Progress in Mathematical Physics Volume 33
Editors-in-Chief
Friedrich W. Hehl Yuri N. Obukhov
Anne Boutet de Monvel, Universite' Paris VII Denis Diderot Gerald Kaiser, The Virginia Center for Signals and Waves
Editorial Board D. Bao, University of Houston C . Berenstein, University of Maryland, College Park P. Blanchard, Universitlit Bielefeld A.S. Fokas, Imperial College of Science, Technology and Medicine C. Tracy, University of California, Davis H. van den Berg, Wageningen University
Foundations of Classical Electrodynamics Charge, Flux, and Metric
Birkhauser Boston Base1 Berlin
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Friedrich W. Hehl Institute for Theoretical Physics University of Cologne 50923 Cologne Germany and Department of Physics & Astronomy University of Missouri-Columbia Columbia, MO 6521 1 USA
Yuri N. Obukhov Institute for Theoretical Physics University of Cologne 50923 Cologne Germany and Department of Theoretical Physics Moscow State University 117234 Moscow Russia
Preface
Library of Congress Cataloging-in-Publication Data Hehl, Friedrich W. Foundations of classical electrodynamics : charge, flux, and metric I Friedrich W. Hehl and Yuri N. Obukhov. p. cm. -(Progress in mathematical physics ; v. 33) Includes bibliographical references and index. ISBN 0-8176-4222-6 (alk, paper) - ISBN 3-7643-4222-6 (Basel : alk. paper) 1. Electrodynamics-Mathematics. I. Obukhov, IU. N. (IUrii Nikolaevich) 11. Title. 111. Series.
20030521 87 CIP AMS Subiect Classifications: 78A25,70S20,78A05, 81V10,83C50,83C22
Printed on acid-frcc paper. 02003 BirkhYuser Boston
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All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhluser Boston, c/o Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 0-8176-4222-6 ISBN 3-7643-4222-6
SPIN 10794392
Reformatted from the authors' files by John Spiegelman, Abbington, PA. Printed in the United States of America.
BirkhBuser Boston Basel Berlin A member qf Berrel.7,p,annSpringer Science+Rusiness Media GmbH
I11 this book we display the fundamental structure underlying classical electrodynamics, i.e., the phenomenological theory of electric and magnetic effects. The book can be used as a textbook for an advanced course in theoretical electrodynamics for physics and mathematics students and, perhaps, for some highly motivated electrical engineering students. We expect from our readers that they know elementary electrodynamics in the coiiventional (1 3)-dimensional form including Maxwell's equations. Moreover, they sholild be familiar with linear algebra and elementary analysis, including vector analysis. Some knowledge of differential geometry would help. Our approach rests on the metric-free integral formulation of the conservation laws of electrodynamics in the tradition of F. Kottler (1922), h. Cartan (1923), and D. van Dantzig (1934), and we stress, in particular, the axiomatic point of view. In this manner we are led to an understanding of why the Maxwell equations have their specific form. We hope that our book can be seen in the classical tradition of the book by E. J. Post (1962) on the Formal Strtlcture of Electmmagnetics and of the chapter "Charge and Magnetic Flux" of the encyclopedia article on classical field theories by C. Truesdell and R. A. Toupin (1960), including R. A. Toupin's Bressanone lectures (1965); for the exact references see the end of the introduction on page 11. The manner in which electrodynamics is conventionally presented in physics courses k la R. Feynman (1962), J . D. Jackson (1999), and L. D. Landau & E. M. Lifshitz (1962) is distinctly different, since it is based on a flat spacetime manifold, i.e., on the (rigid) Poincare group, and on H. A. Lorentz's approach (1916) to Maxwell's theory by means of his theory of electrons. We believe that the approach of this book is appropriate and, in our opinion, even superior for
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vi
Preface
a good understandi~igof the structure of electrodynamics ns a classical field theory. In particular, if gravity cannot be neglected, our framework allows for a smooth and trivial transition to thc curved (and contorted) spacetime of general relativistic field tlicolies. This is by no means a minor merit when one has to treat magnetic fields of the order of 10%esla in the neighborhood of a neutron star where spacetime is appreciably curved. Mathematically, intcgrands in the conservation laws are represented by exterior differential forms. Therefore exterior calculus is the appropriate language in which clectrodynaniics shoulcl be spelled out. Accordingly, we exclusively use this formalism (even in our computer algebra programs which we introduce in Scc. A.1.12). In Part A, :tnd later in Part C, we try to motivate and to supply the ncctwary ~nathematicalframework. Readers who are familiar with this formalism may want to skip thesc parts. Thcy could start right away with tlie physics in Part B and tlicn turn to Part D and Part E. In Part B four axioms of classical clectroclynamics are formulated and the consequences derivctl. Tliis general framework lins to be completed by a specific el~ctromagnetzcspacet~merelation as a fifth axiom. This is done in Part D. The Maxwell-Lorentz theory is then recovered under specific conditions. In Part E, we apply clcctrotlynamics to moving continua, inter alia, which requires a sixth axiom on the formulation of clcctroclynamics inside matter. This book grew out of a scientific collaboration with the late Dermott McCrea (University College Dublin). Mainly in Part A and Part C, Dermott's liandwriting call still be seen in numerous places. There are also some contributions to "our" mathematics from Wojtelc KopczyAslci (Warsaw University). At Cologne University in tlic summclr tc.rm of 1991, Martin Zimzbauer started to teach the thcorctical clcctrotlynamics course by using the calculus of exterior differential forms, and lie wrote up successively improvcd notes to his coursc. One of the a~itliors(FWH) also taught this course threc times, partly based on Zirnbaucr's notes. Tliis influcnccd our way of prescnting electrodynamics (and, we believe, also his way). We are very grateful to him for many discussions. There are many collcagucs ant1 friends who helped us in critically reading parts of our book and wlio maclc srlggestions for in~provementor wlio communicated to 11s their own ideas on electrodynamics. We are very grateful to all of them: Carl Brans (Ncw Orleans), Jeff Flowers (Teddington), Dav~dHartley (Aclelaiclc), Cliristia~~ Hcinickc (Cologne), Yakov Itin (Jernsalcm) Martin Janssen (Cologne), Gerry II(aiser (Glen Allen, Virginia), R. M. Kielin (brmerly Houston), Attay II(ovctz (Tcl Aviv), Claus Lammerzalil (Konstanz/Eremen), Bahram Mnslilioon (Columbia, Missouri), Eckeliard Mielke (Mexico City), WeiTou Ni (Hsin-chu), E. Jan Post (Los Angeles), Dirk Piitzfeltl (Cologne), Guillermo Rubilar (Colognc/Conccpci61i), Yasha Shnir (Cologne), Andrzej Dautman (Warsaw), Arkady Tseytlin (Colu~nbus,Ohio), Wolfgang Weller (Leipsig), and otlicrs. We are particularly grateful to the two reviewers of our book, to Jini Nester (Cliung-li) and to an al~onymfor their numerous good suggcstio~lsand for their painstaking work.
I'reface
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We arc very obliged to Uwe Essmann (Stuttgart) and to Gary Gl~txrnaicr (Santa Cruz, California) for providing beautiful and instructive images. We arc equally grateful to Peter Scherer (Cologne) for his permission to reprint his three comics on computer algebra. The collaboration with the Birkliauser people, with Gerry Kaiser and Ann Kostant, was effective and fruitful. We would like to thank Debra Daugherty (Boston) for improving our English. Pleasc convey critical remarks to our approach or the discovery of mistakes by surface or electronic mail (1iehlQtlip.uni-koeln.de,
[email protected]) or by fax +49-221-470-5159. This project has been supportcd by the Alexander von Humboldt Foundation (Bonn), tlie German Academic Exchange Service DAAD, and the Volkswagen Foundation (Hanover). We are very grateful for tlic unbureaucratic help of these institutions.
Friedrich W. Helil Cologne Yuri N. Obukhov Moscow April 2003
Contents
Preface
v
Introduction 1 Five plus one axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 To~~ological approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Elcct.ronii~gnclticspnccttime relation i ~ fifth s axiom . . . . . . . . . . . . 4 Elcctrodynnmics in lilatter and thc sixth axiorrl . . . . . . . . . . . . . 5 List. of axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 A r c ~ n i ~ i d rElect~rodyriamics r: in 3-tlimcnsional Euclidrcx-ln vect.or calculus 5 On t h r litcrnture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
A
References
11
Mathematics: Some Exterior Calculus
17
Why exterior differential forms?
19
A.1 Algebra 23 A . l . l A real vcctor space and its dual . . . . . . . . . . . . . . . . . . 23 A . 1.2 Tensors of type [y] . . . . . . . . . . . . . . . . . . . . . . . . . 25 A.1.3 @ Ageneralization of tensors: geometric quantities . . . . . . . . 27 A.1.4 Almost colnplex structure . . . . . . . . . . . . . . . . . . . . . 29 A. 1.5 Ext.erior pforrns . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Contents
A.l.G A.1.7 A.1.8 A.1.9 A.1.10 A.l.11 A.1.12
Exterior multiplication . . . . . . . . . . . . . . . . . . . . . . . 30 Interior m~~ltiplication of a vector wit], a form . . . . . . . . . . 33 @Volumeelcments on a vector space, densities, orientation . . . 34 @Levi-Civitasymbols and generalized Kronecker deltas . . . . . 36 The space M F of two-forms in four dimensions . . . . . . . . . 40 Almost complex structure on Me . . . . . . . . . . . . . . . . . 43 Computer algebra . . . . . . . . . . . . . . . . . . . . . . . . . 45
A.2 E x t e r i o r calculus 57 A.2.1 @Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . 57 A.2.2 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 A.2.3 One-form ficlds, differential p-forms . . . . . . . . . . . . . . . . 62 A.2.4 Pictures of vectors and onc-forms . . . . . . . . . . . . . . . . . 63 A.2.5 @Volume forms and orientability . . . . . . . . . . . . . . . . . 64 A.2.G @Twisted forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 A.2.7 Exterior dcrivativc . . . . . . . . . . . . . . . . . . . . . . . . . G7 A.2.8 Frame and coframe . . . . . . . . . . . . . . . . . . . . . . . . . 70 A.2.9 @Mapsof ~nanifoltls:push-forward and pull-back . . . . . . . . 71 A.2.10 @Licderivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 A.2.11 Excalc, a Rcdrlcc package . . . . . . . . . . . . . . . . . . . . . 78 A.2.12 @Closet1ant1 exact forms, de Rham cohomology groups . . . . 83
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87 A.3 I n t e g r a t i o n o n a manifold A.3.1 Integration of 0-forms and orientability of a manifold . . . . . . 87 A.3.2 Integration of 11-forms . . . . . . . . . . . . . . . . . . . . . . . 88 A.3.3 @Inte g rationof pforms with 0 < p < n . . . . . . . . . . . . . . 89 . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 A.3.4 Stoltcs' tlicorc~i~ A.3.5 @DeRhani's thcorcms . . . . . . . . . . . . . . . . . . . . . . . 9G
B
I
References
103
Axioms of Classical Electrodynamics
107
B.l Electric charge conservation 109 B . I . ~ Counting chargcs. Absolute and relative dimension . . . . . . . 109 B. 1.2 spacetit1~cand the first axiom . . . . . . . . . . . . . . . . . . . 114 B.1.3 Elcctrotnagnctic excitation H . . . . . . . . . . . . . . . . . . . 116 B4 Timc-spnce dccom1)osition of the inhomogeneous Maxwell cquat.ion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117 B.2 Lorentz force density 121 ~ . 2 . 1 ~lectromagneticfield strength F . . . . . . . . . . . . . . . . . 121 ~ ~ 2 . Second 2 axiom relatillg ~ecllanicsand electrodynamics . . . . . 123 T'he first t.hrce invariants of the electromagnetic field . . . . . 126 ~ . 2 . 3@
B.3 M a g n e t i c flux conservation 129 B.3.1 Third axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129 B.3.2 Electromagnetic potential . . . . . . . . . . . . . . . . . . . . .132 B.3.3 @AbelianChern-Simons and Kiehn 3-forms . . . . . . . . . . . 134 B.3.4 Measuring the excitation . . . . . . . . . . . . . . . . . . . . . .136 B.4 Basic classical electrodynamics summarized. e x a m p l e B.4.1 Integral version and Maxwell's equations . . . . . . . . . B.4.2 @Len2and anti-Lenz rule . . . . . . . . . . . . . . . . . B.4.3 @Jumpconditions for electromagnetic excitation and field strength . . . . . . . . . . . . . . . . . . . . . . . . B.4.4 Arbitrary local noninertial frame: Maxwell's equations ill components . . . . . . . . . . . . . . . . . . . . . . . 13.4.5 @Electrodynamicsin flatland: 2DEG and QHE . . . . .
