface, and vanishes in the limit of infinite thickness, so that the horizon transmits no sound. About 1/8 of the energy is reflected back out with a strong red shift, the rest being absorbed. The differential cross section per unit area A to do this, i.e., to scatter sound of frequency u back into solid angle d!2 and frequency uf within du/ is

dh

27A

of

2 — cos(6>) (7.39) Ul \ This cross-section is plotted in figure 7.6. Thus the horizon fluoresces red. The horizon becomes much more dissipative at low frequencies if it is hot, and indeed limits to classical critical opalescence. The heat capacity of the horizon is large but finite. The energy density a distance z away from the interface is dfi do/

2

16TT LJ

1-3

1 r° huq q2dq E_ V ~ 2 W 0 exp(f3hujq) - 1'

(7.40)

George Chapline/Robert

Laughlin/David

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193

1

E/E0

05

0 0

1

2

(Mv2JkBTy?2

3

4

Figure 7.7: Thermal energy per unit volume defined by equation (7.40) for a critical surface at temperature T. The divergent Planck law is shown for comparison. where w, is given by equation (7.15) with vs = Z/T0. This is plotted in figure 7.7. It may be seen to limit properly as z —> 0 to ^

7.5

= 0.128 ( ^ ) 3 / 2 (kBTf'\

(7.41)

Discussion

The properties of the black hole event horizon which follow from the quantum many-body description resolves the information paradox. Our proposal differs from most [15, 16], but not all [14], recent condensed matter models for black holes in the key respect that it ascribes black hole behavior at the event horizon to a quantum ground state. Thus in this model there is no Hawking radiation. It is obvious from the example of the Bose superfluid in the vertical tank that at zero temperature there is no radiation. Moreover the horizon has a finite positive heat capacity and can therefore have any temperature it likes. The horizon does not destroy quantum information, but makes entropy the same way black paint does: by scattering the energy into a thermodynamically large number of degrees of freedom. This permits us to greatly strengthen an earlier suggestion [17] that collective effects are the correct explanation for the puzzling paradoxes connected with black holes. It is unfortunately impossible to infer the nature of the vacuum beyond the horizon from existing measurements, since the latter are protected by

194

Emergent relativity and the physics of black hole horizons

7(r)

r/2M

Figure 7.8: Prototype time dilation factor j(r), given by equation (7.43), in the vicinity of a black hole event horizon. principles of universality and therefore contain no information about the underlying microphysics. Any microscopic theory of gravity we would propose would be un-falsifiable speculation until astrophysical observations of black holes improve substantially. However, the simplest guess is that it is literally like the liquid-vapor transition, meaning that the Einstein field equations, like the laws of quantum hydrodynamics, are valid and well defined (free of singularities) in both phases. Thus we are led to propose the metric ds 2 = - d r 2 + r2[d02 + sin2(0)d<£2] - 7d*2,

(7.42)

(r/2M)2 2M/r

(7.43)

where

-{!

7(0 = which satisfies

r < 2M, r> 2M,

1 3 R^ ~^9^R= JJ^, <W (7.44) inside the black hole. Equation (7.43) is plotted in figure 7.8. Note the similarity to figure 7.2. Our solution corresponds physically to a vacuum vessel containing a region of spacetime with a positive cosmological constant. The value of this constant is picked to match the boundary condition at r = 2M.. While we cannot rule out that the proposed phase transition at the black hole horizon is first-order we feel that a continuous phase transition is more likely because it does much less violence to the concept of spacetime

George Chapline/Robert

Laughlin/David

Santiago

195

as we know it. A first-order transition would require a discontinuity in the metric and hence the presence of diverging pressure at the surface of a black hole. Our solution has a surface tension T = — 3ci/32irG2A4 at the infiniteredshift surface, but no divergent pressure. Moreover at finite temperature the local pressure caused by this tension is always negligible compared to any background thermal radiation pressure. While the specific metric we propose identifies the second phase as de Sitter space, it is arguable that the key distinguishing characteristic is not the cosmological constant, per se, but topology. The two phases are distinguished as a practical matter by which is inside the sphere and which outside. This solution has the additional useful feature that the energy inside the black hole sums correctly to M.. This particular solution also has several concrete experimental consequences: • The absolute mass scale M is measured by the heat capacity. With wq defined as in equation (7.15) with va = C7 1 / 2 (r) we have for the total energy per bosonic degree of freedom inside the event horizon (as measured by a distant observer) fGM/c* E

J0

-i/a(r)

L 2TT2

,«

J0

2

q

^

exp(/?K) - 1 aq.

r2dr

"'{fc^raf]}^ ™ Thus the heat content of the black hole is about Mc2/kBT times the volume of empty space of the same radius. This heat capacity is comfortably small for astrophysical objects. If we take M. to be one solar mass and M to be the Planck mass (hc/G)1^2, then the energy in equation (7.45) becomes comparable to Mc2 when T ~ 106 °K. This implies that the temperature of a stellar-mass black hole might well be sufficiently high (> 103 °K) to make it visible against the cosmic microwave background. • The absolute mass scale is also measured by reflectivity. A lightscattering experiment analogous to the sound-reflection experiment shown in figures 7.5 would reveal the horizon to be highly reflectivelike a metal-to light of frequency less than c2 times the black hole mass

196

Emergent relativity and the physics of black hole horizons and to transmit light slightly above this frequency in narrow resonances that depend on the angle of incidence. The reflectance edge is similar to that caused by the convergence of the radial coordinate [18], which is a purely classical effect, but the variation of the resonance energies with transverse momentum measures the masses of free bosonic excitations at the horizon. • High-frequency light impinging on the horizon would also scatter inelastically with a characteristic spectrum terminating at a red shift of 90% for normal incidence, as occurs in figure 7.6. This might be visible [19] if gamma ray photons from a burst echo with appropriate redshift from an apparently empty part of the sky. • If a mass scale M existed at a black hole horizon then it must exist in asymptotically flat spacetime as well and correspond to an absolute velocity scale at which relativity fails and a particle loses its integrity. No such scale has ever been observed. However deviations from the usual relativistic dispersion relation for photons might be observable by studying the time structure of cosmological gamma ray bursts [20] or in high-energy cosmic rays.

Thus the idea that Einstein gravity is emergent in the sense we describe is inherently falsifiable. It predicts that the relativity principle itself must break down at sufficiently high energy scales, and that this breakdown would show up experimentally as spontaneous decay of bosons, such as photons, that otherwise should have integrity.

Acknowledgements: We would like to thank Dr. Domenico Giuliano for critical reading of the manuscript and many helpful suggestions. DIS and RBL were supported by NASA under Collaborative Agreement No. NCC 2-794.

Bibliography [1] L. D. Landau and E. M. Lifshitz, Classical Theory of Fields, 2nd ed. (Addison Wesley, Reading, Mass., 1962). [2] P. G. Bergman, Rev. Mod. Phys. 29, 352 (1957). [3] G. Chapline, E. Hohlfeld, R. B. Laughlin, and D. I. Santiago, Phil. Mag. B 8 1 , 235 (2001). [4] A. D. Sakharov, Sov. Phys. Dokl. 12, 1040 (1968). [5] C. Barcelo, M. Visser, and S. Liberati, Int. J. Mod. Phys. D 1 0 (2001) 799-806 [gr-gc/0106002]. [6] H. R. Glyde, Excitations of Liquid and Solid Helium (New York, Oxford Univ. Press, 1994). [7] M. R. Andrews et al., Phys. Rev. Lett 79, 553 (1997). [8] N. N. Bogoliubov, J. Phys. U.S.S.R. 11, 23 (1947). [9] M. D. Miller, L. H. Nosanow, and L. J. Parish, Phys. Rev. B. 15, 214 (1977). [10] A. Gammal, T. Frederico, L. Tomio, and Ph. Chomaz, Phys. Rev. A 6 1 , 051602(R) (2000). [11] V. L. Ginzburg and L. P. Pitaevskii, Zh. Eksp. Teor. Fiz. 34, 1240 (1958) [Sov. Phys. J E T P 7, 858 (1958)]; L. V. Pitaevskii, ibid. 40, 646 (1961) [ ibid. 13, 451 (1961)]; E. P. Gross, J. Math. Phys. 4, 195 (1963). [12] L. Landau, J. Phys. U.S.S.R 5, 71 (1941); I. M. Khlalatnikov, An Introduction to the Theory of Superfluidity, (Benjamin, New York, 1965), p. 53. 197

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[13] A. S.Eddington, The Mathematical Theory of Relativity (Cambridge U. Press, London, 1965). [14] G. E. Volovik, Phys. Rep. 351, 195-348 (2001). [15] T. A. Jacobson and G. E. Volovik, Phys. Rev. D 58, 064021; ibid., "Effective Spacetime and Hawking Radiation from Moving Domain Wall in Thin Film of 3 He-A", [gr-gc/9811014]; G. E .Volovik, J E T P Lett. 69, 705 (1999). [16] L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 85, 4643 (2000). [17] G. Chapline, Mod. Phys. Lett. A 7, 1959, (1992); G. Chapline in Foundations of Quantum Mechanics, edited by T. D. Black et. al., Singapore: World Scientific (1992). [18] R. H. Price, Phys. Rev. D, 2419 (1972). [19] Ronald J. Adler, private communication. [20] G. Amelino-Camelia, et. al., Nature, 393, p.763 (1998); Sir Martin Rees, private communication.

Chapter 8 Quasi-gravity in branes Brandon Carter Departement d'Astrophysique Relativiste et de Cosmologie Centre National de la Recherche Scientifique Observatoire de Paris, 92195 Meudon France E-mail: [email protected]

Abstract: In contrast with pseudo-gravitational effects that are mathematically analogous to but physically quite distinct from gravity, this chapter deals with a kind of quasi-gravitational effect. This effect arises in an asymmetrically moving brane worldsheet and acts in a manner that approximates (and in a crude analysis might be physically indistinguishable from) the effects that would arise from genuine gravitation. This quasi-gravity can mimic ordinary Newtonian gravity in non-relativistic applications, and scalar-tensor gravity (Jordan-Brans-Dicke gravity rather than pure Einstein gravity) in relativistic applications. 199

200

8.1

Quasi-gravity in branes

Introduction

Most of the talks at the workshop on Analog Models of/for Gravity were concerned with what may be termed pseudo-gravitational effects, meaning effects whose mathematical description is more or less analogous to that of gravity, but whose physical nature is quite distinct. These effects typically involve a second Lorentz-signature metric that coexists with the ordinary spacetime metric (whose deviations from flatness are interpretable as corresponding to true gravity) but which couples to matter in an entirely different way, specifying pseudo light cones that typically govern the propagation not of real light, nor gravity, but of some quite independent excitation such as sound. The purpose of the present contribution is to draw attention to something rather different, what may be described as quasi-gravitational effects, meaning phenomena that affect matter locally in approximately the same physical manner as true gravity, even though their origin and detailed behaviour may be rather different. In the context of non-relativistic Newtonian gravitation theory, the most familiar example of such a quasi-gravitational effect is the centrifugal field attributable to the rotation of the earth that modifies the locally observable Galilean acceleration field. It does so by contributing an extra term that is to be added to the strictly gravitational contribution due to the Newtonian inverse square law attraction due to the terrestrial matter distribution. Although this centrifugal quasi-gravitational contribution is indistinguishable from the truly gravitational contribution in a crude laboratory experiment, the difference is of course detectable, via the Coriolis effect, in more sensitive experiments such as that of the Foucault pendulum. The kind of quasi-gravitational effect I wish to describe here is something that modifies the induced spacetime metric on the (q+1) dimensional worldsheet of a q-brane in a higher dimensional background in a manner that approximates the effect of true (q+1) dimensional gravity, even though its origin and precise nature is essentially different. It is of particular potential interest in the currently fashionable context of models that represent our 4dimensional universe as a 3-brane in a five dimensional background. Though in the kind of scenarios that are most commonly envisaged there, the effect considered here would be excluded by the usual assumption of symmetry between the two opposite sides of the 3-brane. Even if the symmetry assumption were dropped (as has recently been proposed in cases where the q-brane worldsheet is coupled to a background gauge (q+1) form [1, 2, 3]) the

Brandon Carter

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quasi-gravitational effect could still be overwhelmed by much stronger effects of genuinely gravitational origin (just as the terrestrial centrifugal effect is overwhelmed by the centrally directed genuinely gravitational attraction).

8.2

Equation of motion of b r a n e worldsheet

The effect to be considered here is derivable directly by perturbing the general purpose brane worldsheet equation of motion, which is given [4] in terms of the second fundamental tensor K^f of the brane worldsheet and of the corresponding worldsheet stress-energy density tensor T1*" of the brane by T" K„J> =1.^ / " ,

(8.1)

where / M is the external force density, if any, acting on the brane, and IP is the orthogonal projection tensor. The complementary (rank p+1) tangential projection tensor 1$ = ft-K, (8.2) i.e., the first fundamental tensor, defines the tangential covariant differentiation operator VP = 7 ; V „ , (8.3) whose action on the first fundamental tensor defines the second fundamental tensor according to the specification = 1% V„7pff ,

V

(8-4)

which is such as to ensure the Weingarten symmetry condition #„/ = *„/.

(8.5)

as a worldsheet integrability condition, as well as having the more obvious tangentiality and orthogonality properties V 7 /

= 0 = l * , KXv»,

(8.6)

while its trace K" = K»» = Vvlvp ,

(8.7)

inherits the simple worldsheet orthogonality property 7 ^ / ^ = 0.

(8.8)

202

8.3

Quasi-gravity in branes

Perturbed worldsheet configuration

The quasi-gravity effect to be considered occurs (in its simplest form) when the total surface stress-energy tensor T^" is dominated by an isotropic (DiracNambu-Goto type) contribution specified by a large fixed tension T^ say, together with a small additional contribution r'"' arising from the effect of local fields on the brane (representing the observable matter of the universe in brane-world scenarios) in the form V" = -T^-f

+

T^

,

(8.9)

in the presence of an external force of the commonly occurring kind (including a Magnus force on a string and a wind force on a sail) that is automatically orthogonal to the worldsheet, 7 ^ / = 0 so that the orthogonal projection on the right in (8.1) is superfluous. Then if the observable matter contribution T^" were absent, the dynamical equation of motion (8.1) would reduce to the simple form T^KO^T(8-10) Starting from an almost uniform (low curvature) reference configuration of this kind, one can consider an actual configuration that deviates from this due to the presence of a matter distribution rM1/ confined within a lengthscale that is relatively small (compared with the reference curvature scale). For this situation the dominant terms in the dynamical equation, obtained by perturbation of (8.1), can be seen to be given by an expression of the form %. 6K" = !*" K^" .

(8.11)

Here the perturbation 5KP of the curvature is given in terms of the d'Alembertian wave operator • = V V „ of the (q+1) dimensional worldsheet metric, and of the surface orthogonal vector field ^ specifying the displacement of the worldsheet, by an expression of the form 6Kp~nC-

(8.12)

This expression is obtained from the general curvature perturbation formula [4] by retaining only the gradient terms of highest order, which are the ones that dominate in the localised (short length-scale) limit. For a brane worldsheet matter distribution that is approximately specified with respect to the relevant tangent rest frame unit vector u** (uvuv = —1)

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203

by a stress-energy density tensor of the non-relativistic form T"" ~ p u^u",

(8.13)

in terms of a surface mass density p whose space section volume integral determines the corresponding total mass M say, the resulting equation takes the form T^De^P^WK^. (8.14) Here, due to staticity the (hyperbolic) d'Alembertian operator will reduce to a Laplacian operator (of elliptic type), so that for a (q — 2)-dimensional spherically symmetric distribution the solution will be expressible in terms of the radial distance r from the center by an expression that for q > 3 will have the power law form T

-

e

=

-(q-2)^r^a"-

Here the rest frame orthogonal worldsheet acceleration vector aM = is given by a" = u ^ K^o.

(8 15)

-

u"Vvu^ (8.16)

For the familiar, experimentally accessible, case of an ordinary membrane, with q=2, there will be an analogous formula involving radial dependence of logarithmic rather than power law type.

8.4

Quasi-gravitational metric perturbations

Under the conditions described in the preceding section, the brane worldsheet geometry characterised by the fundamental tensor 7M„ will be subject to a corresponding perturbation, h^ — £ 7 ^ that will be given [4] in terms of the second fundamental tensor of the unperturbed reference state by an expression of the form

/V = -^/6-

( 8 - 17 )

This perturbation will have a time component ^00 = u " r V ,

(8.18)

hw = -2a"^p,

(8.19)

given by the formula

Quasi-gravity in branes

204

while the trace Wv will be given by an expression of the analogous form h\ = -2K"

£p .

(8.20)

Evaluating (8.19) explicitly using (8.37), one sees that is reducible to an expression of the standard (dimensionally generalised [5]) Newtonian form 2G h + i] M

K0 = ^ q3- 2 ^ , r

(8-2D

with the relevant generalised Newton constant given by G

^

=

( q - 2 ) n h - ^ ° ^ -

(8 22)

-

Although this mechanism will thus effectively simulate Newtonian type gravitational attraction in so far as its effect on non-relativistic Kepler type orbits is concerned, it leads to a value for the ratio haa/h"v that can be seen from (8.19) and (8.20) to be given by (8.15) as

be- = ^£-.

(8.23)

This will not in general agree with the prediction of Einstein's theory. Although the ensuing prediction for the relativistic behaviour (e.g., of light deflection) will thereby deviate from that of Einstein's purely tensorial theory of gravity, it gives a result that will be shown to be matchable by a more general theory of the Jordan-Brans Dicke type to be described in the next section.

8.5

Jordan-Brans-Dicke type theories

The action integral, I = jc\\g\\1'2S<+1)x,

C = CD + CM

(8.24)

for a Jordan-Brans-Dicke type scalar tensor theory [6] in a (q+1) dimensional space-time, with metric g^v in a Dicke type conformal gauge (meaning one in which the weak equivalence principle is satisfied) is given by a Lagrangian

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density consisting of a Dicke type gravitational contribution £ D involving a dilatonic scalar field $ as well as the metric, and an ordinary matter contribution CM that is independent of $ , with the Dicke contribution given in terms of a coupling constant CJD by an expression of the form £

->

=

2(q-i

1

)nh-ii(

$ f l

-T^^-)-

(8 25)

-

Here R is the Ricci scalar for the metric <7M„ and fifa-1! is the surface area of the unit (q — 1) sphere, which, for an ordinary 4-dimensional spacetime, with space dimension q=3, will be given by Q^ = 4ir. In terms of the trace of the material stress-energy density tensor, T

M" = 2) ^ 5! rL-

£

M^,

(8-26)

the scalar wave equation for such a theory will be given in terms of the d'Alembertian operator D = V„ V" by D^ = aD^-1lTM%.

(8.27)

The dilatonic coupling constant a D is given in terms of the original Dicke constant uiD by — =wD + - ^ - . (8.28) aD q-1 To deal with the gravitational equations, it is convenient to express the dilatonic amplitude $ in terms of some fixed value $ and of a dimensionless scalar field <$> in the form $ = e" 2 * $ (8.29) and to change to what is known as an Einstein gauge by a conformal transformation <7M„ i-»- <7Mi/ that is specified by setting 9v» = &"gv»,

(8-30)

where the field a is given in terms of by by the proportionality relation 20=(q-l)a.

(8.31)

In terms of the Einstein type conformal gauge the action (8.24) will take the form

I = J £ WM1'2 d" +1 >s,

£ = CD + £h

(8.32)

Quasi-gravity in branes

206 with the matter contribution given by £M=e(q+i)CT£M;

(8 .33)

while £ D is given as the sum of an ordinary Einstein-Hilbert type term and a linear scalar field contribution in the form

£

~° = 2 ( q - l W ' ' ( * - ? r ^ ) -

(8 M)

'

Here R is the Ricci scalar for the Einstein metric <7M„ and a D is the constant given by (8.28), while the fixed amplitude 3> now acts as the inverse of the (dimensionally generalised) Newton constant, which can be identified as G

[q+1] = ^ •

(8>35)

In this reformulation there will be a matter stress-energy density contribution given by T% = ^+l)°T»v (8.36) whose trace will act as the source for the linear wave equation for , which will be expressible in the form

• ^=_IQ[q-i]G[q+1]aDfMV

(8.37)

where • is the d'Alembertian operator for the Einstein metric 'g^. The corresponding Einstein type gravitational equations will be expressible as %» - \R%V = — ( 2 0 M 0 „ - 9^9^4>P(t>,) + (q - 1) ^ l 9 " 1 ] G[g+1]fuli„ . (8.38)

8.6

Linearised local scalar tensor field configurations

Let us now consider the weak field, low source density, limit in which the system can be linearised with respect to the dilatonic perturbation field 4> and

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207

the Einstein metric perturbation field h^ denned relative to a flat Minkowski background metric -n^ by setting %v = Vvu, + %v •

(8.39)

The equation (8.37) for is already linear as it stands, while the corresponding linearised Einstein equation for h^ is obtainable from (8.38) in the standard form • V = -2fttq~1] G[q+l] ((q - l)fM/u, - fM% 7 ^ ) .

(8.40)

The gravitational field that is directly measured by the observation of Kepler type orbits will not be given by this Einstein type metric Tj^ (to which, due to the involvement of in the relevant stress-energy tensor T£", the usual equivalence principle does not apply) but by the original Dicke type metric, g^, which will be expressible analogously to (8.39) by 9fiu = Vitv + hp,,

(8.41)

with / V = ' V + 2<7 VIIV , (8.42) to linear order, by (8.30). Using the relation (8.31), it can be seen, by combining the wave equations (8.37) and (8.40), that the directly observable metric perturbation /iMt/ will be given, to linear order, by • V = - 2 ^ [ q " 1 ] G[q+1] ((q - 1) rM/M, + (1 - AD) TM% n^) .

(8.43)

in which the dilatonic deviation constant is given by A n - -, 1 = -^- . D («-lK+q q-l

(8.44)

Due to the presence of the deviation constant A D , the coefficient G[q+1] will not be quite the same as the effective Newtonian coupling constant G[q+1] that will be observed in the static non-relativistic limit for which, in terms of the relevant rest frame unit vector uM (with u^u^ = — 1) the stress-energy density will be approximately of the form

rr = PuV,

(8.45)

Quasi-gravity in branes

208

in which p is the mass density, whose space volume integral will be identifiable in this limit as the total mass M. It can be seen that for a spherically symmetric distribution the time component, hm = v?uv V ,

(8-46)

of the metric perturbation will be given in terms of the radial distance r from the center by an expression of the standard (dimensionally generalised [5]) Newtonian form

/>oo = ^ g - >

(8-47)

but with the effective gravitational coupling constant given by G [q+l] = G [ q + 1 ) (l + ^ ) .

(8.48)

It can be seen that it will be related to the corresponding expression for the trace bP of the metric perturbation by 2-(q+l)AD

*',= q:K-V.2 + T°K A D 0,

(8-49)

(q - 2 R + q - 1 -'»,• 2wD + l

M

which is equivalent to

" . . = " - : ° : ,

8.7

Conclusion

It can be seen that the ratio (8.50) can be matched by the simulation effect leading to the corresponding ratio (8.23) if the relevant reference frame curvature vector K^ and the corresponding acceleration vector oP are related by ^ ( q - ^ R + q - l ^ (851) or equivalently

Brandon Carter

209

This means that, in the linear approximation we have been using, the quasi-gravitational effect arising from the extrinsic curvature of the brane simulates what would be predicted by a Jordan-Brans-Dicke theory with wD given by what is obtained by solving (8.51), namely

WD

_ (q-l)a»Kp-a»gp ~2a"a„-(q-2Ktf„'

(8 53)

'

which corresponds to a dilatonic deviation A D given by

A

2o'a, + ( 2 - g ) a ' i r , ° - (q+lKa. + a ^ '

(8 54)

'

It is to be emphasised that the approximation presented here has been derived only for static configurations in a linearised weak field limit, and cannot be expected to remain accurate when stronger fields or significant deviations from staticity are involved.

Acknowledgments: The author wishes to express thanks to Mario Novello for initiating the organization of this workshop, to Matt Visser for ensuring the completion of the edited proceedings, and to Gilles Esposito-Farese for technical discussions about the simulated worldsheet gravity mechanism in the context of scalar-tensor theories.

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Bibliography [1] A. Kehagias and E. Kiritsis, "Mirage cosmology", J.H.E.P. (1999) 022, [hep-th/9910174].

9911

[2] R.A. Battye and B. Carter, "General junction conditions and minimal coupling in brane world scenarios", Phys. Lett. B (to appear), [hepth/0101061]. [3] B. Carter and J-R Uzan, "Reflection symmetry breaking scenarios with minimal gauge form coupling in brane world cosmology", Nuc. Phys. B (to appear), [gr-qc/0101010]. [4] B. Carter, "Dynamics of cosmic strings and other brane models", in Formation and interactions of topological defects, NATO A.S.I. B349, Newton Inst., Cambridge 1994, ed. R. Brandenberger, A.-C. Davis, pp 303-348 (Plenum, New York, 1995). [hep-th/9611054] [5] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, "Phenomenology, Astrophysics and Cosmology of Theories with Sub-Millimeter Dimensions and TeV Scale Quantum Gravity", Phys. Rev. D59, 086004 (1999). [hep-ph/9807344] [6] R.H. Dicke, "Experimental Relativity", in Relativity, Groups, and Topology (Les Houches 1963), ed B. & C. De Witt, pp 165-313 (Gordon & Breach, New York, 1964).

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Chapter 9 Towards a collective treatment of quantum gravitational interactions Renaud Parentani Laboratoire de Mathematiques et Physique Theorique, CNRS UMR 6083, Universite de Tours, 37200 Tours, France E-mail: [email protected]

Abstract: When considering the backward propagation of Hawking phonons in an acoustic geometry, the blue-shifting effect associated with the horizon stops when the inter-atomic distance is reached. In contrast, in a fixed gravitational black hole geometry, Hawking photons emerge from trans-Planckian configurations because there is no scale to stop the blue-shifting. However, when taking into account the gravitational interactions neglected in the semiclassical treatment, a new UV scale is dynamically engendered and acts as the inter-atomic distance. To describe these non-linear effects, we use the largeN limit, where N is the number of matter fields. This treatment provides the next order contribution after the semi-classical one, and takes into account the vacuum fluctuations of the energy-momentum tensor. In this scheme, the gravitational interactions express themselves in terms of a stochastic ensem213

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ble of metric fluctuations, and propagation of light in this ensemble resembles that of sound in a random medium.

9.1

Outline

In this chapter, we further exploit the analogies between black hole physics in general relativity and certain situations of condensed matter physics. However, the way we shall extend these analogies runs somehow against the main flow: Instead of applying the techniques of GR to condensed matter models exhibiting an effective metric, we shall apply to GR techniques which have been found useful in condensed matter physics. Our aim is to compute the quantum gravitational corrections to Hawking radiation [1]. To this end, we have chosen a statistical treatment of these radiative corrections based on a large-TV limit, where N is the number of copies of the matter field. The reason for this choice is that we do not have a theory of quantum gravity. Therefore in order to go beyond the semiclassical treatment one should adopt an effective approach which allows one to compute (approximatively) the radiative corrections to physical quantities. We recall that the semi-classical treatment is a mean field approximation: The radiation field propagates in a classical geometry which is a solution of Einstein equations driven by the mean value of its own energy-momentum tensor, the one-point function (TM„). To go beyond this treatment means that the gravitational effects driven by the quantum properties of TM„ must be taken into account. In this paper, this will be done approximatively through the use of the connected two-point function (T^fe) Tap(x'))c- In a large-./V limit, this treatment emerges as the next to leading order contribution [2]. Remember that the semi-classical approximation is the leading order contribution [3]. We shall then show that light propagation in this treatment possesses analogies to wave propagation in a random medium [4]. Before proceeding, it is perhaps necessary to explain why such a treatment might give some important effects when applied to Hawking radiation (or to quantum effects induced by the presence of an event horizon). When studying the origin of Hawking quanta one faces a difficulty which is specific to horizon physics: The configurations giving rise to Hawking quanta are

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characterized by ultra-high frequencies when measured by infalling observers near the horizon [5, 6, 7, 8]. Indeed, in the semi-classical treatment, the use of free fields propagating in a given background leads to unbounded frequencies as a direct consequence of the structure of the outgoing null geodesies near the horizon. Therefore, any ultra-violet scale which would signal the breakdown of the semi-classical treatment will be inevitably reached. This is nicely illustrated by considering sound propagation in an acoustic geometry which possesses a horizon: The propagation is dramatically modified with respect to the standard propagation (governed by the d'Alembertian) when the UV scale (the inter-atomic distance) is reached. However, these near horizon modifications leave no imprint on the asymptotic properties of Hawking radiation when the two relevant scales, the surface gravity K and the inverse inter-atomic distance, are well separated [9, 10]. When taking into account the gravitational effects induced by the fluctuations of TM„, one finds that the trans-Planckian correlations which existed in the semi-classical treatment are washed out. Moreover this washing out mechanism leaves the asymptotic properties of Hawking radiation unaffected: the thermal flux receives corrections which scale like {KO)2
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(V) These interactions give rise to a the new length scale which characterizes the spread of the horizon at which the 'focusing' of backward propagated configurations effectively stops. It determines the maximum resolution given the fuzz induced by vacuum fluctuations. This scale is constant in time. This follows from the stationarity of the vacuum (the Green function is a function of the difference t — t' only) and that of the background metric.

9.2

Introduction

In his original derivation [1], Hawking considered the propagation of the radiation in a given background metric, that of a collapsing star. This means that the metric is once and for all determined by the energy of the collapsing star and is therefore unaffected by the quantum processes under examination. In this approximation, the radiation field satisfies a linear equation (in the absence of matter interactions). One then finds that the in-falling and outgoing field configurations are completely uncorrelated near the black hole horizon. In fact the pairs of quanta generated by its formation are composed of two outgoing quanta, one of each side of it. The external ones form the asymptotic flux whereas their partners fall towards the singularity at r = 0, (see [8] for a detailed description of the conversion of vacuum fluctuations into pairs of quanta). Upon tracing over the inner configurations one gets an outgoing incoherent flux described by a thermal density matrix. Nevertheless, there is a precise relationship between the mean values of the in-falling and the outgoing energy fluxes. Indeed, the asymptotic outgoing null 1 flux (T u u (r = oo)) is accompanied by a negative in-falling flux (Tvv) which has, on the horizon r = 2M, exactly the opposite value when one works, as we shall do, in the vacuum, i.e., (Tvv(r = oo)) = 0. This follows from the conservation of the radial flux (= (Tuu) — (Tvv)) in the static metric outside the collapsing body as well as from the fact that (Tuu) vanishes like (r - 2M) 2 when approaching the future horizon [12, 13]. This last property becomes crucial when considering the gravitational reaction to the mean fluxes, i.e., the metric change determined by Einstein's equations driven by (T^). Let us first describe what happens at spatial infinity when {Tvv(r = oo)) = 0. Energy conservation and adiabaticity, 'The null coordinates v and u are given by v = t + r*, and u = t — r*. Here r* = r + 2Mln(r/2M - 1) is the tortoise coordinate [11].

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i.e., AM I At < l, 2 imply that the mass loss will be determined by {Tuu(r = oo)), thereby describing a geometry characterized by a slowly decreasing (Bondi) mass. However, to validate the hypothesis of adiabaticity, a local analysis of the evaporating black hole geometry is required [14, 15, 16]. In this analysis, the vanishing of (Tuu) at r = 2M is crucial. Indeed it is a necessary condition for keeping the regularity of the near horizon geometry during the evaporation process. It also implies that the black hole horizon shrinks because the hole absorbs a negative flux through its future horizon: Einstein's equations tell us that it is the negative (Tvv) which drives in situ the shrinking of the horizon area according to AM _

1

- j ^ - - \Tvv)\r=Thorizon=2M

^~Jp-

\9A)

In this equation M(v) determines the time dependent radius of the apparent horizon. It governs the Vaidya metric As2 = - ( l - ^y^-)

dv2 + 2Av Ar + r2 (A92 + sin 2 6Act>2) .

(9.2)

In brief, during the whole evaporation process (9.2) offers a good approximation of the near horizon region (i.e., \r — 2M\
(9-3)

1 - 2M/r This implies that a wave packet centered along the null outgoing geodesic u had a frequency ui oc XeKU when it emerged from the collapsing star, (K — 2

We work with in Planck units: c = h = Mpianck = 1.

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Towards a collective treatment of quantum gravitational

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l/AM is the surface gravity and fixes Hawking temperature T# = K/2T[.) Unlike processes characterized by a typical energy scale, the relation w ex XeKU shows that black hole evaporation rests, in this scenario, on arbitrary high frequencies. This analysis of wave packets is confirmed by the study of the non-diagonal matrix elements of TM„ which encode the fluctuations of the flux around its mean value. As shown in [8], contrary to the expectation value (the diagonal part) which is regular and of the order of M - 4 , these matrix elements are generically singular on the horizon, i.e., their Fourier content is characterized by frequencies u which grow according to (9.3). As emphasized by 't Hooft [7], this implies that the gravitational interactions between the configurations giving rise to Hawking quanta and in-falling quanta cannot be neglected, thereby questioning the semi-classical description. 3 In questioning its validity, two issues should be distinguished, see section 3.7 in [13]. First, there is the question of the low frequency O(K) changes which can be measured asymptotically, and secondly, that of the high frequency modifications of the near horizon physics. Since all thermodynamical reasonings indicate that the asymptotic properties (namely thermality governed by K and stationarity) should be preserved, the problem is to conciliate their stability with the radical change of the near horizon physics which is needed to tame the trans-Planckian problem. This is not an easy problem: Indeed, a perturbative analysis [17] of near horizon interactions leads to back-reaction effects which also grow like u) in (9.3). This threatens the stationarity of the flux and questions the choice of the treatment which is adequate to compute the gravitational effects neglected in the semi-classical approximation. A new point of view to this problem has been provided by the analogy with condensed matter physics pointed out by Unruh [5] (and revisited by Jacobson [6, 18]). He noticed that sound propagation in a moving fluid obeys a d'Alembertian equation which defines an acoustic metric. Therefore, when the acoustic metric corresponds to that of a collapsing star, thermally distributed phonons should be emitted. However, contrary to photons the dispersion relation of phonons is not linear for frequencies (measured in the rest frame of the fluid) higher that a critical u)c. Since the frequencies to > wc which were solicited in Hawking's derivation are no longer avail3

't Hooft criticizes in particular the above mentioned absence of correlations: "Any decomposition of Hilbert space in terms of mutually non-interacting field quanta will be hopelessly inadequate in this (near horizon) region", see section 9 in the review article [7].

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able, the stationarity of the flux is directly threatened. However this does not occur: when uic ~> K the asymptotic properties of Hawking phonons are unaffected [9, 10, 19] in spite of the fact that the near horizon propagation of the phonon field drastically differs from that of photons when the blue shifted frequency reaches wc. The appealing feature of these models is to provide both, a simple explanation (in terms of adiabaticity which essentially follows from scale separation UJC » K) for the robustness of the asymptotic properties of the flux, and a simple physical reason (a modified dispersion relation) which eradicates the ultra-high frequencies. (It should be pointed out that a similar trans-Planckian problem arises in inflationary models when studying the origin of the spectrum of primordial energy density fluctuations [20, 21, 22]. In that case as well, scale separation and regularity of the metric are sufficient conditions to guarantee that the properties of the spectrum are unmodified [23].) Besides the robustness of the IR properties, the main outcome of these considerations is that a new universality has emerged: for all dispersion relations but the linear one, the blue shifting effect stops. Therefore, the never ending blue shifting effect obtained by using the linear (scale-less and non-dispersive) relation Q, = p now appears as an isolated and unstable behaviour. Thus, instead of asking if Hawking radiation is robust against modifying the dispersion relation, we are let to question whether Q = p is robust or is instead an artifact of free field theory. These considerations suggest that a new UV scale might be engendered by radiative corrections evaluated in a black hole geometry. It would then break the scale invariance of free field propagation and prevent the appearance of trans-Planckian frequencies [8, 13, 19]. To verify this conjecture, one must determine the physical effects induced by the non-linearities engendered by gravitational interactions. When this is done, one can indulge in the luxury of making contact with dumb holes physics by analyzing if/why the effects of non-linearities can be reproduced by an effective linear equation (i.e., a non-trivial dispersion relation) governing outgoing propagation. The central problem is thus to compute the consequences of the nonlinearities introduced by gravitational interactions. This clearly requires us to go beyond the semi-classical treatment. Thus one should adopt a scheme which allows to compute these consequences. As a first step towards a full quantum gravitational treatment, we proposed [2] (inspired by [7, 8, 13, 24, 25, 26, 27, 28]) to analyze the gravitational effects driven by the connected two-point function (T^(x) Tap(x'))c- As already mentioned,

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Towards a collective treatment of quantum gravitational

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this treatment emerges in a large N limit. In this limit, in-falling vacuum configurations act as an environment for the outgoing quanta and their mutual gravitational interactions express themselves in terms of a stochastic ensemble of metric fluctuations. The specification of vacuum at early times determines the statistical properties of this ensemble and this, combined with the non-trivial properties of the black hole metric, fixes the new scale a (in terms of K) and provides a frame which breaks the 2D (local [6]) Lorentz invariance. 4 Then, the main effect of these interactions is to dissipate the trans-Planckian modes near the horizon but without affecting the asymptotic properties of Hawking radiation. The unsolved question concerns the range of validity of the treatment based on the two-point function of T p „. This is a complicated question whose final answer requires a better understanding of quantum gravity. Let us make a few remarks. First, this question closely follows that concerning the validity of the semi-classical treatment which is equally complicated. 5 Secondly, our analysis indicates that the semi-classical treatment fails before our treatment. 'Before' should be understood radially, given the blue shift effect encountered during the backward propagation of configurations specified on J+, see (9.3). What emerges is a kind of Russian doll structure in which gravity progressively dominates the physics. Far away from the hole (r — 2M » 2M) one has outgoing thermal (on shell) radiation. In a first intermediate regime (a = 0, see (9.7), and the Green function is a sum of functions of u — u' and v — v' separately. When including radiative corrections, the dressed propagator no longer possesses this property. The reasons for this breakdown of Lorentz invariance are similar to those which give rise to the fact that the self-energy of an electron immersed in a thermal bath of photons is not Lorentz invariant: In both cases, in the low energy regime the integrands governing loop corrections are not Lorentz invariant because of the non-trivial metric in the black hole case and the thermal bath in the other. 5 The validity of the semi-classical treatment has been often questioned in rather general terms. However, a significant answer requires to find physical quantities (i.e., matrix elements of operators) which are erroneously evaluated in this treatment and to propose improved expressions for the same quantities in order to see the discrepancy. What is known [3] is that the semi-classical treatment is the leading contribution in the large N limit with GN held fixed. The important point for us is that the next order contribution, that is, the set of graphs weighted by powers of G2N, corresponds to our treatment. For more details, see the Appendix.

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differently. It is in this regime that Hawking radiation gets established, see equations (76-83) in [8]. This description based on modes stops to be valid when reaching Jacobson's time-like boundary [18], at r - 2M ~ a, when outgoing modes get progressively entangled to the infalling configurations, thereby loosing their 'mode' quality. The principle aim of this paper is to analyze this transitory regime. Deeper in r — 2M, one has some unknown regime governed by Planckian physics. This physics presumably also occurs around us but stays well hidden inside its Planckian husk in the absence of a good microscope.

9.3

The model

For simplicity, we shall consider only s-waves propagating in spherically symmetric space times. As in [28], we choose the background metric to be that resulting from the collapse of a null shell of mass M0 which propagates along v = 0. Inside the shell, for v < 0, the geometry is Minkowski and described by (9.2) with M = 0. Outside, the metric is also static and given by (9.2) with M = M0. As we shall see, this choice of the collapsing metric will have no influence in what follows since we shall focus on the vacuum interactions occurring near the horizon, outside the collapsing matter. To identify the degrees of freedom involved in these interactions, we first analyze the global properties of the modes. The ingoing massless waves fall into two classes according to their support on J~, the light-like past infinity. The waves in the first class have support only for v < 0, inside the shell, and will be noted ^>_. They propagate inward in the flat geometry till r = 0 where they bounce off and become outgoing configurations. The relationship between the value of u of the geodesic which originates from v on J~ is [28]: V(u) = —4M(1 + e~ku). The first class is thus divided in two subsectors: For v < —4M, the reflected waves cross the in-falling shell with r > 2M and reach the asymptotic region 6 whereas those for 0 > v > —4M cross the shell with r < 2M and propagate in the trapped region till the singularity. The separating light ray i># = —4M becomes the future horizon u = oo after bouncing off at r = 0. The configurations which form the second class live outside the shell, have support only for v > 0 and are noted +. They 6 dudv(f> = 0 is valid for all r only when working in the geometric optics approximation. In the exact d'Alembertian, see (9.7), there is a potential centered around r = 3M which induces partial reflection, a phenomenon irrelevant for our purposes.

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Towards a collective treatment of quantum gravitational

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propagate in the static Schwarzschild geometry, are always in-falling and cross the horizon towards the singularity. In Hawking's derivation of black hole radiation, owing to the linearity of the field equation, these classical (on-shell) properties also apply: The configurations for v < vH give rise to the asymptotic quanta, those for VH < v < 0 to their partners [8] whereas (f>+ plays no role in the asymptotic radiation. For further details concerning the properties of Hawking radiation, we refer to the review [13]. One also finds that the correlations between the asymptotic quanta and their partners follow from the fact that, on J~ and in the vacuum, the rescaled field cj> = \fAiTr2x (where \ is the 4D s-wave) satisfies (cf>(v) 4>(v')) = f Jo

^ - e-^"-"'' = ~ 47T6J

ln(v - v' - it) + constant.

(9.4)

4-7T

Since this equation is valid for all v, v' one might think that there also exist correlations between 4>- and <j>+. However, for late Hawking quanta, they vanish since these quanta and their partners emerge from configurations which are localized extremely close to VH- This follows from the asymptotic (KU ;» 1) behaviour of the relation V(u) V{u) -vH<x

e~KU .

(9.5)

As shown in [1], this exponential is responsible for the thermal radiation at temperature K/2TT. It also shows that the quanta emerge from transPlanckian frequencies on J~ since udV = Adu. In brief, in the absence of gravitational interactions, _ and cj>+ are effectively two independent fields. By independent we mean that by sending quanta described by wave packets built with + only, there is no induced emission of Hawking quanta. Indeed, in order to get induced emission [29] at time u, one should send 0_ quanta localized close to vH as indicated in (9.5) and correspondingly characterized by frequencies u oc XeKU J> K. Let us now analyze more closely how these properties translate in the Fock space of 4> propagating in the background (9.2). In this metric, the action of (j> is

Sa = - [dv dr dv dr +\(l-

^f) (drT

(9.6)

with M(v) = 0 for v < 0 and M(v) — M0 for v > 0. Being interested in the near horizon physics, we have dropped the potential term of s-waves,

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(2Mo/r3)(j)2, since it does not affect the near horizon propagation. This can be seen by using the double null coordinate system u = v — 2r*,v. Using them, the 4D d'Alembertian reads

0

KO-^H^^)]*-

(97)

where <j>i is the rescaled mode of angular momentum I. Thus, as emphasized in [24], the propagation of waves (at fixed angular momentum and even for an arbitrary mass) effectively obeys a 2D conformal invariance in the near horizon geometry. 7 This is confirmed by the fact that, classically, the trace of the 2D part of T^ vanishes off-shell. Thus, in our model for s-waves, TM1/ has only two (classical) components, T„v = (dv)2 and Tuu — (du(j>)2. When considering 0 in second quantization, the 2D conformal invariance implies that the Fock space is composed of tensorial products of in-falling states (on which <j>+ acts) and outgoing states. In a Schrodinger language this means that an initially factorized state (i.e., | * ) = | * + ) ® | * _ ) ) remains factorized when its evolution is governed by (9.6). This absence of entanglement among the two sectors explains the above mentioned absence of induced emission when adding a few (j>+ quanta. In a Heisenberg language, it tells us that any matrix element of 0 is a combination of matrix elements of - and + evaluated separately. This implies in particular that the connected part of the two-point correlation ))c — {Tw{

)> - (Tvv(x))

(Tuu(x'))

(9.8)

identically vanishes for all factorized states. This applies to the "Unruh" vacuum, the state describing Hawking radiation. Physically, the vanishing of (9.8) means that the fluctuations of (Tvv) and (Tuu) around their mean values are uncorrelated. This is just another way to say that there cannot be induced emission of 0_ by (j>+. It should also be clear that this absence of quantum correlations is precisely what is contested by 't Hooft, (see footnote 3). Finally, in spite of this absence of correlation, we mention that the mean value of Tvv and Tuu are related to each other by energy conservation through 7

This invariance leads to the trans-Planckian problem: the steady production rate of outgoing quanta arises from an integral over in-frequencies w whose measure is that of a 2D massless field. Explicitly one obtains that the thermal distribution is multiplied by dw/w = /cdu since u> oc eK" see (9.3). For more details see [17] or equation (2.54) in [13].

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Towards a collective treatment of quantum gravitational

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the 2D trace anomaly [12]. However, this new component of TM„ plays no role in the gravitational interactions we shall analyze because it is a c-number: it does not fluctuate.

9.4

T h e gravitational interactions between _ a n d (/)+

The aim of this section is to obtain the action governing the gravitational interactions between _ and <j>+. In the following sections, we shall study the consequences of these interactions with particular emphasis on the correlations they induce. The generating functional governing the matter-gravity system is Z = \V Vh

i 5 e

[ »+*+s*.»].

(9.9)

In this equation, h is the change of the metric with respect to the background g discussed above and S/,i9 is the Einstein-Hilbert action. Sg+h is the action of <j) propagating in the fluctuating geometry g + h. When the metric fluctuations are spherically symmetric, h can be characterized by two functions ip, \i. Moreover they are completely determined by the energy-momentum tensor of <j>. This is like the longitudinal part of the electro-magnetic field which is constrained to follow the charge density fluctuations, by Gauss' law. The line element in the fluctuating metric can be written as [28] 8

1

C -?)

ds2 = e*

2 2 dvz + 2di; dr + r dQ 2

(9.10)

where M = M0 + fx(v, r) for v > 0. In this new metric, the matter action is 8

This line element differs from that used by Baxdeen [14]: e*(

1

2M 0» + 2/i "\ j I'llB 1 dv2 2

+

2 d v

d r

1

+r2<ml

The ip function is the same whereas, to first order in ip and fis, H = HB - ip{r - 2Mo)/2. The usefulness of our choice is that 4> no longer affects the null geodesies. We also recall that Einstein's equations read dvfiB = Tvv — Tuu and dr-ip = 4Tuu/(r - 2M), when expressing 1 m, in terms of the null fluxes TVV,TUU.

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the same as in equation (9.6):

Sg+h = -

dv dr dvdT + I 1

2M\ {dT<j)f J 2

(9.11)

The new mass function M incorporates the relevant metric change /z. Indeed, Sg+h is independent of t/>, thereby recovering the 2D conformal invariance mentioned earlier. Our aim is to work out the first order corrections due to the gravitational interactions between <j>_ and +. To this end only quadratic terms in h should be kept in Sh,g- The Gaussian integration over h can be performed (this is equivalent to solve the linearized Einstein's equations). It gives rise to a self-interacting field theory described by V<j>elSs+lbini.

(9.12)

/ ' By construction Sint is a non-local quadratic form9 of the energy-momentum tensor of . To identify the relevant part of Sj n ( we first recall that T^u has only two fluctuating components, thanks to the 2D conformal invariance. Thus, in a perturbative treatment (such as the interacting picture) one obtains two types of interaction terms only. First one has self-interaction terms depending on (j>- or + separately. These terms do not destroy the factorizability of the theory and will not be considered in what follows.10 Secondly, one has a cross term coupling _ to +. The essential point is that this term will inevitably break the factorizability of the theory into the ± sectors. Let us analyze it. 9 In the t,r coordinate system, i.e., when gTt = 0, Si„t is given by a linearized version (see equation (90) in [8]) of the BCMN [30] Hamiltonian. 10 On-shell, the (j>+ - 4>- contribution to Sint is more tricky to handle in the advanced coordinates v, r. The reason is that infalling geodesies are affected by the presence of an outgoing flux T u u (as clearly seen when using the coordinates u,r). This modification translates in v,r into a deformation of the description of outgoing geodesies u = u(v,r) and it is this effect which is responsible for the <j>-4>~ contribution to Sint- Let us finally notice that a non-perturbative treatment of the self-interactions of <j>- has been developed in [31]. It leads to small effects 0(K/M) and induces no damping of the waves when approaching the horizon.

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Towards a collective treatment of quantum gravitational interactions

Since infalling configurations obey dr+ = 0 even in the presence of gravitational interactions, the cross term coupling 4>- to <j)+ is, see (9.11), Sint = G [°°dr [°°dv ^ W _ ) 2 (9-13) r Jo Jo where G is Newton's constant. We have introduced it in the front of fi to read more easily in the next equations the order of the interactions between _ and +. n+(v) is the mass fluctuation driven by +. It is given by H+(v) = fdv' Tvv(v') = r dv' {dv,+)2 . (9.14) Jo Jo The reader might be surprised by the fact that we are using on-shell fields (j>± in Sint. In principle indeed, only the off-shell field should be used in the action. However, when calculating perturbatively lowest order corrections in G, this amounts to use (9.13) as it stands. We are now in position to show that the gravitational interactions between <j)+ and <j)_ diverge on the horizon. To this end, let us consider two classical fluxes described respectively by Tvv = £l8(v — v0) and Tuu = \5{u — u0). fi and A are the asymptotic energies measured on J~ and J+ respectively, and ?;o and uo are such that the two spherical shells meet at r0 in the near horizon geometry, for r0 - 2M < 2M. In this case, using r 0 - 2M ~ 2Me*("°~Uo) one obtains

s

-^iGi^hj-

< 9 - 15 >

The action Si„t diverges as r0 —> 2M like u did in equation (9.3). The difference with (9.3) is that Sint is a scalar. Hence the divergence in (9.15) is coordinate (gauge) invariant. Since the gravitational interactions become strong (i.e., can no longer be ignored) when Sint cz 1, (9.15) tells us that the free description breaks down for r0/2M — 1 = GK2 (= 1/M in Planck units) when considering two shells of energy Q = A = K. The same distance will appear in section 6 when considering radiative corrections: in this case a ~ 1/M will indicate where the semi-classical propagation breaks down when working in the vacuum. We should emphasize this last point: even though our approach closely follows that of [7] (it can be considered as an s-wave reduction of it) we shall not study the interactions between (j>+ and _ quanta. Rather we shall focus on the residual interactions when the state of (f)+ is vacuum. For earlier attempts in this direction, we refer to [24, 25]. Before analyzing these second

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quantized effects, it is instructive to solve two preparatory exercises. In the first we shall show that Sint acts only as a shift operator of the asymptotic value of u. In spite of this, in the second exercise, we show that Sint nevertheless engenders an entanglement which prevents the factorizability of the states into ± sectors. This provides an explicit example of a quantum effect induced by (9.13) which is absent in the semi-classical description.

9.5

Non-vacuum gravitational effects

Let us consider the following classical problem: Given (j>°+ on J~ and <jP_ on J+, what is the value of the field amplitude <j> near the horizon? Because of the 2D conformal invariance, the scattered amplitude (f> still decomposes as + + — Then, since dT<j>+ = 0 is exact in our gauge wherein v stays light-like in the presence of gravitational interaction, <j>+{v) = \(v). Thus the infalling flux of energy is unaffected by the energy carried by and it is given by Tvv — (dv4>\)2. Hence, /z+ of (9.14) acts as a given metric change in the equation of motion of -\

an-(i-»* + » » » M ) »

<£_ = 0.

(9.16)

Since this equation is linear in 0_ and first order in the space-time derivatives, its exact solution is M « , r ) =/_(«„(«, r)), (9-17) where uM(?;,r) is the outgoing null geodesic in the modified metric characterized by M0 + n+(v). The modified geodesic u^(w,r) also obeys (9.16) with the boundary condition that it converges to the un-modified value u0(v, r) = v — 2r* for r —> oo. To first order in G, the change Su = u^ — u0 is determined by a non-homogeneous equation n whose solution is Su(v)\uo

= G Hdv' ^ (9.18) Jv r(v')\uo - 2M0 where r(i;)|„0 is obtained by inverting UQ(V, r) = v—2r*. The important point is that, once more, the integral in (9.18) is dominated by the near horizon region where r(v)\uo - 2M0 < 2M0. "Using the fact that 2dv + (1 - 2M0/r)dr defines 2d„|U0 (by definition of u0(v,r) — constant), 5u obeys dv\U0Su = (/i + /r)9 r |„uo thereby giving (9.18).

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The lesson we got from (9.17) is that the eikonal approximation is exact: the scattered value of the field amplitude is given by its asymptotic value evaluated along the modified characteristic Uf,,(v,r). Thus, classically, the gravitational interactions encoded in (9.13) only induce a shift of the argument of field and do not induce non-linearities in the field amplitude. In spite of this, the quantum evolution governed by the total action Sg + Sint dynamically engenders entanglement between the otherwise uncorrelated ± sectors. To see this, let us consider the evolution of an initially factorized wave function | * i n ) = |*™) ® |*™). The infalling part |*™) is specified on J~ for v > 0 whereas the outgoing piece |v]>™) is specified on v = 0 all u, just outside the infalling matter engendering the black hole. Let us also choose |\P+) to be a superposition of two well defined and separated wave packets |*l+") = A \¥?'a) + B |*V"' 6 ).

(9.19)

By well defined and separated we mean that the two fluxes associated with each component, {T^v) = (\I/^M|T„W|,I'+'1) i = a,b, are localized and separated from each other. This implies that the overlap between |\J/™'a) and \V™'b) vanishes. As in classical terms, the dr(j)+ = 0 now viewed as a Heisenberg equation tells us that the evolution in the + sector is trivial. Therefore, in the interacting picture, when applying the evolution operator elSint to the total wave function | ^ ) one obtains two uncorrelated evolutions: one governing the — sector | ^ _ ) in the a-modified metric characterized by the c-number mass M — M 0 + ^+ with na+ = f dv{T£v), and the other one in the 6-modified metric. Thus the final value of the wave function (on the union of the event horizon and J+) is |*)

=

e iS '-'|*
=

A |#' + "' a ) ® eiS?m |*«») + B \yfb)

® eiS™< |**»>.

(9.20)

S\nt are the two interaction Hamiltonians acting on \$>™). They are given by (9.13) with the corresponding the metric changes fi\., i = a,b. The entanglement induced by Sint acts, as usual, as a measurement: Consider for instance the Stern-Gerlach experiment wherein the center of mass motion is determined by the spin projection of the electron which is moving in a magnetic field. The mapping from that situation to the present one is as follows. The two kets representing the spin projections are here played by

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the two infalling states 1$™'*). The center of mass wave function is played by the outgoing wave function | ^ _ ) and the interaction Hamiltonian is Sint of (9.13). The analogy works quite well when the initial outgoing wave function I*!!1) is well peaked. Then, its "image" on J+ would be either a spot at u0 + Sua with probability \A\2, or one at u0 + 5ub with probability 1 — \A\2. UQ is the location of the spot when the gravitational interactions are ignored and the values of the shifts 8ua, Sub are given by (9.18) fed by the mass changes Ha+ and nb+ respectively. This quantum result should be compared with what would have been obtained by using the semi-classical treatment, i.e., by using the mean mass change /2+ = |^4|2/i+ + |-B|2//+ instead of each mass change separately. The semi-classical treatment incorrectly predicts a single spot on J+ located at the "mean" position u0 + 5u with Su = l ^ l 2 ^ - ! - \B\28ub- The validity of the semi-classical treatment rests on the possibility in neglecting the fluctuations in the infalling flux Tvv. This is mathematically controlled by the relative importance of the connected part of two-point function {TVVTVV)C with respect to the mean square to (Tvv)2. Thus when this ratio is not negligible (this is the case when the two fluxes considered above are well-defined and when \A\ ~ \B\), the two evolutions, the exact one and the semi-classical differ, i.e., the semi-classical treatment leads always to incorrect predictions. Instead when \A\ » \B\, most of the time it is correct, since the mean 8u would be very close to 8ua. Before considering second quantized effects, it is also interesting to relate (9.12) to the former treatments of black hole evaporation discussed in the literature: Hawking's approach [1] and the semi-classical treatment. Hawking's approach is recovered by putting G/J.+ = 0 in (9.12). Then Z factorizes as Z+ x Z_ (when ignoring the trace anomaly) and 0_ is a free outgoing field propagating in the (fixed) background geometry g. Thus (j>+ drops out from all matrix elements built with the operator 0_. It should be emphasized that the trans-Planckian problem {e.g., the fact that the in —out Green function is characterized by trans-Planckian frequencies when one of the operator approaches the horizon [8, 28]) encountered in Hawking's approach directly follows from this factorizability. Indeed it is the absence of gravitational coupling between the + and — sectors which permits the unbounded growth of frequencies when probing, near the horizon, configurations specified on J+. The semi-classical treatment can be obtained from the path integral formalism (9.9) by first integrating over at fixed h and then searching for the

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classical extremum of h.n In this approach, by construction, the fluctuations of h are neglected and the near horizon propagation of 0_ is governed by a self-consistent metric governed by mean (n+(v)). This mean change is driven through (9.14) by the (properly subtracted [13]) expectation of Tvv = (dv(j>+)2. When working in the vacuum (9.4), the mean flux is ))|r=2MW = - ^ ( ^ )

,

(9.21)

where K(V) = {AM{v))~l with M(v) = M 0 + G(n+{v)). This flux has the opposite value of a 2D thermal flux and drives black hole evaporation according to (9.1). In this treatment, <^>_ propagates as a free field in the classical "evaporating" metric (2). The only change with respect to the fixed background approach of Hawking is the replacement of M0 by M0 + G(fi+). Thus the propagation of out-going configurations is hardly affected by the evaporation as long as it is slow, i.e., as long as M(v) ~S> MpianckTherefore, in the semi-classical scenario, the trans-Planckian problem stays as in Hawking's approach: The coupling between 0_ and the mean change (^ + ) is incapable to provide a taming mechanism since it does not open new interacting channels. To solve this problem clearly requires to take into account the fluctuating character of the interactions between _ and +, i.e., the possibility of entangling their wave functions, as in the quantum mechanical exercise presented above.

9.6

Modified Green function

Our aim is to show how the two-point Green function of 4>G(Xi,x2)

=

,

(9.22)

where Z is given in (9.12), is affected by the gravitational interactions encoded in Sint when the infalling configurations are in their vacuum state. 13 12

It is less easy to get this approximation from (9.12) since the gravitational interactions between <j>+ and 4>- have been made explicit. Nevertheless, by splitting T><j> as T>+ x T> by integrating freely over (j>+ (i.e., by ignoring its coupling to -), and by retaining only the mean mass change (n+(v)), one obtains the semi-classical description. The outcome of these manipulations can be properly derived by using a large N limit, see footnote 14. 13 Being in search of a new scale induced by vacuum effects, see point V in the outline, we should re-consider the choice of having kept (as in [7, 24]) only the +, - interaction

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The computation of G(x\, x2) can be done in two different approaches. The first consists in splitting V<j> as V<j)+ x £)_ and first integrating over (f>+ so as to determine the influence functional (IF) [32] governing the effective dynamics of _.14 The other consists in developing etSint in both integrands of (9.22) in powers of Smt so as to engender the (connected) Feynman diagrams governing the radiative corrections. The usefulness of the IF approach is that non-linear effects are naturally taken into account through the effective action for _ (the action of the Nth copy of the field in the large N limit, see previous footnote). The same nonlinear effects can of course be reached from the Feynman diagrams approach at the cost of summing infinite subsets of graphs. It is in the identification of these infinite subsets that the large N limit finds its real justification. For a schematic description of these aspects, we refer to the Appendix. In what follows, we shall pursue with the IF approach. When computing the lowest order corrections to the self-energy in (9.22), we can use (9.4), the 'free' propagator of <j>+. This approximation concerning degrees of freedom not directly involved in the matrix elements (i.e., which factorized out in the absence of interactions) is a common procedure both in quantum field theory where it gives the vacuum contribution, see chapter 9 in [32], and in statistical mechanics (e.g., the polaron, chapter 11). In our case, in this approximation, the IF gives rise to a non-local action which is a sum of terms containing (9 r 0_) 2 and kernels given by the Wick contractions of Tvv evaluated with (9.4). 15 To order G2, one obtains i(SintSint)+ where (

= iG2jd2xJd2x'(rr')-l(dA-)2

{(i+(v)n+(v'))

(dr,_)2 , (9.23)

) + means that the expectation value applies to <j>+ only. Using

c.f., the discussions after (9.12). The neglect of the other terms might indeed fallaciously engender the new scale; more on this delicate problem when considering the value of the cut-off, after (9.28). 14 The careful reader would object to the splitting of V(f> since, off-shell, it is illegitimate to decompose as <j>+ + 4>- • To legitimize mathematically what we did, he should consider N copies of and integrate over the JV - 1 ~ JV spectator copies not appearing in the numerator in (9.22). In this case, this integral will give rise to the IF we are talking about with G2 replaced by NG2. 15 In general, when one does not make this approximation, the Wick contractions of <j>+ will give rise to a series in powers of G which starts with (9.24) and whose higher order terms depend on <j>-. Thus (9.24) would have become operator-valued [24] in 4>- > thereby obtaining a situation analogous to that of transition amplitudes when enlarging the phase space so as to take into account recoil effects [35].

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(9.4), the connected two-point function is

FMT„W)c = ^ j ^

W

.

(9-24)

Then, (9.14) gives

l Z"00 = ^ - 2 / dwuexp[-iu)(v-v')].

(9.25)

This equation gives the mean metric fluctuations driven by + in the unperturbed (G = 0) vacuum state, see [27] for a analysis of the 4D twopoint function of induced metric fluctuations in Minkowski vacuum. Notice that (fj,+ (v)n+(v')) is not real. This follows from the quantum vacuum which contains only positive frequencies when hit by (j>, see (9.4). Notice that (9.24) gives the "unsubtracted" value of the connected two-point function. As shown in [34], the counterterms which lead to the renormalized one-point function (9.21) also provide divergent contributions to (9.24) which tame its singular behaviour as v —> v'. Keeping only (9.23) in the IF (or by summing the corresponding infinite set of Feynman graphs, see the Appendix) is equivalent to work with a stochastic {i.e., a classically given) Gaussian ensemble of metric fluctuations. By equivalent we mean that all matrix elements of (/>_, such as (9.22), can be computed from this stochastic theory. Thus as far as the propagation of (/)- is concerned and to lowest order in G, the functional integration over <j>+ in (9.22) effectively engenders a stochastic ensemble of metric fluctuations governed by (9.25). Hence all the techniques developed in [28] apply. In what follows we shall present schematically the main results and we refer to this work for details. Because of the Gaussianity of the ensemble and the conformal invariance of (9.6), one can obtain non-linear corrections to (9.22) from the fluctuating characteristics of (9.16), i.e., the outgoing null geodesies uM(v, r), the nontrivial solutions of ds 2 = 0 in the fluctuating metric (9.10). To determine the effects engendered by these metric fluctuations, it is instructive to analyze the backward in time propagation of asymptotic plane waves representing Hawking quanta. In the absence of metric fluctuations the plane wave e~lXu behaves near the horizon as e-iAoo(*,r)

^ 0( r _ 2 M o ) e-
(9.26)

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It vanishes for r < 2M 0 and possesses an infinite number of oscillations as r —> 2M 0 with increasing momentum pr = —idr. This is the trans-Planckian problem. In a Gaussian ensemble of metric fluctuations the averaged waves are given by / e -**"/»(«>»•)\

=

e -tAu 0 (t;,r) &-\-(6u(v)Su(v))

_

Using (9.18), the fact that rQ{v)\tt0 - 2M 0 ~ 2M0eK(-v-u°\ obtains

(9.27)

(5u{v)\U0 6u{v)\uo)

=

— /

—

and (9.25), one

+l 2

^->—i

The spread 2M 0 . Thus the correlations between asymptotic quanta and early configurations, which existed in a given background as shown in (9.26), are washed out by the metric fluctuations once r - 2M 0 ~CTA~ 1/M0. The reason of this loss of 16

In the simplified treatment we are using, the value of A must be chosen from the outset. Instead, in an improved treatment where all terms are kept in Sint, we believe that the values of a,adlap- shall be unambiguously determined. In that case, when using the properly subtracted [34] two-point function {T^Ta^), only ui ~ K will contribute to the dr's. Indeed in the UV domain all expressions become Minkowskian in character and hence cannot contribute to Lorentz breaking effects such as those engendered by a. This result can be implemented in our truncated model by putting A ~ K in (9.28), thereby obtaining a ~ ad'sp- ~ GK.

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coherence is that the state of _ becomes correlated to that of + [7, 24]. Physically, this loss of coherence implies that induced emission [29] no longer exists when the threshold energy 1/CT is reached. 17 Phenomenologically this loss can be viewed as a dissipation of outgoing waves, and, as in condensed matter [4, 26], it can be described by a non-trivial dispersion relation, see equations (5.9), (5.11) in [28]. We should further explain the physical relevance of these results. To this end, one must identify the matrix elements of <£_ which are highly sensitive to the metric fluctuations (and governed by the ensemble averaged waves (9.27)) and those which are not. The simplest example of an operator which is sensitive is the in-out Green function with one operator at v, r and the other on J+. Indeed since the 'second' point lives on J+ where the out vacuum is defined, the phase of the out-wave function evaluated at fixed u is not affected by metric fluctuations. On the contrary that of the wave function evaluated near the horizon at v,r is sensitive to the metric fluctuations encountered from J+ to where it lives.18 It is this (unusual, see below) discrepancy in the modification of the phase at each point which explains why the ensemble averaged one-point waves (9.27) govern this two-point Green function. Instead usual expectation values, such as for instance the in-in Green function with two points evaluated at fixed u on J+ (or two nearby points close to the horizon), are NOT severely affected by the metric fluctuations because there are not governed by the ensemble averaged waves (9.27). The reason is that the ensemble average is performed after having computed the operator for each member of the ensemble. (This is not a choice: the stochastic ensemble is merely a tool to reproduce quantum mechanical expectation 17 An interesting and unsolved question raised by this loss is whether new correlations are induced by the gravitational interactions as the same time as the old ones are washed out by them. This is presently under study. 18 Our cautious reader might wonder if the effects we are describing are not induced by the choice of working at fixed u or at fixed v, r. To waive his qualms, we recall that Green functions have no physical meaning per se, rather they are elements which appear in integrals describing transition amplitudes (for a discussion of this point in a quantum gravitational context see section 2 in [36]). It is through this channel that one can verify that u is a physically meaningful coordinate on J+ since du| r = dt where dt is the proper time of a particle detector on J+. Thus when additional quantum mechanical systems are coupled to the radiation field, the matrix elements governing transition amplitudes will have, in their integrand, phase factors behaving like e _ i A " in any coordinate system. Similarly, upon questioning what an infalling observer might see when crossing the horizon, v, r are meaningful since dr|„ oc - d r where dr is his proper time.

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values. This quantum origin fixes the rules of ensemble averaging without ambiguity.) In our case, this implies that the shift (9.18) affects coherently the phase at each point, see section IV.A in [28]. This guarantees that the shift drops out in the coincidence point limit. This cancellation in turn guarantees that the asymptotic properties are (almost) unaffected since the Green function possesses the usual Hadamard singularity [33]. By almost we mean that the corrections scale like (na)2 and thus are of order 1/M 4 . It should be pointed out that it is the dynamically induced scale d\ and not the UV cut-off A which governs these corrections. We would like to further emphasize the fact that the metric fluctuations strongly affect the correlations between configurations specified on J+ and near the horizon without modifying the short distance behaviour of the Green function. 19 The radical difference of the impact of vacuum gravitational interactions follows from the fact that asymptotic observers inevitably use out-states to probe the physics. Therefore, the overlaps they consider will be automatically of the in — out type since the Heisenberg state of the field is specified (prepared) before the collapse. It is this two-states formalism giving rise to non-diagonal matrix elements [8] (exactly like in a S-matrix formulation [7]) which is at the origin of the difference: The metric fluctuations cannot coherently affect configurations specified in the 'kef on J~ and in the 'bra' on J+, hence the coherence is lost. (Notice that when computing induced emission probabilities, one automatically considers overlaps between oui-states specified on J+ and some early state specified on J~'. This explains why induced emission probabilities are washed out.) On the contrary, infalling observers have access only to the near horizon behaviour of the Green function in terms of in configurations. Hence coherence is maintained for them.

19 This clearly illustrates that the physics seen by infalling observers completely differs from that reconstructed from observers at large distance from the hole. This is similar to what was advocated in [24]. However, the fact that the in — out Green function obeys (9.27) indicates that the near horizon physics is unaccessible and therefore lost to remote observers. Thus it seems that these two descriptions could not obey the 'complementarity principle' [24]. By complementary to each other, it was meant that the two descriptions are both complete, like the position and momentum representations of the same vector state in quantum mechanics.

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9.7

Towards a collective treatment of quantum gravitational

interactions

Conclusions

We have studied the effects induced by the gravitational interactions governed by (9.13). Even though we worked out only the lowest order in G (aA <* G) we believe that the five results listed in the Outline are robust. We see no reason for higher order terms to suppress the entanglement of (j>- with + so as to giveCTA= 0 thereby recovering trans-Planckian correlations. Indeed higher order modifications to (9.24) and (9.25) should be of the type (Guj2)n and therefore will not affect the low frequency behaviour of (9.25) thereby leaving the effective spread _ with + is unavoidable and universal: a similar entanglement would be also unavoidable when considering the coupling of <j> to other quantum fields such as massive ones. The entanglement will then prevent the unbounded growth of frequencies encountered in the free field theory and will be accompanied by the reorganization of the description of vacuum. By this we mean that the usual states of the free field theory, giving rise to the notion of on-shell asymptotic particles, provide bad approximations of the true eigenstates (still characterized by the Killing energy A since the situation is stationary) of the interacting theory when r —• 2M. It is the growing discrepancy which leads to the dissipation of the amplitude in (9.27). We also claim that the radiative corrections encoding dissipation are finite because only low frequency ui ~ K infalling configurations contribute to them. Indeed, in the UV regime for both infalling and outgoing configurations, the expressions coincide with those evaluated in the tangent plane and are Minkowskian in character. Hence the high frequency regime cannot contribute to the effects which break Lorentz invariance. (This still needs to be confirmed by an explicit calculation and is presently under examination.) We would like to conclude this work by several remarks on related aspects of Quantum Gravity and Black Hole Physics. First, we point out the similarity between the above effects induced by the metric fluctuations representing gravitational interactions in the vacuum and those attributed [37] to 'foam'. By foam we mean quantum gravitational configurations which radically affect the smoothness of space-time at short distance. (They might arise from gravitational instantons [38] or even stringy

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effects [39].) In all cases, the replacement of free field propagation in a fixed background by the appropriate interacting model might lead to very similar (universal?) deviations when analyzing the departure from the free field description that all models possess at large distances. Therefore, these first deviations might be described by some effective mesoscopic theory of spacetime properties which would essentially signal the existence of a minimal resolution length [40], the equivalent of our a, in the otherwise local field theory. Secondly, we conjecture that a (properly computed so as to include the contribution of higher angular momentum modes) should also be the length scale which enters in the entanglement description of the black hole entropy [41]. We recall that when using free field in a given space time, the entanglement diverges due to the unbounded character of the reservoir of high energy modes. To get a finite entropy density per unit area, some cutoff should be introduced. We propose that the cut-off defining the black hole entropy should be the dynamically induced length-scale a(N), i.e., the length scale at which correlations between configurations on J+ and the near horizon region get lost when N quantum fields contribute to the entropy and therefore to the near horizon gravitational interactions.