143
. . . .143 . . . .146 . . . .150
. . . .151 . . . . 152
163 B.5 Electromagnetic e n e r g y - m o m e n t u m c u r r e n t a n d action B.5.1 Fourth axiom: localization of energy-momentum . . . . . . . . 163 B.5.2 Energy-momentum current, electric/magnetic reciprocity . . . 166 B.5.3 Time-space decomposition of the energy-momentum and the Lenz rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .174 B.5.4 @Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .177 B.5.5 @Couplingof the energy-momentum current t o the coframe . . 180 B.5.6 Maxwell's equations and the energy-momentum current it1 Excalc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 .
C
References
187
More Mathematics
193
C . 1 Linear connection 195 C.1.1 Covariant differentiation of tensor fields . . . . . . . . . . . . . 195 C.1.2 Linear connection 1-forms . . . . . . . . . . . . . . . . . . . . .197 C.1.3 @Covariant differentiation of a general geometric quantity . . . 199 C.1.4 Parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . .200 C.1.5 @Torsionand curvature . . . . . . . . . . . . . . . . . . . . . .201 (2.1.6 @Cartan'sgeometric interpretation of torsion and curvature . . 205 C.1.7 @Covariantexterior derivative . . . . . . . . . . . . . . . . . . .207 C.1.8 @Theforms o(a), connl (a,b), torsion2(a). curv2(a.b) . . . . . . 208 C.2 M e t r i c 211 C.2.1 Metric vector spaces . . . . . . . . . . . . . . . . . . . . . . . . 212 C.2.2 @Orthonormal. half-null. and null frames. the coframe statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .213 C.2.3 Metric volume 4-form . . . . . . . . . . . . . . . . . . . . . . .216
Content.s
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E.4.7 Rotating ohscrvcr . . . . . . . . . . . . . . . . . . . . . . . . . .363 E.4.8 Accelerating observer . . . . . . . . . . . . . . . . . . . . . . . . 364 E.4.9 The proper reference frame of the nonincrtial observer ("nonincrtial frame") . . . . . . . . . . . . . . . . . . . . . . . .366 E.4.10 Universality of the Maxwell-Lorcntz spacetirnc relation . . . . SG8
References
371
@Outlook 375 How does gravity affect electrodynamics'? . . . . . . . . . . . . . . . . . 376 Rcissner-Nordstroln solution . . . . . . . . . . . . . . . . . . . . 377 Rotating source: Kerr-Ncwnian solution . . . . . . . . . . . . . . 379 Electrodynamics outside black holes and neutron stars . . . . . . 381 Force-free elcctrodynarnics . . . . . . . . . . . . . . . . . . . . . .383 Re~narkson topology and electrodyl~amics. . . . . . . . . . . . . . . . 385 Superconductivity: Rcnlarks on Ginzb~~rg-Landall theory . . . . . . . . 387 Cl;~~sical (first quantized) Diriic field . . . . . . . . . . . . . . . . . . .388 On thc quantum Hall effect and thc colnpositc ferlnion . . . . . . . . . 390 On quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . . 390 On elcctrowcak unification . . . . . . . . . . . . . . . . . . . . . . . . . 391 References Author Index
397
Subject Index
403
Foundations of Classical Electrodynamics Charge. Flux. and Metric
Introduction
Five plus one axioms I11 this book we display the structure underlying classical electrodynamics. For this purpose we formulate six axioms: conservation of electric charge (first axiom), existence of the Lorentz force (sccond axiom), conservation of magnetic flux (tliird axiom), local energy-momentum distribution (fourth axiom), existence of an electromagnetic spacetime relation (fifth axiom), and finally, the splitting of the electric current into material and external pieces (sixth axiom). Tlic axioms expressing the conservation of electric charge and magnetic flux arc formulated as integral laws, whereas the axiom for the Lorentz force is reprcsentcd by a local expression basically defining the electromagnetic field strcngth F = (ElB) as force per unit charge and thereby linking electrodynamics to mechanics; here E is the electric and B the magnetic field strength. Also the energy-momentum distribution is specified as a local law. The fifth axiom, the Maxwell-Lorentz spacetime relation is not as unquestionable as the first four axioms and extensions encompassing dilaton, skewon, and axion fields are cliscussed and nonlocal and nonlinear alternatives mentioned. Wc want t o stress the fundamental nature of the Frst axiom. Electric charge conscrvation is experimentally firmly established. It is valid for single elementary particle processes (like P-dccay, n -,p+ ei7, for instance, with n as neutron, p as proton, e as electron, and i7 as electron antineutrino). In other words, it is a microscopic law valid without any known exception. Accordingly, it is basic to electrodynamics to assume a new type of entity called electric charge, carrying a positive or negative sign, with its own physical
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2
I I I
II I
II
Introduction
(lilnension, independent of tlic classical fundamental variables mass, length, and time. Furthermore, electric charge is conscrvcd. In an age in which single electrons ailtl (anti)protons arc counted and caught in traps, this law is so deeply ingrained in our tliinking that its explicit formulation as a fundamental law (and not only as a consrqrlcnce of Maxwell's equations) is often forgotten. We show that this first nrzom yields the inhomogeneous Maxwell equation together with a clefinition of the electrolnagnetic excitation H = (X, '23); here 3-t is the excitation ("magnetic field") and 2) the electric excitation ("electric displacemc~~t"). Thc cxcit,ation H is a microscopic field of an analogous quality as thc ficltl strengtli F . There exist operational definitions of the excitations 2) and 3-t (via Maxwcllian clouble plates or a compensating superconducting wire, rcspectivcly). Thc second axiom for the Lorcntz force, as mentioned above, leads to the llotion of the field strength and is thereby exhausted. Thus we need further axioms. The only conservation law that can be naturally formulated in terms of the field strength is the conservation of magnetic flux (lines). This thzrd axiom has the liornogeneous Maxwcll equation - that is, Faraday's induction law and t,hc vanisliing divergence of the magnetic field strength - as a consequence. Moreover, with the help of these first three axioms we arc led, although not completely ~~niqucly, to the elcctlolnngnctic e n ~ r q y - m o m e n t u mcurrent (fourth axiom), which stibsumes the energy and momentum densities of the electromagnct,ic field ant1 their corresponding fluxes, and to the action of the elcctrolnagnct,ic field. In this way, the basic structure of electrodynamics is set up, including the complete set of Maxwell's equations. To make this set of electrodynamic equations wrll determined, we still have to add the fifth axiom. Magnetic nlonopolcs arc alien to the structure of the axiomatics we arc using. In our axiomatic. framrwork, a clear as?ymmetm~is built in between electricity ; ~ l l c l magnetism in tlic scnse of Oersted and Ampbre wherein magnetic effects arc clentcd by moving electric charges. This asymmetry is characteristic for and intrinsic to Maxwell's theory. Therefore the conservation of magnetic flux and llot that of magnetic charge is postulated as the third axiom. The existence of a magnetic charge in violation of our third axiom would have far-reaclling consequences: First of all, the electromagnetic potential A would llot exist. Accordingly, in Hamiltoninn mechanics, we wol~ldhave to give up tllc c o u l ~ l i of ~ ~ag chargcd particle to the clcctrornapnetic field via TI = p - p A. Morcov("~the sccontl axiom on the Lorcntz force could he invalidated since one would IlaVe t,o supplement it with a term carrying the magnetic charge density. By im~)lic:ttion,an extension of the fifth axiom on the energy-momentum current w()tlltl br ncccssary. I11 otIICk words, if ever a magnetzc monopole1 were found, our axiomatics would 11'" its coherence, its compactness, and its plausibility. Or, to formulate / --
'Ollr. ,!'Rull7cnts rcfcr only t o Abclian gauge theory. In non-Abclian gauge theories the situat,ion '"different,. There monopoles seem to be a Inust, a t least if a IIiggs field is present.
Topological approach
3
it more positively: Not long ago, He [22], Abhott et al. [I], and Kalbfleisch et al. [32] determined experimentally new improved limits for the nonexistence of (Abelian or Dirac) magnetic monopoles. This ever increasing accuracy in the exclusion of magnetic monopoles speaks in favor of the axiomatic ap~>roach in Part B.
Topological approach Since the notion of metric is a complicated one, which requires measurements with clocks and scales, generally with rigid bodies, wlliclt themselves are systems of great complexity, it seems undesirable to take metric as fun,damental, particularly for ph,enomena which are simpler and actually independent of it. E. Whittaker (1953) The distinctive feature of this type of axiomatic approach is t,hat one only needs minimal assumptions about the structure of the spacetime in which these axioms are formulated. For the first four axioms, a 4-dimensional differentiable manifold is required that allows for a foliatzon into 3-dimensional hyp~.rsurfaccs. Thus no connection and no metric are explicitly introduced in Parts A and B. The Poincard and the Lorentz groups are totally ignored. Nevertheless, we recover Maxwell's equations already in Part B. This shows that elcctrodynan~ics is not as closely related to special relativity theory as is usually supposed. This mininialistic topologzcal type of approach may appear contrived a t first look. We should rccognizc, however, that the metric of spacetime in tlie framcwork of general relativity theory represents the gravitational potential and, similarly, the conncction of spacetime (in the viable Einstrin-Cartan throry of gravity, for example) is intimately linked to gravitational propert,ics of matter. We know that we really live in a curved and, perhaps, contorted spacetime. Consequently our desire should be to formulate the foundations of clcct,rodynamics such that the metric and the connection don't interfere or interfere only in tlie least possible way. Since we know that the gravitational field permeates all the laboratories in which we make experiments with electricity, we should take care that this ever present field doesn't enter the formulation of the first principles of electrodynamics. In other words, a clear separatzon between pure clectrodynamic effects and gravitational effects is desirable and can indeed be achieved by means of the axiomatic approach to be presented in Part B. Eventually, in the spacetime relation (see Part D), the metric does enter. The power of the topological approach is also clearly indicated by its ability to describe the phenomenology (at low frequencies and large distances) of the quantum Hall effect successfully (not, however, its quantization). Insofar as the macroscopic aspects of the quantum Hall effect can be approximately understood in terms of a 2-dimensional electron gas, we can start with (1
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Electrodynamics in 3-tlimensional Euclidean vcctor calculus 2)-dimensional clcctrodynamics, tlic formulation of which is straightforward in our axiomatics. It is then a mcrc finger exercise to show that in this specific casc of 1 2 clinicnsions thcre cxists a lznear constitutivc law that docsn't require a metric. As a conscquencc the action is metric-free too. Tlius the formulation of tlie quantum Hall effect by Incans of a topological (Chern-Simons) Lagrangian is imnlincnt in our way of loolting at clcctrodynamics.
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Electromagnetic spacetime relation as a fifth axiom Let us now turn to that domain wherc tlic metric docs enter tllc 4-dimensional electrodynarnical formalism. Wlicn thc Maxwclliarl structure, including t,hc Lorcntz force and tlic action, is sct up, it docs not rcprcscnt a concrete physical tlicory yct. What is missing is tlic elcctroniagnctic spacctimc relation linking tlic excitation to tlic ficld strength, i.c., V = V ( E , B ) , 3-1 = X(B, E), or written 4-diniensionally, H = H(F). Trying the sinlplcst approach, we assume localzt?/ ancl 11nrnnty bctwt.cn tlie excitation H and the field strcngtli F , tliat is, H = tc(F) witli tlic linear operator tc. Together witli two more "teclinical" assumptions, namely that H = tc(F) is clcctric/magnetic wczprocal and tc symmetric (these propertics will be discussed in detail in Part D), we are able to denve tlie metric of spacetimc from H = u ( F ) up to an arbitrary (conformal) factor. Accordingly, the light cone structure of spacctime is a consequence of a linear electromagnetic spacptirne relation witli the additional properties of reciprocity ancl symmetry. In this sense, tlic light cones arc dcrivcd from electrodynamics. Elcctrotlynalnics docsn't live in a preformed rigitl Minkowslti spacetimc, Rather it ha9 an arbitrary (1 3)-dimensional spacctime manifold as its habitat wliicli, as soon as a linear spacctimc relation witli rcciprocity nncl symrnctry is supplied, is equipped witli local light cones cvcrywherc. With the light concs it is possiblc to definc the Hodge star operator * that maps pforms to (4-11)-forms and, in particular, H to F according to H N * F . Tlius, in the end, that property of spacetimc that tlescribcs its local "constitutivc" structure, namcly the metric, cntcrs the formalisnl of electroclynamics and lnakcs it into a complctc thcory. Onc merit of our approach is tliat it docsn't lnattcr whetllcr it is the rigid, i.e., flat, Minkotuskz lnctric of special relativity or tllc "flexible" Rlemannzan mctric ficlti of general relativity tliat cllangc's from point to point according to Einstein's field equation. In this way, thc traditional discussion of how to translate clectrodynarnics from special to general relativity loses its sense: Thc Maxwell equations remain tlic same, i.c., the exterior dcrivat,ivcs (the " c o m n ~ a .in~ ~coorclinatc language) arc kept and arc not sul~stitutcclby something "covariant" (tlic "scmicolons"), and tlie spacetimc relation H = Xo *Flooks the sanlc (Ao is a suitable factor). However, the Hodge star "feelsv the tlifference in rcfcrring either to a constant or to a spacetimcdependent metric, respectively (see [51, 231).
+
5
Our formalism can accommodate generalizations of classical electrodynamics, including those violating Lorcntz invariance, simply by suitably modifying the fifth tixiom while keeping the first four axioms as indispensable. Then a scalar dllnton field X ( 2 ) , a pseudoscalar axion field a ( x ) , and/or a tensorial traceless skewon field $,J can come up in a most straightforward way. Also tlie Heiscnberg-Eulcr and the Born-Infeld electrodynamics are prime examples of such possiblc modifications. In the latter cases, the spacetime relation becomes cffcctivcly nonlinear, but it still remains a local expression.
Elcctrodynamics in matter and the sixth axiom Evcnt,ually, we have to face thc problem of formrllating clcctrodynamics inside matter. We codify our corresponding approach in the sixth axiom. The total electric cuncnt, cntcring as tlle source in the inlion~ogcneousMaxwell equation, is split into bound charge and free charge. In this way, following Truesdcll & Tolipin [63] (scc also the textbook of Kovetz [34]),we can develop a consistent theory of clcctrotlynamics in matter. For simple cases, wc can amend the axioms by a llnrar constzt~~tive 1au1.Since in our approach (%, V) are n~icroscopicficlds, like (E,R ) , wc bclicve that the conventional theory of elcctrodynanlics inside matter ncctls to be redesigned. I11 ordcr to demonstrate the effcctivencss of our formalism, we apply it t o the clcctrodynamics of moving matter, thereby returning to the post-Maxwellian era of tlic 1880s wlicn a relativistic version of Maxwell's theory had gained momentum. In this contcxt, wc discuss and analysc the cxpcriments of Walkcr & Walkcr ant1 Jnlncs and tliosc of Rontgcn-Eiclicnwalcl aricl Wilson & Wilson.
List of axioms 1. Conscrvation of clectric charge: (B.1.17). 2. Lorentz forcc tlcnsity: (B.2.8). 3. Conscrvation of lnagnctic flux: (B.3.1). 4. Localization of cnergy-mol-ncntum: (B.Ti.7). 5. M:ixwcll-Lorcntz spacct,imc relation: (D.G.13).
6. Splitting of the clectric current in a conscrvcd matter piece and an external piccc: (E.3.1) and (E.3.2).
A rernindcr: Elcctrodynamics in 3-dimensional Euclidcan vector calculus Bcforc wcxstart to clcvclol> elcctrotlynamics in 4-dimensional spacetime in the fralneworlt of the calculus of exterior diffcrential forms, it may be useful to re-
Introduction
6
mind ourselves of electrodynamics in terms of conventional 3-dimensional Euclidean vector calculus. We begin with the laws obeyed by electric charge and current. If 6= (a,, D,, V,) denotes the electric excitation field (historically "electric displacement") and p the electric charge density, then the integral version of the Gauss law, 'flux of V through any closed surface' equals 'net charge inside' reads
with d% as area and dV as volume element. The Oersted-Ampkre law with the magnetic excitation ficld " ;i (X,, X,, 'Hz) (historically "magnetic ficld") and the electric current density j = (j,, j,, j,) is a bit more involved b c c a ~ ~ sofc the presence of the Maxwellian electric excitation current: The 'circulation of 3-1 around any closed contour' equals '$ flux of V tlrrougll surface spanned hy
On the literature
7
Note the minus sign on its right-hand side, which is chosen according t o the Lenz rule (following from energy conservation). Finally, the 'flux of through any closed surface' equals 'zero', that is,
I?
The laws (1.4) and (1.5) are inherently related. Later we formulate the law of magnetic flux conservation and (1.4) and (1.5) just turn out to be consequences of it. Applying the Gauss and the Stokes theorems, the integral form of the Maxwell equations (I.l), (1.2) and (I.4), (1.5) can be transformed into their differential versions:
-4
(
I
contour)' plus 'flux of
4
throagh surface' ( t =time):
-4
Here & is the vectorial line element. Here tllc dot . always denotes the 3dimensional ~netric-dependentscalar product, S denotes a 2-dimensional spatial surface, V a 3-dimensional spatial volume, and BS ant1 BV the respective boundaries. Later we recognize that both, (1.1) and (I.2), can bc derived from the charge conservation law. The homogeneous M,axwcll cquations are formulated in terms of the elcctric field strength l? = (E,, E,, E,) and the magnetic field strength I? = (B,, B1,,B,). They arc defined opcrationally via the expression of the Lorentz force $. An electrically chargecl particle with charge q and velocity ;experiences the force
-.
Here the cross x denotes the 3-dimensional vector product. Then Faradny '.s induction law in its integral version, namely 'circulation of E around any closed contour' equals 'minus $ flux of B through surface spanned by contour ' reads:
(
1
Additionally, we have to specify the spacetime relations V = EO 3 , l? = po 'I?, and if matter is considered, the constitutive laws. This formulation of electrodynamics by means of 3-dimensional Euclidean vector calculus represents only a preliminary version since the 3-dimensional -. the vector products and, in particular, the difmetric enters the scalar and ferential operators div V. and curl V x , with V as the nabla operator. , ~ counting procedures enter, In the Gauss law (1.1) or (I.6)1, for i n ~ t a n c e only namcly counting of elementary charges inside V (taking care of their sign, of course) and counting of flux lines piercing through a closed surface dV. No length or time measurements and thus no metric are involved in such processes, as described in more detail below. Since similar arguments apply also to (1.5) or (1.7)1, respectively, it should be possible t o remove the metric from the Maxwell equations altogether.
-
-
-.
-.
On the literature Basically not too much is new in our book. Probably Part D and Part E are the most original. Most of thc material can be found somewhere in the literature. What we do claim, however, is some originality in the completeness and in the appropriate arrangement of the material, which is fundamental to the structure electrodynamics is based on. Moreover, we try to stress the phen.omena underlying the axioms chosen and the operational interpretation of the quantities introduced. The explicit derivation in Part D of the metric of spacetime from pre-metric electrodynamics by means of linearity, reciprocity, and symmetry, although considered earlier mainly by Toupin [61], Schonbcrg [53], 2 ~ h sr~bscript c 1 refers to the first equation in (1.G).