Acknowledgements: I wish to thank D. Arteaga, R. Balbinot, C. Barrabes, R. Brout, L. Ford, V. Frolov, L. Garay, B.L. Hu, T. Jacobson, S. Liberati, S. Massar, E. Verdaguer and G. Volovik for useful discussions. I also thank the organizers of the 5th and 6-th Peyresq meetings on Quantum Spacetime, Brane Cosmology and Stochastic Effective Theories, those of the first Rio meeting on Analog models of General Relativity, those of the Black Hole III conference held in Canada and those of the ESF-COSLAB workshop held in London in July 2001 where the topics presented here have been discussed. This work was supported by the NATO Grant CLG.976417.

9.8

Appendix: T h e large TV limit

We briefly mention several interesting features of the large N limit which illuminate the problems we addressed.

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The semi-classical description of quantum processes occurring in a curved background is based on the following equations G^

=

8TTG(*|TM1/|*)

Ag = 0

(9.29) (9.30)

In (9.30) the field operator propagates in the classical geometry g = g*v which is a solution of (9.29) driven by the expectation value {^[T^]^) evaluated in the state \ty) using (9.30). In this sense, (9.29, 9.30) is a self-consistent (Hartree) approximation. It is quite reasonable that this approximation correctly predicts the time evolution of certain quantities in certain circumstances, e.g., the rate of mass loss of a large black hole. However, the criteria which characterize the validity range of the predictions obtained from (9.29,9.30) are not known. An obstacle in finding these criteria is the identification of the "small parameter(s)" which control the deviations between the exact evolution and that predicted by the semi-classical equations. A rather formal answer to these questions is provided by considering a large N limit, where N is the number of copies of the field. The simplest way to understand why the semi-classical equations govern the large N limit is by considering the path integral approach of matrix elements. By duplicating N times <j> in (9.9) and first integrating over them with a fixed metric h, the 1-loop effective action for h contains an overall prefactor N when Newton's constant G is expressed as G = £/N. Therefore, in a large N limit, the path integral over h can be done at the saddle point approximation. The stationary phase condition then gives rise to (9.29) and the spread around the saddle point scales like A7"-1/2. In this vision, the validity of the semi-classical equations purely relies on a statistical argument, as in a thermodynamic limit. The weakness of this very general argument is the absence of a hierarchy of the length scales governing the processes under examination, unlike what was found [42] when a Born-Oppenheimer treatment is applied to quantum gravity. More interestingly the large N limit makes also predictions beyond the semi-classical equations. For instance, in the limit N —> oo with GN fixed, the short distance behaviour of the graviton propagator is modified, see [34, 27]. In Minkowski vacuum, these modifications are of course Lorentz invariant. However, in a thermal bath or a curved background, the correction terms will no longer possess the Lorentz invariant form. Hence they

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can induce the effects we are seeking: a dynamically induced scale which breaks the (local) Lorentz invariance that the non-interacting theory possessed. Moreover, only low energies (i.e., energies comparable to the temperature) can contribute to this new scale because in the UV limit all expressions tend to their Minkowski vacuum, and hence Lorentz-invariant, form. Hence there should not be any additional UV divergences in the expressions giving rise to the new scale; as is the case (for instance) for the corrections to the self-energy of an electron immersed in a thermal bath. To further strengthen the relations with what we did in section 6, it is instructive to see how the semi-classical treatment emerges from (9.22) viewed as generating perturbatively the connected Feynman diagrams when expanding elSint in powers of Si„t. In this description, one finds that the Green function is a double sum of powers of N and G which possess the following properties. • The power of N is equal or inferior to that of G. • The semi-classical treatment corresponds to the leading series: the set of graphs weighted by (GN)n. Upon summing this series, one identically recovers the Green function evaluated in the 'mean' geometry
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(Planckian) energies? We conjecture that this is the case: the sorting out of graphs in terms of m is effectively an expansion in the energy of the processes involved in the matrix element under consideration. This is what seems to emerge from our analysis. As indicated by (9.27, 9.28), the semi-classical description of the correlations breaks down when r — 2M ~ a, i.e., when the energy of a mode pr = \/{r/2M — 1) reaches the new scale 1/a.20 We thus find, as in [42], that the validity of the semi-classical equations relies on a hierarchy of energy scales. Indeed, for a large black hole, a
20

Notice that a scales as N ll2(GN/M). Writing the scale in this manner emphasises the fact that in the large N limit at fixed GN, i.e., with a iV-independent Hawking flux, <7 -> 0 like AT -1 / 2 , thereby verifying that in this limit the scale which signals the breakdown of the semi-classical description indeed vanishes.

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[36] R. Parentani, Nucl. Phys. B492 (1997) 475. [37] L. Garay, Phys. Rev D58 (1998) 124015, and references therein. [38] S. Hawking, Nucl. Phys. B144 (1978) 349. [39] D. Amati et a/., Phys. Lett. B216 (1989) 41. [40] A. Kempf and G. Mangano, Phys. Rev. D55 (1997) 7909, and R. Brout et a/., Phys. Rev. D59 (1999) 044005. [41] T. Jacobson, Black Hole Entropy and Induced Gravity, gr-qc/9404039. [42] S. Massar and R. Parentani, Nucl. Phys. B513 (1998) 375.

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C h a p t e r 10 Role of sonic metric in relativistic superfluid Brandon Carter Departement d'Astrophysique Relativiste et de Cosmologie Centre National de la Recherche Scientifique Observatoire de Paris, 92195 Meudon France E-mail: [email protected]

Abstract: After recapitulating the basic principles of ordinary relativistic superfluid mechanics, and in particular the irrotational barotropic case that is relevant to superfluidity at zero temperature, this article provides a self contained introduction to the natural relativistic generalisation of the Landau two constituent model for a superfluid at finite temperature, with special emphasis on the role of the Unruh type sonic metric in the low temperature limit. 245

246

10.1

Role of sonic metric in relativistic

superfluid

Introduction

The present discussion will be limited to scalar (spin 0) superfluid models such as are appropriate for the familiar case of Helium 4 (though not Helium 3) and for the inter-penetrating neutron superfluid fluid in the crust of a neutron star, which is the part most relevant to the analysis of pulsar frequency variations. The first three sections consist of a recapitulation from modern standpoint of the long well known essentials of the relativistic version of the single constituent kind of superfluid model that is appropriate for the description of Helium 4 at zero temperature, with special attention to the theory of small perturbations, in which an Unruh type sonic metric plays a role that is more directly important than that of the background spacetime metric. The fourth section provides a presentation of the more recently developed generalisation [1, 2, 3] to a two constituent model (of the kind whose non-relativistic analogue was originally developed by Landau) in which the second constituent represents a gas of phonon excitations. Quite apart from the fact that it provides a clearer picture of the dynamics of superfluid Helium 4 in an ordinary laboratory context (in which many aspects are complicated, not simplified, by the introduction of an artificially preferred rest frame such is necessary for the traditional kind of non-relativistic treatment) the development of relativistic superfluid models is motivated by the need to treat neutron superfluidity in the cores and particularly the lower layers of the crusts of neutron stars, in order to account for pulsar frequency data, in which the most salient feature is the common occurrence of glitch discontinuities. The generalized Landau model described here is not by itself sufficiently elaborate to account for such effects: its construction is just an intermediate step in a long and still far from complete program of development of the more sophisticated models that will be needed. The way to combine the strictly fluid kind of model described here with the kind of elastic solid model that is appropriate for the outer crust in order to obtain the rather elaborate kind of hybrid model needed for the mesoscopic description of the inner neutron star crust layers (at densities exceeding the neutron drip threshold) has been outlined elsewhere [4]. This issue will not be discussed here, and anyway it is to be noted that the validity of such a mesoscopic solid-superfluid hybrid model is limited to scales small compared with the inter-vortex spacing which is much less than a millimetre in typical pulsars. For a macroscopic description of the inner crust of a neutron star, on scales larger than the inter-vortex spacing an elegant conservative model

Brandon Carter

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providing a fully relativistic description allowing for the anisotropic stress resulting from the averaged effect of the quantised vortices (that must be present wherever there is rotation) has recently been developed [5] and is described [6] elsewhere. Such anisotropy has been neglected in a cruder non conservative model that has been developed even more recently [8] for the purpose of providing a global description of the long term evolution.

10.2

Single constituent perfect fluid models

Our purpose here is to offer a self contained presentation of the basic principles of the relatively simple models that suffice for the familiar laboratory case of superfluid Helium 4 (though not for Helium 3, which is so complicated [10] that a relativistic description is not yet available), starting with the kind of single constituent model (for which I have provided a more detailed discussion in the lecture notes of a recent Les Houches school [6]) that is applicable to the zero temperature limit. This zero temperature limit will be describable by an ordinary barotropic fluid model subject to the restriction that the vorticity, meaning the exterior derivative of the relevant momentum covector (whose definition in a relativistic context will be recapitulated below) should vanish, so that the momentum covector itself has the form of the gradient of a scalar. The meaning of the term barotropic is simply that the perfect fluid pressure scalar P should be determined by an equation of state as a function only of the fluid mass density p. The barotropic equation of state determines a corresponding speed c, say, of ordinary "first" sound, that will be given by the familiar formula c,2 = dP/d/9,

(10.1)

and that must be subluminal, c 2 < c2 (where c is the speed of light) in order for the usual causality requirement to be respected. Before proceeding it is desirable to recapitulate the essential elements of the relativistic kinematics and dynamics that will be required. This is particularly necessary in view of the regrettable tradition in non-relativistic fluid theory — and particularly in non-relativistic superfluid theory — of obscuring the essential distinction between velocity (which formally belongs in a tangent bundle) and momentum (which formally belongs in a cotangent bundle) despite the fact that the distinction is generally respected in other branches of non-relativistic condensed matter theory, such as solid state

248

Role of sonic metric in relativistic

superBuid

physics, where the possibility of non-alignment between the 3-velocity va, and the effective 3-momentum pa of an electron travelling in a metallic lattice is well known. Failure to distinguish between contravariant entities, such as the velocity va, and covariant entities, such as the momentum pa, is something that one can get away with in a non-relativistic treatment only at a price that includes restriction to strictly Cartesian (rather than for example cylindrical or comoving) coordinates. In a relativistic treatment, even using Cartesian coordinates x11 «-> {t, xa} of Minkowski type, with a flat spacetime metric g^ whose components are of the fixed standard form diag{—c2, 1, 1, 1}, the necessity of distinguishing between raised and lowered indices is inescapable. Thus for a trajectory parametrised by proper time r, the corresponding unit tangent vector da;'' u" = ^ (10.2) dr is automatically, by construction, a contravariant vector: its space components, ua = 7^° with 7 = (1 — v2/c2)~ll2 will be unaffected by the index lowering operation u^ i-*- uM = g^u", but its time component u° = dt/dr = 7 will differ in sign from the corresponding component u0 = —7c2 of the associated covector u^. On the other hand the 3-momentum pa and energy E determine a 4-momentum covector p,v with components ixa = pa, p0 = —E that are intrinsically covariant. The covariant nature of the momentum can be seen from the way it is introduced by the defining equation, dL *, = ^ ,

(10.3)

in terms of the relevant position and velocity dependent Lagrangian function L, from which the corresponding equation of motion is obtained in the well known form

^

= ¥- •

(10-4)

K dr dx" ' In the case of a free particle trajectory, and more generally for fluid flow trajectories in a model of the simple barotropic kind that is relevant in the zero temperature limit, the Lagrangian function will have the familiar standard form L = \pg^uv - \PLC2 , (10.5) in which (unlike what is needed for more complicated chemically inhomogeneous models [11, 4]) it is the same scalar spacetime field \i that plays the

Brandon Carter

249

role of mass in the first term and that provides the potential energy contribution in the second term. The momentum will thus be given by the simple proportionality relation H„ — puv , (10.6) so that one obtains the expressions E = 7/JC 2 , pa = p'jVa, in which the field p is interpretable as the relevant effective mass. In the case of a free particle model, the effective mass p will of course just be a constant, p = m. This means that if, as we have been supposing so far, the metric g^ is that of flat Minkowski type, the resulting free particle trajectories will be obtainable trivially as straight lines. However the covariant form of the equations (10.2) to (10.6) means that they will still be valid for less trivial cases for which, instead of being flat, the metric g^v is postulated to have a variable form in order to represent the effect of a gravitational field, such as that of a Kerr black hole (for which, as I showed in detail in a much earlier Les Houches school [12], the resulting non trivial geodesic equations still turn out to be exactly integrable). In the case of the simple barotropic perfect fluid models with which we shall be concerned here, the effective mass field p will be generically nonuniform. In these models the equation of state giving the pressure P as a function of the mass density p can most conveniently be specified by first giving p in terms of the corresponding conserved number density n by an expression that will be decomposible in the form p = mn+^,

(10.7)

in which m is a fixed "rest mass" characterising the kind of particle (e.g., a Cooper type neutron pair) under consideration, while e represents an extra compression energy contribution. The pressure will then be obtainable using the well known formula P = (np- p)c2, (10.8) in which the effective dynamical mass is defined by M= ^ = m + ~ . (10.9) an cs an It is this same quantity p (sometimes known as the "specific enthalpy") that is to be taken as the effective mass function appearing in the specification

250

Role of sonic metric in relativistic

superSuid

(10.5) of the relevant Lagrangian function (on what is formally identifiable as the tangent bundle of the spacetime manifold). When one is dealing not just with a single particle trajectory but a spacefilling fluid flow, it is possible and for many purposes desirable to convert the Lagrangian dynamical equation (10.4) from particle evolution equation to equivalent field evolution equations [11,4]. Since the momentum covector \iv will be obtained as a field over spacetime, it will have a well defined gradient tensor V p /t„ that can be used to rewrite the right hand side of (10.4) in the form d ^ „ / d r = upVpnu. Since the value of the Lagrangian will also be obtained as a scalar spacetime field L, it will also have a well defined gradient which will evidently be given by an expression of the form „

r

dL

dL„

We can thereby rewrite the Lagrangian dynamical equation (10.4) as a field equation of the form upV„/i„ + jJpV„u' = V„L. (10.10) An alternative approach is of course to start from the corresponding Hamiltonian function, as obtained in terms of the position and momentum variables (so that formally it should be considered as a function on the spacetime cotangent bundle) via the Legendre transformation H = HvUv-L.

(10.11)

In this approach the velocity vector is recovered using the formula

and the associated dynamical equation takes the form ^ = ~ . (10-13) v dr dx" ' The consideration that we are concerned not just with a single trajectory but with a space-filling fluid means that, as in the case of the preceding Lagrangian equations, so in a similar way this familiar Hamiltonian dynamical equation can also be converted to a field equation which takes the form 2u'V[,/i1,] = -V„tf >

(10.14)

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251

with the usual convention that square brackets are used to indicate index antisymmetrisation. On contraction with u" the left hand side will evidently go out, leaving the condition u"V„Jf = 0,

(10.15)

expressing the conservation of the value of the Hamiltonian along the flow lines. The actual form of the Hamiltonian function that is obtained from the simple barotropic kind of Lagrangian function (10.5) with which we are concerned will evidently be given by # = ^ / ' 7 M V + ^-,

(10-16)

in which it is again the same scalar spacetime field // that plays the role of mass in the first term and that provides potential energy contribution in the second term. In order to ensure the proper time normalisation for the parameter r, the equations of motion (in whichever of the four equivalent forms (10.4), (10.10), (10.13), (10.14) may be preferred) are to be solved subject to the constraint that the numerical value of the Hamiltonian should vanish, H = 0,

(10.17)

initially , and hence also by (10.15) at all other times. This is evidently equivalent to imposing the standard normalisation condition «"uM = - c 2 ,

(10.18)

on the velocity four vector. In more general "non-barotropic" systems, such as are needed for some purposes, the Hamiltonian may be constrained in a non uniform manner [11, 4] so that the term on the right of (10.14) will be non zero, but in the simpler systems that suffice for our present purpose the restraint (10.17) ensures that this final term will drop out, leaving a Hamiltonian equation that takes the very elegant and convenient form «"«;„„ = 0.

(10.19)

in terms of the relativistic vorticity tensor that is defined as the antisymmetrised ("exterior") derivative of the momentum covector, i.e., wvp

= 2V^jUp].

(10.20)

252

Role of sonic metric in relativistic superQuid

It is an evident consequence (and, as discussed in greater detail in the above cited Les Houches notes [6], would still be true even if (10.17) were not satisfied) that if w^ is zero initially it will remain zero throughout the flow, which in this case will be describable as "irrotational". In cases for which the vorticity is non-zero, the "barotropic" dynamical equation (10.19) is interpretable as requiring the flow vector M*4 to be a zero eigenvalue eigenvector of the vorticity tensor w^, which is evidently possible only if its determinant vanishes, a requirement that is expressible as the degeneracy condition W^Wpa] = 0 . (10.21) Since the possibility of it having matrix rank 4 is thus excluded, it follows that unless it actually vanishes the vorticity tensor must have rank 2 (since an antisymmetric matrix can never have odd integer rank). This means that the flow vector uM is just a particular case within a whole 2-dimensional tangent subspace of zero eigenvalue vorticity eigenvectors, which (by a well known theorem of differential form theory) will mesh together to form well defined vorticity 2-surfaces as a consequence of the Poincare closure property, V[M
(10.22)

that follows from the definition (10.20). Although it has long been well known to specialists [16], the simple form (10.19) of what is interpretable just as the relativistic version of the classical Euler equation is still not as widely familiar as it ought to be, perhaps because its Hamiltonian interpretation was not recognised until relatively recently [11, 4]. It does not constitute by itself the complete set of dynamical equations of motion for the perfect fluid, but must be supplemented by a particle conservation equation of the usual form for the particle number current n" = nu" ,

(10.23)

which must of course satisfy the condition Vvn" = 0 .

(10.24)

A much more widely known, but for computational purposes (particularly in curved spacetime) less useful form of the perfect fluid dynamical equations is to express them in terms of the stress-momentum-energy density tensor,

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253

which is given in terms of the mass density p and the pressure P by T"" = (p + ^ ) «"u" + Pa"" ,

(10.25)

and which must satisfy a so called conservation law of the standard form V„T"" = 0.

(10.26)

Although it is conveniently succinct, a disadvantage of this traditional formulation is that it is directly interpretable as a law of conservation of momentum and energy in the strict sense only in the case of a flat (Minkowski type) spacetime, but not in a curved background such as that of a neutron star. The possibility in the barotropic case (i.e., when P is a function only of p) of decomposing the combined set of dynamical equations (10.25) as the combination of the convergence condition (10.24) (obtained by contracting (10.26) with Ufj) and the relativistic Euler equation (10.19), which can be written out more explicitly as n*V[„/ip] = 0,

(10.27)

has the advantage that these are interpretable as genuine conservation laws — for particle number flux and vorticity respectively — even in an arbitrarily curved spacetime background.

10.3

Single constituent superfluid models

The simplest superfluid models, namely those pertaining to the zero temperature limit, are just ordinary perfect fluid models subject to the restraint of irrotationality, with a momentum covector given as the gradient M„ = V „ S ,

(10.28)

of a periodic scalar field S that is to interpreted as being proportional to the angle of the mesoscopic phase factor, e8* say, of an underlying bosonic condensate that might consist of Helium 4 atoms or Cooper type neutron pairs) in which the phase angle is given according to the usual correspondence principle by 4>^S/h. (10.29)

254

Role of sonic metric in relativistic

superBuid

In a multiconnected configuration of a classical irrotational fluid the Jacobi action field S obtained from (10.28) might have an arbitrary periodicity, but in a superfluid there will be a U(l) periodicity quantisation requirement that the periodicity of the phase angle should be a multiple of 2n, and thus that the periodicity of the Jacobi action S should be a multiple of 27rh. The simplest configuration for any such superfluid is a uniform stationary state in a flat Minkowski background, for which the phase will have the standard plane wave form S/h = kaxa - cut, (10.30) from which one obtains the correspondence pv •<-> {—HCJ, hka}, which means that the effective energy per particle will be given by E = 7/xc2 = huj and that the 3-momentum will be given by pa = pjva = hka. It is to be remarked that for ordinary timelike superfluid particle trajectories the corresponding phase speed cj/k of the wave characterised by (10.30) will always be superluminal, a fact of which most laboratory handlers of Helium 4 are blissfully unaware, and can usually safely ignore, since what matters for most practical purposes is not the phase speed but the group velocity of perturbation wave packets. In the irrotational case characterised by (10.28) the Euler equation (10.19) is satisfied automatically, so the only dynamical equation that remains is (10.24). When the phase scalar is subject to a small perturbation, 5(f> = (p say, it can be seen that the corresponding perturbation of the conservation law (10.24) provides a wave equation of the form D^ = 0

(10.31)

in which D is a modified d'Alembertian type operator that is constructed from an appropriately modified space-time metric tensor g^" in the same way that the ordinary d'Alembertian operator • = V V ^ is constructed from the ordinary spacetime metric tensor g***'. The appropriately modified spacetime metric, namely the relativistic version of what is known in the context of Newtonian fluid [13, 14] and superfluid [15] mechanics as the Unruh metric, can be read out in terms of the light speed c and the (first) sound speed c, given by (10.1) as flr = ^ ( c l 7 ^ - c I - V t i , ' ) ) (10.32) n where 7 ^ is the spatially projected (positive indefinite) part of the ordinary space time metric, as defined by y " = gT + c" V u " . (10.33)

Brandon Carter

255

The quantum excitations of the linearised perturbation field

1= fc d$<4>,

dS<4> - M ^ d 4 x

t

(10.34)

for some suitable scalar Lagrangian functional £. There are several available procedures for doing this for a generic perfect fluid with rotation, involving radically different choices of the independent variables to be varied: although they are all ultimately equivalent "on shell" the "off shell" bundles over which the variations are taken differ not only in structure but even in dimension. These methods notably include the worldline variation procedure (the most economical from a dimensional point of view) developed by Taub [17], and the Clebsch type variation procedure developed by Schutz [18], as well as the more recently developed Kalb-Ramond type method [19, 6] that has been specifically designed for generalisation [5]

256

Role of sonic metric in relativistic

superSuid

to allow for the anisotropy arising from the averaged effect of vortex tension in the treatment of superfluidity at a macroscopic level. None of these various methods is sufficiently simple to have become widely popular. The problem is much easier to deal with if, to start off with, one restricts oneself to the purely irrotationall case characterised by (10.28), which is all that is needed for the description of zero temperature superfluidity at a mesoscopic level. In this case a very simple and well known procedure is available. In this procedure, the independent variable is taken to be just the Jacobi action S, or equivalently in a superfluid context, the phase

(10.35)

with the 4-momentum itself given by the relation (10.28) that applies in the irrotational case, i.e., /i„ = J*V„0. (10.36) Thus setting C = P,

(10.37)

and using the standard pressure variation formula 8P = c2n6fi,

(10.38)

one sees that the required variation of the Lagrangian will be given by 8C = -nvbnv

= -hn"Vv{5(j>).

(10.39)

Demanding that the action integral (10.34) be invariant with respect to infinitesimal variations of ip = 5<j> then evidently leads to the required conservation law (10.24). It is to be noted that this variational principle can be reformulated in terms of an independently variable auxiliary field amplitude $ and an appropriately constructed potential function V{$} as a function of which the action takes the desirably fashionable form C = - ^ - $ 2 ( V ^ ) W - V{$}

(10.40)

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257

which is interpretable as the classical limit of a generalised Landau-Ginzburg type model. In this formulation, as discussed in greater detail elsewhere [19, 6], the auxiliary amplitude is to be identified as being given by the formula n

*

_ /n\!/2

y/p+P/c* -(=f. ~ V

•

<"•«>

while the prescription for the corresponding potential energy density function is that it should be given by V =

P

-^-

.

(10.42)

Having evaluated V as a function of <J> one can recover the effective mass H, number density n, mass density p and pressure P of the fluid using the formulae ^

=

^ % '

n

= *2^

(10 43)

-

and P= i * V + ^,

F = 5 ^ V - 7 ,

(10.44)

which are derivable from (10.8) and (10.9). It is to be remarked that the covariant inverse of the generalised Unruh metric tensor (10.32) is expressible in this notation as Tj* = ^(c^^u

- cIc_2uMu„)

(10.45)

A particularly noteworthy example is the conformally invariant special case [6] characterised by a potential function that is homogeneously quartic, V oc $ 4 , which is what is obtained for a radiation gas type equation of state of the familiar form P = pc2/3, and for which the (first) sound speed is given by c,2 = c 2 /3.

10.4

Landau-type two-constituent superfluid models

As an intermediate step between the very simple single constituent superfluid models described in the previous section and the more elaborate models

258

Role of sonic metric in relativistic

superBuid

needed in the context of neutron stars, the purpose of this section is to describe the relativistic version of the category of non dissipative 2-constituent superfluid that was originally developed by Landau for the description of ordinary superfluid Helium 4 at non-zero temperature. As well as the relevant conserved particle number current n,i (representing the flux of Helium atoms in that particular application) such a model involves another independently conserved current vector, s^ say, representing the flux of entropy. In the single constituent case characterised by the variation rule (10.39) we saw how the current vector n" was associated with a dynamically conjugate covector \i,v that is interpretable as representing the effective mean 4-momentum per particle. In a similar way in a 2-constituent model the second current vector s" will be analogously associated with its own dynamically conjugate 4-momentum covector 0 „ . The earliest presentations of the generic category of non-dissipative 2-constituent superfluid were, on one hand, a generalisation [1] of the relativistic Clebsch formulation [18] based on the variation of a generalised pressure function ^ depending on the 4-momentum covectors \xv and 0 „ according to the partial differentiation rule d * = -n" d/x,, - s" d 6 „ ,

(10.46)

and, on the other hand, a generalisation [20, 4] of the world line variational formulation due to Taub [17] based on the variation of a master function A depending on the currents n11 and s" according to the partial differentiation rule dA = /i I / dn" + e „ d s ' / . (10.47) Although they were originally developed independently these alternative formulations were subsequently shown to be equivalent to each other and to an intermediate mongrel version [2] based on a Lagrangian density C = * + s"e„ = A - n"/i„ ,

(10.48)

depending on the particle 4-momentum covector \iv and the entropy current sv according to the partial differentiation rule d £ = 6 1 / ds' / - n " d / i „ .

(10.49)

All of these variational formulations are subject to the complication that the allowable field variations are not free but must be suitably constrained to

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259

avoid giving overdetermined field equations. Although it violates the symmetry between the two kinds of conserved current n" and sv that are involved, the mongrel formulation characterised by (10.49) is the one that allows the simplest specification of the constraints required to get the appropriate dynamical equations for the superfluid case. In this formulation [2] the constraint on the particle 4-momentum covector is simply that it should have the same phase gradient form (10.28) as in the zero temperature limit in which the entropy constituent is absent, namely H„ = HV^.

(10.50)

The corresponding constraint on the current vector s" of the "normal" constituent is the not quite so simple Taub type requirement that its variation should be determined by the displacement of the flow lines generated by an arbitrary vector field C" say, which means [4] that it must have the form ds" = C V p s " - s"V,C + s"VpC,

(10.51)

whose derivation is obtainable by a procedure that will be explained more explicitly in the next section. Demanding invariance of the volume integral of C with respect to infinitesimal local variations of the phase variable then gives the usual particle conservation law in the same form (10.24) as for the single constituent limit. Demanding invariance for an arbitrary local displacement field £ gives not only the analogous entropy conservation law V„*" = 0,

(10.52)

*"V [ v e p ] = 0.

(10.53)

but also the dynamical equation

This governs the evolution of the thermal 4-momentum covector in a manner analogous to that whereby the relativistic Euler equation (10.27) governs the evolution of the momentum covector in an ordinary perfect fluid. These dynamical equations entail (but unlike the single constituent case are not entirely contained in) an energy momentum pseudo-conservation law of the usual form (10.26) for a stress-momentum-energy density tensor that can be written in the form V^rrne + sTQe + W,,

(10-54)

260

Role of sonic metric in relativistic

superBuid

which will in fact (although it is not obvious in this particular expression) be automatically symmetric, T^ — 0. The category of models characterised by the preceding specifications for various conceivable forms of the equation of state specifying £ as a scalar function of /i„ and s" is very large. The use of what is interpretable [21] as a special subcategory therein, on the basis of a particular kind of separation ansatz, was proposed in early work of Israel [22] and Dixon [23] and has been advocated more recently by Olsen [24]. Unfortunately however, the simplification provided by the Israel-Dixon ansatz (effectively the relativistic generalisation of the obsolete Tisza-London theory that was superseded by that of Landau) is incompatible with the kind of equation of state that is needed for even a minimally realistic treatment of a real superfluid. A satisfactory treatment of what goes on at temperatures high enough for non-linear "roton" type excitations to be important is not yet available, but in the low temperature "cool" regime, in which only linear "phonon" type excitations are important, it is not difficult to provide a straightforward analytic derivation of the kind of equation of state that is appropriate. Following the lines developed in a non-relativistic context by Landau himself [25] the relativistic version of the appropriate "cool" equation of state has recently been derived [3] by considering perturbations of the single constituent model - with equation of state specified as a pressure function, P{n} - that describes the relevant zero temperature limit. The result is obtained in an analytically explicit form that (despite the fact that it is not of the separable Israel-Dixon kind) can be given a very simple expression in terms of what we referred to as the "sonic" metric. This is specifiable by the conformal relation Qf" = ^ c - 1 ^ (10.55) in terms the Unruh phonon metric (10.32) that is associated with the relevant zero temperature limit state as specified by the relevant momentum covector \iu. This in turn, by (10.35), determines the relevant value of the scalar \i and hence (via the zero temperature equation of state, using the formalism of section 10.2) also of the relevant phonon speed c, and field amplitude $ . While the Unruh metric is more convenient for many purposes, the advantage of the conformal modification we have used, namely gp° = gp° + (c~2 - c-2)upua

,

(10.56)

is that its spatially projected part agrees with that of the ordinary space metric, from which it differs only in the measurement of time.

Brandon

Carter

261

The result that is obtained [3] is given by a Lagrangian of the form £ = P-3tp

(10.57)

in which the deviation from the zero pressure limit value P{p} is given as a function not just of the particle 4-momentum covector \xv but also of the entropy flux s" (postulated to be sufficiently weak to be constituted only of phonons) by the formula ^ = \c-^\G-ys^l\

(10.58)

where h is identifiable to a very good approximation with the usual DiracPlanck constant h, its exact value being given by h

^QMh-

=^(-j)

(1

°-59)

This is equivalent to taking the generalised pressure function to be * = P + xp,

(10.60)

V' = | ( ^ ) 3 ( a " f f e p e < r ) 2 ,

(10.61)

with

in which the effective thermal 4-momentum per unit of entropy is given (according to the partial differentiation formula (10.49) by QP = f\c^ysT1/3Q-py,

(10-62)

with 2

G~Pa = 9po + ( l - ^ ) | / i > , | - V p / ^ •

(10-63)

An concrete illustration, allowing the explicit evaluation of the relevant quantities, is provided by the polytropic case, as characterised by a (single constituent) equation of state giving the mass density p as a function of the number density n in terms of a fixed ("rest") mass per particle m, a scale constant K and a fixed dimensionless index 7 in the form p — mn + /en7 «=> p. — m + K7n 7 _ 1 ,

(10.64)

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Role of sonic metric in relativistic

superBuid

which corresponds to taking the pressure to be given by P = /cc 2 ( 7 -l)n>

= «:c2(7 - 1) (0—^)7/(7_1),

(10.65)

while the corresponding sound speed will be given (independently of K) by 2 Ci

= (7-l)(l-^)c2.

(10-66)

Acknowledgments: The author wishes to thank David Langlois and Isaac Khalatnikov for conversations and collaboration.

Bibliography [1] V.V. Lebedev, I.M. Khalatnikov, Sov. Phys. J.E.T.P.

56, 923 (1982).

[2] B. Carter, I.M. Khalatnikov "Equivalence of convective and potential derivations of covariant superfluid dynamics" Phys. Rev. D45, pp 45364544 (1992); [3] B. Carter, D. Langlois, "The Equation of state for cool relativistic two-constituent superfluid dynamics", Phys. Rev. D 5 1 , pp 5855-5864 (1995).[hep-th/9507 058] [4] B. Carter, "Covariant Theory of Conductivity in Ideal Fluid or Solid Media", in Relativistic Fluid Dynamics (C.I.M.E., Noto, May 1987) ed. A.M. Anile, & Y. Choquet-Bruhat, Lecture Notes in Mathematics 1385 pp 1-64 (Springer - Verlag, Heidelberg, 1989). [5] B. Carter, D. Langlois, "Kalb-Ramond coupled Vortex fibration model for relativistic Superfluid dynamics", Nuclear Physics B454, pp 402-424 (1995). [hep-th/9611082] [6] B. Carter, "Relativistic dynamics of vortex defects in superfluids" in Topological defects and the non-equilibrium dynamics of symmetry breaking phase transitions (Les Houches 99), ed. Y.M. Bunkov and H. Godfrin, pp 267-301 (Kluwer, 2000). [gr-qc/9907039] [7] M. Ruderman, "Millisecond pulsars and low mass X-ray binaries", Astroph. J. 366, 261 (1991). [8] D. Langlois, D. M. Sedrakian, B. Carter, "Differential rotation of relativistic superfluid in neutron stars" Mon. Not. Roy. Astr. Soc. 297, pp 1198-1201 (1998). [astro-ph/9711042] 263

264

Role of sonic metric in relativistic

superfluid

[9] B. Carter, D. Langlois, D. M. Sedrakian, "Centrifugal buoyancy as a mechanism for neutron star glitches" Astron. Astroph., 361, pp 795-802 (2000). [astro-ph/0004121]. [10] G.E. Volovik, "Exotic properties of superfluid helium 3", (World Scientific, Singapore, 1992). [11] B. Carter, "Perfect fluid and magnetic field conservation laws in the theory of black hole accretion rings", B. Carter in Active Galactic Nuclei, ed. C. Hazard & S. Mitton, pp 273-300 (Cambridge U.P., 1979). [12] B. Carter, "Black hole equilibrium states", in Black Holes (proc. 1972 Les Houches Summer School), ed. B. & C. DeWitt, pp 57-210 (Gordon and Breach, New York, 1973). [13] W. Unruh, "Experimental black hole evaporation", Phys. Rev. Letters 46, pp 1351-1357 (1981). [14] W. Unruh, "Dumb holes and the effects of high frequencies on black hole evaporation", Phys. Rev. D51, pp 2827-2838 (1995). [gr-qc/9409008] [15] C. Barcelo, S. Liberati, M. Visser, "Analog gravity from Bose-Einstein condensates". Class. Quantum Grav. 18 pp 1137-1156 (2001). [grqc/0011026] [16] A. Lichnerowicz. Relativistic Hydrodynamics and ics (Benjamin, New York, 1967).

Magnetohydrodynam-

[17] A.H. Taub, Phys. Rev. 94, 1469 (1954). [18] B. Schutz, Phys. Rev. D 2 , 2762 (1970). [19] B. Carter, Class. Quantum Grav. 11, pp 2013-2030 (1994). [20] B.Carter, in A Random Walk in Relativity and Cosmology, Proc. Vaidya-Raychaudhuri Festschrift, IAGRG 1983, ed. N. Dadhich, J. Krishna Rao, J.V. Narlikar, C.V. Vishveshwara, pp 48 -62 (Wiley Eastern, Bombay, 1985). [21] B. Carter, I.M. Khalatnikov, "Momentum, Vorticity, and Helicity in Covariant Superfluid Dynamics", Ann. Phys. 219, pp 243-265 (1992).

Brandon Carter

265

[22] W. Israel, Physics Letters A86, 79 (1981); Physics Letters A92, 77 (1982). [23] W.G. Dixon, Arch. Rat. Mech. Anal. 80, 159 (1982). [24] T.S. Olsen, Physics Letters A149, 71 (1990). [25] L.D. Landau, E.M. Lifshitz, Statistical Physics (trans. E. and R.F. Peierls), section VI.