Introduction
8
and Jadczyk (281, is new and rests on recent results of Fukui, Gross, Rubilar, and the authors [42, 24, 41, 21, 521. Also the generalization encompassing tlie dilaton, the axion, and/or tlie sltcwon field opens a new perspective. In Part E the electroclynamics of moving bodies, including tlie discussions of some classic experiments, colltains much new material. Our marn sources are the works of Post [46, 47, 48, 49, 501, of Truesdell & Toupin [G3], and of Toupin [GI]. Historically, the metric-free approach to elcctroclynarnics, based on integral conservation laws, was pioneered by Kottler [33], h. Cartan [lo], and van Dantzig [64]. The article of Einstein [16] and the books of Mic (391, Weyl (651, and Sornmerfeld [58] should also be consulted on these niatters (see as well the recent textbook of Kovetz [34]).A description of the corresponding historical development, with references to the original papers, can be found in Whittakcr (661and, up to about 1900, in the penetrating account of Darrigol [12]. The driving forces and the results of Maxwell in his research on electrodynamics arc vividly presented in Everitt's [17] concise biography of Maxwell. In our book, we consistently use exterior calculus13including de Rharn's odd (or twisted) differential forms. Telrthooks on electrodynamics using exterior calc~ilusare scarce. In English, we know only of Ingarden & Jamiolkowski [26], in German of Mcetz & Engl (381 and Zirnbauer (671, and in Polish, of Janrewicz [31] (see nlso [30]). Howcvrr, as a discipline of mathematical physics, corresponding presentations can be found in Bamberg & Sternberg [4], in Thirring [GO], and as a short sketch, in Piron [44] (see also [5, 451). Bambcrg & Sternberg arc particularly easy to follow and present clcctrodynamics in a very transparent way. That clcctrodyn~micsin the framework of exterior calculus is also in the scopc of clcctrical engineers can be sccn from Dcschamps [14], Bossavit[8], and Baldolnir & Haliimond [3]. Prcscntations of exterior calculus, partly togcthcr witli applications in physics and electrodynamics, were given amongst many others by Burke [9], ChoquetBruliat ct al. [ll],Edelcn (151, Flanlders [19], Rankel [20], Parrott [43], and ~lebodziriski[57]. For differential geometry we refer to tlie classics of de Rllam [13] ant1 Schouten [54, 551 and to n a u t m a n [G2]. What clsc influenced the writing of our book? The ariomatics of Bopp [7] is different but relatctl to ours. In the more microphysical axiomatic attempt of LRmnierzalll et al. Maxwell's equations (351 (and tlie Dirac equation (21) arc dediiccd from direct experience with electromagnetic (and matter) waves, inter alia. The clcar separation of drffrrrntzal, afine, and metnc structures of spacetime is nowlicrc more 11ron011nced than in Schrodingcr's [56] Space-time stmlcturr. A fiirtlicr presentation of elcctrodynamics in this spirit, somewhat similar 3)-decomposztzon to that of Post, has been given by Stachel (591. Our (1
+
-
%nylis [GI nlso ndvocntes a gcolnctric nppronch, using Clifford algcbrm (see also Janccwicz [20]). In such a. frnmcwork, howcvcr, nt least the way Baylis does it, the metric of 3-dimensional space is i r ~ t r o d ~ ~~.ight c ~ dfrom the beginning. 111this sense, Baylis' Clifford algebra approach is cornplcmrntnry to our metric-free electroclynaniics.
of spacetime is based on the paper by Mielkc & Wallncr [40]. More recently, Hirst (251 has shown, n~ainlybased on experience with neutron scattering on magnetic structures in solitls, that magnetization M is a microscopic quantity. This is in accord witli our axiornatics which yieltls the magnetic excitation 3.1 rrs a microscopic quantity, quite analogously to the field strength B , whereas in conventional texts M is only defined as a niacroscopic average over niicroscopically fll~ctuatingniagnctic fields. Clearly, 'H and also the electric cxcitat,iotl D ,i.e., the electromagnetic cxcit,at,ion H = (3.1,V)altogether, olight to be a microscopic field. Sections and subsections of the book that can be skipped ilt by tllc symbol are ~i~arltcd @.
ii
first reading
References
[I] B. Abbott et al. (DO Collaboration), A search for heavy pointlike Dirac monopoles, Phys. Rev. Lett. 81 (1998) 524-529. [2] J. Audrctsch and C. Lammerzahl, A new constructive axiomatic scheme for the geometnj of space-time In: Semantical Aspects of Space-Time Geornetqy. U. Majer, H.-J. Schmitlt, cds. (BI Wissenschaftsverlag: Mannheim, 1994) pp. 21-39.
[3] D. Baldomir and P. Hammond, Geometry and Electromagnetic Systems (Clarcndon Press: Oxford, 1996). [4]
P.Balnberg and S. Sternberg, A Course in Mathematics for Students of Physics, Vol. 2 (Cambridge University Press: Cambridge, 1990).
[5] A.O. Barut, D.J. Moore and C. Piron, Space-time models from the electromagnetic field, Helv. Phys. Acta 67 (1994) 392-404.
[6] W.E. Baylis, Electrodynamics. A Modern Geometric Approach (Birkhauser: Boston, 1999). [7] I?. Bopp, Prinzipien der Elektrodynamik, 2. Pt~ysik169 (1962) 45-52. [8] A. Bossavit, Differential Geometnj for the Student of Numerical Methods in Electromagnetism, 153 pages, file DGSNME.pdf (1991) (see h t t p : //www. l g e p . supelec . f r/mse/perso/ab/bossavit . html),
[9] W.L. Burlte, Applied Dzflerential Geometry (Cambridge University Press: Cambridge, 1985).
[lo] fi. Cartan, O n Man,ifold.s luith an A f i n e Connection and the Theory of General Relativity, English translation of thc Frcnch original of 1923124 (Bibliopolis: Nagoli, 1986).
1241 F.W. Helil, Yu.N. Ob~~lthov, and G.F. Rubilar, Spacetime metric from linear ele~trod?~n.am.ics II. Ann. Pl~ysik(Lcipzig;) 9 (2000) Spccial issue, SI71-SI-78.
[ll] Y. Choquct-Bruh;Lt, C. DeWitt-Morcttc, ancl M. Dillard-Blcick , Analysis, Manijo1d.s and Pl~?j.sic.s, revised cd. (North-Holland: Amsterdam, 1982).
[25] L.L. Hirst, Th,e m,icroscopic m.agnetizntion: concept and application, Rev. Mod. Pl1.y~.69 (1997) G07-627.
(121 0. Darrigol, ELectrorl~~nam.ics from Ampere to Einstein (Oxford University Press: New York, 2000).
[26] R.. Ingardcn and A. Jamiolkowski, Classical Electrodynamics (Elsevier: Amsterdam, 1985).
[13] G. de Rham, Differentiable Man.ifolds: Forms, Currcnts, Harmonic Forms. Transl. from thc F'rcncli original (Springer: Berlin, 1984).
[27] J.D. Jacltson, Clas.sical Electrodynnm.ics, 3rd cd. (Wiley: Ncw York, 1999).
[14] G.A. Descliamps, Electrom.agn,etics an.d differential fo~m.s,Proc. IEEE 69 (1981) 676-696. [15] D.G.B. Etlelen, Applied Exterior Ca1culzi.s (Wiley: New York, 1985) [16] A. Einstein, Einc neue form.ale Deutun.g rler Maxwellsclt~en.Feldgleicl~ungen dcr Elcktrodyn.am.ik, Sitzungsber. IGnigl. Preuss. Aka.d. Wiss. Bcrlin (1916) p p 184-188; scc also T1j.e collected papers of Albert Ei,n..stein. Vol.6, A..J. Kox et al., eds. (1996) pp. 263-269. [17] C.W.F. Evcritt, James Clerk Maxwell. Physici.st a.ntl Nat1~ra.1Philosoplrer (Charlcs Sribncr's Sons: New York, 1975). [18] R.P. Feynmnn, R.B. Lcighton, ant1 M. Santls, The Feynman Lect~~res on Pl~.~/sics, Vol. 2: Mainly Electromagnetism and Mattcr (Addison-Wesley: Reatling, Mass., 1964). [19] H. Fla~itlcrs,Dzfferentiol Forms with Applications to tlie Physical Sciences. (Academic Press: Ncw York, 1963 and Dovcr: New York, 1989). [20] T. F'rankcl, Tlre Geonletry of Physics: A n In.trod.uction (Cambritlge University Prcss: Calnbridg;e, 1997). (211 A. Gross ancl G.F. Rubilar, O n the derivation of the spacetime metric from linear electrodynnmics, Pl~ys.Lett. A285 (2001) 267-272. [22] Y.D. He, Search for a Dirac m.agnetic nlonopole i n high enwrgy n.u.cleus-nuclet~sco1lisi0n.s~Pllj~s.Rev. Lctt. 79 (1997) 3134-3137. [23] F.W. HcIll and y~1.N.Obukliov, I-Iow does the electromngnctic ficld couplc to grnvit,y, in particular to metric, nonmet,ricity, torsion, ant1 cllrvature? In: Gyros, Clocks, Interferometers . . . : Testing Relativi.stic Gmvit?j i n Space. C. Liimmerzahl et al., eds. Lecture Notes in Physics Vo1.5G2 (Springer: Berlin, 2001) pp. 479-504; see also Los Alanios Eprint Archive gr-qc/0001010.
[28] A.Z. Jadczyk, Electromagnetic permenhilit!/ of the uacu.7~rnand light-conx structure, Bull. Acad. Pol. Sci., Sbr. sci. phys. et mtr. 27 (1979) 91-94. [29] B. Janccwicz, Mvltivector.~and Clzfford Algebra i n Electrodynamzcs (Worlcl Scientific: Singapore, 1989). [30] B. Janccwicz, A variable metric electrodynamics. The Coulomb and BiotSauart laws i n anisotropic media, Ann. Phys. ( N Y ) 245 (1996) 227-274. [31] B. J;xncewicz, Wielko.s'ci skierotuane 111 elektrodynam,ice (in Polish). Directed Quantities in Electrodynamics. (University of Wroclaw Press: Wroclaw, 2000); an English version is u~ldcrpreparation. [32] G.R.. I
14
Introduction
[55] J.A. Schoutcn, Tensor Anal?lsis for Physicists, 211d ccl, rel>rintctl (Dover: Mincola, Ncw York 1989).
[40] E.W. Miclkc ntl R.P. Walll~er,Mass and spin of double dual sol.utions in Poincark gauge tll,eo~hNIIOVO Cimciito 101 (1988) 607-623, erratum B102 (1988) 555.
[5G] E. Schrodingcr, Space-Time Stn~cture(Can~britlgeUniversity Press: Cambridge, 1954).
1421 Yu.N. Obukl~ovand F.W. Hehl, Space-time metric from. linear electrodynam.ics, Plrys. Lett. B458 (1999) 466-470. [43] S. Parrott, Relatit~istic Electrodynamics and Differential Geometry (Springer: New York, 1987). [44] C. Piron, ~lectrodyn.ami~ue et optique. Coursc givcn by C. Piron. Notes edited by E. Pittet (University of Geneva, 1975).
I
1451 C. Piron ant1 D.J. Moorc, New aspects of field theory, Turk. J . Phys. 19 (1995) 202-216.
I
1461 E.J. Post,, Formril Stm~ctureof Electromagnetics: Genwral Covariance and Electromagnetics (North Hollnncl: Amsterdam, 1962, and Dover: Mineola, New York, 1997).
1 'I1
[47] E.J. Post, Tire constituti~iemap and some of its ramifications, Annals of Pll.ysics (NY) 71 (1972) 497-518. I
1481 E.J. Post,, I(ot,tler-Cartan-~~n7zDan,tzig (KCD) an.rl noninertial systems, Forlnd. Pllys. 9 (1979) (319-640. (491 E.J. Post, Plt,ysical dim.ension and covariance, Found. Phys. 12 (1982) 169-195.
1
~
[57] W. ~lcbodzilislti,Exterior Forms and Their Applications, Rcvisecl translat,ioll from the French (PWN-Polish Scientific Publishers: Warszawn, 1970). [58] A. Somn~crfeltl,Elektrodynam7k. Vorlcsungen uber Thcorctiscl~cPhysik, Band 3 (Dieterich'schc Verlagsl~ucl~l~ancll~~~~g: Wicsbatlen, 1948). English translation: A. Sommcrfeld, Electrodynamzcs, Vol. 3 of L r c t u r c ~in Tlicoretical Pllysics (Academic Press: New York, 1952). [59] J . Stnchcl, Th.e generally covariant form of Maxwell's equa,tions, in: J.C. Maxrvcll, the Scscluicciltenninl S.yinposiurn. M.S. Berger, cd. (Elscvicr : AIRsterdani, 1984) pp. 23-37.
[GO] W. Tliirring, Classical Mnthernc~ticnlPlrysics: Dynamical S?l.stems nntl Field Tlzeories, 3rd cd. (Springer: New York, 1997). [GI] R.A. Toupin, Elasticity and electro-magnetics, in: Non-Linear Coi~tirir~r~rn Tt~cories,C.I.M.E. Conference, Brcssanonc, Ita1,y 1.965. C. Truesdell and pp. 203-342. G. Grioli ~oordinat~ors, [62] A. Trautman, Diffemn.tial Geometry for Pii,?lsicists,Stony Brook Lccturcs (Bibliopolis: Napoli, 1984). [63] C. Txucsdcll and R.A. Toupin, The classical field theories, in: Handbuch tlcr Physik, Vol. 11111, S. Fliiggc ctl. (Springer: Berlin, 1960) pp. 226-793. [G4] D. van Dantzig, Tlre fiindamen,tal equatior~sof eleclromagnetism, hdcpen(1en.t of m.etrical geometry, Proc. C;tmbridgc Phil. Soc. 30 (1934) 421-427.
[50] E..J. Post, Quant~r~ri, Reprogramming: Ensembles and Single Sy.stems: A Two-Tier Approacl1 to Q ~ ~ a n t uMecl~anics m (Kluwer: Dordrccht, 1995).
(651 H. Wcyl, Rnurn,, Zcit, Materie, Vorlcsungcn iibcr Allgcrneinc R.clntivitiitstlicorie, 8th ctl. (Springer: Berlin, 1993). Engl, trallslation of tlic 4th cd.: Space-Tinre-Matter (Dover: New York, 1952).
1511 R.A. Puntignm, C. Liimmcrzahl and F.W. Hehl, Maxwell's theom on a post-Riemannian spacetime an.d tile eq~~ivalence principle, Class. Quantum Grav. 14 (1997) 1347-1356.
[GG] E. Whittaker, A Ifi.stor?l of tlre Tlleories of Aether and Electricity. 2 volumes, rcprintetl (Hulnanitics Prcss: Ncw York, 1973).
I
I
15
1391 G. Mic, Lelrrbuch dcr Elektrizitri't u.nd des Ma,gn,etismus, 2nd ed. (Enke: Stuttgart 1941).
1411 Yu.N. Obukhov, T. Fukui, and G.F. Rubilar, Wave propagation in linear electrod?ln,am.ics,Pliys. R,ev. D62 (2000) 044050, 5 pages.
I
References
[52] G.F. Rubilnr, Yu.N. Obukhov and F.W. Held, General covariant Resnel equ,ntion and tlre emergen,ce of the light cone structure in pre-metric electrodyn,amics, 1i1t. J , Mod. Phys. D l 1 (2002) 1227-1242. [53] M. ScliOnbcrg, Elc~tromagnetis7n.nand gravitation, Rivista Brasileira de Fisicn 1 (1971) 91-122. [54] J.A. Schoutc?n, Ricci-Calculus, 211d cd. (Springer: Berlin, 1954).
1671 M.R. Zirnbaucr, Elektrorl?lnamzk. Tcx-script July 1998 (Springer: Bcrlili, to be publishctl).
Part A
Mathematics: Some Exterior Calculus
Why exterior differential forms?
In Part A and later in Part C, we arc concerned with assembling the geometric conccpts tliat are needed to formulate a classical field theory like elcctrodynnmics ancilor the theory of gravitation in the langrlagc of differential forms. T h e basic gconictric struct,urc untlerlying such a theory is that of a spaceti~necontinuum or, in mathematical terms, a 4-dimcnsionnl di#eren,tiable manifold Xq.T h e charact,cristics of t,hc gra~it~ational field will be tlct,crmincd by the nature of thc addit,ional gcon~ct,ricst,ructllrcs that arc? supcrilnj)os(?d on this "bare manifoltl" X4. For instance, in Einstein's genornl rclativity theory (GR.), the ~nallifoldis endowctl with a. metric together with a torsion-free, metric-compatible connection: It is a 4-dimensional Riemannian spacctimc V4. Thc mcanillgs of these terms arc cxplainecl in dctail in wliat follows. I11 Maxwell's t,lieory of electrodynamics, under most circumstances, gravity can be safely ncglcctcd. Then thc Riemannian spacctimc becomes fZa,t, i.e., its curvature vanishes, antl we liavc thc (rigid) Minkowskian spacctilnc M,, of special relativity thcory (SR). Its spatial part is the ordinary 3-dimensional Euclidean space R3. Howcvcr, and this is one of tlic messclgcs of the book, for the f~uid~rncnt,al a.xioms of clcctrodynamics we tlon't need to take into accolmt, the metric structure of spncctimc antl, even more so, wc! sliould not takc it into account,. Tliis hclps to keep clcctrodynamical strl~ctllrcscleanly ~cparttt~cd from gravitt~tionalolios. Tliis sol~nr;ttionis part,icularly dccisivc for a proper untlerstnndilig of tlie emergcncc of the light con,e. On the one side, hy its very definition, it is an electrodynamical concept in tliat it deterrnincs tlie front of a propagating elcctromagnetic dist,urbance; on tlie othcr hand it constitutes the main (conformally invttrittnt) art of the lnctric tensor of spacctirne and is as such pttrt of the grav-
Part A. Mathcrnatics: Some Exterior Calculus
20
itational potential of GR. Tliis complicatetl interrelationship we try to untangle in Part D. A central role in tlic formulatioll of classical electrodynamics adopted in the present work is plnyc~lby the conservation laws of electric charge and magnetic flux. Wc start from their intcgrnl formulation. Accorclingly, therc is a necessity for an adcquatc unclerstantling of the coiicept,~involved when one writes down an inteqrnl over some do~nainon a differentiable manifold. Specifically, in the Euclidean space R3 in Cartesian coordinates, one encountcrs integrals like the electric tcr~sioii(voltage)
Why ext.crior clifferential forms'? Coilsitlcr tlic iritegrnl
S
p(z171, z) dn: dl/ dz
and nlaltc a change of variables:
For simplicity and only for thc prcscnt purpose, lct us supposc that the Jacobian deteriniliallt
cvaluatcd along n (I-dimensional) curvc C, thc magnetic flux
J(B,
~ I dz J
+ B, dz dn: + B, cin:dy)
is positive. Wc obtain (A4
S
ovcr a (2-dimensional) surface S , and tlic (total) charge
/v
I
I
p d r dydz
This suggest.^ that wc should writc
intcgrntctl ovcr a (3-dimensional) volume V. The funtlnmcntal result of classical integral calcr~lusis Stokes' tlicorem which relates an intcgrnl over tlic houndnry of a region to one takcn over the region itself. Falniliar cxnrnples of this tlicorcln are providcd by the expressions
If we sct n: = y or n: = z or y
= z tlie determinant has equal rows and llcnce vanishcs. Also, an odd permutation of n:, 11, z cli~~ngcs thc sign of thc determinant whilc an cvcn pcrmutation lcnves it unchanged. Hencc, we havc
dn:dr=O,
dzdz=O,
(A. 11)
dn: d?/dz = dy dz dx = dz rlx d?j = -dy dn: dz = -dz dz dy = -dz d?/dn: . (A.12)
and
I
dydy=O,
(B, dy d t
It is this rzlternatzng alqebrnic str7~ct1~r.e of inteqrnnds that gave risc to tllc development of exterior algebra and cnlcult~swhich is bccomiiig morc and niorcl recognized as a powerful tool in mathematical physics. In general, an cxtcrior pforni is an expression
+ B1/dz dn. + B, dn: dy) =
n'v
(A.13) wlierc d S ant1 dV are t l ~ cbountlaries of S and V, respcctively. The right-hand sitlcs of thcsc cqunt,ions corrcspontl to Jcurll?. dfand S d i v B d ~ respectively. ,
s
v
where tlie components w,, ,,, arc completely aiitisylnmctric in thc inclicrs and im = 1,2,3. Furtliermorc, sulnmat,ion from 1 to 3 is understood ovcr rcpcatcd
Part A. Mathematics: Some Exterior Calculus
22
indices. Tlicn, when trarislating (A.5) in cxterior form calculus, we recognize B as a 2-form
/ /v
av
B=
Bliil
alxBijl
dB.