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Chapter 11 Effective geometry in nonlinear field theory (Electrodynamics and Gravity) Mario Novello Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro Brazil E-mail: [email protected]

Abstract: The electromagnetic force a photon is subject to in a nonlinear regime can be geometrized, that is, written in terms of an "effective geometry". This is a rather unexpected result and, at the same time, a beautiful consequence of the analysis of the behavior of the discontinuities of nonhomogeneous nonlinear electromagnetic fields. We will show how such geometrization comes about. By the same token we will show that this property is not restricted to spin-1 fields (such as the photon) but on the contrary it is a rather general property of nonlinear field theories. We will limit our analysis here to the cases of spin-1 and spin-2. 267

268

11.1

Effective geometry in nonlinear held theory

Introduction

Modifications of light propagation in nonlinear media, and in non-standard quantum vacuum states, has recently become a subject of considerable interest. Investigations have shown that, under distinct non-trivial quantum vacua (related to such circumstances such as temperature, boundary conditions, quantum polarization, etc.), the motion of light can be viewed as though we were dealing with electromagnetic waves propagating through a classical refractive medium. The medium induces modifications to the photon equations of motion, which are described in terms of nonlinearities of the field. As a particular application of such a medium interpretation we consider modifications of electrodynamics due to virtual pair creation. In this case the nonlinear effects can be simulated by an effective Lagrangian which depends only on the two gauge invariants F and G of the electromagnetic field [1, 2]. One of the main achievements of these investigations is the understanding that, in such a nonlinear framework, photons propagate along geodesies that are no longer null in the physical Minkowski spacetime but are instead null in an induced "effective geometry". Although the basic understanding of this fact — at least for the specific case of Born-Infeld electrodynamics [3] — has been known for a long time [4], it has been scarcely noticed in the literature. Moreover, its consequences were not exploited in any systematic manner. In particular, we emphasize the general application of this idea, and the corresponding consequences of the method of the effective geometry as outlined below. Examination of photon propagation in situations that go beyond Maxwell electrodynamics has a rather diverse history: It has been investigated in curved spacetime, as a consequence of non-minimal coupling of electrodynamics with gravity [5, 6], and in nontrivial QED vacua, as an effective modification induced by quantum fluctuations [7, 1, 2]. As a consequence of these investigations some rather unexpected results appear. Just to point out a particularly striking example, we mention the possibility of faster-than-light photons. 1 The general approach of all these models is based on gauge invariant effective actions, which take into account the modifications of Maxwell elecJ The meaning of such expression is that the wave propagates along spacelike characteristics in the Minkowski background.

Mario Novello

269

trodynamics induced by many different sorts of processes. Such a procedure is intended to deal with the quantum vacuum as if it were a classical medium. Another important consequence of such a point of view is the possibility of interpreting all such vacuum-induced modifications — at least with respect to the photon propagation — as an effective change of the spacetime metric. This allows one to appeal to an analogy with the electromagnetic wave propagation in curved spacetime due to gravitational phenomena. It is this analogy, relating nonlinear media and gravitational fields, that is the central theme of this chapter.

11.2

Nonlinear electrodynamics

11.2.1

Definitions and notation

We call the electromagnetic tensor FMl/, while its dual F*„ is

Kp = \*°r Fr,

(n.i)

where eQ/3M„ is the completely antisymmetric Levi-Civita tensor; the Minkowski metric tensor is represented by its standard form rfv. The two standard invariants constructed with these tensors are defined as F = F^F^,

(11.2)

G E E F " " ^ .

(11.3)

Since the modifications of the vacuum which will be dealt here do not break the gauge invariance of the theory, the general form of the modified Lagrangian for electrodynamics may be written as a functional of these invariants, that is, L = L(F,G). (11.4) We denote by Lp and LG the derivatives of the Lagrangian L with respect to the invariant F and G, respectively; and similarly for the higher order derivatives. We are particularly interested in deriving the characteristic surfaces which guide the propagation of the field discontinuities. Let E be a surface of discontinuity for the electromagnetic field. Following Hadamard [8] we assume that the field itself is continuous when crossing S, while its first derivative exhibits a finite discontinuity. We accordingly set [F^h

= 0,

(11-5)

Effective geometry in nonlinear field theory

270 and

[dxFlw\E = fllvkx,

(11.6)

[ J ] E = lim (J\x+s - J\z-s)

(H.7)

in which the symbol <5-v0+

represents the discontinuity of the arbitrary function J through the surface E characterized by the equation E(xM) = constant. The tensor f^ is called the discontinuity of the field, and kx = dx-£

(11.8)

is the propagation vector.

11.2.2

One-parameter Lagrangians

In this section we will investigate the effects of nonlinearities on the equation of evolution of electromagnetic waves. We first consider the simple class of "one-parameter" gauge invariant Lagrangians defined by L = L{F).

(11.9)

From the least action principle we obtain the field equation 3M (LF F"v) = 0.

(11.10)

Applying conditions (11.5) and (11.6) for the discontinuity of the field equation (11.10) through E we obtain LFff,uK

+ 2LFF£F>wku

= 0,

(11.11)

where £ is defined by £EEF<^/Q/3.

(11.12)

The consequence of such discontinuity in the cyclic identity is Uukx + /„**„ + fx»K = 0.

(11.13)

In order to obtain a scalar relation we contract this equation with kar]aXF,il'', resulting in £ kvk„ rT + 2F"" fvx kxk„ = 0. (11.14)

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271

Let us consider the case in which £ does not vanish. 2 From equations (11.11) and (11.14) we obtain the propagation equation for the field discontinuities as given by (LF rfv - ALFF F^ Fav) k„kv = 0. (11.15) Expression (11.15) suggests that one can interpret the self-interaction of the background field F**", insofar as it concerns the propagation of electromagnetic discontinuities (11.6), as if it had induced a modification to the spacetime metric rj^, leading to the effective geometry C

= LF rfv - 4 LFF F\

Fav.

(11.16)

A simple inspection of this equation shows that only in the particular case of linear Maxwell electrodynamics does the discontinuity of the electromagnetic field propagate along null paths in the physical Minkowski background. The general expression of the effective geometry can be equivalently written in terms of [the vacuum expectation value (VEV) of] the energymomentum tensor, given by (11.17) where F is the effective action TEE [ d*xy/=yL,

(11.18)

and 7M„ the is Minkowski metric written in an arbitrary coordinate system; 7 is the corresponding determinant. In the case of one-parameter Lagrangians, L = L(F), we obtain T^ = -4LF F^ Fav - Lr,^,

(11.19)

where we have chosen an Euclidean coordinate system in which 7M„ reduces to 77P„. In terms of this tensor the effective geometry (11.16) can be re-written 2

For the case in which £ = 0, the quantity /MI/ is a singular two-form. Following Lichnerowicz [9], it can be decomposed in terms of the propagation vector k^ and a spacelike vector a,, = at^ orthogonal to k^, in which e^ is the normalized polarization vector. Hence, we can write f^ = k^a„ - k^a^ on E. From equation (11.11) it follows that f^K = 0, and contracting (11.13) with ^k,, yields ^vr\a^kak& = 0. Therefore, such modes propagate along standard null geodesies in Minkowski spacetime.

272

Effective geometry in nonlinear field theory


) rT + ^

TT.

(11.20)

We remark that, once the modified geometry along which the photon propagates depends upon the energy-momentum tensor distribution of the background electromagnetic field, it is tempting to search for an analogy with the corresponding behavior of photons in a gravitational field. We will return to this question later on. Therefore, the field discontinuities propagate along null geodesies in an effective geometry which depends on the field energy distribution. Let us point out that, as it is explicitly shown from the above equation, the stressenergy distribution of the field is the fundamental quantity responsible for the deviation of the geometry, as felt by photons, from its original Minkowski form.4 In order to show that the photon path is actually a geodesic curve, it is necessary to know the inverse g^ of the inverse effective metric gvX. That is, we want to calculate the covariant components g^ of the effective metric which are defined by

srglfx = S"x.

(n.21)

The calculation is simplified if we take into account the well known properties: F;vFvX = -l-GS\,

(11.22)

and F;x F*XV

- F„A Fx» = \ F 8 \ .

(11.23)

Thus the covariant form of the metric can be written in the form: 9^ = a V + bT^,

(11.24)

in which a and b are given in terms of the Lagrangian and its corresponding derivatives by:

a 3

= -b(JjrF+L+lT)>

(n-25)

For simplicity, we will denote the effective metric as g1"' instead of g^ from now on. For Ty,v — 0, the conformal modification in (11.20) clearly leaves the photon paths unchanged. 4

Mario Novello

273

and b = 16 ^

[(F 2 + G2) L2FF - 16 (L F + FLFFf

]~\

(11.26)

where T = T" is the trace of the energy-momentum tensor. Similarly, a related alternative representation is 9^ = Arj^ + B F^x FXu

(11.27)

A=^(LF

(11.28)

where A and B are + 2FLFF),

* = n f ,

(11-29)

and R is defined by R = LF2 + LFF (2FLF-

11.2.3

G2).

(11.30)

Effective null geodesies

The geometrical relevance of the effective geometry goes beyond its immediate definition. Indeed, in the following it will be shown that the integral curves of the vector kv (i.e., the photon trajectories) are in fact geodesies. In order to derive this result we will require an underlying Riemannian structure for the manifold associated with the effective geometry. In other words, this implies a set of Christoffel connection coefficients r%„ = Tavil, by means of which there exists a covariant differential operator VA (the covariant derivative) such that v A < r =
(11.31)

From (11.31) it follows that the effective connection coefficients are completely determined from the effective geometry by the usual Christoffel formula. Contracting (11.31) with k^K results in KK
PVA f .

(11.32)

Differentiating (11.50) we have 2AM,AW" + *».W,',A =

0

-

(11.33)

274

Effective geometry in nonlinear Geld theory

Inserting (11.32) for the last term on the left hand side of (11.33) we obtain

aTk^xK =
(n.34)

As the propagation vector k^ — £iM is an exact gradient one can write fcwA = k\.tll. With this identity, and defining k11 = g^k,,, equation (11.34) reads */.iA*A = 0, (11.35) which states that k^ is a geodesic vector. By remembering that it is also a null vector (with respect to the effective geometry g^"), it follows that its integral curves are therefore null geodesies.

11.2.4

Two-parameter Lagrangians

In this section we will go one step further and deal with the general case in which the effective action depends upon both electromagnetic invariants, that is L = L(F, G). (11.36) The equations of motion are d„ {LF F"" + LG F*"") = 0.

(11.37)

Our aim is to examine the propagation of the discontinuities in such a case. Following the same procedure as presented in the previous section one gets [LF /"* + 2 A F"" + 2BF*fi"]kl/ = 0,

(11.38)

In this expression we define A = 2^LFF

+ CLFG),

(11.39)

B = 2(^LFG

+ <:LGG),

(11.40)

and introduce a new quantity £, which complements the previously defined

£,by C = F<"%. a

(11.41) a

Contracting equation (11.37) with F ^ka and with F* llka, yields £LF + \BG

ifvkVLkv-2AFvaFaiikvkVi

=§

respectively, (11.42)

Mario Novello

275

and QLp-BF

+

^AG rT K k» - 1BF\

F^kykp

= 0.

(11.43)

In order to simplify our equations it is worth defining the quantity U = C/£. From equations (11.42) and (11.43) it follows that Q 2 fi 1 + fiQ2 + Q 3 = 0,

(11.44)

with the quantities fi;, i = 1,2,3 given by fii

=

—LFLFG

fi2

=

(LF + 2GLFG)(LGG-LFF)

Q3

=

LFLFG

+ 2FLFGLGG

+ 2FLFFLFG

(11.45)

+ G{LGG-LFG),

+ 2F(LFFLGG + G(LFG-LFF).

+

LFG),(UM) (11.47)

The quantity £1 is then given by the algebraic expression

where A = (fi 2 ) 2 - 4fii

fi3-

(11.49)

Thus, in the general case we are concerned here, the photon path is kinematically described by $""fcMfc„= 0, (11.50) where the effective metric g'"' is now given by gT

=

LF V"" - 4 [(LFF + SILFG) F " A F A " + (LFG + nLGG)F»xF*Xl'}.

(11.51)

When the Lagrangian does not depend on the invariant G, expression (11.51) reduces to the form (11.16). From the general expression of the energy-momentum tensor for an electromagnetic theory L = L(F, G) we have TM„ = -4LF F„a Fav -(L-G

LG) r / ^ .

(11.52)

The scale anomaly is given by the trace T = 4{-L

+ FLF + GLG).

(11.53)

276

Effective geometry in nonlinear field theory

We can then re-write the effective geometry in a more appealing form in terms of the energy momentum tensor, that is, gT = Mifv + NT»\

(11.54)

where the functions M. and H are given by M

= LF + G (LFG + Q LGG) + ^-(LFF + fl LFG) (L - G LG), (11.55) LF N = -^(LFF + CILFG). (11.56) LF As a consequence of this, the Minkowski norm of the propagation vector k^ reads

iT Kkv = —j^ T>1V K^v11.2.5

(n.57)

Exceptional Lagrangians

Equation (11.51) contains a remarkable result: the velocities of the photon are, in general, doubled. (That is, the system exhibits birefringence?) There are some exceptional cases, however, for which the uniqueness of the photon path is guaranteed by the equations of motion. Such uniqueness occurs for those dynamics described by Lagrangian L that satisfy the condition A = 0.

(11.58)

The most well-known example of such uniqueness for the photon velocity in a nonlinear theory is the situation in Born-Infeld electrodynamics [3]. Let us pause for a while in order to make the following remark: In the case of the Born-Infeld theory all three quantities fij, i = 1,2,3 vanish identically. Hence, in this situation we cannot obtain the effective geometry from equation (11.51). In this very exceptional case we proceed as follows. Let us return to the original equation (11.38). Now, the Lagrangian for the Born-Infeld model is provided by the expression

L=yJb* + \vF-^G-b2.

(11.59)

Substituting this form of L into equation (11.38) we obtain the unique characteristic equation

H*

b2 + =;F ) rT + F^x FXu k„kv = 0,

(11.60)

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277

thus yielding (for A ^ 0) 5 ^ = (ft2 + 1 F ) T/*" + F " A F*",

(11.61)

which does not show birefringence. This formula was obtained for the first time by Plebanski.

11.2.6

Electromagnetic traps

The possibility of writing the characteristic equation for nonlinear electrodynamics in terms of an effective modification of the spacetime metric has some unexpected and wide-ranging consequences. One of these concerns is the existence of trapped photons in a compact domain. Such a configuration is made possible due to the nonlinearity of electrodynamics, and has a striking resemblance to gravitational black holes, although not presenting all properties of the latter. We will concentrate here on a toy model, just to exhibit the possibility of new phenomena induced by the nonlinearities. Let us start by considering a static and spherically symmetric field for the case L = L(F). The source is an electric charge [electric monopole] located at the origin of the spherical coordinate system (t, r, 8, ip). The only non-zero components of the electromagnetic field are Ftr = ~Frt = E(r).

(11.62)

The equations of motion are easily solved, yielding LFE

= %.

(11.63)

The corresponding effective geometry (11.16) is given by gtt = -g"~ = LF-4LFFE\

(11.64)

while the remaining non-zero components of the inverse metric have values proportional to those of Minkowski geometry, 9M

=

~-2LF,

(11-65)

9W

=

—T^L"rz sin 0

(1L66)

Effective geometry in nonlinear field theory

278

From equation (11.64) it follows that it is possible to envisage the existence of a region T> defined by some finite radius r — rc such that gTT(rc) vanishes. The metric component gtt also vanishes at V. Then, coordinates r and t interchange their roles when crossing V, that is gtt(r > rc) > 0, grr{r > r c ) < 0, and gtl(r < rc) < 0, gTr{r < rc) > 0. Let us note, however, that the existence of such rc implies that there is a further undesirable consequence concerning the value of the electric field. Indeed, equation (11.63) can be written as r 2 = Q/(Lp E); then differentiating with respect to E and using (11.64) yields

r

m - mm •"•

<"67)

Thus it follows from this that, at T>, there is a possibility of the existence of a field barrier. That is, a limitation of the domain of the solution's existence due to the possibility of the inverse function r = r(E) having an extremum. In order to avoid such a limitation of the domain of definition for the electric field, the theory must be such that the second derivative vanishes at rc, that is dgr = 0. (11.68) dE v Additionally, one must impose the supplementary condition d2grr dE2

^ 0.

(11.69)

Thus, if the theory L = L(F) allows this kind of solution, then three typical properties of a trapped region appear: • There exists a null surface V, defined by r = rc in the effective geometry. • Coordinates t and r interchange their role when crossing V. • Light cones inside the region bounded by V are directed toward the origin of the r-coordinate which plays the role, in this domain and only for photon propagation, of a time-like coordinate. These theories should be further examined, since from what we have seen above, they may be the germ of the existence of the electromagnetic version of the gravitational black hole. Let us now turn our examination to a very similar situation inside material media.

Mario Novello

11.3

279

Nonlinear dielectric media

It is also possible to describe wave propagation in dielectrics, governed by ordinary Maxwell electrodynamics, in terms of a modification of the underlying spacetime geometry by using a variant of the framework developed above for nonlinear electrodynamics. In a dielectric, the electromagnetic field is represented by two antisymmetric tensors, the electromagnetic field FM„ and the polarization P^. These tensors are decomposed, in the standard way, into their corresponding electric and magnetic parts as seen by an observer which moves with four-velocity v^. We can write: Fp, = Ellvv~

EvuM + e ' V v p H a , + e'K'lll,vpBir.

P^ = D^vv-Dvvll

(11.70) (11.71)

Following Hadamard, we consider the discontinuities in the fields as given by [VA-^MIE

=

[VA-DJE

=

[VA#/JE

=

[VA^MIE

=

fc

AeM, k d

\m

kxhp, fe

A*V

(n-72)

For the simplest linear case, in which we have Da

— tEa, _ Ha

(11.73)

i

(11.74)

da = e e a ,

(11.75)

K

(11.76)

Ba it follows that

After a straightforward calculation one obtains *„ M 7 * " + ( « / * " I K « " ] = 0 -

(11-77)

Thus photon propagation is governed by an effective metric, the so-called Gordon metric Gordon = 7"" + ( € / , - ! ) « " « " .

(11.78)

Effective geometry in nonlinear field theory

280

Let us generalize this situation for nonlinear dielectrics. The suitably generalized Maxwell equations are given by 0 " / £ , = <),

(11.79)

a"/v = o.

(n.80)

For electrostatic fields inside an idealized "isotropic" dielectric P*"* and Ft"/ are related by PM„ = e(E) F^. (11.81) Here e is the electric susceptibility, and the magnetic susceptibility takes the somewhat unusual form fi = — 1/e so that e (i = 1. In the general case, for e = e (E) we simplify our calculation if we note that we can relate the equation of wave propagation to the previous analysis on vacuum polarization (11.9) by means of the identification LF —• e,

(11.82)

LFF —> - ^ ,

(11.83)

which implies

in which e' = d e/d E. Therefore, the simple class of effective Lagrangians (11.9) may be used as a convenient description of Maxwell's theory inside isotropic nonlinear dielectric media; conversely, results obtained in the latter context can as well be similarly restated in the former one. In a nonlinear dielectric medium the polarization induced by an external electric field is described by expressing the scalar function e as a power series in terms of the field strength E: e = Xi+X2E + X3E2 + X4E3 + ...

(11.84)

where the constants Xn are known as the n'th-order nonlinear optical susceptibilities. Note that we are using the standard convention [4] which relates Xn to the expansion of the polarization vector. For this case the effective geometry is given by

tf» = crr + i F"<* F°r

( 1L85 )

E It can also be re-written in the form ^

= e rfv - 1 (£" E" - E2 v» vv) ,

(11.86)

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281

where E2 = — EaEa > 0. In other words, in the rest frame of the medium, where uM = 6^t — (1;0) we have gu = e + e'E,

(11.87)

E This shows that the discontinuities of the electromagnetic field inside a nonlinear dielectric medium propagates along null cones of an effective geometry which depends on the characteristics of the medium given by equation (11.85). It is worth investigating under what conditions for the e dependence on E a kind of horizon-like barrier might appear for the photon inside such a nonlinear dielectric. We will return to this problem elsewhere.

11.4

Moving dielectrics

For the case of a moving dielectric medium, now generalizing by permitting t[i ^ 1, we proceed in a manner similar to that in the previous section. We set [P^h = 0, (11.89) and [dxP^h = P^kx,

(11.90)

It is again convenient to project these tensors in the framework of a real observer endowed with normalized four-velocity «**, thus denning the corresponding electric and magnetic vectors in the 3-dimensional rest-space of the observer v11: F^ = EltvvEvv„ + e";vvpBy, (11.91) and P»v = D^vv-

Dvv^ + e £ v p H a .

(11.92)

Accordingly we decompose the discontinuity tensors into their corresponding electric and magnetic parts: /M„ *-» (e^,6M) and pM„ «-> {d^h^j. The equations of motion are: a „ P ^ = 0. (11.93) and 0"F*=O.

(11.94)

282

Effective geometry in nonlinear Geld theory

Following the definitions and procedure presented previously, one gets from the discontinuity of the equations of motion [equations (11.93) and (11.94)]: (b?vu -bvVnt^""vfie„) k„ = 0, (11.95> and (dp vv - dv Vp - tyj" vp ha) kv = 0.

(11.96)

In the present section we shall focus our analysis on the case in which the polarization tensor is such that Da = eEa and Ba = fiHa. Additionally, we take the dielectric permittivity to be a real function of the electric field, that is e = e(E). In contrast \i is taken to be a fixed but unvarying constant, which is a good approximation for most physical dielectrics. Multiplying the equation of discontinuity by v1* we find that b^fcM= 0 and d^ W = 0. Thus, it follows that h^k^ = 0 and

in which e = de/dE. From the discontinuity equation we get 1 K = H(k . . „ .a„v. ). N W r v P e " k"a

(11-98)

Substituting this expression into the other discontinuity equation and multiplying by E^ we get after some algebraic manipulations

rT + Vivv(jie-l

+ ne'E>j -

-^=E»Ev kpkv = 0.

(11.99)

It then follows that the photon path is kinematically described by 9^klikv

= 0,

(11.100)

where the effective metric g^" is given by gr = rr + v*vv{ut-\ + ntE\ -—tft.

(11.101)

Mario Nbvello

283

Here i^ is the unit vector in the direction of the electric field. Two limits are of note: In the particular case in which e is a constant, this formula goes over to the reduced Gordon geometry Cordon = rT + « " « " ( M 6 - 1) .

(11.102)

Furthermore, if we pick a field configuration for which t\i — 1, we recover (up to an irrelevant conformal factor) the metric of the previous section [equation (11.86)]. The (covariant) metric g^, defined by
(11.103)

is given by

%»= v - «„«„ (I - ^I

+ Q) + j

where we have set £ = e E/t. The velocity of the photon \svph = kliv'i/\k\,

^ V*>

(n-104)

where

|Jfc| = (17""-«"«")*„*„

(11.105)

is the 3-dimensional norm of the wave-vector in the rest frame of the medium. Because there is no frequency-dependent dispersion, this is both the phase and group velocity. Then 1

l+gCQS^fl

t =

* ^V~ T K~'

(1L106)

in which 6 is the angle between the direction of the electric field and the propagation of the photon. Note that in the limit case in which £ vanishes, the photon velocity coincides with the square-root of the determinant of the effective metric. Indeed, for a geometry given by g^v = t]^ + a^ where a^v is symmetric, we have det g^

=

l + a+-a2

+

1 1 1 1 - (add) — - (dddd) - - a (da) + - a (dad) O

-

\a2(aa) 4

+ -a3+

T:

+ l(aa)2, o

—a4 - ~{aa)

Ci

O

(11.107)

Effective geometry in nonlinear field theory

284 where

a = a£, aa = a£ a^ aaa = a1^ a£ a" V "a "/? "i/.

(11.108)

Using this property it follows that the determinant of the effective metric is given by

det

^=^(iW

(ILIO9)

In the case that e does not depend on the electric field, the photon velocity can be written in terms of the determinant of the effective metric vph = y/g.

11.4.1

(11.110)

Discussion

An inspection on the effective metric for the particular restricted case examined by Gordon allows us to make the following complementary statements: • The constitutive relations can be easily displayed in terms of the Gordon geometry. Indeed, if we set pi" = -&"", A*

(11.111)

where Pv = gT ?" Pep,

and

F^ = sT gvfi FafS.

(11.112)

Then, PapyP — Da = tEa and correspondingly, Ba — fiHa. (The precise index placement is vitally important here, as is the way in which the Gordon metric is used to bootstrap the definition of P and

• The dynamics of the electromagnetic field in a linear medium can be written in terms of the Gordon geometry.

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This is nothing but a consequence of the following fact: The dynamical equations of the electromagnetic field in a medium (e, fi) are *Wa] = °>

(11.113)

P Q % = 0.

(11.114)

This last equation, written in terms of the tensor F'"'', yields

(fif*)^'.

dun)

where the symbol ; stands for the covariant derivative in the Gordon geometry, and F is defined by raising indices using the Gordon metric. These nice properties, that the metric determines the constitutive relations and the constitutive relations determine the metric, are restricted to the particular case of the Gordon metric and are unfortunately no longer valid in more general nonlinear media in which the dielectric function e depends on the electric field. Thus, the fact that the Gordon metric has a double role — that is, that it both permits us to re-write the dynamical equations of the electromagnetic field in a polarized medium and guides the evolution of the discontinuities of the field — is just a special case miracle for linear media and cannot be generalized for arbitrary non-linear structure. Note however that such generalization procedure is still possible for some particular cases. A remarkable example of this case is Born-Infeld electrodynamics [3]. We will analyze this later on.

11.4.2

Example I:

In order to show the efficiency of the method presented above we shall apply the previous analysis to display some peculiar properties of photon propagation in a specific field-dependent dielectric medium. This would be highly difficult to achieve using any other formulation. Indeed, we will show that the effective metric can mimic some properties of the geometry discovered by Godel in Einstein's general relativity, where closed paths in spacetime occur. The background Minkowski geometry written in a (t, r, tp, z) coordinate system is ds2 = dt2 - dr2 - r2 dip2 - dz2. (11.116)

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Effective geometry in nonlinear Geld theory

The physical system we analyse consists of a variable time-dependent magnetic field that induces an electric field such that the component F 0 2 does not vanish. For our purposes here we do not need to specify the field further. We choose the observer that is comoving with the dielectric medium endowed with a normalized velocity v^ = 7 (l, 0, £, 0), where 7 = f 1 — ^ ) and v = r<j>. It then follows that the non-null electric components are £•(0) = _ Ejr

2 7

2

- —

7

w,

9U = - 1 , 9

22

= - 1 + (fie - 1 + fie E) M

(11.117) (11.118)

2

g02 = (fie - 1 + fie E) M

- ^ 7 2

2

,

- — A ,

933 = - 1 ,

(11.119)

(11.120) (H.121)

Let us consider a curve defined by the equations t = constant, r = constant, and z = constant. The length element of this curve is ds2eff = g22d
(11.122)

This curve can be a photon path if the condition #22 = 0 is satisfied. In other words, if the velocity of the dielectric is such that 4 = V^-

(11.123)

It is actually possible for this velocity to be achieved by real physical materials (i.e., v < c) when the relative permittivity is less than one, that is, when the electric susceptibility x> defined as e = eo (1 + x) is negative. An example of this is found in materials where the dielectric response of induced dipoles is a resonant phenomenon. In this case, for frequencies above the characteristic frequency of the material OJ,, the electric susceptibility is negative and the equation for the velocity becomes: y2 2 / —1 = c fi0 e0 c \

47riVeffe2 ^ -. r , m uji(w - (Ji))

(11.124)

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where (m, e) are the mass and charge of a free electron respectively and iVeff is the oscillator strength times the total number of electrons per unit volume. 11.4.3

E x a m p l e II:

With the methods developed above we can also transpose part of the behavior of photons from the well-known combined Maxwell-Einstein framework to the nonlinear case of electrodynamics. (That is, we are no longer dealing with dielectrics but instead with nonlinearity in the fundamental electrodynamic Lagrangian.) An example of a physical situation where this can be realised will be dealt next. It again concerns the possibility of the existence of closed paths for photons in spacetime. The physical system we will analyse consists of a charged wire running through a solenoid. The flat Minkowski background geometry is written in a (t, r, 2 - dz2.

(11.125)

The non-null components of the Maxwell tensor F'"' compatible with the symmetry properties of the system are F 0 1 = E(r) and F12 = B(r). In this case the equation of motion of the system reduces to rLFF0l rLFF

= Q,

(11.126)

= /i,

(11.127)

12

where Q and // are constants that determine the charge density of the wire and the current carried through the solenoid respectively. We are interested in the analysis of the propagation of electromagnetic waves in the region inside the solenoid. Following our previous treatment we can assert that the photons propagate as if the metric structure of spacetime was changed into an effective Riemannian geometry. From equation (11.51) we obtain the following components of the effective metric g00

-=

gn -= g02 == g22

---

1-ipE2 2 2

-l-i>{B r -E ) ipEB

- ( i + VB2) T

g33

2

-= - 1 ,

(11.128) (11.129) (11.130) (11.131) (11.132)

Effective geometry in nonlinear field theory

288 where ip is given by

ip = 4

•>FF

(11.133)

The photon paths are null geodesies in this effective geometry. Consider a curve defined by the equations t = constant, r — constant, and z = constant. The length element of this curve is given by dSeff = g22d(p2.

(11.134)

This curve can become a photon path if there is a radius r = rc such that 922(rc) = 0. In terms of the contravariant components of the effective metric listed above this condition requires (1 - i>E2)

0.

(11.135)

The solution for this equation is 1/2

2Q 2Q(LFF\ r, = L \L F F

)

(11.136)

which implies that jpp

>0.

(11.137)

In order to present a simple example which exhibits such closed curves, we work with a Born-Infeld-like Lagrangian (11.138) Note that although highly speculative, this Lagrangian has the Maxwell limit for weak fields (F « /32). It also has an interesting property in the situation we are examining. Substituting the Lagrangian (11.138) into equation (11.135) we find that the magnetic field in this case takes the large value

*•-£

(11.139)

Other nonlinear Lagrangians such as the Euler-Heisenberg Lagrangian for QED, cannot be analyzed with the formalism presented here since they

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depend on both invariants F and G. This is only a technical limitation, not a fundamental one, and those kinds of systems will be discussed elsewhere. The real interest in this phenomenon is that it might be possible to be observed in the laboratory. This possibility rests in the analogy between the propagation of photons described by nonlinear Lagrangians in vacuum and that of photons described by Maxwell's theory in the presence of a dielectric.

11.4.4

Comments

It has been known from more than half a century that gravitational processes seem to allow the existence of closed paths in spacetime. This led to the belief that this strange situation occurs uniquely under the effect of gravity. In the present section we have shown that this is not the case. Indeed, we have demonstrated here that photons can apparently follow closed curves due to electromagnetic forces in a non-linear regime. We presented a specific example of a theory in which closed photon curves exist. In the limiting case in which the non-linearities are neglected, the presence of closed photon curves is no longer possible. Thus we can claim that this new property depends crucially on the non-linearity of the electromagnetic processes and it does not exist in Maxwell's theory. To close, we conclude that the existence of closed photon curves is not an exclusive property of the gravitational interaction: it appears also in pure electromagnetic processes, depending on the non-linearities of the background field. The existence of such closed photon curves in both gravitational and electromagnetic processes demonstrates the need for a fundamental review of the notion of causal structure as displayed by photon geodesies.

11.5

Non-trivial quantum vacua

There have been some comments in the literature regarding a possible connection between the change of light velocity in modified quantum vacua and the scale anomaly (the conformal anomaly). The first to point out such possible connection was Shore [2]. Dittrich and Gies [1] have examined this issue in connection with the Euler-Heisenberg effective action [10]. Using the method of effective geometry we are able to show that there is no deep correlation between the existence of the velocity shifts induced by the nonlinear quantum fluctuations and the scale anomaly. We will consider

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this problem in a general framework. In other words, let us analyse the photon velocity in an arbitrary case, without specifying a particular quantum theory. From what we have learned above one could reasonably suspect that nonlinearity induced by the modification of the electrodynamic vacuum is the necessary and sufficient condition to modify the photon velocity. However, this is not true in general, as it will be seen below.

11.5.1

The conformal vacuum

An important example of modification of the action occurs for the case in which the quantity M vanishes. [See equations (11.54), (11.55), and (11.56).] In this case the net effect of this action on the photon velocity is indistinct from the one produced by the classical vacuum. Could this occur for the case in which T ^ 0? This is answered by the following lemma. Lemma 1 There exist nonlinear modifications of electrodynamics having a non-zero scale anomaly (conformal anomaly) such that the field discontinuities nevertheless propagate along Minkowskian paths. The proof is immediate. From equation (11.54), the general condition for such statement to hold is expressed by LFF + nLFG

= 0.

(11.140)

Any nonlinear action which satisfies this condition is such that the photon propagation occurs in an effective geometry g'"', which is conformal to the Minkowskian one. In this theory, the photon presents the same light-cone structure as it does in linear Maxwell electrodynamics. The important point to consider here is that this situation occurs even in the presence of a nonvanishing scale anomaly. Inserting Q = —LFF/LFG from (11.140) into equation (11.44) we obtain, after some algebra, the condition (LFF LQQ — LFG) [LF LFQ + G \LFF LQG — LFG)\

= 0.

(11.141)

It remains to show that the spectrum of common solutions of (11.140) and (11.141) does not imply T = 0. This can be explicitly shown for the particular class of nonlinear Lagrangians given by L = -±F

+ f(G),

(11.142)

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for which LFF = 0 and LFG = 0. The scale anomaly (11.53) for the class shown in (11.142) takes the form T = 4

(
(11.143)

which vanishes only if f(G) is a linear function of G. Expression (11.48) yields Q = 0 and fi = {AG d2 f / d G2)~l, from which (11.55) gives M = - 1 / 4 and M = 0, respectively. The apparent singularity in (11.54) for M. = 0 can be circumvented by carefully returning to the original expression (11.38), which reduces in this case to /""*:„ = ^•F*lu'kv.

(11.144)

It follows that rfk^k,, — 0, thus adequately describing both solutions of (11.142). Thus a non-zero scale anomaly is not a sufficient condition for modifying photon propagation.

11.5.2

The non-anomalous vacuum

There is no better way to demonstrate the complete independence of the concepts of light velocity modification and the scale anomaly than to consider the converse situation of the previous lemma, a situation for which there is no anomaly at all. This is provided by the following Lemma 2 In the absence of scale anomaly the effective metric is not necessarily conformally flat. Indeed, let us set T = 0.

(11.145)

L = FLF + GLG,

(11.146)

FLFF + GLGF = 0,

(11.147)

GLGG + FLFG = 0.

(11.148)

Prom (11.53) it follows that

which leads to and

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Effective geometry in nonlinear Geld theory

In order for the effective metric to be conformally flat the Lagrangian should obey, besides the above relations, the condition (11.140). Then, it is straightforward to show that Lagrangians satisfying all these requirements must satisfy the condition LFF = 0. (11.149) Hence, when there is no scale anomaly, the unique case in which the effective geometry coincides with the Minkowski one is the linear Maxwell theory. This property allows us to conclude that in the framework of the effective action there is no need for a scale anomaly to induce light velocity shifts. It remains to show that the spectrum of solutions of equation (11.145) does not necessarily reduce to the linear case. It is interesting to point out that not only one but a particular set of nonlinear Lagrangians can be obtained. Indeed, it is immediately seen that L = Gf(F/G)

(11.150)

satisfies equation (11.146) for arbitrary functions f(F/G). Thus a non-zero scale anomaly is not a necessary condition for modifying photon propagation.

11.5.3

Euler-Heisenberg vacuum

The effective action for electrodynamics due to one-loop quantum corrections was calculated by Heisenberg and Euler. For the low-frequency limit v
(11.151)

with 45 \mecj mecz and where a is the fine-structure constant. The trace of the corresponding modified energy-momentum tensor reads

H

T = n[F* + -G*\.

(11.153)

The coefficients M and Af of the associated metric tensor are M M

= = --\\ ++ liF, nF,

(11.154)

M

= -2fi.

(11.155)

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We note that only in the case where both invariants F and G vanish does the scale anomaly (trace anomaly) disappear. In this case the above coefficients become M — - 1 / 4 and J\f = -1[i. The effective geometry is ffEuler-Heisenberg = ( " \ + ^ )

11.5.4

> T " 2/iT'"'.

(11.156)

Photon paths

From what we have learned above it follows that the equation of motion of the photon in a nonlinear regime is given by the variational principle J ds = 0,

(11.157)

in which the fundamental length is constructed with the effective metric, ds 2 = g^ dx*1 dx". Let us remark that if the physical background geometry is not Minkowski the effective metric has the same structure as previously exhibited with the natural substitution of r)^ by the corresponding gravitational metric. In the particular case which will be examined in the next section concerning the static and spherically symmetric configuration the photon path becomes S I (gtt i2 + grr r2 + gee 62 + gw
(11.158)

in which a dot means derivative with respect to the fundamental length variable s. The equation for the angular variable 6 shows that we can conveniently choose the initial condition such that 6 remains constant. From an analogy with planetary motion we set 0 = ir/2. The corresponding equations of the remaining variables are r V = h„,

(11.159)

gtti = E0,

(11.160)

where h0 and E0 are constants of motion. It is rather convenient to obtain the equation for r by making use of the fact that we are dealing with a null curve, and set 9tti2 + grrf2 + gvvLp2 = 0.

(11.161)

Effective geometry in nonlinear field theory

294

Thus, using the above equations for the evolution of t, 9 and

(11.162)

in which the potential V(r) takes the form V(r) =

Z

^ 9tt

11.5.5

+ - ^

9tt rz

+ E2

(11.163)

Circular orbits

The above set of equations allows the possibility of the existence of circular orbits r = r0 = constant for the photon. In this case, it is sufficient for the value of the tt—component for the effective metric at the point r„ to take the value 9tt(r0) = ^r2o,

(11.164)

in which we have, for comparison with the planetary motion, defined the impact parameter b = h0/E0. It is a rather simple and straightforward matter to show that such orbits are unstable, as one should suspect.

11.5.6

Dyadosphere

Recently Ruffini has analysed the consequence of the existence of a region just outside an event horizon of a non-rotating black hole in which the electromagnetic field exceeds the critical value predicted by Heisenberg and Euler. He called such a region a dyadosphere. This can be used as a good example of a nonlinear process that yields an effective geometry. Let us make a simple exercise and examine the consequences of such nonlinearity on the bending of light in the dyadosphere. The first step is to find the background geometry generated by the electromagnetic field using Einstein's general relativity. From the minimal coupling principle the dynamics of the electromagnetic field is given by: ( L F F " % = 0.

(11.165)

We search for the geometry of the spacetime in a static and spherically symmetric configuration, that is: ds 2 = A(r) dt2 - Air)'1

dr 2 - r 2 d0 2 - r2 s i n 2 ( 0 ) d / .

(11.166)

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Using equation (11.151) one obtains, in first order in /i, the expression for the metric coefficient A:

_?=

A(r) = 1

+

^ _ ^

(11.167)

From what we have shown in previous sections, the wave propagation is characterized by the modified light condition: kakp gaf} = - 4 ^p-

F»a F\ fcM k„

(11.168)

Lip

in which the background metric physical g^ is provided by equations (11.166) and (11.167). Put another way, the effective metric is now:

gT K K = (sT + 4 ^

F"° F"}j fc„ kv = 0.

(11.169)

In the case we are considering here we have: g°° = A(r)-l[l n

~g

+ 8nf(r)2] 2

= -A(r)[l 1

f

+ 8vf(r) }

(11.171)

22

(11.172)

= g

33

9

(11.170)

33

=9

(11-173)

Thus the photon propagates in the modified geometry given by ds 2 = [1 - 8/x/(r) 2 ] [A(r)dt2 - A{r)~l&r2] - r 2 d0 2 - r 2 sin2 0dy>2. (11.174) The photon follows a geodesies in this geometry, that is: [1 - 8nf(r)2][A(r)i]

= constant = h0;

2

r

r2 =

(11.175) (11.176)

2

h

l A{r) 2

(U 177) [l-8M/(r) ] r [l-8/i/(r)2]; ' in which a dot means the derivative with respect to the parameter s. Note that we have adjusted the initial conditions 6 = f and 8 = 0. Changing variables using r = l/v we have after some manipulations the radial equation for the photon propagation:

v" + v = 3mv2 - ( l - 3 2 ^ f

2

Q ) K
+ 0 ( M 2 , v4).

(11.178)

Note that the contribution to photon path deflection coming from one-loop QED can be of the same order of magnitude as the classical ReissnerNordstrom charge term.

296

11.6

Effective geometry in nonlinear field theory

Preliminary synthesis

The preceding analysis demonstrates that the propagation of discontinuities of the electromagnetic field in a nonlinear regime (as occurs, for instance, in dielectrics or in modified QED vacua) can be described in terms of an effective modification of the geometry of spacetime. Such an interpretation is an immediate consequence of the analysis we presented here. In particular, as a strange example, we have shown that the characteristic surfaces along which photons propagate in nonlinear electrodynamics, could appear as space-like hypersurfaces in Minkowski spacetime. This description also allows us to recognize a striking analogy between photon propagation in nonlinear electrodynamics and its behavior in an external gravitational field. In both cases the geometry is modified by a nonlinear process. It is clear that such an analogy can not be pushed very far, since in the gravitational case the modified geometry is observed by all kinds of matter and energy (including gravitational energy — at least in general relativity 5 ) and in the electromagnetic case this modified geometry is observed only by the nonlinearly coupled photons. This effective metric analogy certainly deserves further examination, since it may provide for the existence of an electromagnetic analogue of the gravitational black hole. We have also applied this formalism to generate closed paths for photons in non-linear electrodynamics, following the procedure used by Godel in general relativity [11]. It seems worth to make some comments in order to avoid possible misunderstandings. The analysis that Godel carried out in general relativity will in this electromagnetic context only make sense for the nonlinear photons. These propagate following geodesies of the effective metric. All other particles, interactions and observers propagate in the Minkowski background. A class of synchronized inertial observers on a hypersurface x° = constant, that contains the closed curve we are considering, will see this photon path as a closed spacelike curve. It should be pointed out that the locus of these points on the manifold is the same for all possible class of inertial observers. Only the causal orientation can be modified. As is well known from special relativity [12], a continuous sequence of points that belong to a spacelike curve can have its causal orientation modified according to the different classes of observers that observe it. There are 5 For the recent proposal of NDL theory of gravity, as we shall see, this would not be the case.

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three distinct types of possible orientations (each of them corresponding to a particular class of observers): (a) forward in time, (b) simultaneous and (c) backward in time. We are considering a closed curve, so in cases (a) and (c) this classification will split into two orientations since the curve can be divided into two segments of curve connecting the two points of the curve that have maximal separation in time. In one of the segments of the curve the orientation will be forward in time and in the other it will be backward in time. It has been known from more than half a century that gravitational processes allow the existence of closed paths in spacetime. This led to the belief that this strange situation occurs uniquely under the effect of gravity. As we have shown this is not the case: photons can also follow closed curves due to electromagnetic forces in a non-linear regime. We presented a specific example of a theory in which closed photon paths exist. In the limiting case in which the non-linearities are neglected, the presence of closed photon paths is no longer possible. This new property depends crucially on the non-linearity of the electromagnetic processes and it does not exist in Maxwell's theory. A further lesson that we can extract from the above analysis is that the existence of closed photon paths is not an exclusive property of the gravitational interaction: It appears also in pure electromagnetic processes, depending on the non-linearities of the background field. The existence of such closed curves in both gravitational and electromagnetic processes suggests the need for a careful reanalysis of the meaning of the causal structure as displayed by the geodesies of the photons. Let us close this section with a remark concerning the further analysis of the effective geometry with regard to the dynamics of the field: It is well known that the exceptional property of the Born-Infeld theory reposes on the fact that its dynamics can be obtained in terms of the determinant of a function of the field. Indeed, we have from equation (11.59):

'•='ll14,-s(?-f'

(11179)

This form was re-written by Born and Infeld in terms of the Minkowski metric of the background and the antisymmetric field tensor in a very appealing way by means of the identity

dQt

{^ + lF^)=1

+ F

^ -T^G2-

(1L180)

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Effective geometry in nonlinear field theory

The important point we would like to point out here is that this Action can be written also directly in terms of the effective geometry. Indeed, we have the identity det (Ar,^ + B F^ Fav) = {A2 - | W)2,

(11.181)

where W = AF + \BG2. This allows us to recognize that the Born-Infeld theory is written exclusively in terms of the effective geometry: LBI = b2 det L^

- 1 F„a F°v) * - b2.

(11.182)

To close this section we emphasize that the existence of closed photon paths is not an exclusive property of the gravitational interaction: it appears also in pure electromagnetic processes, depending on the non-linearities of the background field. The existence of such closed photon paths in both gravitational and electromagnetic processes asks for a deep review of the causal structure as displayed by the geodesies of the photons.

11.7

T h e case of spin 2 (gravity)

It is rather well-known that non-linear theories of electromagnetism allow for the phenomenon of birefringence. The origin for this can be traced to the fact that the photon, in non-linear electrodynamics, propagates in space-time as if its metric structure were no longer Minkowskian but instead an effective Riemannian one, the structure of which depends on the background electromagnetic field. Thus, one should emphasise that nonlinear electrodynamics deals with two metrics: the background metric and a second effective metric that is felt only by the photons. We emphasize that this duplication phenomenon of the metric structure of spacetime is just a pure electromagnetic effect. Although it is not mandatory, its appearance serves to simplify the description of photon propagation (in the regime of high frequency.) Here we would now like to analyse the corresponding problem: • Does non-linear gravity exhibit a similar phenomenon of birefringence? It is well-known that standard general relativity does not present birefringence for gravitational waves. Due to the recent renewal of interest in more

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general field theories of gravity, this question should be treated in a wider context, and be analysed in terms of the massless spin-two formalism. We will do this and shall prove that the phenomenon of birefringence is typical for almost all non-linear spin-two field theory. Let us anticipate and make a comment concerning the existence of two distinct metrics for the nonlinear theory. This occurs for all integer spin fields. In particular we shall see it occurs also for the spin-two field, since the crucial property is not its spin but its nonlinearity. A warning should be made explicit here: The relativity community is accustomed to considering the presence of two metrics as some sort of twotensor gravity theory. This is not the case here since the extra, effective metric that appears in nonlinear theories — be it of spin 1 or 2 — depends on the structure of the background field. We examine here only those theories of gravity that are described uniquely in terms of a single spin-two field, which we will represent by (p^.

11.7.1

Notation and Definitions

We define a three-index tensor FQ/g^, which we will call the "gravitational field", in terms of the symmetric variable tp^ (which will be treated as the potential), tp^ will be assumed to describe a spin-two field, and F ^ to be given by the expression6 Fapn = -(W[";/J] + -F[«7/%i),

(11.183)

where Fa represents the trace Fa = Faii„rv

= ¥>,« - V W 7 " " -

(H-184)

Prom the above definition it follows that F^ is anti-symmetric in the first pair of indices and obeys the cyclic identity, that is Faiw + Fmv = 0, F^ 6

+ F^a + FVQIi = 0.

(11.185) (11.186)

We are using the anti-symmetrization symbol [x, y] = xy — yx and the symmetrization symbol (x, y) = xy + yx. Note that indices are raised and lowered by the background metric 7MI/. The covariant derivative is denoted by a semicomma ';' and it is constructed with this metric.

Effective geometry in nonlinear Held theory

300

The most general non-linear theory must be a function of the invariants one can construct with the field. There are three of them, which we represent by A, B, and W. That is: A =

FaiiVFa»v

B

=

FMF"

W

=

Faf,xF*a0x

= FapxF'mX^llv

(11.187)

We will deal here only with the two invariants U = A — B and W. The reason for this rests on the linear limit. Indeed, in order to obtain the standard Fierz linear theory — as is required of any candidate to represent the dynamics of a spin-two field — only the combination U is required. 7 This is the case, for instance in Einstein's general relativity theory. Under this condition, the general form of the Lagrangian density is given by L = L(U,W). (11.188) In this way, the gravitational action is expressed as S=

/d4j;/=7L,

(11.189)

where 7 is the determinant of the flat spacetime metric 7M„ written in an arbitrary coordinate system. From the Hamilton principle we find the following equation of motion in the absence of material sources: L

FKn")

+

Lw

F*K»»)

= 0.

(11.190)

Lx represents the derivative of the Lagrangian with respect to the invariant X, which may be U or W. In a recent paper [16] a modification of the standard Feynman-Deser approach [13, 14, 15] to the field theoretical derivation of Einstein's general relativity, which led to a competitive gravitational theory, was presented. The main lines of such NDL approach can be summarized as follows: • Gravity is described by a symmetric second rank tensor <£M„ that satisfies a non-linear equation of motion; 7

The linear term in W does not contribute for the dynamics since it is a topological invariant, related to the Pontryagin index.

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• Matter couples to gravity in an universal way. In this interaction, the gravitational field appears only in the combination 7 ^ 4- ip^, inducing us to define a quantity g^ = 7M„ + <^„. This tensor g^v acts as an effective metric tensor of the spacetime as seen by matter or energy of any form except gravitational energy; • The self-interaction of the gravitational field breaks the above universal modification of the spacetime geometry.

11.7.2

The case of gravitational waves

One-parameter Lagrangians Our main purpose in this subsection is to investigate the effects of nonlinearities in the equation of evolution of gravitational waves. We will first restrict the analysis in this section to the simple class of Lagrangians 8 defined by L = L(U).

(11.191)

Prom the least action principle we obtain the field equation [LuFX(-"^],x = 0.

(11.192)

Applying the gravitational analogue of conditions (11.5) and (11.6) for the discontinuity of the field equation (11.192) through £ we obtain W ) ^

+ 2 ^ F

M

M

)

F

= 0

(11.193)

where £ is now defined by e = F^"/^M-F"FM.

(11.194)

The consequence of such a discontinuity in the cyclic identity (11.186) yields after some algebraic steps

fr^fc"*" + 8

2Fa^f0tiUkakff Faffxfak0kx

The NDL theory is contained in this class.

+ Fafpkak^

= O.

(11.195)

Effective geometry in nonlinear field theory

302

From these equations we obtain the propagation equation for the field discontinuities as given by

LurTKK

+

ALUUIF^F"^

_ I ^ M Faliv - ^ F " J kMkr = 0.

(11.196)

Expression (11.196) suggests that one can interpret the self-interaction of the background field F^va, insofar as it concerns the propagation of the discontinuities, as though it had induced a modification on the spacetime metric rj^, leading to the effective geometry

g% = Lv rT + 4 Lvu (F^

F\0

- l-Fa^ F a / - i w ) . (11.197)

A simple inspection of this equation shows that only in the particular case of the linear theory does the discontinuity of the gravitational field propagate along null paths in the Minkowski background. Two-parameter Lagrangians In this section we will go one step further and deal with the more general case in which the effective action depends upon both invariants, that is L = L{U,W).

(11.198)

The equations of motion are given by 11.190. Our aim is to examine the propagation of the discontinuities in such a case. We have the two basic equations {2AFa^

+ AF^

+ BF^

+ 2L„faiu>) ka = 0,

(11.199)

and {&„ - 2F^faiaf

+ Fbetafa)

kakp = 0.

(11.200)

In these expressions we have, in conformance with the previous electrodynamic calculation, set A = 2{^LUU

+ 2(LUW),

(11.201)

B = 2(ZLUW

+ 2CLWW),

(11.202)

Mario Novello

303

and £ is now defined by C = Faf)» / • / .

(11.203)

Following the same procedure as presented in the previous section we arrive at kpfca

£LuVa0 + A(2F^vFatlv

- l-F^F^"

-

FaF^

= 0,(11.204)

and ™8™a

(r}Lv - BU)r)a0 + +

tp^v0p

a v

B^F^F* Itu

_ p<*pP _ pxpK*&)\ = 0.

(11.205)

Considerable simplifications can be made if we take into account the following identities F ^ T V - F^FaiiV = - ^FSPa, (11.206) * T V

= - F%XFX + ^F^F*^.

(11.207)

In order to simplify our equations it is again worth defining the quantity Q = C/£, in analogy with the situation in nonlinear electrodynamics. The quantity Q is then given by the algebraic expression n± =

L

WW-LUV±^^

( n

208)

2Luw where A = (Luu - Luwf + 4 ( W ) 2 (11.209) Thus, in the general case we are concerned with here, the photon path is kinematically described by sTk^kv

= Q,

(11.210)

where the effective metric g^v is given by ST

= Lur1>"' + 2{Luu + x (2F^vFallv

n±Luw) - ^F^

F^" - Fa F0j .

(11.211)

When the Lagrangian does not depend on the invariant W, expression (11.211) reduces to the form (11.197).

304

11.8

Effective geometry in nonlinear field theory

Conclusions

As we have seen, the propagation of waves in a given nonlinear field theory can be described in terms of a modification of the geometrical properties of the background spacetime in which the waves propagate. In the cases of spin 1 and spin 2 we have shown that this modification can be associated with a Lorentzian (pseudo-Riemannian) structure. To be specific we have seen that the waves propagate along the null geodesies of an effective geometry that depends on the background metric and the distribution of the associated field. This property is not specific to a given class of theories, but is completely general. For the case of spin 2 we have considered a class of models that depend on the invariants U and W as defined above. This class does not contain ordinary general relativity, which is a spin-2 field theory constructed in a rather distinct and specific manner, as been demonstrated many times in the literature (see for instance the book of Feynman [13] for a rather didactic exposition). One should ask if some sort of spin 2 field theory could be constructed that has similar properties to those of the Born-Infeld (BI) theory for the case of spin 1. That is, is it possible to construct the dynamics in such a way that it is expressed solely in terms of the effective geometry? The answer is "yes" and it is left for the reader to show that a similar Born-Infeld like spin 2 theory can indeed be constructed.

Bibliography [1] W. Dittrich and H. Gies, "Light propagation in non-trivial QED vacua," Phys. Rev. D 58 (1998) 025004 [hep-ph/9804375]. [2] G. M. Shore, '"Faster than light' photons in gravitational fields: Causality, anomalies and horizons," Nucl. Phys. B 460 (1996) 379 [grqc/9504041]. [3] M. Born, "Modified field equation with a finite radius for the electron", Nature 132 (1934) 282. M. Born, "Cosmic rays and the new field theory", Nature 133 (1934) 63. M. Born and L. Infeld, "Foundations Of The New Field Theory," Proc. Roy. Soc. Lond. A 144 (1934) 425. [4] J. Plebansky, Lectures on Non-linear Electrodynamics, hagen, 1968).

(Nordita, Copen-

[5] I. T. Drummond and S. J. Hathrell, "QED Vacuum Polarization In A Background Gravitational Field And Its Effect On The Velocity Of Photons," Phys. Rev. D 22 (1980) 343. [6] M. Novello and S. D. Jorda, "Does There Exist A Cosmological Horizon Problem?," Mod. Phys. Lett. A 4 (1989) 1809. [7] J. I. Latorre, P. Pascual and R. Tarrach, "Speed of light in nontrivial vacua," Nucl. Phys. B 437 (1995) 60 [hep-th/9408016]. [8] J. Hadamard, Legons sur la propagation des ondes et les equations de I'hydrodynamique, (Ed. Hermann, Paris, 1903). [9] A. Lichnerowicz, Geometrie des groups de transformations, Paris, 1958). 305

(Ed. Dunod,

306

Effective geometry in nonlinear field theory

[10] W. Heisenberg and H. Euler, "Consequences Of Dirac's Theory Of Positrons," Z. Phys. 98 (1936) 714. [11] K. Godel, "An Example Of A New Type Of Cosmological Solutions Of Einstein's Field Equations Of Gravitation," Rev. Mod. Phys. 21 (1949) 447. [12] J.L. Anderson, Principles of Relativity Physics, (Academic Press, 1973). [13] R. P. Feynman, F. B. Morinigo, W. G. Wagner and B. Hatfield, "Feynman lectures on gravitation", Reading, USA: Addison-Wesley (1995) 232 p. (The advanced book program). [14] S. Deser, "Self-interaction And Gauge Invariance", Gen. Rel. Grav. 1 (1970) 9. [15] L. P. Grischuck, A. N. Petrov, and A .D. Popova, "Exact theory of the (Einstein) gravitational field in an arbitrary background spacetime", Commun. Math. Phys. 94 (1984) 379. [16] M. Novello, V. A. De Lorenci and L., R. de Freitas, "Do the gravitational waves travel at light velocity?", Anrials Phys. 254 (1997) 83.

Chapter 12 Non-inertial quantum mechanical fluctuations Haret R o s u

Instituto Potosino de Investigacion Cientifica y Tecnologica Apartado Postal 3-74 Tangamanga, San Luis Potosi Mexico E-mail: [email protected]

Abstract: Zero point quantum fluctuations as seen from non-inertial reference frames are of interest for several reasons. In particular, because phenomena such as Unruh radiation (acceleration radiation) and Hawking radiation (quantum leakage from a black hole) depend intrinsically on both quantum zero-point fluctuations and some appropriate notion of an accelerating vacuum state, any experimental test of zero-point fluctuations in non-inertial frames is implicitly a test of the foundations of quantum field theory, and the Unruh and Hawking effects. 307

308

12.1

Non-inertial quantum mechanical

fluctuations

Introduction

Analogue or not, the ultimate goal of the physics described in this book is to find clear evidence for gravitational and non-inertial vacuum radiation. Recognized as some of the most important paradigms of present-day theoretical physics, the Hawking and Unruh effects are (as yet) not much more than academic results which, because of the scales of the required energies/ accelerations/ masses, are not easy to implement in real laboratory experiments. Indeed, from the experimental standpoint, they might seem to be merely exotic interpretations for what could be explained by far more mundane physical effects in quantum electrodynamics, quantum optics, and hydrodynamics. As counterpoint, in the case of the Unruh effect the so-called detector method provides a well-defined radiation pattern that may be thought of as vacuum noise, and should be kept under consideration for possible experimental detection in clean analogue experiments. In this chapter, old results of Letaw on scalar vacuum radiation patterns are used to emphasize the radiometric nature of the various Frenet-Serret invariants for certain classes of "stationary" worldlines. This formalism is an extension of, and alternative to, the usual notion: that of adopting the thermal interpretation of the vacuum excitations as seen by a uniformly accelerated quantum detector (Unruh's interpretation). I focus next on the electromagnetic vacuum noise, surveying the HacyanSarmiento approach for calculating physical quantities in the electromagnetic vacuum. The application of this approach to circular worldlines led Mane to propose the identification of Hacyan-Sarmiento zero-point radiation with the ordinary synchrotron radiation; but here I provide some simple counterarguments. I also briefly discuss Bell and Leinaas' proposal of considering electrons in storage rings as prototypes for an Unruh-DeWitt spin polarization detector. In the final section, I sketch the similarity between the Unruh effect and the so-called "anomalous Doppler effect". A separate observation of the latter would mean a confirmation of the possibility of the first.

12.2

Vacuum Field Noise — V F N

The quantum vacuum field noise (VFN) [1] that is recorded by a detector moving along some classical trajectory will in general depend on that trajec-

Haret Rosu

309

tory. For certain restricted classes of worldline trajectories x"(s), most usefully parametrized in terms of proper time s, the observed power spectrum is stationary (time-independent). Experimental observation of the power spectrum of the vacuum field noise is then an important diagnostic tool that can be inverted to extract information about the form of the trajectory — specifically, the curvature invariants (Frenet-Serret invariants) of the worldline. As usual, I model an idealized detector as a simple two-level quantum system [usually known as an Unruh-DeWitt detector]. For scalar quantum field vacua there are six broad classes of trajectory that lead to stationary noise spectra. Basic results were derived by Letaw [2] some time ago, and are reviewed below. They might be of direct experimental interest in the acoustic analogy. On the other hand, one should also keep in mind that nonstationary vacuum noises are not completely beyond experimental reach, and can be analyzed by related mathematical methods which I briefly comment on.

12.2.1

T h e d e t e c t o r m e t h o d in q u a n t u m field t h e o r y

For the idealized Unruh-DeWitt detector, the interaction between the detector [endowed with a monopole moment Q(s) ] and the scalar field (s) = (x(s)) is the field along the worldline of the detector and A is a small coupling constant that I re-scale to A = 1 (since it is not important for current considerations). Detecting particles in the Unruh-DeWitt apparatus requires one to adopt adiabatic switching appropriate for the perturbative approach. At s = —oo, the detector is in the ground-state \Eo) and the field is in the Minkowski vacuum \0M)- After the detector-field interaction is switched on, the detector would not remain in the state \E0), but would make a transition to l-Ei). It is said that the detector "detects" some particles. Then, the transition amplitude for the detector field system to be found in \Ei,tp) at s = +00 is given by first-order perturbation theory as A = i~IEUII> \f°° ds Q(s) (s) E0,0U\

.

(12.2)

In order for first-order perturbation theory to apply one has to assume that the matrix element of Q is sufficiently small. On the other hand, from the

310

Non-inertial quantum mechanical

fluctuations

time evolution of the operator Q in the Heisenberg picture Q(s) = eiHD s'h Q(0) e~iHD

s/h

,

(12.3)

where Hp is the detector Hamiltonian, one immediately obtains oo dsei(£l-£oWA(V,|0(s)|oM).

/

(12.4)

oo

After summation over all final states of the field 1^), the transition rate, i.e., the transition probability per unit proper time from E0 to E\ is ^H<£i|Q(0)|£o)|2SM, where UJ = {E\ — E0)/h

(12.5)

and oo

/

d(a - s') e-iu(s-s,)

g(s - s').

(12.6)

'00

The integrand g is the Minkowski vacuum expectation value of the autocorrelation function (the Wightman function) g(s - s') = (0M\(8) (s')\0M).

(12.7)

Thus, S looks like a response function (or power spectrum) and g as the "quantum noise" in the Minkowski vacuum along the worldline x{s). The peculiar feature of this argument is that the quantum detector performs an "up" transition and at the same time sees ('emits') a 'radiation' spectrum. From the phenomenological point of view such a situation can also be encountered in the case of the anomalous Doppler effect (ADE) as has been remarked by Frolov and Ginzburg [3] (see section 12.4 below).

12.2.2

Six types of stationary scalar V F N

In general, the scalar quantum field vacuum does not possess a stationary vacuum excitation spectrum (abbreviated as SVES) for all types of classical relativistic trajectories on which the Unruh-DeWitt detector could move. Nevertheless, linear uniform acceleration is not the only case with that property. This was shown by Letaw, who extended Unruh's considerations, obtaining six broad classes of worldlines with SVES for an Unruh-DeWitt monopole

Haret Rosu

311

detector (SVES-1 to SVES-6, see below). The line of argument is the following: The Unruh-DeWitt detector is effectively immersed in a scalar bath of vacuum fluctuations. Its rate of excitation is determined by the energy spectrum of the scalar bath, which can be expressed as the density of states times a cosine-Fourier transform of the Wightman correlation function of the scalar field. Since the Wightman function is directly expressed in terms of the inverse of the geodesic interval what one needs to calculate is a Fourier transform of the inverse of the geodesic interval / ds = / y^dx^. Moreover, stationarity means that the Wightman function depends only on the proper time interval. As shown by Letaw, the stationary worldlines are solutions of some generalized Frenet-Serret (FS) equations on which the condition of constant curvature invariants is imposed. That is, one is interested in worldlines of constant curvature K, torsion r , and hyper-torsion (bi-torsion) v, respectively. These curvature invariants can be easily built from the tangent, normal, and binormal vectors and their derivatives. They have physical interpretation in terms of the observer's acceleration and angular velocities. Notice that one can employ other frameworks, such as the Newman-Penrose spinor formalism as recently invoked by Unruh [4], but the Frenet-Serret framework is in overwhelming use throughout physics. It is worth remarking that before Letaw, the Frenet-Serret invariants were discussed by Honig et al. [5] in their study of the motion of charged particles in homogeneous electromagnetic fields. Honig et al. discovered an interesting connection with the two Lorentz invariants of the electromagnetic field, E2 — H2 and E • H: E2-H2OCK2-T2-

EH

i/2,

<XKV.

(12.8)

(12.9)

It is amusing to note that chirality (handedness) in the electromagnetic sense (E • H ^ 0) is proportional to chirality in the worldline sense (nonzero hypertorsion; v ^ 0.) The six stationary scalar VFN can be classified according to the curvature scalars of the corresponding worldlines: 1. K = T = v = 0 - * inertial, uncurved worldlines (constant

velocity).

SVES-1 is a trivial cubic spectrum Sl(E)

E3 = —2.

(12.10)

312

Non-inertial quantum mechanical

fluctuations

This can be interpreted as a vacuum state of zero point energy E/2 per mode, with density of states E2/2TT2. 2. K ^ O , r = i/ = 0 - > hyperbolic worldlines (constant rectilinear acceleration). SVES-2 is Planckian allowing the interpretation of K/2TT as 'thermodynamic' temperature. In the dimensionless variable eK = E/K the vacuum spectrum reads

The physically observed spectrum would be a linear combination of SVES-1 and SVES-2. 3. \K\ < \T\, v = 0, p2 = r 2 — K2 —> helical worldlines. SVES-3 is a complicated analytic function corresponding to the case 4 below only in the limit K^$> p S3(ep)K,-^°S4(eK).

(12-12)

Letaw plotted the numerical integral S3(ep), where ep — E/p, for various values of K,/p. 4. K — r, v — 0 —>• the spatially projected worldlines are the "semicubical parabolas", y oc K X3/2, containing a cusp (at x = 0) where the direction of motion is reversed. SVES-4 is analytic, and since there are two equal curvature invariants (K = T) one can use the dimensionless energy variable eK — E/K Siie.) = ^ ^ e -

2

^ -

(12-13)

It is worth noting that £4, being a monomial times an exponential, is rather close to the Wien spectrum Sw oc e 3 e - c o n s t - £ .

313

Haret Rosu

5. \K\ > \T\, v = 0, a 2 = K2 - r 2 —> the spatially projected worldlines are catenaries, curves of the type x = «;cosh(y/r). In general, SVES-5 cannot be found analytically. It is an intermediate case, which for r/a -> 0 tends to SVES-2, whereas for rja -> oo tends toward SVES-4 52(Q) ° ^

S5(ea) ^

S 4 ( e > ).

6. v ^fiQ, K and r arbitrary —• rotating worldlines uniformly normal to their plane of rotation.

(12.14) accelerated

SVES-6 forms a three-parameter set of curves. The corresponding trajectories are a superposition of the constant linearly accelerated motion and uniform circular motion. SVES-6 has not been calculated by Letaw, not even numerically. Thus, only the hyperbolic worldlines, having just one nonzero curvature invariant, allow for a Planckian SVES. Only that case allows for a strictly oneto-one mapping between the curvature invariant K and the 'thermodynamic' temperature in the celebrated form Tv = K/2-K. The vacuum field noise of semicubical parabolas can be fitted by a Wien-type spectrum, the radiometric parameter then corresponding to both curvature and torsion. The other stationary cases, being nonanalytic, lead to the approximate determination of the curvature invariants defining locally the classical worldline on which the relativistic quantum detector moves. One very general and important statement regarding the universal nature of the kinematical Frenet-Serret parameters occurring in various important quantum field model problems can be formulated as follows: There exist accelerating classical trajectories (worldlines) on which moving ideal (two-level) quantum systems can detect the scalar vacuum environment as a stationary quantum field vacuum noise with a spectrum directly related to the curvature invariants of the worldline, thus allowing for a radiometric interpretation of those invariants.

314

Non-inertial quantum mechanical

fluctuations

According to these results, it seems more appropriate to replace the thermal interpretation of Unruh by the radiometric interpretation of the FrenetSerret invariants. The latter is more general and describes in a more precise way the physical situation to which the Unruh effect refers, that of a quantum particle moving along a classical relativistic trajectory.

12.2.3

Explicit formulae for the spectra

One can calculate the spectrum of vacuum field noise by means of the following general formula p22 E

p r°°

e~iEs ds [xp(s) - *„(())] [*"(*) - x"(0)]

ljl 4^ 47T3

J

1

-6' (12.15) where ^ ( s ) is an arbitrary point on the worldline and 3^(0) is the initial point. The signature of the Minkowski metric is rj^ = ( 1 , - 1 , - 1 , - 1 ) . I confirm Letaw's results by sketching the calculation of the integrals Ij for the six stationary cases. Simple details that have been skipped by Letaw can be found here. 1. The recta. The worldline is xM(s) = (s, 0,0,0); with initial condition ^ ( 0 ) = (0,0,0,0). The integral is +oo

/

e-iEs

— j - ds. oo

(12.16)

s

It can be evaluated by using Cauchy's residue theorem plus the expansion e~'Es = (1 - iEs + ...). The value of the integral is ni(—iE) = nE, and therefore one gets the cubic spectrum. This inertial zero-point cubic spectrum will appear in all the other five stationary spectra as an additive background and therefore one may take into account only the non-cubic contributions as a measure of non-inertial vacuum effects. 2. The hyperbola. The worldline is x^(s) = K~1 (sinh/ts, cosh KS, 0,0); with the initial condition ^ ( O ) = K _ 1 ( 0 , 1,0,0). Now the integral is +oo

/

„-ieKu

. 5 i ^ n y d»-

< 12 - 17 >

Haret Rosu

315

Writing e te"u = cos eKu — i sin eKu, one makes use of formula 3.983.3 at page 505 in the fourth edition of the Table of Gradshteyn and Ryzhik (GR) to get "+0° cos(ax) , ,, . . v / ' -dx = -7racoth(7ra). (12.18) cosh x — 1 0 Jo The sine integral can be evaluated using Cauchy's theorem f>+0

° sin(eKx)dx = vreK. cosh x — 1

"J

(12.19)

Thus h

= -7reKcoth(7re(t) + ^

(12.20)

= ireK [1 - coth(7reK)] - -£•

= -^^ri-T 1 '

(12.21)

( 12 - 22 )

where the first term leads to the Planckian spectrum and the latter to the cubic zero-point contribution. 3. The helix. The worldline is x^{s) = p~2 {rps, K COS ps, K sin ps, 0); with initial condition 2^(0) = p _2 (0, K, 0,0). The integral reads +oo

p-iipU

^

7T-7T-n 2 du,

/ -oo * ^ j ( c O S U - 1) +

where

K2

1 - -5 =

^U

L

(12.23)

( 12 - 24 )

According to Letaw this integral is non-analytic and indeed I was not able to find any helpful formula in the GR Table. 4. The semicubical parabola. The worldline is x^{s) = (s + |K 2 S 3 , \KS2, | K 2 S 3 , 0 ) ; with initial condition x"(0) = (0,0,0,0). The integral reads lt = Kr~

f - f ,«,,_, p f f l * ./.,„ u ! (l + i u ' ) 7_„ (12 + n J )

(12.25)

Non-inertial quantum mechanical fluctuations

316

Of interest is only the second integral that can be found in the GR Table at page 359 e

f.

ax

a2 + x2

= JL e - M ,

(12.26)

for a > 0 and p real. Thus one gets +oo e

/.

-itKu

J.. d

(12+ u2)

vl2

e"^-.

(12.27)

The final result is

5 4 = _^£L e-^c«

(12 28)

4TT2\/12

Interestingly, for a horizontal storage ring (guiding magnetic field in the vertical z direction) the orbit in the moving frame can be approximated for laboratory times such that JLJQ\t\ = 0(1) by the following semicubical parabola

where R® is the instantaneous radius of curvature of a particle's arbitrary trajectory.1 5. The catenary. The worldline is xt'(s) = <7~2(Ksinhcrs,Kcoshers,TCTS,0); with initial condition xM(0) = <J" 2 (0, K, 0,0). The integral is of the type r+00

e-itoU

1

^rr- ,

d u

""

,2 , ,

-00 2^(coshu - 1) - £

(12-30)

•c

where ^ — ^ = — 1. This integral turns into I2 and 74 in the limits mentioned in the text, respectively, but again there is no helpful formula in the GR Table, and thus I5 appears to be non-analytic. 1

See figure 2 and equation 2 in [22].