(A.14)
v
a
Notc tliat := (B,, - B,,)/2, C(,,) := (C,, +C,,)/2, etc. Accordingly, in R3, wc havc tlie magnetic ficld B as a 2-form aiitl, from (A.l), tlic electmc field as a 1-form. Thc charge p in (A.3) turns out to bc a ,?-form. In tlic 4-dimension:il Mznkowskz spacc A44, the clcctric current J, likc p in R,, is rcprcscnted by ii 3-form. Since tlic action f~~nctional of tlic electromagnctic field is defined in terms of a 4-clinicnsional integral, the integrand, the Lagrangian L, is a 4-form. Tlic coupling term in L of the current J to the potciitial A, constructed as the c.xtcrior product w J A A, identifies A as a l-form. In the inliomogcncous Maxwell equation J = d H , the 3-form character of J requires thc excitation H to hc a 2-form. If, eventually, we cxecute a gauge transforniation A -+ A df, F --+ F , wc will encounter a 0-form f . Consequrntly, a (gauge) ficld tlicory, starting from a conserved current 3-form (herc tlic clcctric currcnt J), gcneratcs in a straightforward way forms of all ranks P I 4. We know from classical calculus that if tlie Jacobian determinant (A.8) above lias ncgativc values, i . ~ .the , two coordinate systcms do not havc tlie same oricnt,ation, then, in (A.9) and (A.10), thc .lacobian dctcrlniliant must be replaced by its absolute valuc. In part,icular, instcad of (A.10), we get the general formula
Algebra
+
Tliis bcliavior under a cliangc of coordinates is typical of what is known as a d ~ n s l t y Wc . shall see tliat, if wc wish to drop thc rcquircmcnt that all our cooicllnatc systeliis should havc tlic samc orientation, thcn dcnsities beconic iniportaiit and thcsc latter arc closely rclatcd to the twzst~dd?ffercntzal forms that oric has to introducc, in addition to the. ordinary differential forms, since tlic clcctric currcnt, for cxaml~lc,is of such a twistcd type. I11 Cliaptcr A.1, wc first ~onsidera vcctor spacc ant1 its dual and study tlie algebraic aspcct of tensors and of geometrical quantities of a morc general type. Then wc turn our attention to cxterior forms and their algcbra and to a corresponding compliter algcbra program. Since tlic taiigcnt space at eve1y point of a differcntiablc manifold is a linear vcctor spacc, we can associate an cxteiior algcbra with each point and define differentiable fields of exterior fornis or, morc concisely, differentzal forms 011 the nianifold. Tliis is donc in Chapter A.2, whilc Chapter A.3 deals with integration on a manifold. It is important to notc tliat in Part A, we are dealing with the "bare manifold". Tlic lincar conncctioll and metric will be introduced in Part C, after the basic axiomatics of electrodynamics are laid down in Part B.
A.l.l
A real vector space and its dual Our considerations are based o n an n-dimensional real vector space V. One-forms are the elements of the dual vector space V* defined as linear maps of the vector space V into the real numbers. The dual bases of V and V* transform re:iprocall?/ to each other with respect to the action of the linezr group. The vectors and I-forms can be alternatively defined by their components witl~a specified transformation law.
Lct V be an n-dimensional rcal vector space. We can depict a vector v E V by an arrow. If we multiply v by a factor f , the vector has an f-fold size (see Fig. A.l.1). Vectors are addcd according to the parallelogram rule. A linear map w : V 4 R is called a 1-form on V. The set of all 1-fornis on V can be given the structure of a vector spacc by dcfining the sum of two arbitrary :-forms w and cp,
aiicl thc product of w by a real number X E R, (Xw)(v) = X (w(v)),
v E V.
(A.1.2)
This vector space is denoted V* and called tlie dual of V. The d i ~ e n s i o nof V* is equal to the dimension of V. The identification V** = V holds for finite dimensional spaces.
A . 1 . Algebra
24
,,1.2 Tensors of type
[d]
25
Figure A . l . l : Vectors as arrows: their multiplication by a factor, their addition (by the parallelogram rulc). In accorclallce witli (A.1.1), (A.1.2), a 1-form can bc rcpresclited by a pair of ordcrcd hyperplanes, namely ( n - 1)-dimensional subspaccs (sce Fig. A.1.2). The nearer the hyperplanes arc to each otlier, the stronger is thc 1-form. In Fig. A.1.3, tlic action of a 1-form on a vector is depicted. Denotc by e, = { e l , . . . , e,,) a (vector) basis in V. An arbitrary vector v can 1x2 drcomposed witli respect to such a basis: v = v" e,. Summatioll from 1 to n is understood ovcr repcntcd indiccs (Einstein's sulnrnation convention). Thc n real numbers v", a = 1 , .. . , n , arc called con~ponentswitli respect to the given basis. With a basis c , of V wc. can associatc its dual 1-form, or covcctor basis, tlic so-callctl cobasis 19" = { d l , .. . ,d") of V*. It is dctcrminctl by tlic relation
Hero 6; is tlic I
w,=w(e,).
===+
(A.1.4)
A transformatioll from a basis e, of V to another one ("alpha-prinic" basis) e , ~ = { e i , .. . , e:,) is described by a matrix L := (La/") E GL(n, R) (general linear real 1%-dimensional group):
= L,"'
29,
,
where (L,"')
is the inverse matrix to (La,"), i.e., L,"'
L,tP =
6. Syrnbol-
ically, wc may also write el = L r and 19' = ( L ~ ) - ' 19, Hcrc T denotrs thc transpose of tlic matrix L. Consequent,ly, ollc can view a vector v E V as n ordcrctl numbcrs v N that transform under a char~gc(A.1.5) of thc basis as v"' = Lam'va ,
(A.1.7)
whereas a 1-form w E V* is dcscribcd by its components w, with thc transformation law w,, = La," w,, .
(A.1.8)
The similarity of (A.1.7) to (A.1.G) and of (A.1.8) to (A.1.5) ant1 the fact tliat the two lnatxiccs in thcsc formulas arc contragradient (i.c., invorsc and trnnsposed) to each otlier cxplnins thc old-fashioned llalues for vectors ant1 1-forms (or covcctors): contravaria~ltand covariallt vectors, rcspectivcly. Nevcrthcless, one slio~~ltl 1x2 carcful: (A.1.7) rcprcscllts tlrc transformation of n componcnts of one vector, whcreas (A.1.6) cllcrypts the transformation of n diffcrcnt 1-forms.
A. 1.2 Tensors of type
k]
A tensor is a multilinear map of a product of vector and dual
Tlic corresponding cobascs are thus collnccted by 19"'
Figurc A.1.2: Onc-forms arc rcpresentcd as two parallel hyperplanes (straight, lines in 12 = 2) with a direction; tlicir multiplication by a factor.
(A.1.G)
vector spaces into the real num,bers. An alternative definition of tensors specifies the transform.ation law of their cornpone7t.t~ tuitll, respect to a change of the h i s .
A.1. Algebra
A.1.3 @"Agcncralizalion of 1.ensors:geometric quant,itics
for any 1-forms w and any vectors 1). Tensors of type [F] that liavc the form vl @ . . . QD v, QD wl QD . . . @ w, are called dccomposnble. Each tensor is a linear combination of dccomposnble tcnsors. More precisely, using tlic clefinition of the colnponents of T according to (A.1.11), one can prove that
T
= Tml...(*l,
p
,...p,, e e l @ . + . Q D ~@, d, ,P 1 @... @ d P ~ .
(A.1.12)
Therefore tensor protlucts of basis vectors e, and of basis 1-forms 190 constitute a basis of t,hc vcctor spacc q,?'of tcnsors of type [F] on V. Thus the dimension of this vcctor space is nPtq. Elementary cxanlplcs of the tensor spaces arc given hy thr original vcctor space ant1 its d i d , Vd = V ant1 V: = V*. The I
[;I,
Figure A.1.3: A 1-form acts on a vector. Here we have w(u) = 1 , w(v) x 2.3; d(u)x 0.3, $(v) x 4.4; $(w)FZ - 2.1. A 1-form can be understood as a machine: You input, a vcctor and the output is a number which can be read off from our imagcs. The related concepts of a vc.ctor and a 1-form can be generalized to objects of I~ighcrrank. Tllc 11rototypc of such an object is tlic stress tensor of continuum mechanics, A tellsor T on V of typc [jl] is a multi-lincar rnap
--
T : V* x
+ . .x
V* x V x . . . x V
I'
IW.
(A.1.9)
rl
It can be clcscribetl as a geometrical quantity whose components with respect to thc col~asis19" and tlie basis ep arc given by T(19"1,.. . ,$"ll; e n l l . .. , ep,,) .
T"1~.."1301,,,P,l =
(A.1.10)
A.1.3 @A gcnernlization of tensors: geometric quantities A geom.etric quantit?~is defined b?y tlre actior~of tlre general linear group on a certnin set of elements. Importa,rrt czanj,ples are tcn,sor-valued forms, the o~ientntion.,nn.d tloisted ten,sors. In ficltl theory, tensors arc not the only objects needed for thc. dc~cript~ion of nilturc. Twistetl fortns or vector-valuetl forms, for cxamplc, rccluirc a Inore gc~icri~l definition. As wc ~ I L V Oseen ;L~)OVC, there are two ways of dealing with tensors: citlicr wc can describe them as elerne~itsof the abstract tcnsor spacc V : or as components, i.c., clcnlcnt s of Rn"' ', that, l l i t ~a ~prcscribctl trimsfornlatiol~law. These obscrvations can be gcncralizctl as follows. Let IY bc a set, ant1 let p be a left action of tlic group GL(n,IW) in tlie set W , i.c., to cacli clcment L E GL(n, IW) wc attach a nlap p~ : W -+ M/ in such a way that
Tlic tra~isfor~nat,ion law for tcnsor components can be deduced from (A.1.5) and (A.1.G):
If we havc two tensors, T of type [F] and S of type [:I, we can construct their as follows tensor protluct, the tensor T 8 S of type
[i::],
Denote by P ( V ) the space of all I)ases of V and consitlcr tllc Cartesian procliict, W x P ( V ) . The for~nula(A.1.5) provides us with a left action of G L ( n , R ) in P ( V ) which can l,e compactly wlittcn as P' = L e . Then L1 (L2c) = (LlL2)c holds. Thus wc can dcfinc tlic lcft action of GL(Iz,R) on the product 1Y x P ( V ) :
An orbit of this action is called a geometric quan.tity of t,ype p on V. In otllcr words, a. geometric quantity of typc p on V is an equivnlencc class [ ( w ,c)].
A.1. Algebra
28
Two pairs (w, e) and (III', el) arc equivalent if and only if thcre exists a matrix L E GL(n, R) such that w1 = PL(w)
and
el = L e .
(A. 1.15)
In many physical applications, thc sct W is an N-dimensional vector space R N and it is requircd that the inaps p~ are linear. In other words, p~ is a representation p~ : GL(n, R) -, G L ( N , R ) of the linear group GL(n, R) in thc vector space W by N x N matriccs p~ = p n R ( ~ E) G L ( N , IW), with A, B, . . . = 1 , . . . , N. Let us denote as en the basis of the vector space W. Then, we can represent thc geometric quantity ~u = ? u A en by mcans of its componcnts w A with rcspcct to the basis. T l l ~actioll of the group G L ( n , R) in W rcsults in a linear transform a t 1011 '
-
ant1 accordingly, tlic components of tho geornrtric quantity transform wA
wA' = p B A ' ( ~ - ' )wB.
29
A.1.4 Almost complex structure
A.1.4 Almost complex structure An even-dim,ensional vector space V with n = 2k can be equipped with an additional structure that has many interesting applications in electrodynamics and in other physical theories. We say that a real vector space V has an almost is defined on it that has the property complex structure' if a tcnsor J of type
[i]
With respect to a chosen frame, this tensor is represented by the components J,O and the above condition is then rewritten as J,'
JTP = -6:.
(A.1.19)
By means of a suitable choicc of the basis e,, the complex structure can be brought into the canonical form a 9
(A.1.17) Here Ik is thc k-dimensional unity matrix with k = n/2.
Examples:
1) We can takc W = yF and p~ = idw. The corresponding geometric quantity is thcn a tcnsor of typc
[,PI.
2) We can takc = IW~'''' ' and choose p in s1ich a way that (A.1.15) induces (A.1.11) togcther with (A.1.5). This type of geometric quantity is also a For instance, if we take W = R7' and either p~ = L or tensor of typc pl, = ( L T ) - ' , then from (A.1.16) ant1 (A.1.17) we get vectors (A.1.7) or 1-forms (A.1.8), respcctivcly.
A.1.5 Exterior p-forms Exterior forms are totally antisymmetric covarian.t ten,sors. Any tensor of type [:] defines an exterior p-form by means of the alternating map in,volving the generalized Kronecker.
g].
3) The two cxanlplcs above can be combined. Wc can takc W = Rn"+" VVT , and p as in cxarnplc 2). That mcnns th:it wc can consider objects with componcnts T"' "z1p, p,, belonging to V,T which transform according to the rule (A.1.11). This mixture of t,wo approacl~.sseems strange at first, sight, hut it appears ~~roductivc if, instcad of V l , we take spaces of sforms AqV. Such tensor-valued fornls turn out to be useful in differential geometry and physics. 4) Let W = ($1, -1) and pl, = sgn(c1etL). This geonlctric quantity is an omentation in thc vcctor space V. A framc e E P ( V ) is said to have a positive orientation if it forms a pair with +1 E W. Each vcctor space has two different orient a t ions. ' 5) Combine the examples 1) and 4). Lct W = V t and pr, = sgn(dctL) idw. This geometric quantity is callcct a twisted (or odd) tensor of type on V. Particularly useful arc twistcd exterior forms since they can bc intcgratcd even on a manifold which is nonorientable.
k]
As we saw a t the beginning of Part A, exterior pforlns play a particular role as intcgrantls in field theory. We now turn to their general definition. Once again, let V be an n-dimensional linear vector space. An exterior pform w on V is a real-valued linear function
such that
for all vl, . . . ,up E V and for all a , p = 1 , . . . , p. In other words, w is a completcly antisymmetric tcnsor of type In terms of a basis e,, of V and the cobasis 6" of V*, the linear function w can be expressed as
.I:[
A.1.6 Exterior m~lltiplication
where each cocfficicnt w,,...,,, := ~ ( e , ,, . . . , ear,) is completely antisymmctric in all its indices. The spacc of red-valued plinear functions on V was denotcd by Then, wit11 for any p E
5:.
31
Table A . l . l : Numbcr of components of p-forms in 3 nrld 4 dimensions and rxamples from electrodynamics: p electric chnrgc and 3 elcct,ric current density, D electric and 3-1 magnetic excitation, E elcctric and B magnetic field, A covector potential, cp scalar potcntial, and f gauge function. L is the Lagrangian, J = ( p , ~ )El, = ( V , X ) , F = ( E , B ) , A = ( p , A ) . Tlie forms J and H a r e twistcd forms (see Scc. A.2.6).
wc can define a corresponding (t~ltcrtiating)cxterior p-form Alt cp by Alt P = (PI,,
,,.cYl,]
1901
@ . . @ dO1' .
(A.1.25)
Here we have
with tlic gc:ncralizcd Kroneckcr tleltn if ,Dl,... ,PP is nn even permutation of a l l. . . , cup, -1 i f p l , . . . ,,Dl,isanodd permutation of a ] , . . . , aP, 0 otherwise, tl
PI ...PI,
.-
(A.1.27)
whcre cul, . . . ,a,, are p different numbers from the set 1, . . . , n. Provitlcd p,, is alrcady antisyrnmctric in all its indices, thcn
,,
,,,,
Thc set of extcrior p-forms on V forms an (:)-dimcnsionnl si~hspacrof 1/,0 which we denote by A" V*. Hcre rcprescnt the binonlial coefficients. In particular, for p = 0 and p I= 1, we have
(I:)
AOV*
=
A' V* = V* .
R,
For n = 4, t,hc dinrcnsions of t h r spaccs for pforms are
p = 0, 1 , 2, 3, 4
( 4p ) =
1, 4, 6, 4, 1 dimensions,
In order to handle cxtcrior forms, we hnvc to tlefinc their multiplication. Thc exterior protluct of tlze p 1-forms w', . . . , w" V*, talcen in that ordcr, is a pform tlefincd by
spoken as "oln~ga-one wedge . . . wc~lgc.omega-p". It follows that, for ally set of vectors ~ 1 ., .. , u p E V,
(A.1.29) 4! ol. ~'(4-p)'
(A.1.30) Given w E AP V*
A.1.6 Exterior multiplication Tlle exterior product defines a ( p + q)-form, for every pair of pan,d q-forms. The basis of the space of p-forms is then n,att~rally con.stmrcted as the p-th exterior power- of the 1-form basis. The exterior product converts the direct s u m of all forms into an. algebra.
SO
that
A.1. Algebra
32
33
A.1.7 Interior multiplication of a vector with a form
A.1.7 Interior multiplication of a vector with a form The interior product decreases the rank of an exterior form by one.
(i)
-
Since, in addition, t,he p-forms (6"' A . . . A d o p , 1 L a1 < a 2 < . . < ap5 n ) are lincarly inclepcndcnt, it follows that they constitute a basis for Ap V*. Equation (A.1.35) may also be writt,cn as
The intliccs PI < p2 < . . . < ,LIT,are called strongly ordered. Furthermorc, it is clcar from (A.1.35) that a p-form with p > n is equal to zcro. Thc exterior proclrlct of two arbitrary forms is introducccl as a map
as follows: Lct Ij, E AI'V* ant1 4 E AflV*. Tllcn $
4 E A"+'IV*
is defined by
By exterior multiplication, wc increase the rank of a form. Besides this "constructive" operation, we need a "destructive" operation decreasing the rank of a form. Here interior multiplication comes in. For p > 0, the interior product is a map
which is introduced as follows: Let v E V and AP-' V* is defined by
4
E
A"*.
Then
(VJ
4)
E
for all 211,. . . ,up-l E V. We say "v in @'. In the literature sometimes the interior product of v and 4 is alternatively abbreviated as i,$. For p = 0,
In terms of a 1-form basis 19" of V*, we sl~allllave
4)
=
1
p!
,,
.a, 190'
A
.
*
A #@,I ,
(A.1.39)
Note that if p = 1, the definition (A.1.44) implies
Thc following propertics of interior nlultiplication follow ilnrncdiat,cly frorn the definitions (A.1.44), (A.1.45):
tmd their cxt,erior product rei~tls
1) v ~ ( d , $) f =
V J ~ + U J ~ !
2) ( v + u ) J ~ = v J ~ + u J ~ l3om tlic clefillition (A.1.31), it is a straightforward matter to derive the following propcrtics of t?xt,crior multiplication:
1) (A
+
11,)A
v=XAv
+
/L
A
I/
[distributive law],
2) (aX) A v = X A (nv) = n(X A v )
[multiplicativc law],
3) ( X A I / ) A W = X A ( ~ A W )
[associative law],
4) A A
I)
= (-1)r'q (1) A A)
[(auti)commutative law],
w l i ~ r cA, LI E AI'V*, v E ArlV*, w E ATV*,and n E R. With the extorior nlult,iplicatio~lintroduced, the direct sum of t,he spaces of all forms
bcconlcs :ui algcbra. 0vc.r V*. This is usually called the exterior algebra.
3) (av) J
4)
VJ
[linearity in a vcctor],
[multiplicative law],
4 = a(v J 4)
V J ~ L d, J = -UJ
[distributive law],
4
[anticommutativc law],
where 4, 41 E A W * , w E AqV*, V , 21 E V, and a E R. Let e, be a basis of V and 19" the cobasis of V*. Then, by (A.1.46),
Hence, if wc apply the vector basis ea to the p-form
A.1. Algebra
34
1.1.8 @Volume clcmcnts on a vector space, densities, orientation
35
i.e., ep J 4 , the11 the propcrtics listed above yield with L-
If we multiply this formula by do, we fintl tlic identity
' = d e t ( l p a ' ) and, convcrscly,
The geometric quantity with transformation law given by (A.1.55) is called a scalar density. It can easily be generalized. The geometric quantity S with
transformation law
A.1.8 @Volumeelements on a vector spacc, densities, orientation A volunte e1enten.t is a form of m.axirnal rank. T/~,vs,it has one n.ontiivia1 com.ponen,t. Unrler the action of the lin,ear group, this compon,ertt is a densit?/ of weight + I . O~ien~tation is an eq~~ivalen,ce class of 1101umeforms related b y a po.sitive real factor. The choice of an o~ientation,i.s cqui7)alent to the selectioi~, of sirnilarl?j orien.tcd bases in V. The space A n V* of exterior n-forms on an n-dimensional vector space V is 1-dimensional, ant1 for w E ATLV*, we have
Tlie lionzero clcn~cntsof A'LV* are callccl 71olurne elrmrn tar. Collsidcr a linear transformation (A.1.5) of the basis C, of V. The corrcsponding t,r:insform~tioi~ of tlic cohasis d" of V* is given by (A.l.6). Lct L := det(L,,O). Tllcn dl' A . . .
19"'
L,,,
=
1'
. . . L,,,"'
1'
=
I
79"1
. . . Len"'
,. . . , 21,"
601 . , 0 " )
l.,,,L
61 A
..
,
A 1111.