Haret Rosu

317

6. The helicoid (helix of variable pitch). This is the most general case. The worldline is x"(s)

=

("5T5T sinh(# + s), ——

cosh(R+s),

CO R S

RIJT

^ ~ ^ R^R-

sin

CM); ( 12 - 31 )

while the initial condition reads x p (0) = (0, ^ - , ^ 5 ^ 3 , 0 ) . denned:

I have

A2

=

)-{R2 + K2 + r 2 + i/2);

(12.32)

4

=

(K2 + T2 + U2)2-4K2T2;

(12.33)

R

2

R+

=

B?_ =

2

\{R

2

2

2

+ K - T - u );

I(ija_re2

+ r 2 + I/ 2).

(12.34) (12.35)

The following integral is obtained:

I

-nf"

+0

e~itRU du

° 2

2 ( ^ ) [ c o s h ( ^ « ) - 1] + 2 ( ^ f - ) 2 [ c o s ( ^ ) - 1]' (12.36) This is the most complicated non-analytic stationary case, with no helpful formula in the GR Table. J —0

12.2.4

Non-stationary vacuum field noise

Non-stationary vacuum field noise has a time-dependent spectral content requiring joint time and frequency information, i.e., one needs generalizations of the power spectrum concept. One can think of (i) tomographical processing and/or (ii) wavelet transforms. For instance, the recently proposed non-commutative tomography (NCT) transform M(s; /x, v) [6] seems to be an attractive way of processing non-stationary signals. In the definition of M, s is just an arbitrary curve in the non-commutative time-frequency plane, while /i and v are parameters characterizing the curve. The most simple

318

Non-inertial quantum mechanical

Suctuations

examples are the axes s = fit + uu, where fi and v are linear combination parameters. The non-commutative tomography transform is related to the Wigner-Ville quasi-distribution W(t, u) by an invertible transformation and has the following useful properties

M(t;l,0) M(w;0,l)

= =

\f(t)\2,

(12.37) 2

|/M| ,

(12.38)

where / is the analytic signal which is simulated by M. Furthermore, employing M leads to an enhanced detection of the presence of signals in noise which has a small signal-to-noise ratio. The latter property may be very useful in detecting VFNs, which are very small 'signals' with respect to more common noise sources. On the other hand, since in the quantum detector method the vacuum autocorrelation functions are the essential physical quantities, and since according to various fluctuation-dissipation theorems they are related to the linear (equilibrium) response functions to an initial condition/vacuum, the fluctuation-dissipation approach has been developed and promoted by Sciama and collaborators [7]. In principle, the generalization of the fluctuationdissipation theorem for some classes of out of equilibrium relaxational systems, such as glasses, looks also promising for the case of non-stationary vacuum noise. One can use a so-called two-time fluctuation-dissipation ratio X(t,t') and write a modified fluctuation-dissipation relationship [8]

TeS(t, t') R(t, f) = X(t, t') ^ j p - ,

(12.39)

where R is the response function and C the autocorrelation function. The fluctuation-dissipation ratio is employed to perform the separation of scales. Moreover, Teg are timescale-dependent quantities, making them promising for relativistic VFNs, which correspond naturally to out of equilibrium conditions.

ffaret Rosu

12.3 12.3.1

319

Circular electromagnetic vacuum noise Introduction

The circular electromagnetic vacuum noise, which in principle is more promising experimentally,2 has been first discussed for specific purposes by Candelas and Deutsch, and by Bell and Leinaas. However, here we will pay more attention to the approach of Hacyan and Sarmiento (HS) [12] who in 1989 introduced a clear-cut and general method for calculating the main electromagnetic vacuum spectral quantities and applied it to the basic cases of linear acceleration and uniform rotation. In the latter case, they obtained a nonzero energy flux in the direction of motion of the detector. It was this result that prompted Mane [13] to suggest a connection with synchrotron radiation. In principle, the circular vacuum noise power spectrum Sc could be calculated via the residue theorem, but the equation for the zeros of the denominator x2 = v2 sin2 x (see below) is not analytically solvable. Nevertheless, for v > 0.85 one can expand the sine to find the zero with the smallest imaginary part, besides x = 0 [14].

12.3.2

The Hacyan—Sarmiento approach

Starting with the expression for the electromagnetic energy-momentum tensor Tp,, = ^

( 4 F ^ F? + i)v, F\& F*) .

(12.40)

Hacyan-Sarmiento define the electromagnetic two-point Wightman functions as follows D+,(x, x') = i (4F
(12-41) (12.42)

2 Rogers [9] proposed to study the motion of a single electron in a Penning trap (geonium) to detect the circular electromagnetic vacuum noise. For two-level atoms in circular motion the reader is referred to Audretsch et al. [10], whereas in the analogue style approach Calogeracos and Volovik [11] considered the quasiparticle radiation from objects rotating in superfluid vacuum.

Non-inertial quantum mechanical fluctuations

320

This may be viewed as a variant of the "point-splitting" approach advocated by DeWitt. Moreover, because of the properties rTD^

= 0,

D% = D%,

dvD? = 0,

(12.43)

the electromagnetic Wightman functions can be expressed in terms of the scalar Wightman functions as follows D%{x,x') = cdlldv D±(x,x'),

(12.44)

where c is in general a real constant depending on the case under study. This shows that from the standpoint of their vacuum fluctuations the scalar and the electromagnetic fields are not so different. Now introduce sum and difference variables 8= — ;

a = —j-.

(12.45)

Using the Fourier transforms of the Wightman functions oo

/

dae-iua

D±(s,a),

(12.46)

-oo

where u is the frequency of zero-point fields, the particle number density of the vacuum seen by the moving detector and the spectral vacuum energy density per mode are given by "(w, s) = - V

\D+(^

s

) ~ D'(oJ, sj\ ,

¥- = — \D+{u>,s) + D-(u,,sj\. dw 7r L J

(12.47) (12.48)

The most important application of these results is to a uniformly rotating detector whose proper time is s and angular speed is LJQ in motion along the circular world line x<x s { ) — (7S> -^o COS(OJOS), Ro sin(w0s), 0), (12.49) where Ro is the rotation radius in the inertial frame, 7 = (1 — v2)~ll2, and v = uioRo/j- 3 In this case there are two Killing vectors ka = (1,0,0,0) and 3

This is correct for Galilean electromagnetism and works well at low velocities and/or in gradient index (lens) media. For full Lorentz covariant electrodynamics, one should use the Trocheris-Takeno nonlinear relationship v — tanh(wi?o/c). See, e.g., [15].

Haret Rosu

321

ma(s) = (0, — Rosm(ujos),R0cos(u)os,0). Expressing the Wightman functions in terms of these two Killing vectors, HS calculated the following physically observable spectral quantities (i.e., those obtained after subtracting the inertial zero-point field contributions): • The spectral energy density de 73 d^ = 2 ^ 3

(J 2 + ( 7 ^ Q ) 2

-i

v3w2 w2 + ( 2 7 , ) 2 M « 0 .

(12-50)

• The spectral flux density dp

73

du

2n3R%

UJ2 +

(JVUJO)2

UJ2

4v4 k,{w), 7

(12.51)

• The spectral stress density ds _

3

" 2 + (7^o)2

7

V3W2

, .

,

.

Here (UJ2 + (,yvu}o)2)/uj2 is a density-of-states factor introduced for convenience and hy(w), ky(w), and j1('w) are the following cosine-Fourier transforms . . . f°° ( Nh(x,v) 3 2 7 V\ h^(w) = / o • 2 i3 - x "4 + — 2 ~ cos(wx) dx; X v sm x x Jo \T[ \ J

*»--r(T^'^XP-I-*?)°-<«)di; /•°° /

'>)S/„

1

The numerators Nh(x,v) Nh(x,v) Nk(x,v)

1

2rV\

l7V-^sin^p-^ + ^ j and Nk(x,v)

= (3 + v2)x2+ 2

2

2

C

°

S (

^

) d

"

(12.53)


'

(12 55)

'

are given by (v2 + 3v4)sm2x-8v2xsmx; 2

= x + u sin a; - (1 + w )xsina;.

The employed variables are iu = ^ and x = £ | a .

(12.56) (12.57)

Non-iaertial quantum mechanical fluctuations

322

Of special interest are the ultra-relativistic and nonrelativistic limits. In the first case, 7 >• 1, the quantities V IV

^

=

V tV

^TT2W

M H

*f

= 4l,4

*»'

^ = w 2 + (2 7 u)» i7(tt,) ' (12.58)

have the following scaling property Xky(kw) = k3X7(w),

(12.59)

where A; is an arbitrary constant, and X — H,K, J. This is the same scaling property as that of a Planckian distribution with a temperature proportional to 7. A detailed discussion of the nonrelativistic limit has been provided by Kim, Soh, and Yee [16], who used the parameters v and a>0, and not acceleration and speed as used by Letaw and Pfautsch for the circular scalar case [17]. They obtained a series expansion in velocity de _ du>

CJ3 7T

2

V2U (n )2 + U T ) {T( }l ) * - * - - "H nV . k /; h^+lh *!(2n-*)! { 7 uJ n=0

jfc=0

(12.60) where H is the usual Heavyside step function. Thus, to a specified power of the velocity many vacuum harmonics could contribute; making the energy density spectrum quasi-continuous.

12.3.3

Synchrotron radiation as electromagnetic vacuum fluctuations ?

In 1991, Mane used the Hacyan-Sarmiento formula for the energy flux to argue that its time component is related to synchrotron radiation. The Hacyan-Sarmiento Poynting flux is directed along the Lorentz boost from the laboratory frame to the rest frame of the observer, which is taken as the y axis. It can be written

" V - I i S H ^ I

8 0

- ^ .

(12 61

' '

Note that py is proportional to h and therefore becomes zero in the classical limit. However, for electrons which couple to this flux via the fine structure

Haret Rosu

323

constant a = e2/kc, the radiation effect looks totally classical. The recoil induced by the flux of the vacuum fluctuations on the four-momentum of the particle per unit proper time is aApycx^t,

(12.62)

where A « i?^c2/(i>274) is the transverse interaction area between the electron and the electromagnetic field. In the laboratory frame, the energy loss of the particle per unit laboratory time is given by the Larmor formula I = \^^v)\

(12.63)

This is related to the damping force F in the form I = F • v and therefore the recoil induced by synchrotron radiation on the four-momentum of the particle per unit proper time is again proportional to e^uftv/c? as in equation (12.62). Therefore, the order of magnitude of the recoil of the particle induced by the a coupling to the vacuum flux is equal to that derived by the Larmor formula in the ultra-relativistic limit. If one goes as far as accepting the idea that synchrotron radiation is due to noninertial electromagnetic vacuum fluctuations, one should reproduce in this approach all the many basic features of synchrotron radiation that are known from both theory and measurements at storage rings. Recall, for example, that the Schwinger spectral intensity of the magneto-bremsstrahlung in the synchrotron regime [18] is proportional to the so-called shape function W„ oc F (—)

,

(12.64)

where wm is given in terms of the cyclotron radian frequency wc as wm = CJC73, and the shape function F is given by F(Q = ^-C l°°K5/3(z)dz,

(12.65)

where K is the MacDonald (modified Bessel) function of the quoted fractional order. The small and large asymptotic limits of the synchrotron shape function are as follows F « < 1) « 1.33 C1/3, (12-66)

324

Non-inertial quantum mechanical

fluctuations

and F(( > 1) « 0.78 C 1/2 e" c ,

(12.67)

with a maximum (amount of radiation) to be found at the frequency w m / 3 . An examination of the Hacyan-Sarmiento asymptotic limits shows that there are clear differences between the Hacyan-Sarmiento and synchrotron energy density spectrum. Neither of the two limits coincide, neither the HacyanSarmiento spectrum divided in two equal parts by its peak frequency as in the case of synchrotron radiation. Moreover, the well-defined polarization state of synchrotron radiation that can be calculated in closed form in terms of the squares of Bessel Kx/3 and K2/3 functions in electrodynamics would prove really difficult to obtain in the vacuum approach. Finally, an expansion in velocity powers of the synchrotron radiation does not coincide with that in equation (12.60). In the opinion of the author, the circular electromagnetic vacuum noise should be considered as only a radiation signal embedded in the synchrotron radiation background.

12.3.4

Electron beam polarization at storage rings: Spin flip synchrotron radiation versus circular electromagnetic vacuum noise

Electromagnetic circular vacuum noise is interesting not only because of the Hacyan-Sarmiento results and Mane's suggestion but also as being responsible, according to Bell and Leinaas [19], for electron depolarization at storage rings. This famous proposal was put under intense focus in 1998 at the Monterey conference organized by Pisin Chen, where one of the most authoritative contrarians, Professor J.D. Jackson declared [20]: Avoid the indiscriminate appeal to Unruh in order to "understand" something amenable to a simpler explanation. The following is a brief introduction to this problem. It has been included here as an illustration of the confrontation of theoretical ideas with the experimental market; a confrontation eagerly awaited for the attractive analogue proposals presented in this book. The Bell-Leinaas proposal relies on the spin degree of freedom of the electron in an external magnetic field B0 along the z axis. The spin may be thought to have two (quasi)stationary states corresponding to az = ± 1 , with an energy splitting A = 2\fi\ \B0\. This approximation is valid when

Haret Rosu

325

a second term in the effective spin-field interaction Hamiltonian, due to the so-called Thomas precession, is not included. Thus, the electron looks in a first approximation like an Unruh-DeWitt detector. The transitions between the two spin states induced by the radiation field are then written in terms of first-order time-dependent perturbation theory, and a thermal ratio is obtained as if produced by the equilibrium ratio of populations of the upper and lower levels. The effect of the Thomas precession term in the effective Hamiltonian does not alter the shape of the polarization curve, and only shifts it horizontally when plotted as a function of the magnetic moment. However, there is a simple quantum electrodynamical explanation of the polarization effect at storage rings in terms of the so called spin-flip synchrotron radiation that has been proposed by Sokolov and Ternov in 1963 [21]. The spin-flip radiated power is very small with respect to ordinary synchrotron radiation, becoming of the same order only at 7 S / = (mcRo/h)1/2, 6 which for a common storage ring is around 7 « 6 x 10 . This is more than two orders of magnitude higher than the actual 7 < 104 of electrons in current storage rings leading to a spin-flip radiated power representing only 10 - 1 1 of the usual (non spin-flip) synchrotron emitted power. It is only because the spin flip accumulates over a time scale of tens of minutes to a few hours that one gets the observed asymptotic polarization Pum = 8 / 5 ^ 3 = 0.924. It is either this non-stationarity of the spin-flip synchrotron radiation, or the fact that the orbiting electrons are actually more complicated interacting systems than simple Unruh-DeWitt detectors, that lead only to frequencydependent effective temperatures, which in the opinion of most accelerator physicists are not useful parameters. The description of radiative polarization in terms of spin levels came under the scrutiny of Professor Jackson long ago [22]. He showed that the spacing between orbital levels is very small compared to the magnetic dipole Ml transition energy, and therefore the Ml transition will involve some changes in the orbital quantum number. In 1973, Derbenev and Kondratenko [23] obtained, in a quasi-classical approach, a formula for the equilibrium polarization in which spin-orbit effects are included through a spin-orbit coupling function. Their formula is considered to be the standard result for the transverse polarization at storage rings. According to the Derbenev-Kondratenko formula, for the range 0 < g < 1.2 one of the levels is preferentially populated with respect to the other one. This effect cannot be reproduced in the Bell-Leinaas approach without resorting to time-dependent couplings and frequency-dependent 'temperatures' [14]. In 1987, Bell and Leinaas pub-

326

Non-inertial quantum mechanical

fluctuations

lished a more detailed analysis of their proposal, in which (still assuming a thermal spectrum of the spin excitations) they took into account the fluctuations in the orbital motion. They obtained a polarization formula rather close to the standard one with some differences only close to a narrow depolarizing resonance. The claim is that when passing through resonance the polarization falls from 92% to -17% followed by an increase to 99% before settling again to 92%. Thus, the confirmation of their calculation, and of the thermal vacuum bath, as opposed to the Sokolov-Ternov limiting polarization would require precise experimental measurements of a transient passing through a depolarizing resonance, an experiment that is still to be performed.

12.4

U n r u h effect versus anomalous Doppler effect

The concept of the anomalous Doppler effect (ADE) was introduced in classical electrodynamics by Frank in 1942 [26]. ADE refers to the waves emitted within the Cherenkov cone by a "superluminal" oscillator moving in a refractive medium. (Frank's example is an electric dipole harmonically oscillating at angular frequency Q.) By definition, these waves exhibit an anomalous Doppler shift in the sense that their frequencies (with respect to O) have a negative Doppler directivity factor D (see below) and are given by OJADE = il/D. In the quantum version of this phenomenon, as discussed by Frolov and Ginzburg, one uses the energy-momentum conservation law for massless Bose radiation from a superluminal two-level detector to get the energy formula (12.68) below. As we have already mentioned in section (12.2.1), when studied with the detector method the Unruh effect for a detector with internal degrees of freedom is in some ways very similar to this anomalous Doppler effect (ADE), since in both cases the quantum detector is radiating 'photons' while passing into the upper level and not on the lower one [3] (see figure 12.1). This is refrained in the well-known conclusion of Unruh and Wald [24] when they considered the uniformly accelerated quantum detector looked upon from the inertial reference frame: When the observer places himself in an inertial reference frame then he is able to observe both the excited quantum detector (fur-

Haret Rosu

327

I

I

Figure 12.1: The normal and anomalous Doppler effects and the corresponding transitions. nishing at the same time energy to it) and the 'photons'. By writing down the energy-momentum conservation law he will be inclined to say that the 'photons' are emitted precisely when the detector is excited. Neglecting recoil, absorption, and dispersion (a completely ideal case) the elementary radiation events for a two-level detector with the change of the detector proper energy denoted by 8e are classified according to the photon energy formula [3] hu = --£(12.68) L/7 where 7 is the relativistic velocity factor (7 > 1) and D is the Doppler directivity factor £> = l - ( — ) c o s 0 .

(12.69)

The discussion of signs in equation (12.68) implies 3 cases as follows: 1. D> 0 for normal Doppler effect (NDE, 6e < 0, 9 > 9C) 2. D = 0 for Cherenkov effect (CE, 6e — 0, undetermined case, 9 = 9C) 3. D< 0 for anomalous Doppler effect (ADE, Se > 0, 6 < 9C).

328

Non-inertial quantum mechanical

fluctuations

Consequently, for a quantum system endowed with internal degrees of freedom the stationary population of levels is determined by the probability of radiation in the ADE and NDE regions. The possibility of inducing population inversion by means of the ADE has been extensively discussed in the literature [25]. Bolotovsky and Bykov [27] have studied the space-time properties of ADE in the simple case of a superluminal dipole (v > c/ri) propagating in uniform rectilinear motion in a non-dispersive medium. These authors claim a positive theoretical result with regard to the separate observation of the ADE phenomenon for this case. It is not, unfortunately, a realistic case and requires a special equation of motion of the dipole. Theoretical and experimental investigations of the possible manifestation of ADE in dispersive and/or lens media is an important task for the future. A direct experimental evidence of ADE would be highly valuable as being equivalent to a test of the Unruh effect. The acoustic ADE is another challenge for the future [28].

12.5

Summary

As is the case for all claimed quantum vacuum effects (including the mechanical Casimir effect), the stationary radiative spectra surveyed in this chapter can be attributed equally well to radiation reaction fields, i.e., to solutions of the inhomogeneous Klein-Gordon or Maxwell equations evaluated at the source [29]. However, the main point we want to emphasize is different, namely: These radiation patterns (if they really exist — remember the calculations are all performed in first-order perturbation theory, or in order a for electrodynamic radiation reaction fields) can be used to extract radiometric information related to the Frenet-Serret geometric invariants of the trajectories of the relativistic corpuscules. Finally, we emphasise that over the last few years the physics community has become aware of many interesting similarities/analogies between the Hawking/Unruh effects and shock-type effects in material media. A very promising line of research could be the study of the Cherenkov effect, and the associated anomalous and normal Doppler effects of relativistic dipoles propagating in strongly dispersive substances. Potentially realistic laboratory configurations for examining these effects are, for example, Cherenkov-type experiments with bunches of electric dipoles (polarization pulses) created by

Haret Rosu

329

femtosecond optical pulses in electro-optic materials [30]. In addition, radiation from vortices in two-dimensional annular Josephson junctions [31], or even from other 'relativistic' defects in condensed-matter physics should be taken into account from the perspective of this book.

Acknowledgement s: The author would like to thank the organizers of the workshop for inviting him to this interesting event.

This page is intentionally left blank

Bibliography [1] For review, see S. Takagi, "Vacuum noise and stress induced by uniform acceleration", Prog. Theor. Phys. Suppl. 88, 1 (1986). [2] J.R. Letaw, "Stationary world lines and the vacuum excitation of noninertial detectors", Phys. Rev. D 23, 1709 (1981). [3] V.P. Frolov and V.L. Ginzburg, "Excitation and radiation of an accelerated detector and anomalous Doppler effect", Phys. Lett. A 116, 423 (1986). [4] W. Unruh, "Radiation reaction fields for an accelerated dipole for scalar and electromagnetic radiation", Phys. Rev. A 59, 131 (1999). [5] E. Honig, E.L. Schucking, and C.V. Vishveshwara, "Motion of charged particles in homogeneous electromagnetic fields", J. Math. Phys. 15, 774 (1974). For higher-dimensional generalization, see B.R. Iyer and C.V. Vishveshwara, "The Frenet-Serret formalism and black holes in higher dimensions", Class. Quant. Grav. 5, 961 (1988). [6] V.I. Man'ko and R. Vilela Mendes, "Noncommutative time-frequency tomography", Phys. Lett. A 263, 53 (1999) [physics/9712022]. [7] P. Candelas and D.W. Sciama, "Irreversible thermodynamics for black holes", Phys. Rev. Lett. 38, 1372 (1977). [8] L.F. Cugliandolo, J. Kurchan and L. Peliti, "Energy flow, partial equilibration and effective temperatures in systems with slow dynamics", Phys. Rev. E 55, 3898 (1997) [cond-mat/9611044]; R. Exartier and L. Peliti, "Measuring effective temperatures in nonequilibrium systems", Eur. Phys. J. B 16, 119 (2000), [cond-mat/9910412]. 331

332

Non-inertial quantum mechanical

fluctuations

[9] J. Rogers, "Detector for the temperature-like effect of acceleration", Phys. Rev. Lett. 6 1 , 2113 (1988). [10] J. Audretsch, R. Miiller, and M. Holzmann, "Generalized Unruh effect and Lamb shift on arbitrary stationary trajectories", Class. Quant. Grav. 12, 2927 (1995). [11] A. Calogeracos and G.E. Volovik, "Rotational quantum friction in superfluids: Radiation from object rotating in superfluid vacuum", Pis'ma Zh. Eksp. Teor. Fiz. 69, 257 (1999). [12] S. Hacyan and A. Sarmiento, "Vacuum stress-energy tensor of the electromagnetic field in rotating frames", Phys. Rev. D 40, 2641 (1989); Phys. Lett. B 179, 287 (1986). [13] S.R. Mane, "Comment on HS", Phys. Rev. D 43, 3578 (1991); See also, T. Hirayama and T. Hara, "A calculation on the self-field of a point charge and the Unruh effect", Prog. Theor. Phys. 103, 907 (2000) [gr-qc/9910111]. [14] W.G. Unruh, "Acceleration radiation for orbiting electrons", Phys. Rept. 307, 163 (1998) [hep-th/9804158]. [15] S. Kichenassamy and R.A. Krikorian, "Note on Maxwell's equations in relativistically rotating frames", J. Math. Phys. 35, 5726 (1994); R.D.M. De Paola and N.F. Svaiter, "A rotating vacuum and a quantum version of Newton's bucket experiment", Class. Quant. Grav. 18, 1799 (2001) [gr-qc/0009058]. [16] S.K. Kim, K.S. Soh, and J.H. Yee, "Zero-point field in a circular-motion frame", Phys. Rev. D 35, 557 (1987). [17] J.R. Letaw and J.D. Pfautsch, "Quantized scalar field in rotating coordinates", Phys. Rev. D 22, 1345 (1980). [18] J. Schwinger, "On the classical radiation of accelerated electrons", Phys. Rev. 75, 1912 (1949). [19] J.S. Bell and J.M. Leinaas, "Electrons as accelerated thermometers", Nucl. Phys. B 212, 131 (1983); "The Unruh effect and quantum fluctuations of electrons in storage rings", Nucl. Phys. B 284, 488 (1987).

Haret Rosu

333

[20] J.D. Jackson, "On effective temperatures and electron spin polarization in storage rings", in Quantum aspects of beam physics, ed. Pisin Chen, World Scientific (1999) pp. 622-625 [physics/9901038]. [21] A.A. Sokolov and I.M. Ternov, "On polarization and spin effects in the theory of synchrotron radiation", Dokl. Akad. Nauk 153, 1052 (1964) [Sov. Phys. Dokl. 8, 1203 (1964)]. [22] J.D. Jackson, "On understanding spin-flip synchrotron radiation and the transverse polarization of electrons in storage rings", Rev. Mod. Phys. 48, 417 (1976). For a recent review see, D.P. Barber, "Electron and proton spin polarization in storage rings — an introduction", in Quantum aspects of beam physics, ed. Pisin Chen, World Scientific (1999) pp. 67-90 [physics/9901038]. [23] Ya. S. Derbenev and A.M. Kondratenko, "Polarization kinetics of particles in storage rings", Zh. Eksp. Teor. Fiz. 64, 1918 (1973) [Sov. Phys. J E T P 37, 968 (1973)]. [24] W.G. Unruh and R.M. Wald, "What happens when an accelerating observer detects a Rindler particle", Phys. Rev. D 29, 1047 (1984). [25] See section 11 and corresponding references in H. Rosu, "Hawking-like and Unruh-like effects: Toward experiments?", Gravitation k Cosmology 7, 1 (2001) [gr-qc/9406012]. [26] I.M. Frank, Izv. Akad. Nauk SSSR, Ser. Fiz. 6, 3 (1942). [27] B.M. Bolotovski and V.P. Bykov, "On the theory of ADE", Radiofizika 32, 386 (1989). [28] M.E. Goldstein, Aeroacoustics, (McGraw-Hill, New York, 1976). [29] A.O. Barut and J.P. Dowling, "Quantum electrodynamics based on self fields: On the origin of thermal radiation detected by an accelerated observer", Phys. Rev. A 4 1 , 2277 (1990). [30] D.H. Austin, K.P. Cheung, J.A. Valdmanis, and D.A. Kleinman, "Cherenkov radiation from femtosecond optical pulses in electro-optic media", Phys. Rev. Lett. 53, 1555 (1984).

334

NoD-inertial quantum mechanical

fluctuations

See also, T.E. Stevens, J.K. Wahlstrand, J. Kuhl, R. Merlin, "Cherenkov radiation at speeds below the light threshold: Phonon-assisted phase matching", Science 291, 627 (2001) and references therein. [31] See, e.g., V.V. Kurin, A.V. Yulin, LA. Shereshevskii, and N.K. Vdovicheva, "Cherenkov radiation of vortices in a two-dimensional annular Josephson junction", Phys. Rev. Lett. 80, 3372 (1998).

Chapter 13 Phonons and forces: M o m e n t u m versus pseudomomentum in moving fluids Michael Stone University of Illinois Physics Department Urbana, Illinois 61801 USA E-mail: [email protected]

Abstract: I provide a pedagogical introduction to the notion of pseudomomentum for waves in a medium, and show how changes in pseudomomentum may sometimes be used to compute real forces. I then explain how these ideas apply to sound waves in a fluid. When the background fluid is in motion, the conservation laws for pseudomomentum and pseudoenergy are most easily obtained by exploiting the acoustic metric and the formalism of general relativity. 335

336

13.1

Phonons and forces: Momentum

versus

pseudomomentum

Introduction

The workshop on which this book is based was devoted to the physics of waves moving through a medium which affects them as would a background metric. There is therefore a natural analogy with waves propagating in a gravitational field — but we should take care not to push the analogy too far. These systems differ from real general relativity in that the medium constitutes a physical aether. While we may ignore the aether for many purposes, occasionally it is important. For example, if we wish to compute forces exerted by the waves, we must take into account any stress transmitted by the background medium. The natural tool for computing forces by tracking the flux of energy and momentum is the energy-momentum tensor (stress-energy tensor). This is best defined as the functional derivative of the action with respect to the background metric — but we have two metrics at our disposal: the spacetime metric and the acoustic or other metric which we are exploiting for our GR analogy. Despite the temptation to believe otherwise, we should remember that it is only by differentiating with respect to the "real" metric that we obtain "real" energy and "real" momentum. When we differentiate with respect to the analogy metric, we obtain the density and flux of other quantities. These are usually the pseudoenergy and the pseudomomentum [1]. Failure to distinguish between real energy and momentum and pseudoenergy and pseudomomentum has caused much confusion and controversy over the years. Consider the formula for the momentum of a photon in a dielectric: should the refractive index go in the numerator, where it was placed by Minkowski, or in the denominator, as argued by Abrahams? This dispute was not resolved until Blount [2] identified Minkowski's expression with the pseudomomentum and Abrahams' with the true momentum, including the mechanical momentum of the dielectric [3, 4]. Although I will not address the problem here, my initial motivation for thinking about these topics was the fear that a similar confusion lies behind some recent controversies [5] involving the Iordanskii force. This force, which has an appealing GR analogue in the gravitational Aharonov-Bohm effect of a spinning cosmic string [6], is supposed to act on a vortex in a superfluid when it moves relative to the normal component of the fluid. A related issue, and one that I will address, lies at the heart of the twofluid model for a superfluid or Bose-Einstein condensate. It is one of the fundamental assumptions of the two-fluid model that the phonons in a fluid

Michael Stone

337

possess momentum hk, and that, unlike that of phonons in a solid, this momentum is true Newtonian momentum, mv. This assumption is essential because we wish to identify the phonon momentum density with the mass current, both being equal to p\. The desired identification is supported by the approximate solution of Bogoliubov's weakly interacting Bose gas model, in which the phonon creation operator, a'k, appears to be create bona-fide momentum, so it is quite unnerving to discover that in the literature of fluid mechanics the attribution of real momentum to a sound wave is regarded as a naive and dangerous fallacy. A particularly forceful statement of this opinion is to be found in the paper [7] On the Wave Momentum Myth, by Michael Mclntyre. In the present article we will focus on the difference between the true momentum and the pseudomomentum associated with the acoustic metric. The first part is a general pedagogical account of the distinction between momentum and pseudomomentum, and the circumstances under which the latter may be used for computing forces. The second part will discuss the energy and momentum associated with sound waves in a background flow [8].

13.2

Momentum and pseudomomentum

The distinction between true momentum and pseudomomentum is especially clear when we consider the problem of transverse vibrations on an elastic string. The action S = | d z d i { ^

2

- |

y

'

2

}

(13.1)

gives rise to the familiar wave equation p y - T y " = 0,

(13.2)

with c = \jTjp being the wave speed. By manipulating the wave equation we can establish two local conservation laws. The first,

l ^ + M + l;!-^ 0 -

<13-3>

is immediately recognizable as an energy conservation law, with the flux -Tyy' being the rate of doing work by an element of the string on its neigh-

338

Phonons and forces: Momentum versus pseudomomentum

bor. The second,

is slightly more obscure in its interpretation. In a relativistic system the appearance of —pyy' = (energy flux)/c2 as the local density of a conserved quantity would not be surprising. The symmetry of the energy-momentum tensor requires that T j0 = T° J , so the energy flux T j0 is also (after division by c2) the density of 3-momentum. Here, however, we are dealing with non-relativistic classical mechanics — and with transverse waves. Whatever the quantity —pyy' may be, it is not the density of the x component of the string's momentum. It is instead the density of pseudomomentum. To understand the origin of pseudomomentum, observe that our elastic string may be subjected to two quite distinct operations either of which might be called "translation in the x direction": • An operation where the string, together with any disturbance on it, is translated in the x direction. • An operation where the string itself is left fixed, but the disturbance is translated in the x direction. The first operation leaves the action invariant provided space is homogeneous. The associated conserved quantity is true Newtonian momentum. The second operation is a symmetry only when both the background space and the string are homogeneous. The conserved quantity here is pseudomomentum. Such a distinction between true and pseudo- momentum is necessary whenever a medium (or aether) is involved. Adding to the confusion is that, although the pseudomomentum is conceptually distinct from the true momentum, there are many circumstances in which changes in pseudomomentum can be used to compute real forces. As an example consider a high speed train picking up its electrical power from an overhead line. The locomotive is travelling at speed U and the pantograph pickup is exerting a constant vertical force F on the power line. We make the usual small amplitude approximations and assume (not unrealistically) that the

Michael Stone

339

Figure 13.1: A high-speed

train.

line is supported in such a way that its vertical displacement obeys an inhomogeneous Klein-Gordon equation py-Ty"

+ Pn2y =

F6(x-Ut),

(13.5)

with c = s/T/p, the velocity of propagation of short wavelength disturbances. If U < c, the vertical displacement relaxes symmetrically about the point of contact. Once U exceeds c, however, the character of the problem changes from elliptic to hyperbolic, and an oscillatory "wake" forms behind the pantograph. As with all such wakes, the disturbance is stationary when viewed from the frame of the train. With this in mind, we seek a solution to (13.5) of the form y — y{x — Ut). Since the overhead line is undisturbed ahead of the locomotive, we find y=^sin[^(Ut-x)},

V = 0,

x Ut.

(13.6)

Here 7 = (U2/c2 — 1) _ 2 is the Lorentz contraction factor modified for tachyonic motion. The condition that the phase velocity, uj/k, of the wave constituting the wake be equal to the forward velocity of the object creating it is analogous to the Landau criterion determining the critical velocity of a superfluid. There are no waves satisfying this condition when U < c, but they exist for all U > c. In the wake, the time and space averaged energy density

(£)

\irf + \ry'2 + \W

(13.7)

340

Phonons and forces: Momentum

versus

pseudomomentum

is given by

<£> = ^ 7 4 ( j r ) V

(13.8)

The expression for the pseudomomentum density for the Klein-Gordon equation is the same as that for the wave equation, and the average pseudomomentum density is

i-pyy') = \nA ( f ) u.

(13.9)

Because energy is being transfered from the locomotive to the overhead line, it is clear that there must be some induced drag, F d , on the locomotive. This is most easily computed from energy conservation. The rate of work done by the locomotive, FdU, must equal the energy density times the rate of change of the length of the wave-train. Thus FdU where

= {£) (U - Ua)

(13.10)

dw c2k c2 U„ = i~dk r == ~LJ' — Z== 77 U

(13-11)

is the group velocity of the waves, and so the speed of the trailing end of the wake wave-train. After a short calculation we find that the wave-induced drag force is FA = Y ^ -

(13.12)

Since the average pseudomomentum density turned out to be the average energy density divided by U, we immediately verify that we get exactly the same answer for Fa if we equate the wave drag to the time rate of change of the total pseudomomentum. For the sceptic we note that we may also obtain the same answer by a more direct evaluation of the force required to deflect the overhead wire. From the solution (13.6) we see that the force F is related to the angle of upward deflection, 0, by 0 ss t a n 0 = -y'(Ut,t) = j2F/T. By balancing the acceleration of the power line against the force and the tension in the line, we see that the force cannot be exactly vertical, but must be symmetrically disposed with respect to the horizontal and deflected parts of the line. The force exerted by the pantograph thus has a small horizontal component F sin 0/2. The wave drag is therefore Fd = ^F2j2/T, as found earlier.

Michael Stone

341

Figure 13.2: Force on overhead wire. Since there is a real horizontal force acting on the wire, true Newtonian momentum must also be being transfered to the wire. Indeed the wire behind the train is being stretched, while that in front is being compressed. A section of length 2ci ong (i — i 0 ), where Ciong is the velocity of longitudinal waves on the wire and (i — to) is the elapsed time, is in uniform motion in the x direction. Since usually ci ong » c, this true momentum is accounted for by an almost infinitesimal motion over a large region of the wire. The pseudomomentum and the true momentum are to be found in quite different places — but the divergence of their flux tensors, and hence the associated forces, are equal.

13.3

Radiation pressure

In this section we will consider the "radiation pressure" exerted by sound waves incident on an object immersed in the medium. This is a subject with a long history of controversy [9, 10]. The confusion began with the great Lord Rayleigh who gave several inequivalent answers to the problem. Our discussion will follow that of Leon Brillouin [11] who greatly clarified the matter. We begin by considering some analogous situations where the force is exerted by transverse waves on a string. Consider a standing wave on a semi-infinite elastic string. We have restricted the vibration to the finite interval [0, L] by means of a frictionless bead which forces the transverse displacement of the string to be zero at x = L, but allows free passage to longitudinal motion, and so does not affect the tension. Suppose the transverse displacement is 7TTICC

y = A sinujt sin —r—

(13.13)

342

Phonons and forces: Momentum

versus

pseudomomentum

Figure 13.3: A vibrating string exerts a force on a bead. with u) = cnn/L.

The total energy of the motion is

E = Jdx {i^ 2 + \Tyi} = \^A2L.

(13.14)

If we alter the size of the vibrating region by slowly moving the bead, we will alter the energy in the oscillations. This change in energy may be found by exploiting the Boltzmann-Ehrenfest principle which says that during an adiabatic variation of the parameters of a harmonic oscillator the quantity E/u) remains constant. Thus 6(E/LJ) = 0 or 8E = (||) 5LJ. TO apply this to the string, we note that . Su}

r

SL OL -u,T.

/cn-K\

= * ( — ) =

(13.15)

The change in energy, and hence the work we must do to move the bead, is then SE

-(f)

SL.

(13.16)

The "radiation pressure" is therefore E/L = {£), the mean energy density. This calculation can be confirmed by examining the forces on the bead along the lines of figure 13.3. The average density of pseudomomentum in each of the two travelling wave components of the standing wave is ± | ( £ ) / c . The radiation "pressure" can therefore be accounted for by the 2 x c x | {£) /c rate of change of the pseudomomentum in the travelling waves as they bounce off the bead. Thinking through this example shows why pseudomomentum can be used to compute real forces: On a homogeneous string the act of translating the bead and the wave together, while keeping the string fixed, leaves the action invariant.

Michael Stone

343

The associated conserved quantity is the sum of the pseudomomentum of the wave and the true momentum of the bead. Keeping track of pseudomomentum cannot account for all forces, however. There is another plausible way of defining the radiation pressure. This time we use a finite string and attach its right-hand end to a movable wall

5L

Figure 13.4: Another way to define radiation

pressure.

Now, as we alter the length of the string, we will change its tension, and so alter the value of c. We must take the effect of this into account in the variation of the frequency SUJ

6L 6c - w — + u)—. L c

fcrni \ \-L)

(13.17)

The change in the energy of the vibrating system is therefore 6E = -(E)

1 -

dine dlnL

5L.

(13.18)

The radiation pressure is thus given by P={£)

1-

dine dlnL

(13.19)

This force is in addition to the steady pull from the static tension T in the string. The generality of the Boltzmann-Ehrenfest principle allows us to apply the previous discussion with virtually no changes to compute forces exerted by a sound wave. We need no explicit details of the wave motion beyond it being harmonic. The two ways of defining the radiation pressure for a vibrating string correspond to two different experimental conditions that we

344

Phonons and forces: Momentum versus pseudomomentum

SL

Figure 13.5: Tie ftayieigb radiation pressure, might use for measuring the radiation pressure for sound waves. The movable end condition corresponds to what is called the Rayleigh sound pressure. We establish a standing wave in a cylinder closed at one end and having a movable piston at the other. Moving the piston confining the sound wave changes both the wavelength of the sound and the speed of propagation, producing a sound radiation pressure

'-»H5£;).)-

(13.20)

The subscript S on the derivative indicates that it is being taken at fixed entropy. As with the string, this radiation pressure is in addition to the equilibrium hydrostatic pressure, Po, on the piston. The analogue of the string with the sliding bead leads to the Langevin definition of the radiation pressure. Here we insert a bypass so that moving the piston confining the sound wave does not change the density or pressure of the fluid. The radiation pressure on the piston is simply {£).

• ' "1

j

I_$L 1

Figure 13.6: Laxigevin Radiation Pressure. The difference in the two definitions of radiation pressure arises because, if we keep the mean pressure fixed, the presence of a sound wave produces an 0(A2) change in the volume of the fluid. If, instead, the mean density

345

Michael Stone

is held fixed, as it is in the Rayleigh definition, then this volume change is resisted by an additional hydrostatic pressure on the walls of the container. The most common way of measuring sound pressure involves a sound beam in an open tank of fluid. Since the fluid is free to expand, this corresponds to the Langevin pressure.

Figure 13.7: The usual experimental

situation.

The radiation pressure is, in reality, a radiation stress X..-(£)

(hh

+ Sli(^^]

).

(13.21)

The anisotropic part depends on the v/ave-vector k of the sound beam, and may be accounted for by keeping track of the pseudomomentum changes. The isotropic part cannot be computed from the linearized sound-wave equation since it requires more information about the equation of state of the fluid medium than is used in deriving the wave equation. The extra information is encapsulated in the parameter ( f g ) , a fluid-state analogue of the Graneisen parameter which characterizes the thermal expansion of a solid. When the experimental situation is such that this isotropic pressure is important, the force associated with the sound field cannot be obtained from the pseudomomentum alone.

13.4

Mass flow and the Stokes drift

Further confusion involving momentum and pseudomomentum in acoustics is generated by the need to distinguish between the Euler (velocity field at a

346

Phonons and forces: Momentum

versus

pseudomomentum

particular point) description of fluid flow, and the Lagrangian (following the particles) description. Suppose the velocity field in a sound wave is vw(x,

t) = A cos{kx - cut).

(13.22)

Using the continuity equation dxpv + dtp — 0, setting p = p0 + pi, and approximating pv « Po^(i)> we find that px = -p0kAcos(kx

- ut) + 0{A2).

(13.23)

The time average of the momentum density pv is therefore (pv) = <

W )

k

> =

-^-\A\

(13.24)

to 0(A2) accuracy. This Newtonian momentum density is clearly non-zero, and numerically equal to the pseudomomentum density. Here, unlike the case of the elastic string, there is only one velocity of wave propagation and so the pseudomomentum and true momentum, although logically distinct, are to be found in the same place. Further (pv) is both the momentum density and the mass-current. A nonzero average for the former therefore implies a steady drift of particles in the direction of wave propagation, in addition to the back-and-forth motion in the wave. We can confirm this by translating the Eulerian velocity field i>(i) = Acos(kx — ojt) into Lagrangian language. The trajectory £(t) of a particle initially at x0 is the solution of the equation (

^ = v{1)(Z{t),t)

= Acos{kt-u)t),

e(0) = x 0 .

(13.25)

Since the quantity £ appears both in the derivative and in the cosine, this is a nonlinear equation. We solve it perturbatively by setting f (*) = x0 + A Sx(t) + A2 E2(t) + •••.

(13.26)

We find that Ei(t) =

sin(fcx0 - ut).

(13.27)

d"2 k " = - sin2(fc£o - w<). dt u

(13.28)

Substituting this into (13.25) we find

Michael Stone

347

Thus S 2 has a non-vanishing time average, k/(2w), leading to a secular drift velocity vi = \kA2ju that is consistent with (13.24). This motion is called the Stokes drift. We also see why there is no net Newtonian momentum associated with phonons in a crystal. The atomic displacements in a harmonic crystal are given by rin = Acos(k(na) - ut), (13.29) so the crystal equivalent of (13.25) is ^ = Acos{k(na) - ut). (13.30) at Thus r] does not appear on the right-hand side of this equation. It is a linear equation and gives rise to no net particle drift. So, a sound wave does have real momentum? — But wait! The momentum density we have computed is second order in the amplitude A. The wave equation we have used to compute it is accurate only to first order in A. We may expand the velocity field as v = v ( i) + v (2 ) + • • •,

(13.31)

where the second-order correction V(2) arises because the equations of fluid motion are non-linear. This correction will possess both oscillating and steady components. The oscillatory components arise because a strictly harmonic wave with frequency Wo will gradually develop higher frequency components due to the progressive distortion of the wave as it propagates. (A plane wave eventually degenerates into a sequence of shocks.) These distortions are usually not significant in considerations of energy and momentum balance. The steady terms, however, represent 0(A2) alterations to the mean flow caused by the sound waves, and these may possess energy and momentum comparable to that of the sound field. For example, we may drive a transducer so as to produce a beam of sound which totally fills a closed cylinder of fluid. At the far end of the cylinder a second transducer with suitably adjusted amplitude and phase absorbs the beam without reflection. Since the container is not going anywhere, it is clear that the average velocity of the center of mass of the fluid must be zero, despite the presence of the sound wave. An exact solution of the nonlinear equation of motion for the fluid must provide a steady component in V(2) and this counterflow completely cancels the (piv^)) term. Indeed in

348

Phonons and forces: Momentum versus pseudomomentum

\l k Figure 13.8: Momentum Bux without m&ss-Sow,

Lagrangian coordinates the fluid particles simply oscillate back and forth with no net drift. The wave momentum and its cancelling counterflow are simply artifacts of our Eulerian description. This is one reason why professional fluid mechanics dislike the notion of momentum being associated with sound waves. The outlook for the two-fluid model is not entirely bleak, however. The V(2) corrections do not always exactly cancel the momentum. Any non-zero value for V • piv(1) — such as occurs at the transducers at the ends of the cylinder — will act as a source or sink for a v(2) counterflow, but its exact form depends on the shape of the container and other effects extrinsic to the sound field. For transducers immersed in an infinite volume of fluid, for example, the counterflow will take the form of a source-sink dipole field, and, although the total momentum of the fluid will remain zero, there will be a non-zero momentum density. Because it is not directly associated with the sound field, in the language of the two-fluid model the induced V(2) counterflow is not counted as belonging to the normal component of the fluid, i.e., to the phonons, but is lumped into the background superflow. It is determined by enforcing mass conservation,

V • ( f t v a ) + V • (Pnvn) = 0.

(13.32)

The difference of opinion between the physics and fluid mechanics communities over whether phonons have real momentum reduces, therefore, to one of different accounting conventions.

Michael Stone

13.5

349

The Unruh wave equation

Now we will examine the energy and momentum in a moving fluid. The flow of an irrotational fluid is derivable from the action [12]

S = J d'x {/> + \p{V<)>)2 + u(p)} .

(13.33)

Here p is the mass density, (j> the velocity potential, and the overdot denotes differentiation with respect to time. The function u may be identified with the internal energy density. Varying with respect to <j> yields the continuity equation p + V • {pv) = 0,

(13.34)

where v = V. Varying p gives a form of Bernoulli's equation 0 + i v 2 + Mp) = O,

(13.35)

where p(p) = du/dp. In most applications p would be identified with the specific enthalpy. For a superfluid condensate the entropy density, s, is identically zero and p is the local chemical potential. The gradient of the Bernoulli equation is Euler's equation of motion for the fluid. Combining this with the continuity equation yields a momentum conservation law dt(f>Vi) + dj(pvjVi) + pdip, = 0. (13.36) We simplify (13.36) by introducing the pressure, P, which is related to p by P(p) = / pdp. Then we can write dt{pvi) + djUji = 0,

(13.37)

Flij = p Vi Vj + Sij P.

(13.38)

where Hij is given by This is the usual form of the momentum flux tensor in fluid mechanics. The relations p = du/dp and p = dP/dp show that P and u are related by a Legendre transformation: P — pp — u(p). From this and the Bernoulli equation we see that the pressure is equal to minus the action density: - P = p 0 + i p ( V 0 ) 2 + «(p).

(13.39)

350

Phonons and forces: Momentum versus pseudomomentum

Consequently, we can write Ilij = P drfdj(j> - Stj Lj> + |p(V0) 2 + u(p)\.

13.6

(13.40)

The acoustic metric

To obtain Unruh's wave equation we set 0 = 00 + 01 P = Po + Pi-

(13.41)

Here 0O and po define the mean flow. We assume that they obey the equations of motion. The quantities 0i and pi represent small amplitude perturbations. Expanding S to quadratic order in these perturbations gives S = S0 + Jd4x

L ^ ! + l- l - \

p\ + ^Po(V0O2 + piv • V0x | .

(13.42)

Here v = V(0) = V0o- The speed of sound, c, is defined by

or more familiarly c2 = ^ . (13.44) dp The terms linear in the perturbations vanish because of our assumption that the zeroth-order variables obey the equation of motion. The equation of motion for pi derived from (13.42) is /»i = ~ { 0 i + v - V 0 1 } . (13.45) & Since p\ occurs quadratically, we may use (13.45) to eliminate it and obtain an effective action for the potential 0X only S2 = jd'x

j-ipo^)

2

+ 0 ( 0 \ + v • V0x) 2 } .

(13.46)

Michael Stone

351

The resultant equation of motion for 4>\ is [13, 14]

1ft +

V

' V ) % (o\

+ V V

' )

01 = V

^°

V <

( 13 - 47 )

^

This can be written as 1

'-9 where

^(V=ffP'"'^
(13.48)

-9r = pi V,1,

(13.49) VV T — C2I c1 We find that ^/~5 = \/det g^v = pl/c, and the covariant components of the metric are 2 2 vT A> c — v , (13.50) 9iu> v, -I The associated space-time interval can be written ds2 = ^ L{c 2 di 2 c

^•(dx* - vidt)(dxj

- vjdt)}

,

(13.51)

Metrics of this form, although without the overall conformal factor po/c, appear in the Arnowitt-Deser-Misner (ADM) formalism of general relativity [15]. There, c and — vl are referred to as the lapse function and shift vector respectively. They serve to glue successive three-dimensional time slices together to form a four-dimensional space-time [16]. In our present case, provided po/c can be regarded as a constant, each 3-space is ordinary flat R 3 equipped with the rectangular Cartesian metric gf? = dij — but the resultant space-time is in general curved, the curvature depending on the degree of inhomogeneity of the mean flow v. In the geometric acoustics limit, sound will travel along the null geodesies defined by g^. Even in the presence of spatially varying p0 we would expect the ray paths to depend only on the local values of c and v, so it is perhaps a bit surprising to see the density entering the expression for the Unruh metric. An overall conformal factor, however, does not affect null geodesies, and thus variations in p0 do not influence ray tracing. ' I use the convention that Greek letters run over four space-time indices 0,1,2,3 with O s t , while Roman indices refer to the three space components.

352

13.7

Phonbns and forces: Momentum

versus

pseudomomentum

Second-order quantities

We are going to derive various energy and momentum conservation laws from our wave equation. Before we do, let us consider what sort of quantities we would want them to contain. It is reasonable to define the momentum density and the momentum flux tensor associated with the sound field as the second order averages j(phonon)=+v(/02))

(13.52)

and n (phonon)

_ p0^VwiV{1).^+Vi^plV(1)j^+Vj^piV{1)i^+Sij(P2)

+

ViVj(p2).

(13.53)

In these expressions I have taken no account of any steady part of V(2). This is not a quantity intrinsic to the sound field and has to be found by methods outside the purely acoustic. The other second-order quantities P2 and p2 can be computed in terms of first-order amplitudes. For Pi we combine Ap

1

A2p

AP = ^ A / x + ^ ( A / i )

2

+ 0((Anf)

(13.54)

and Bernoulli's equation in the form AM = - ^ I - ^ ( V ^ 1 ) 2 - V - V 0

1 )

(13.55)

with dP

d2p

d^ = '•

d^

d

P

=

d^

=

P ?•

m ^ (13>56)

Expanding out and grouping terms of appropriate orders gives Pi = - / 3 o ( 0 i + v - V 0 1 ) = c2/91,

(13.57)

which we already knew, and Pi = - P o ^ ( V ^ ) 2 + ^ ( 0 ! + v • V0O 2 .

(13.58)

We see that P2 = \f--gL where L is the Lagrangian density for our sound wave equation. For a plane wave (P 2 ) = 0

Michael Stone

353

The second order change in the density, p2, may be found similarly. It is P2 = hn-LA(%r) • (13-59) c2 p0 \dlnpjs For a plane wave the time average {P2) = 0, but because p2 contains p\, the time average of this quantity is non-zero. The resulting change of volume of the fluid, or, if the volume is held fixed, the resulting pressure change, is the origin of the isotropic terms in the radiation stress tensor discussed earlier.

13.8

Conservation laws

Because the linear wave equation does not have access to information about counterflows or second-order density changes, we will not be able to derive the real energy and momentum fluxes from its solution. We can still derive from the wave equation what look superficially like conservation laws for these quantities, however, and these laws give us insight into the behaviour of the solutions. The conserved quantities are of course the pseudoenergy and pseudomomentum. We begin by defining a (pseudo)-energy-momentum tensor T
(13.60)

Let us recall how such a tensor comes to be associated with conservation laws. Suppose that we have an action S( (p + Sip and g^ —> g^ + Sg^, where Sip = e^dpcp 6g„v = D^

+ DuCn.

(13.61)

Here DM is the covariant derivative containing the Riemann connection compatible with the metric. The variation in the metric comes from the computation ds 2

= g^(x) dx" dx" -> 9^{xa + ea) d(x" + e") d{x" + tv) = ( e ° d a V + gaud^t" + g^dvta) dx" Ax" = {D^ + D^dx^dx".

(13.62)

354

Phonons and forces: Momentum

versus

pseudomomentum

The assembly of the terms into covariant derivatives in the last line is most easily established by using geodesic coordinates and the fact that 8gau is a tensor. The combination D^ev + Z)„eM is the Lie derivative, Ceg^, of the metric with respect to the vector field e1*. Since a mere re-coordinatization does not change the numerical value of the action, we must have

0 = 8S = /dW=S ((IV„ + D^)-F=P-

+ ^d^)^f-)

•

(13.63) Now the equations of motion state that S is unchanged by any variation in
0= [d*Xy/=g &&)-?=•??- = J " v^gog^ J

ftfx^^Dj*^-), VV^PHW

(13.64) where, in the last equality, we have integrated by parts by using the derivation property of the covariant derivative and the expression ZV"

= -^d^y/^J")

(13-65)

for the divergence of a vector. Since (13.64) is true for arbitrary eM(a;), we deduce that £>M (-T=r^-)

= D»TIW

= °-

(13-66)

Although (13.66) has the appearance of a conservation law, and has useful applications in itself, we have not yet exploited any symmetries of the system — and it is symmetries that lead to conserved quantities. To derive a genuine local conservation law we need to assume that the metric admits a Killing vector, rf. This means that the particular reparameterization eM = rf is actually an isometry of the manifold and so leaves the metric invariant C^Qiu, = D»Vv + DvVn - °-

(13.67)

Combining (13.67) with (13.66) and using the symmetry of TM" we find that D^T^) = 0. (13.68)

Michael Stone

355

Using (13.65), this can be written d^V^gT^r,,,) = 0.

(13.69)

Thus the 4-vector density Q* = sf--9 T^r),, is conventionally conserved, and Q= fd3xQ°

(13.70)

is independent of the time slice on which it is evaluated. Since (13.46) can be written as the usual action for a scalar field

s2 = J d 4 * ^ ^
(i3.7i)

we have T»" = d'facTfa - sT Q ^ a ^ i ^ i ) •

(13.72)

The derivatives with raised indices in (13.72) are defined by 0 % = 9°" d^

= ^ ( 0 ! + v • V0i), Poc

(13.73)

and aVi J* ddpfa dVi = 9* u4n = — (viifa + v • V&) - 0%^)

.

(13.74)

Thus

PIW)

=

c2„

7r-

(13.75)

The last two equalities serve as a definition of £r and fo. The quantity £r is often described as the acoustic energy density relative to the frame moving with the local fluid velocity [17]. It is, of course, more correctly a pseudoenergy density.

Phonons and forces: Momentum

356

versus

pseudomomentum

We can express the other components of (13.72) in terms of physical quantities. We find that

=

c° -3 (PiV(i)i + Vip2) . Po

(13.76)

The first line in this expression shows that, up to an overall factor, Tl° is an energy flux — the first term being the rate of work done by a fluid element on its neighbour, and the second the advected energy. The second line is written so as to suggest the usual relativistic identification of (energy-flux)/c 2 with the density of momentum. This interpretation, however, requires that p2 be the second-order correction to the density, which, sadly, it is not. Similarly c2 T13 = ~~3 (PoV(i)iV(i)j + ViPiv{l)j + UjPi«(i)i + SijP2 + ViVjfe). Po

(13.77)

We again see that if we were only able to identify p2 with P2 then T*J' has precisely the form we expect for the second-order momentum flux tensor. Although it comes close, the inability of the pseudomomentum flux to exactly capture the true-momentum flux is inevitable as we know that computing the true stresses in the medium requires more information about the equation of state than is available to the linearized wave equation. We can also write the mixed co- and contra-variant components of the energy momentum tensor T*v = T^gxu in terms of physical quantities. This mixed tensor turns out to be more useful than the doubly contravariant tensor. Because we no longer enforce a symmetry between the indices p and v, the quantity ST is no longer required to perform double duty as both an energy and a density. We find yf^g T°0

=

(Sr + piv ( 1 ) • v)

y/=gT0

=

( — + v - v ( i ) J (p0V{i)i + piv{0)i),

and I

T-iO

(13.78)

Michael Stone

357 V ^ p Tj

=

- (poU(i)iW(i)j + ViPivWj + dijPi).

(13.79)

We see that p2 does not appear here, and all these terms may be identified with physical quantities which are reliably computed from solutions of the linearized wave equation. Now we turn to the local conservation laws. In what follows I will consider only a steady background flow, and further one for which p0, c, and hence \f-g = PI/C can be treated as constant. To increase the readability of some expressions I will also choose units so that p0 and c become unity and no longer appear as overall factors in the metric or the four-dimensional energymomentum tensors. I will, however, reintroduce them when they are required for dimensional correctness in expressions such as poV(i) or £ r / c 2 . From the acoustic metric we find the Christoffel symbols 1

00

=

^(v-v)M2

1

iO

=

--di

p1 t 00

=

^(vV)M2-^M2

=

-(diVj

pt 1 J"0

=

-^Vidj

r*

=

-ViidjVk

\v\2 + -VjidiVj

-

djVi)

+ djVi) \v\2 + -{djVk - dkVj){vkVi

-'

C%k)

(13.80)

+ dkVj).

We now evaluate

=

3MT"° + T ^ T""..

(13.81)

After a little algebra we find C

T"" = l-{diVj + djVi) {povwvWj

+ SijPt).

(13.82)

Note the non-appearance of pi and p2 in the final expression — even though both quantities appear in T^". The conservation law therefore becomes dt£T + $(PiV(i)i + Vi£r) + ^ij{diVj

+ djVi) = 0,

(13.83)

358

Phonons and forces: Momentum

versus

pseudomomentum

where £ y = PoV(i)iV(i)j + 5ijP2.

(13.84)

This is an example of the general form of energy law derived by LonguetHiggins and Stuart, originally in the context of ocean waves [18]. The relative energy density, £T = T00, is not conserved. Instead, an observer moving with the fluid sees the waves acquiring energy from the mean flow at a rate given by the product of a radiation stress, Sy, with the mean-flow rate-of-strain. This equation is sometimes cited [19] as an explanation for the monstrous ship-destroying waves that may be encountered off the eastern coast of South Africa. Here long wavelength swell from distant Antarctic storms runs into the swift southbound Agulhas current and is greatly amplified by the opposing flow. We now examine the energy conservation law coming from the zeroth component of the mixed energy-momentum tensor. After a little work we find that the connection contribution vanishes identically and the energy conservation law is therefore dt {£T + PiV(i) •v)+di((

— + \- V(i) J (poV(i)i + PiU(o)i) J = 0.

(13.85)

We see that the combination £T + PiV^j • v does correspond to a conserved energy — as we should have anticipated since a steady flow provides us with a Killing vector e 0 = dt. This conservation law was originally derived by Blokhintsev [20] for slowly varying flows, and more generally by Cantrell and Hart [21] in their study of the acoustic stability of rocket engines. Finally the covariant conservation equation D^T^ = 0 reads dtPiV(i)j + dt (poV(i)iV(1)j + ViPivWj + 5ijP2) + P\V(i)idjVi = 0.

(13.86)

Here connection terms have provided a source term for the momentum density. Thus, in an inhomogeneous flow field, momentum is exchanged between the waves and the mean flow.

13.9

P h o n o n s and conservation of wave action

The conservation laws we have derived in the previous section may all be interpreted in terms of the semiclassical motion of phonons. As noted by

Michael Stone

359

Mclntyre [7], the existence of such an interpretation is a sure sign that the conservation laws in question are those of pseudomomentum and pseudoenergy. To make contact with the semiclassical picture we observe that when the mean flow varies slowly on the scale of a wavelength, the sound field can locally be approximated by a plane wave (j)(x, t) = A cos(k • x - wt).

(13.87)

The frequency w and the wave-vector k are related by the Doppler-shifted dispersion relation u = wr + k • v, where wr = c\k\, is the frequency relative to a frame moving with the fluid. A packet of such waves is governed by Hamilton's ray equations da;1 _ du dkl _ du) ~dT ~ W1' ~dt ~ ~~dx{' In other words the packet moves at the group velocity V 9 = x = c ^ + v.

(13.88)

(13.89)

and the change in k is given by %• = -kip:.

(13.90)

In this equation the time derivative is taken along the ray: | - |

+

V,.V.

(13.91)

This evolution can also be derived from the equation for null geodesies of the acoustic metric [8]. The evolution of the amplitude A is linked with that of the relative energy density, £r, through (£r) = \A2fH^.

(13.92)

For a homogeneous stationary fluid we would expect our macroscopic plane wave to correspond to a quantum coherent state whose energy is, in terms of the (quantum) average phonon density N, Em = (Volume) (£T) = (Volume) N huT.

(13.93)

360

Phonons and forces: Momentum

versus

pseudomomentum

Since it is a density of "particles", N should remain the same when viewed from any frame. Consequently, the relation Nh=^-

(13.94)

should hold true generally. In classical fluid mechanics the quantity (£T)/wr is called the wave action [22, 17, 23]. The time averages of other components of the energy-momentum tensor may be also expressed in terms of N. For the mixed tensor we find (y^T0,,)

=

(v^n)

- ((£

(£T + vp1vll))

= Nhu)

+ v V(1)

'

) ( w ) i + pi«»)) = wM^)i

(-V^T")

-

(piV(i)i) = Nhki

{-yf^g

=

(p0v{1)iv{1)j

Tj)

+ viPlv{l)j

+ 8ijP2) = Nhkj(Vg)i.

(13.95)

In the last equality we have used that (P 2 ) = 0 for a plane progressive wave. If we insert these expressions for the time averages into the Blokhintsev energy conservation law (13.85), we find that dNhw

+ v

(^y^

= 0

(13 g6)

at We can write this as Nh(

'dw „, _ \ — +Vg-Vu)+huj(

.

fdN — + V- (NVg)

) = 0.

(13.97)

The first term is proportional to dw/di taken along the rays and vanishes for a steady mean flow as a consequence of the Hamiltonian nature of the ray tracing equations. The second term must therefore also vanish. This vanishing represents the conservation of phonons, or, in classical language, the conservation of wave-action. Conservation of wave action is an analogue of the adiabatic invariance of E/w in the time dependent harmonic oscillator. In a similar manner, the time average of (13.86) may be written

^

+ V.(* W +

**.£

Michael Stone

361

= A r +V V +

+

+ V (iVV9)

( ^ ^^ ^£) ^(^ '

We see that the momentum law becomes equivalent to phonon-number conservation combined with the ray tracing equation (13.90).

13.10

Summary

When dealing with waves in a medium it is essential to distinguish between the conceptually distinct quantities of momentum and pseudomomentum. Under suitable circumstances either may be used for computing forces — but there is no general rule for determining those circumstances. In the case of non-dispersive sound waves, momentum and pseudomomentum often travel together, and are therefore easily confused — but the conservation laws derived by manipulating the wave equation are those of pseudomomentum and pseudoenergy. The GR analogy provided by the acoustic metric provides a convenient and structured route to deriving these laws when the background fluid is moving.

Acknowledgements: This work was supported by US National Science Foundation. I would like to thank the staff and members of NORDITA in Copenhagen, Denmark, for their hospitality while this paper was being written.

)-

(13

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Bibliography [1] Sir Rudolf Peierls, Surprises in Theoretical Physics, (Princeton, 1979). [2] E. I. Blount, Bell Telephone Laboratories Technical Report 38139-9 (1971). [3] J. P. Gordon, Phys. Rev. A 8 (1973) 14. [4] D. F. Nelson, Phys. Rev. A 44 (1991) 3985. [5] M. Stone, Phys. Rev. B 61 (2000) 11780. [6] G. E. Volovik, Pis'Ma Zhu. Eksp. Teor. Fiz. 67 (1998) 841 [JETP Letters 67 (1998) pp.881], (cond-mat/9804308) [7] M. E. Mclntyre, On the Wave-momentum Myth, J. Fluid. Mech. 106 (1981) 331. [8] M. Stone, Phys. Rev. E 62 (2000) 1341. [9] E. J. Post, J. Acoustical Soc. Amer. 25 (1953) 55. [10] R. T. Beyer, Radiation Pressure: the history of a mislabelled tensor, J. Acoustical Soc. Amer. 63 (1978) 1025. [11] L. Brillouin, Annales de Physique, 4 (1925) 528 (in French). [12] A. M. J. Schakel, Mod. Phys. Lett. BIO (1996) 999. (cond-mat/9607164) [13] W. Unruh, Phys. Rev. Lett. 46 (1981) 1351. [14] W. Unruh, Phys. Rev. D 51 (1995) 282 363

364

Phonons and forces: Momentum

versus

pseudomomentum

[15] R. Arnowit, S. Deser, C. W. Misner in Gravitation: an Introduction to Current Research ed. L. Witten (Wiley N.Y. 1962), pp227-265; C. Misner, K. Thome, J. Wheeler, Gravitation (W. H. Freeman, San Francisco 1973). [16] For a picture see: C. Misner, K. Thome, J. Wheeler, op. cit. p504. [17] Sir James Lighthill, Waves in Fluids (Cambridge University Press 1978) [18] M. S. Longuet-Higgins, R. W. Stuart, J. Fluid. Mech. 10 (1961) 529. [19] M. S. Longuet-Higgins, R. W. Stuart, Deep Sea Res. 11 (1964) 529. [20] D. I. Blokhintsev, Acoustics of a Non-homogeneous Moving Medium. (Gostekhizdat, 1945). [English Translation: N. A. C. A. Technical Memorandum no. 1399, (1956)] [21] R. H. Cantrell, R. W. Hart, Interaction between sound and flow in Acoustic cavities, mass, momentum and energy considerations, J. Acoustical Soc. Amer. 36 (1964) 697. [22] C. J. R. Garrett, Proc. Roy. Soc. A299 (1967) 26. [23] D G. Andrews, M. E. Mclntyre, J. Fluid. Mech. 89 (1978) 647.

Chapter 14 Coda

Matt Visser Victoria University School of Mathematics and Computer Science Wellington New Zealand E-mail: [email protected]

Abstract: To wrap things up I discuss the extent to which the occurrence of an effective metric should be expected to be generic, give some background on the notion of induced gravity, and consider the possibility of the breakdown of Lorentz invariance at ultra-high (Planck scale) energies. Finally I close by reiterating one of the key points of this book: experimental tests of analogue Hawking radiation are highly desirable and (while certainly experimentally challenging) are technologically tantalizingly close to being feasible. 365

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366

14.1

Introduction

As we have seen from the preceding chapters, the idea of building analogue models of, and possibly for, general relativity is currently attracting considerable attention. One of the driving forces behind this is the extreme difficulty (and inadvisability) of working with intense gravitational fields in a laboratory setting. Because of this, interest has now turned to investigating the possibility of simulating aspects of general relativity — though it is not a priori expected that all features of Einstein gravity can successfully be carried over to the condensed matter realm. In wrapping up the book I wish to make a number of points: • The occurrence of "effective metrics", analogue models of gravity, and consequently of analogue black holes is in some sense generic. • Analogue models for gravity, meaning analogue models in which one tries to obtain some approximation to the Einstein equations, are a trickier proposition — but still well worth looking at. • Analogue models provide useful templates for thinking about the possible breakdown of Lorentz invariance at extremely high (Planck scale) energies, a possibility currently under investigation within the particle physics community.

14.2

Effective metric techniques

Although we have not gone into all the possibilities in detail, numerous rather different physical systems have now been seen to be useful for developing analogue models of general relativity. The reason for devoting so much attention to the Bose-Einstein condensates and quasi-particle systems of the first few chapters is a very practical one — these seem to be the best systems for developing analogue black holes for which one has a hope of experimentally detecting Hawking radiation. But the use of "effective metric" techniques, that is, analogue models of general relativity, is a much more generic phenomenon. A doubtless still incomplete list of the possibilities includes the following. • Dielectric media: A refractive index can be reinterpreted as an effective metric, the Gordon metric. (Gordon [1], Skrotskii [2], Balazs [3],

Matt Visser

367

Plebanski [4], de Felice [5], and many others.) 1 • Acoustics in flowing fluids: Acoustic black holes, aka "dumb holes". (Unruh [6], Jacobson [7], Visser [8], Liberati et al [9], and many others; see the introductory chapter of this book.) • Phase perturbations in Bose-Einstein condensates: Formally similar to acoustic perturbations, and analyzed using the nonlinear Schrodinger equation (Gross Pitaevskii equation) and Landau-Ginzburg Lagrangian; typical sound speeds are centimetres per second to millimetres per second. (Garay et al [10], Barcelo [11] et al; see the chapter by Luis Garay.) • High-refractive-index dielectric fluids ("slow light"): In dielectric fluids with an extremely high group refractive index it is experimentally possible to slow lightspeed to centimetres per second or less. (LeonhardtPiwnicki [12], Hau et al [13], Visser [14], and others; see the chapter by Ulf Leonhardt.) • Quasi-particle excitations: Fermionic or bosonic quasi-particles in a heterogeneous superfluid environment. (Volovik [15], Kopnin-Volovik [16]. Jacobson-Volovik [17], and Fischer [18]; see the chapters by Ted Jacobson and Grigori Volovik.) • Nonlinear electrodynamics: If the permittivity and permeability themselves depend on the background electromagnetic field, photon propagation can often be recast in terms of an effective metric. (Plebanski [19], Dittrich-Gies [20], Novello et al [21]; see the chapter by Mario Novello.) • Linear electrodynamics: If you do not take the spacetime metric itself as being primitive, but instead view the linear constitutive relationships of electromagnetism as the fundamental objects, one can nevertheless reconstruct the metric from first principles. (Hehl, Obukhov, and Rubilar [22, 23, 24].) • Scharnhorst effect: Anomalous photon propagation in the Casimir vacuum can be interpreted in terms of an effective metric. (Scharnhorst [25], Barton [26], Liberati et al [27], and many others.) 'This is the Gordon of the Klein-Gordon equation.

368

Coda

• Thermal vacuum: Anomalous photon propagation in QED at nonzero temperature can be interpreted in terms of an effective metric. (Gies [28].) • "Solid state" black holes. (Reznik [29], Corley and Jacobson [30], and others.) • Astrophysical fluid flows: Bondi-Hoyle accretion and the Parker wind [coronal outflow] both provide physical examples where an effective acoustic metric is useful, and where there is good observational evidence that acoustic horizons form in nature. (Bondi [31], Parker [32], Moncrief [33], Matarrese [34], and many others.) • Other condensed-matter approaches that don't quite fit into the above classification [35, 36]; see the chapter by Robert Laughlin, George Chapline, and David Santiago. A literature search as of August 2001 finds well over a hundred scientific articles devoted to one or another aspect of analogue gravity and effective metric techniques. The sheer number of different physical situations lending themselves to an "effective metric" description strongly suggests that there is something deep and fundamental going on. Typically these are models of general relativity, in the sense that they provide an effective metric and so generate the basic kinematical background in which general relativity resides; in the absence of any dynamics for that effective metric we cannot really speak about these systems as models for general relativity. However, as we will discuss more fully bellow, quantum effects in these analogue models might provide some sort of dynamics resembling general relativity. The common theme underlying these various "effective metric" techniques seems to be this: Start with some complicated field theory, and linearize around some background in order to define quasi-particles; that is perform field-theory "normal modes" analysis [37]. The quasiparticles will propagate and describe wavelike excitations if and only if the linearized field equations are hyperbolic. But hyperbolic field equations imply that the characteristic surfaces (the Monge cone) has a structure qualitatively similar to the light cone of general relativity. From there it is a small step to defining first a conformal class of metrics and then some notion of effective spacetime metric. To illustrate this consider a single scalar field (j) whose dynamics is governed by some first-order Lagrangian £(c^0, (/)). (By "first-order" we mean

Matt Visser

369

that the Lagrangian is some arbitrary function of the field and its first derivatives.) We want to consider linearized fluctuations around some background solution <^0(i, x) of the equations of motion, and to this end we write e2

{t,x) = Mt,x) + eMt,x)

+ - 0 2 ( f , f ) + 0(e 3 ).

(14.1)

Now use this to expand the Lagrangian around the classical solution <po(t, x): £(<^,

) = £ ( d ^ o , 0o) + e

dc

e2

+— 82C +0(e3).

(14.2)

It is particularly useful to consider the action S[}= fdd+ixC(d^,cl>),

(14.3)

since doing so allows us to integrate by parts. (Note that the Lagrangian £ is taken to be a tensor density, not a scalar.) We can now use the EulerLagrange equations for the background field 8C a

» < ^ ) - d - = 0'

(14.4)

to discard the linear terms (remember we are linearizing around a solution of the equations of motion) and so we get 8(8^)

+

f d2C Kd<j>d

d(8„4>).

Maro}W i | + ° ( e 3 ) - ( 1 4 - 5 )

Having set things up this way, the equation of motion for the linearized fluctuation is now easily read off as

^({a(a^w)}^O"(^~a4a(||^})01 = o(14.6)

370

Coda

This is a second-order differential equation with position-dependent coefficients (these coefficients all being implicit functions of the background field o). Following an analysis developed for acoustic geometries (Unruh [6], Visser [8], Liberati et al [9], Barcelo et al [11]), which we now apply to this much more general situation, the above can be given a nice clean geometrical interpretation in terms of a d'Alembertian wave equation — provided we define the effective spacetime metric by

<147)

^ ^ = ^={5i5&)}L-

'

Suppressing the <po except when necessary for clarity, this implies [in (d+1) dimensions, d space dimensions plus 1 time dimension]

'--{^{»mm?i})

•

<148)

Therefore „ vUs_(_A 9

\

d2C

-tx-V^-Dl

f

(0 O )-^ ^\d{dii(j))d{dv(t))))

d2C

\\

\IKW)WA))\* (14.9)

And, taking the inverse

/

U\-(-A

d2c

^y/("-*> |

/

a2£

y'\

(14.10) We can now write the equation of motion for the linearized fluctuations in the geometrical form [A(5(o))-m>)]<£i = 0,

(14.11)

where A is the d'Alembertian operator associated with the effective metric g{a), and V((f>o) is the background-field-dependent potential

v

™-^{£kM«B°i})-

(1412)

Thus V((j>0) is a true scalar (not a density). Note that the differential equation (14.11) is automatically formally self-adjoint (with respect to the measure vS?dd+1:r).

Matt Visser

371

It is important to realise just how general the result is (and where the limitations are): It works for any Lagrangian depending only on a single scalar field and its first derivatives. The linearized PDE will be hyperbolic (and so the linearized equations will have wave-like solutions) if and only if the effective metric g^ has Lorentzian signature ±[—,+d]. Observe that if the Lagrangian contains nontrivial second derivatives you should not be too surprised to see terms beyond the d'Alembertian showing up in the linearized equations of motion. Specific examples of this in special cases are already known: for example this happens in the acoustic geometry when you add viscosity (Visser [8]; not really a Lagrangian system but the general idea is the same), or in the quantum geometry of the Bose-Einstein condensate if you keep terms arising from the quantum potential (Barcelo et al [11]). As a specific example of the appearance of effective metrics due to Lagrangian dynamics we mention that inviscid irrotational barotropic hydrodynamics naturally falls into this scheme (which is why, with hindsight, the derivation of the acoustic metric was so relatively straightforward) [6, 8, 11]. In inviscid irrotational barotropic hydrodynamics the lack of viscosity (dissipation) guarantees the existence of a Lagrangian; which a priori could depend on several fields. Since the flow is irrotational v = W is a function only of the velocity potential, and the Lagrangian is a function only of this potential and the density. Finally the equation of state can be used to eliminate the density leading to a Lagrangian that is a function only of the single field # and its derivatives. Specifically, start with the action for inviscid irrotational barotropic hydrodynamics <S = [dtddxlpd

+ ^-p{Vd)2 + V{p)\.

(14.13)

(Compare with the penultimate chapter by Michael Stone.) Here p is the mass density, -d the velocity potential, and the overdot denotes differentiation with respect to time. The function V(p) denotes the internal energy density. Varying with respect to p gives the algebraic equation h(p)=tf+]-(Vd)2.

(14.14)

Here h(p) — dV(p)/dp denotes the specific enthalpy; and dh = dp/p. This equation can be inverted algebraically (i.e., no integrations or differentiations

Coda

372 required)

p = /i-i(tf + I(Vtf) 2 ).

(14.15)

Resubstitute back into the action, using p(p) = p h{p) -

V{p).

(14.16)

It is then easy to see that

S =

=

fdtddxp /^(tf + ^ w ) 2 )

(14.17)

fddxpU+^(vd)A.

(14.18)

This has eliminated p completely and reduced everything to a problem of an action depending only on a single scalar field d. The completely general analysis given above then applies and guarantees the existence of an effective Lorentzian metric governing the fluctuations. Note that in all these cases the (fundamental) dimensionality of spacetime is put in by hand — in the present formalism there is no way to determine the fundamental dimensionality dynamically. (Of course in a Kaluza-Klein framework the effective dimensionality can change if some dimensions become small for dynamical reasons.) Also note that d = 1 space dimensions is special, and the present formulation does not work unless det(/' u / ) = 1. This observation can be traced back to the conformal covariance of the Laplacian in 1 + 1 dimensions, and implies (perhaps ironically) that the only time the procedure risks failure is when considering a field theory defined on the world sheet of a string-like object. The message to take from the above is this: The emergence of an "effective metric", in the sense that this notion is used in the so-called "analogue models" of general relativity, is a rather generic feature of the linearization process. While the existence of an effective metric by itself does not allow you to simulate all of Einstein gravity, it allows one to do quite enough to be really significant — in particular it seems that the existence of an effective Lorentzian metric is really all that is in principle needed to obtain simulations of the Hawking radiation effect [6, 38, 39]. In this regard, the major technical limitation of the current analysis is that it is limited to a single scalar field. Extensions of this idea involve both some technical subtleties and some new

Matt Visser

373

physics, and that scenario will be discussed more fully elsewhere. The major piece of additional physics is the possible presence of birefringence, or more generally "multi-refringence", with different normal modes possibly reacting to different metrics. The Eotvos experiment [the observational universality of free fall to extremely high accuracy] indicates that all the physical fields comprising ordinary bulk matter "see" [to high precision] the same metric. It is this observational fact that permits us to formulate the Einstein Equivalence principle and speak of the metric of spacetime. Thus in extending the notion of effective geometry to a system with many degrees of freedom, experiment tells us that we should seek conditions that would naturally serve to suppress birefringence. Only in that case would it make sense to speak of a unique spacetime metric (or at worst, of multiple almost-degenerate metrics).

14.3

Analogue models for Einstein gravity

When it comes to considering analogue models for Einstein gravity, variants on Sakharov's idea of "induced gravity", it is useful to remember that the last few years have seen an increasing number of indications that Einstein gravity (and even quantum field theory) may not be as "fundamental" as was once supposed: • In "induced-gravity" models a Id Sakharov [40] the dynamics of gravity is an emergent low-energy phenomenon that is not fundamental physics. In those models the dynamics of gravity (the approximate Einstein equations) is a consequence of the quantum fluctuations of the other fields in the theory. (In induced gravity models gravitation is not fundamental in exactly the same sense that phonons are not fundamental: phonons are collective excitations of condensed matter systems. Phonons are not fundamental particles in the sense of, say, photons. But this should not stop you from second quantizing the phonon field as long as you realise you should not take phonons seriously at arbitrarily high momenta.) • Donoghue [41] has strongly argued that quantum gravity itself should simply be viewed as an effective field theory, in the same sense that the Fermi theory of the weak interactions is an effective field theory — it still makes sense to second quantized in terms of gravitons [42], but the high-energy physics is likely to be rather different from what

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374

could be guessed based only on observing low-energy excitations, and you should not necessarily take the gravitons seriously at arbitrarily high momenta. • Indeed, even strictly within the confines of particle physics, attitudes towards effective theories seem to be changing — they are now much more likely to be used, at least as computational tools. As long as one has a clear understanding of when to stop taking them seriously effective theories are perfectly good physics even if they are not "fundamental" [43, 44]. Once (following for instance the ideas of the preceding section) one has developed the notion of a derived "effective metric" based on a linearization procedure, we can certainly consider the effect of second quantizing the linearized fluctuations. At one loop the quantum effective action will contain a term proportional to the Einstein-Hilbert action — this is, in modern language, a key portion of Sakharov's "induced gravity" idea [40]. Thus one can argue (modulo non-trivial technical difficulties concerning the details of how the Einstein equations are to be recovered) that the occurrence of not just an "effective metric", but also an "effective geometrodynamics" closely related to Einstein gravity might be a largely unavoidable feature of the linearization and second quantization process.

14.4

Approximate Lorentz symmetry

On a related front, the particle physics and relativity communities have also seen Lorentz symmetry emerge as a low-energy approximate symmetry in several distinct physical situations having nothing to do with relativity per se: • As an infra-red fixed point of the renormalization group in certain nonLorentz invariant quantum field theories; (Nielsen et al [45]). • As a low-momentum approximation to acoustic propagation in the presence of viscosity; (Visser [8]). • As a low-momentum approximation to quasi-particle propagation governed by the Bogoliubov dispersion relation; (Barcelo et al [11]).

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375

• In certain other quasi-particle dispersion relations (Volovik [46]). This feeds back into the question of whether or not Lorentz symmetry eventually breaks down at Planck scale energies — the various analogue models and their associated effective metrics provide concrete physical examples of different ways in which the breakdown of Lorentz invariance can occur. Thus, because the condensed matter physics on which the analogue models are often based is generally reasonably well understood, analogue models can serve as templates for investigating symmetry breakdown.

14.5

Last words

This book has covered a lot of ground, from classical hydrodynamics to quantum field theory to general relativity. We, the editors, trust that the readers have been both entertained and enlightened — we strongly wish to encourage experimental participation in developing these ideas, specifically in bringing one or more of the proposed analogue models for Hawking radiation to experimental realization.

Acknowledgements: The editors wish to thank all the contributers for their enthusiasm and the hard work involved in writing up their respective chapters. We also wish to thank Nick Yunes for proofreading and pedagogical suggestions, though remaining errors are the editors' sole responsibility. The striking cover art was kindly provided by Enrique Arilla, and we are very grateful to him for permitting its reproduction. Finally, the editors also wish to thank Daniel Cartin and the editorial staff at World Scientific for their support and patience.

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Coda "The velocities of light in modified QED vacua", Ann. Phys. (Leipzig) 7, 700-709 (1998) [hep-th/9810221].

[26] G. Barton, "Faster-than-c light between parallel mirrors. The Scharnhorst effect rederived", Phys. Lett. B 237, 559-562 (1990); G. Barton and K. Scharnhorst, "QED between parallel mirrors: light signals faster than c, or amplified by the vacuum", J. Phys. A 26, 20372046 (1993). [27] S. Liberati, S. Sonego and M. Visser, "Scharnhorst effect at oblique incidence", Phys. Rev. D 63, 085003 (2001) [quant-ph/0010055]. [28] H. Gies, "Light cone condition for a thermalized QED vacuum", Phys. Rev. D 60, 105033 (1999) [hep-ph/9906303]. [29] B. Reznik, "Trans-Planckian tail in a theory with a cutoff", Phys. Rev. D 55, 2152 (1997) [gr-qc/9606083]. "Origin of the thermal radiation in a solid-state analog of a black hole", Phys. Rev. D 62, 0440441 (2000) [gr-qc/9703076]. [30] S. Corley and T. Jacobson, "Hawking Spectrum and High Frequency Dispersion", Phys. Rev. D 54, 1568 (1996) [hep-th/9601073]; S. Corley, "Particle creation via high frequency dispersion", Phys. Rev. D 55, 6155 (1997); "Computing the spectrum of black hole radiation in the presence of high frequency dispersion: An analytical approach", Phys. Rev. D 57, 6280 (1998) [hep-th/9710075]. [31] H. Bondi, "On spherically symmetric accretion", Mon. Not. Roy. Astron. Soc. 112, 195-204 (1952). [32] E.N. Parker, "Dynamical properties of stellar coronas and stellar winds V. Stability and wave propagation", Ap. J. 143, 32 (1966). [33] V. Moncrief, "Stability of a stationary, spherical accretion onto a Schwarzschild black hole", Ap. J. 235, 1038 (1980). [34] S. Matarrese, "Perturbations of an irrotational perfect fluid", in Proceedings of the 4 'th Italian conference on general relativity and the physics of gravitation, (Florence, 10-13 October 1984), pp 283-287;

Matt Visser

381

"Phonons in a relativistic perfect fluid", in Proceedings of the 4 'th Marcel Grossmann meeting on General Relativity, R. Ruffini (ed.), (Elsevier, 1986), pp 1591-1595; "On the classical and quantum irrotational motions of a relativistic perfect fluid. I. Classical Theory", Proc. R. Soc. Lond. A 401 53 (1985). [35] G. J. Stephens and B. L. Hu, "Notes on black hole phase transitions", gr-qc/0102052. [36] G. Chapline, E. Hohlfeld, R. B. Laughlin, and D. I. Santiago, "Quantum phase transitions and the breakdown of classical general relativity", grqc/0012094. [37] C. Barcelo, S. Liberati, and M. Visser, "Analogue gravity from field theory normal modes?", gr-qc/0104001; Classical and Quantum Gravity (in press). [38] S. W. Hawking, "Particle Creation By Black Holes", Commun. Math. Phys. 43, 199 (1975). [39] M. Visser, "Hawking radiation without black hole entropy", Phys. Rev. Lett. 80, 3436 (1998) [gr-qc/9712016] "Essential and inessential features of Hawking radiation," hepth/0106111. [40] A. Sakharov, "Vacuum quantum fluctuations in curved space and the theory of gravitation", Soviet Physics Doklady, 12, 1040 (1968). [41] J. F. Donoghue, "Quantum general relativity is an effective field theory", PRINT-96-198 (MASS.U.,AMHERST) Talk presented at the 10th International Conference on Problems of Quantum Field Theory, Alushta, Ukraine, 13-17 May 1996; "Gravity and Effective Field Theory: A Talk for Phenomenologists", hep-ph/9512287; "Introduction to the Effective Field Theory Description of Gravity", gr-qc/9512024; "The Ideas of gravitational effective field theory", hep-th/9409143; "General relativity as an effective field theory: The leading quantum corrections", Phys. Rev. D 50, 3874 (1994) [gr-qc/9405057].

382

Coda

[42] R. P. Feynman, F. B. Morinigo, W. G. Wagner and B. Hatfield, "Feynman lectures on gravitation", Reading, USA: Addison-Wesley (1995) 232 p. (The advanced book program). [43] S. Weinberg, "What is quantum field theory, and what did we think it was?", hep-th/9702027. [44] S. Weinberg, The quantum theory of fields, (Cambridge, England, 1999). [45] H. B. Nielsen and I. Picek, "Lorentz Noninvariance", Nucl. Phys. B 211, 269 (1983). S. Chadha and H. B. Nielsen, "Lorentz Invariance As A Low-Energy Phenomenon", Nucl. Phys. B 217, 125(1983). [46] G. E. Volovik, "Induced gravity in superfluid He-3", J. Low. Temp. Phys. 113, 667 (1997) [cond-mat/9806010]; "Superfluid analogies of cosmological phenomena", gr-qc/0005091; "Reentrant violation of special relativity in the low-energy corner", hepph/0101286.

Appendix: Elements of general relativity M a t t Visser

General relativity is formulated in a four-dimensional spacetime with coordinates xv = (t,x) — (t,xl). The fundamental quantity is the spacetime metric 1 ds 2 = g^ dx" dxv = 0, (1) which is used to measure invariant distances and proper times. Particles, unless they are acted on by other external forces, follow geodesies of the spacetime. That is, /"

/

dx^ fa" (2)

/

ds = / dA \Jg^{x{\)) — —, is an extremum. The parameter A is at this stage arbitrary. The EulerLagrange equations of motion coming from extremizing this integral are d2^ „ „ dxa dxP dx" + r dA^ ^dA--dA 0C dA-'

,„. (3)

where the P * ^ are the so-called "Christoffel symbols" r ^ a / 3 = ^ ST (5
(4)

Here as usual gaf} with both indices up denotes the inverse of the matrix ga$ with both indices down, and gaptl denotes the partial derivative dg^/dx1. x

The Einstein summation convention implies repeated indices axe to be summed so that g^v(x) dx* dx" is defined to mean £}f=0 H3j=o9^"(x) ^x* ^x" • 383

Artificial Black Holes

384

By changing the parameterization of the curve A —> A(A), and going to an "affine parameter" it is possible to simplify the geodesic equations to d 2 x"

dF

dx<* dx? + r

(5)

^HdA=°-

Tidal forces are governed by Jacobi's equation of geodesic separation: If two initially parallel geodesies are separated by a transverse distance As'' then2 d2(A^) _ dx» - — ^ - R ^ - ^ - A x

dx» -^-.

(6)

Here dx^/dX is the tangent along either spacetime geodesic (they start out parallel to each other); Ax1* is the transverse separation between the geodesies, and R^vap is the Riemann curvature tensor which is defined in terms of second derivatives of the metric R^vap ~

—T^va.P + r ^ . a

+ T1*aa VVQ

— T^^p

Vua.

(7)

All the above can easily be copied over to the analogue models; the bit that is difficult to copy over is the Einstein equations of general relativity. Define the Ricci tensor and Ricci scalar Ra0

= R°'ao0',

R = ga

Rap

= 9°

R" aap-

(8)

Then the Einstein equations are RaP — yR

9aP = 87rC? New ton Taff.

(9)

These equations relate part of the curvature of spacetime to the distribution of matter (which is encoded in the stress-energy tensor Tap). Backtracking through the various definitions, the Einstein equations result in complicated non-linear second-order partial differential equations for the metric components in terms of the distribution of matter.

2

In particular, the ocean tides here on Earth are ultimately driven by the Riemann tensor generated by the Moon causing differential acceleration between various parts of the ocean.

Index Abraham momentum, 336 absolute horizon, 10, 24 acoustic black hole, xi, 36-55, 88, 367 difficulty in constructing, 110-123 acoustic geometry, xiii, 8, 10, 12, 1819, 215, 218, 350-351 definition, 9 acoustic horizon, 22, 36, 111 acoustic metric, 37,134, 137,144,147, 153, 155, 157, 246, 254, 257, 260, 351, 359 acoustic momentum, xiv aether, xii, 181, 182, 336, 338 affine parameter, 384 Agulhas current, 358 Aharonov-Bohm effect, 81, 336 analogue gravity, 2 analogue horizon, 2, 82 apparent horizon, 10, 11, 13, 22-24,

black hole, 180, 181, 183, 185, 187, 193-196 Blokhintsev energy law, 358, 360 Bogoliubov dispersion relation, 374 Bogoliubov equations, 38, 39, 42, 43, 49, 50, 53-55 Bogoliubov frequency mixing, 123 Bogoliubov method, 138,140,182,188, 337 Bogoliubov solution, 185 bolometer, 117 Boltzmann distribution, 116 Boltzmann-Ehrenfest principle, 342, 343 Bondi-Hoyle accretion, 12, 368 Born-Infeld Lagrangian, 12, 268, 276, 285, 288, 297, 298, 304 Born-Oppenheimer approximation, 238 Bose-Einstein condensate, xi, 2, 12, 36-55, 62, 122, 139, 143, 184, 253, 336, 366, 367, 371 braneworld cosmology, xiii, 200-209

217 Arnowitt-Deser-Misner formalism, 351 baby universe, 90, 100, 102 back-reaction, 218 barotropic, 15,16,18,19, 21, 23, 247249, 251-253, 371 Bekenstein entropy, 13, 27-28, 237 Bernoulli equation, 114, 349, 352 Bianchi identity, 131, 162 birefringence, xiii, 12, 276, 277, 298, 299, 373 black hole, 195

Cartesian coordinates, 3, 6, 62, 248, 351 Casimir effect, 130, 131, 136, 154, 158, 163-165, 167-173, 328 catenary, 316 Cauchy surface, 101 centrifugal field, 200, 201 chemical potential, 38, 41, 48, 136, 145, 147, 149-152, 154, 156, 165, 166, 349

385

386 Cherenkov effect, 327, 328 chiral fermions, 129, 137, 141 chirality, 311 Christoffel symbols, 4, 6, 273, 357, 383 Clebsch potential, 255, 258 coherence, 234, 235 coherence length, 92, 94, 95, 180, 183, 190 condensate, 93, 96, 104 conformal anomaly, see scale anomaly conformal invariance, 225, 227, 232 continuity equation, 16 Cooper pair, 91, 92, 249, 253 core, 16, 93, 95-97, 102, 246 Coriolis effect, 200 cosmological constant, 129, 131, 133, 134, 147, 149, 152, 153, 156, 157, 162, 173, 182, 194, 195 cosmology, xiii, 148,154,157-161, 200209 counterterms, 145, 232 critical depth, 185 critical equation, 185 critical frequency, 218 critical opalescence, 192 critical point, 180, 182, 183, 185, 190, 191 critical pressure, 185 critical surface, 183, 186, 187, 190 critical temperature, 97 critical value, 169, 170, 188, 294 critical velocity, 82 cutoff, 42, 43, 89, 90, 122, 133, 163, 170 insensitivity of Hawking effect, 112 d'Alembertian, 18, 202, 203, 205, 206, 215, 218, 220, 221, 223, 254, 370, 371

Artificial Black Holes dark state, 73-78 de Laval nozzle, see Laval nozzle Debye temperature, 147, 153, 164 dielectric, 2, 9, 12, 165, 168, 279-282, 285-287, 289, 296, 336, 366, 367 Dirac-Nambu-Goto stress-energy, 202 dispersion, 64, 65, 67-69, 74, 80, 89, 168, 233, 283, 327, 361 dispersion relation, 50, 51, 65, 96,102, 112, 117, 119, 122, 134, 185, 188-190, 196, 218, 219, 234, 359, 375 domain wall, 90, 93-96, 98, 102, 104, 105 Doppler detuning, 82 Doppler effect, 80, 82, 97, 142, 327, 359 anomalous, 310, 326-328 dragging of inertial frames, 21 dumb hole, see acoustic black hole dyadosphere, 294-295 effective action, see effective Lagrangian effective field theory, 78, 83, 145, 146, 148, 373, 374 effective geometry, 90, 268, 269, 2 7 1 274, 276-278, 280, 281, 286290, 292-294, 297, 298, 302, 304 effective Lagrangian, 79, 268, 271, 274, 280, 289, 292, 302 effective metric, xiii, 2, 37, 81, 82, 110, 134, 137, 142, 144, 155, 160, 173, 214, 272, 275, 279, 282-285, 287, 288, 291-294, 296, 298, 299, 301, 303, 366373 eikonal, 228 Einstein equation, 18, 90, 131, 132,

Mario Novello/Matt

Visser/Grigori

173, 214, 216, 217, 224, 225, 373 defined, 384 Einstein gravity, 131, 133-135, 145, 152, 161, 172, 204, 285 basic definitions, 383-384 Einstein summation convention, 4, 383 Einstein tensor, 19, 133, 160, 173 Einstein-Hilbert action, 27, 28, 131, 133, 134, 206, 224, 374 EIT, xi, 69-73, 77-80, 82 elliptic, 203, 339 energy gap, 91, 92, 94, 171, 188, 190 energy-momentum tensor, see stressenergy tensor entanglement, 90, 215, 221, 223, 227, 228, 230, 236, 237 enthalpy, 249, 349, 371 entropy, xii, 88-90, 98, 141, 160, 174, 237, 258, 259, 261, 344, 349 ergoregion, 9-12, 20-22, 91, 98, 104 Euler equation, 16-18, 21, 25, 252254, 259, 349 Euler-Heisenberg Lagrangian, xiii, 12, 288, 289, 292-293 Euler-Lagrange equations, 4, 369 event horizon, 9-13, 20, 36, 214-218, 220, 222, 225, 226, 228, 229, 234 extrinsic curvature, see second fundamental tensor Fermat's principle, 3 Fermi energy, 92 Fermi momentum, 91, 92 Fermi velocity, 92 Feynman diagrams, 231, 232, 239 Feynman rules, 181 flowing fluid, 2, 8, 9, 12, 15, 92, 367 fluid

Volovik

387

non-relativistic, 247 Fock space, 222, 223 Foucault pendulum, 200 Fourier component, 42, 65, 139, 218 Fourier space, 42 Fourier transform, 64, 65, 67, 92, 311, 320, 321 Frenet-Serret invariants, 308, 309, 311, 313, 314, 328 Priedmann-Robertson-Walker universe, 7, 161, 163 Galilean relativity, 96 gamma ray bursts, 196 gauge invariance emergent, 137 Gaussian ensemble, 232, 233 Gaussian integration, 225 general covariance, 137 general relativity, 285 basic definitions, 383-384 geodesic, 3-5, 215, 217, 221, 224, 225, 227, 232, 249, 268, 271-274, 286, 288, 289, 295-298, 304, 351, 359, 383 geometrical acoustics, 9, 351 geometrical optics, 6 Godel solution, 285, 296 Gordon metric, 12, 279, 283-285, 366 gravity emergent, 181, 182, 196 Green function, 216, 220, 229, 230, 234, 235, 239 Gross-Pitaevskii equation, 38, 41, 42, 44, 45, 47, 184, 367 group velocity, 121, 123 Hadamard discontinuities, 269, 279 Hadamard singularity, 235 Hamilton ray equation, 359 Hamilton's principle, 300

388 Hawking effect, 14, 37, 88-91, 93, 9 7 99, 102, 104, 105 Hawking modes, 89 Hawking pair, 98, 104 Hawking photons, xiii Hawking quanta, 89, 104, 214-218, 220, 222, 223, 232 Hawking radiation, xi, 13-15, 23, 28, 29, 36-55, 214, 215, 217-223, 232, 240, 308, 328, 372, 375 estimates, 26-27 Hawking temperature, 89-91, 93, 97, 98, 110 acoustic, 111 Heisenberg ferromagnet, 187, 190 Heisenberg picture, 223 helicoid, 317 Helium 3, xii, 128, 135-137, 141, 152, 246, 247 Helium 3-A, 91, 129, 135, 137, 141, 156, 159-161 Helium 3-B, 156 Helium 4, 91, 128, 129, 135-138, 141, 144, 148, 150, 152, 155, 157, 158, 246, 247, 253, 254, 258 solidification, 120 Hilbert space, 89 horizon, 180, 181, 183, 185, 187, 190196 hydrodynamics, 371 emergent, 182 two-fluid, 135 hyperbolic, 18,104,110, 203, 339, 368, 371 hypersonic flow, 113-117 index gradient method, 4, 6 induced gravity, xii, 28, 373-374 infinite-redshift surface, 180, 183,190, 195

Artificial

Black

Holes

influence functional, 231, 232 information loss, 89, 90, 100, 105 information paradox, xii, 181 invariant distance, 383 Iordanskii force, 336 irrotational, 15, 16, 252-254, 256 Ising phase, 187, 190 Jacobi action, 254, 256 Jacobi equation, 5, 384 Jordan-Brans-Dicke theory, 204-209 Josephson junctions, 329 Kalb-Ramond potential, 255 Kepler orbit, 204 Kerr geometry, 21 Killing energy, 236 Killing temperature, 97 Killing vector, 320, 321 Klein-Gordon equation, 328, 339 Landau model, 135, 246, 258, 260 Landau velocity, 43, 91, 117, 339 Landau-Ginzburg Lagrangian, 367 Landau-Ginzburg model, 257 Langevin pressure, 344, 345 Laplacian, 203 lapse function, 351 large-N limit, 214-237 details, 237-240 lattice, 89 Laval nozzle, 19, 113, 115, 117 Legendre transformation, 250 leptons, 141 Levi-Civita tensor, 269 Lie derivative, 354 Lindblad operator, 64, 65 Lindblad's theorem, 63 line element, 95, 96, 99, 101 Longuet-Higgins-Stuart energy law, 358

Mario Novello/Matt

Visser/Grigori

Lorentz invariance, 140, 238 breakdown, 16, 28, 147, 220, 233, 236, 239, 374-375 emergent, 137 Lorentzian geometry, 6-9, 15, 18, 28 Lorentzian signature, 18, 371 massless relativistic field, 91 master equation, 63 Maxwell action, xiii Maxwell electrodynamics, 268, 271, 279, 280, 288-290, 292, 297 generalized, 268 Maxwell equation, 328 generalized, 280 Maxwell fisheye, 7 mean-field approximation, 183 minimally coupled massless scalar, 18 Minkowski momentum, 336 mode conversion, 89 near horizon physics, 215-223, 226, 227, 229, 230, 232, 234-237 neutron drip, 246 neutron star, 246, 253, 258 neutron superfluid, 246 nonlinear dielectrics, 279-281 moving, 281-289 nonlinear electrodynamics, 12, 269278, 289-298, 367 nonlinear field theory, 268-304 nonlinear media, 268, 285 normal modes, 368 ocean tides, 384 optical black hole, xii, 82 optical distance, 3, 4 optical event horizon, 62 order parameter, 184, 187 out-coupler, 39, 40, 49 Painleve-Gullstrand coordinates, 25

Volovik

389

pantograph, 338-340 Parker wind, 12, 368 Pauli principle, 98 permittivity, 282, 286 phase transition, 180-183, 187, 188, 190, 194, 195 phase velocity, 121, 123 phonon, 111, 115-118, 134, 137, 141, 145, 146, 148, 153, 157, 171, 246, 255, 260, 261 physical acoustics, 9 Planck regime, xiii, 88 Planck scale, 128, 129, 132, 133, 138, 141, 142, 146-148, 151, 153155, 157, 158, 162-165, 170, 172, 173, 215, 217-223, 226, 229, 230, 233, 236, 240 planetary-scale mass, 1 polarization tensor, 279, 282 power law, 203 primordial black holes, 110 proper time, 383 pseudo-energy, 355, 359 pseudo-gravity, 200 pseudo-momentum, 139, 145, 336-361 pseudo-Riemannian geometry, 6-9, 18 pulsar, 246 quantum field theory effective, 182 renormalizable, 182 quantum gravity, 105, 214, 219, 220, 234, 236, 238 quantum liquid, 128, 129, 133-138, 142, 148-149, 152-157, 159, 162-165, 170-173 quantum potential, 371 quantum solid, 137 quarks, 141 quasi-gravity, 200-209

390 quasiparticle, xii, 10, 13, 28, 47, 5 3 55, 91-98, 101, 102, 104, 128, 130, 134, 137, 140-145, 147, 153-155, 157, 163, 164, 166, 167, 171, 173, 367 quintessence, 129, 158, 173 Rabi frequency, 66, 75 radiation pressure, 341 345 random medium, 214 Rayleigh, 341 Rayleigh pressure, 344, 345 recta, 314 redshift, 112 refractive index, 3, 7, 8 regularization, 163, 164 Reissner-Nordstrom geometry, 295 renormalizable, 182 Ricci scalar, 131, 384 Ricci tensor, 19, 384 Riemann tensor, 5, 6, 160, 384 Riemann zeta function, 163 Riemannian geometry, 4, 5 roton, 117-119, 260 Russian doll structure, 220 S-matrix, 181, 235 Sakharov gravity, xii, 28, 133, 134, 173, 373, 374 scale anomaly, 275, 289-293 Scharnhorst effect, 367 Schrodinger picture, 223 Schwarzschild geometry, 6, 7, 10, 15, 24-26, 160 Schwinger Lagrangian, 12 second fundamental tensor, 201, 203, 209 second quantization, 38, 43, 53, 131, 136, 223, 226, 229, 373, 374 semiclassical gravity, 214, 215, 217— 220, 226, 227, 229, 230, 238-

Artificial

Black

Holes

240 shift vector, 351 slow light, xi, 2, 10, 61-83, 121-123, 367 pulse, 77-78 slow light black hole difficulty in constructing, 121 soliton, 93 sonic metric, see acoustic metric sonoluminescence, 23 spacetime basic definitions, 17 spacetime foam, 236 Stefan-Boltzmann constant, 116 stiff matter, 159 Stokes drift, 345-348 stress anisotropic, 247 radiation, 345 stress-energy tensor, 131, 132, 201 203, 205-207, 214, 224, 225, 252, 259, 336, 338, 353, 357, 358, 360, 384 superfluid, 91-93, 96-98, 104, 105, 180, 182-185, 187, 191, 193 non-relativistic, 247 relativistic, 246-262 single constituent, 253-257 two-constituent, 257-262 superluminal, 11, 95, 97, 98, 101, 102, 104 supersonic cavitation, 22-24 surface gravity, 14, 27, 215, 218 synchronized proper time, 180, 181 synchrotron radiation, xiv, 308, 319, 322-325 texture, 93, 94, 96, 104, 105, 159-161 thermal phonons, 89 Thomas-Fermi approximation, 38

Mario Novello/Matt Visser/Grigori throat, 101 time non-universal, 181 universal, 180, 181 trace anomaly, see scale anomaly trapped region, 2, 10, 277, 278 universality class, 187 Unruh effect, xiii, 308, 314, 326, 328 Unruh metric, see acoustic metric Unruh vacuum, 223 Unruh-DeWitt detector, 308-311, 325 vacuum noise, 308-326 non-stationary, 317-318 stationary, 310-317 Vaidya metric, 217, 225 Van der Waals, 183 viscosity, 16, 371 vortex, 16, 19-22, 119, 246, 247, 256, 336 vorticity, 16, 21, 119, 247, 251-253 wave action, 360 wave equation, 15-19 wave momentum, 337, 348 Weingarten condition, 201 white hole, 40, 42, 44, 45, 55, 96, 101, 102, 104 Wick contractions, 231 Wightman function, 310, 311, 319321 winding number, 41, 42 worldsheet, 200-203, 209 zero-point fluctuations, 129, 131, 132, 145, 146, 148, 151, 153, 154, 173, 308-329

A

R

T

I

F

I

,r, C

I

A

L

BLACK HOLES Physicists are pondering the possibility of simulating black holes in the laboratory by means of various "analog models". These analog models, typically based on condensed matter physics, can be used to help us understand general relativity (Einstein's gravity); conversely, abstract techniques developed in general relativity can sometimes be used to help us understand certain aspects of condensed matter physics. This book contains 13 chapters — written by experts in general relativity, particle physics, and condensed matter physics — that explore various aspects of this two-way traffic.

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