Hcncc, for t,lie voli1111ce l e i l ~ e n twe ~ , liavo:
is called scalar den..sity of weiglrt 7u. The generalization to tensor densities of weight tu in terms of components, see (A.1.11), reads
wherens twisted tensor densities of weight w on the right hand side pick up an extra factor sgn det(L,lfi). Let w atid p he two arbitrary volume forms on V. We say that volumes are equivalent if a, posit,ivc real numbcr n > 0 exists such that p = nw. This definition divitles the space AILV*into tlie two equivalence classes of n-forms. One calls each of the equivalence classes an orientastion on V because a specification of a. vol~uil~e form uniquely determines a class of oriented bases on V and convcrscly. Tliis can be demonstrated as follows. On tlic intuitive lcvcl, in the case of a straight line, one speaks of an oricnta.tion by disting~lishingbetween positive and llcgativc directions, whereas in a plane oiic cliooscs positivc ant1 negative values of an angle, see Fig. A.1.4. The gcncralization of these intuitjive ideas t o an n-dimensional vector space is contaiiictl in tlic concept of orientation.: one says that two bases e, and h, of V are sirni1(1,rl?/oriented if /I,, = ep with dct (/top) > 0. Tliis is clearly a11 cquivaleilce relat,ion tliat divides the space of all bases of V into two classes. An orien.tation, of tlte vector space V is an equivalencc class of ordered bases, and V is callccl oiicn.tetl! vector space when a choice of orientation is made. Now we rctiirii t o the volume elemcnt,~in V. Given the volu11ie w we call define the funct,ion o,(e) := sgn w(el,. . . , e,,)
Since, in terms of the basis
19'"
,
w =~
...
,
1 ~ dl' . . A , ~ ~A ~dT1'
it follows from (A.1.51) and (A.1.52) tliat 2 ~ o the r defitlitiori of R volutl~(?clement wztho~rtthe use of a Schiltl 1321, ant1 in p n r t i c ~ ~ l aLr ,a t ~ r o i t[l4].
metric, see also Synge
ant1
(A.1.58)
~ two vali~es:+1 and -1, and accordingly, on t l ~ csct of all bases of V. It l l i only we have a division of the set of all bases into the two subsets. One class is constitlited of the bases for which o,(e) = 1, and the other for which o,(e) = -1. I11 cilch sribsct, the bases are sin~ilarlyoriented. I11 ortlcr t,o show this, let 11s assume that a volume form (A.1.51) is chosen and fixed, and let us take an arbitrary cobasis 0". The value of the volume form on the vectors of the basis (J, reads w ( e l , . . . , en) = wl ,(79' A . . . A 217')(~11. . . , c,,) = w1 Suppose tliis numher is positive; tlicn in accordance with (A.1.58) we have o,(e) = $1. For a different basis It, = A,p ep we obtain
A.1. Algebra
36
A.1.9 @Levi-~ivita symbols and generalized Kronecker deltas
37
Volume forms provide a natural definition of very important tensor densities, the Levi-Civita symbols. In order to describe them, let us choose an arbitrary cobasis da and consider the form of maximal rank
We call this an elementary volume. Recall that the transformation law of this form is given by (A.1.52), which means that E is the n-form density of weight -1. By simple inspection it turns out that the wedge product
is either zero (when a t least two of the wedge factors are the same) or equal to E up to a sign. The latter holds when all the wedge factors are different, and the sign is determined by the number of permutations that are needed for bringing the product (A.1.60) to the ordered form (A.1.59). This suggests a that has similar symmetry properties. natural definition of the object That is, we define it by the relation Figure A.1.4: Distinguishing orientation (a) on a line and (b) on a plane: The vector bases e, and h, are differently oriented.
s one can immediately check, the Levi-Civita symbol terms of the generalized Kronecker symbol (A.1.27)~:
can be expressed
I
w(h1, . . . , h,,) = det (A,B) w(e1, . . . , en) = det (A,P) wl,.,,,. Consequently, if the basis h, is in the same subset as e,, that is o,(h) = +1, then det (A," > 0, which means that the bases e, and h, are similarly oriented. Conversely, assuming det (AnP) > 0 for any two bases h, = A,Pep, we find that (A.1.58) holds true for both bases. Clearly, every volume form that is obtained by a "resealing" wl,,., --+ cpl,,,,, = awl..,, with a positive factor a will define the same orientation function (A.1.58): o,(e) = o,,(e). This yields the whole class of equivalent volume forms that we introduced a t the beginning of our discussion. The standard orientation of V for an arbitrary basis e, is determined by the volume form d1 A . . . A d n with cobmis d o . A simple reordering of the vectors (for example, an interchange of the first and the second leg) of a basis may change the orientation.
A.1.9 @Levi-Civitasymbols and generalized Kronecker deltas The Levi-Civita symbols are numerically invariant q~~antities an.d close relatives of the volume form. They can arise by applying the exterior product A or the interior product -I n times, respectively. Levi-Civita symbols are totally antisymmetric tensor densities, and their products can be expressed in terms of th,e generalized Kronecker delta.
In particular, we see that the only nontrivial component is c ' . . . ~= 1. W ith respect to the change of basis, this quantity transforms as the [I;]-valued 0-form density of weight + l :
Recalling the definition of the determinant, we see that the components of the Levi-Civita symbol have the same numerical values with respect to all bases,
... a: -
...a,,
(A.1.64)
They are + I , -1, or 0. Another fundamental antisymmetric object can be obtained from the elementary volume < with the help of the interior product operator. As we have learned from Sec. A.1.7, the interior product of a vector with a pform generates a (p - 1)-form. Thus, starting with the elementary volume n-form and using a vector of basis we find an (n - 1)-form
3See Sokolnikoff (291.
A.1. Algebra
38
I
The trarisforniation law of this ol~jectdefines it as a couector-~inlued(n- 1)-form density of w~igll~t -1: {,I
= CIC~(L,,")-
La," 6,. A
(A.1.66)
39
A.1.9 @Levi-Civitasymbols and generalized Kronecker deltas
Furthermore, let us talce an integer q < p. T h e contraction of (A.1.70) over the (n- q) indices yields the same result (A.1.71) witli p replaced by q. Comparing the two contractions, we then deduce for the generalized Kroncckers:
Applying once more the interior product of the basis t o (A.1.65), one obtains an ( n - 2)-form, anel so 011. Thus, we can construct chain of forms: In particular, we find n! cv1...a,, 6CI1...CIp- ( n - p)! ' T h e last object is a 0-form. Property 4) of the interior product forces all these epsilons to be totally antisy~nmetricin all their indices. Similarly to (A.l.GG), we can verify that for p = 0, . . . , n the object 2,, , is a -valued ( n-p)-form density of tlie weight -1. These forms { i , i,, i, , . . . , i,, ,,, ), alternatively to {d*, 6"' A I ? " ~, . . . ,19"l A . . . A79(Y7~}1 can be used as a basis for arbitrary forms in the exterior a1g~l)riiA*V. In particr~lar,we fintl that (A.1.68) is the [:]-valued 0-form density of weight -1. Tliis quantity is also called the Levi-Civita sy~nholbccause of its c v i d ~ n t sirnilality t o (A.l.61). Analogously t o (A.1.62) we can express (A.1.68) in terms of the generalized Kio~iecltersynlbol (A.1.27):
Thus we find again that the only nontrivial component is il..,,, = $1. Note that dcspit,~the d ~ e psimilarity, wc cannot idclitify the two Levi-Civita symbols in the ahsence of tlic metric; hence the different notation (with and without hat) is nppropriatc. I t is wort.1iwhilc t,o dcrivc a useful idcntity for the product of thc two LcviCivitn symbols: 6 * ~.. n , ,
2"'"'"" ep,, _J . - . J eo, E e ~ ,J, . . . J ep, (29,' A . . . A 29,") - (19"' A . . . A dflVL) (epl , . . . , ep,,)
Let us collect for a vector space of 4 dimensions the decisive formulas for going down the p-form ladder by starting from the 4-form density i and arriving a t the 0-form i,p,a:
i = Zap,a
=
e,
=
ep _r i, =
dP
Elopy6
1 9 ~ ga/3! ,
d Y d6/2! ,
(A.1.74)
fapy = ey J cap = Zapy6 d 6 iolpy6 = e6-]tn/3y Going up the ladder yields: 29''
+
ipy6,L = 6; i o y 6 - 6; iPyIL6; ?paLb- 6; d" igy6= 6; iPY 6; 6; i,s , d a A t p y = 6;ip - 6;iy,
+
19"~ip=
+
iy6,L,
(A.1.75)
6;;.
One can, with respect t o thc ?-system, definc a (prc-mct,ric) duality opcrator 0 t h a t establishes an equivalence bct,wecn pforlns and totally antisynimttric tensor dcnsitics of weight +1 and of type [ n i p ] , In terms of the bases of the corresponding linear spaces, this operator is introduced as
~ p ~ , , , f i= ,,
-
Tlir wholc derivatio~lis bascd just on tlie use of the corresponding definitions. Namely, we use (A.1.68) in t,he first line, (A.1.61) in the second line, (A.1.44) in the third line, anel (A.1.32) in the last line. Tliis identity Iiell~sa lot in calc~llationsof the. different contractions of t,he Levi-Civita symbols. For cxamplc, we easily oI)tain from (A.1.70):
Consequently, given an arbitrary p-form basis as
the map 0 defines a tensor density
expanded with respect t o the 2-
by
For examplc, in n = 4 wc have O i , = e, and 02 = 1. Thus, every 3-form cp = cp" i, is mapped into a vector density Ocp = cp" e,, whereas a 4-form w yields a scalar density Ow.
40
A.1. Algebra
A.l.10 The space M%f two-forms in four dimensions
A.1.10 The space ~
Correspondingly, taking into account that 19' A F^ =: Vol is the elementary 4volumc in V, wc find
Electromagnetic excitation and field strength, are both 2-fomn.s. On the 6-dim.en.sion.alspace of 2-form,s, t11,ere exists a natliral 6-metric, which i s an im,portant property of this space.
i
(A.1.83) (A.1.84)
p a / \ p t ) = 0, ?,/lib 2,r\,Ob /
Lct e, be an arbitrary basis of V with a , P,. . . = 0 , 1 , 2 , 3 . In later applications, the zcrotli leg eo can bc rclatctl t o the time coordinate of spacetinie, but this will not always be tlie case (for the ~lrlllsymlnctric basis (C.2.14), for instance, a11 the re's have thc sanic status witli rcspcct to time). Tlie three rcniaining legs will be clcnotctl by c , with a , 0,.. . = 1 , 2 , 3 . Accordingly, tlic dual basis of V* is reprcsentetl by .19" = (d o , d n ) . I11 tlic linear spacc of 2-forms A 2 V*, every clcliielit can 1>c decomposed according t o p = p,,p d" A d o . Tllc basis 19" A d P consist,^ of six simple 2-forms. Tliis 6-plct can bc altcrnativcly nunil)errtl by a collrctive intlrx. Accordingly, we enumerate the antisymmetric index pairs 01,02,03,23,31,12 by uppercase lcttcrs I, J , . . . from 1 to G:
41
% two-forms f in four dimensions
,A
A
8'
0, = 6:vol, =
=
(A.1.85)
(- 6: 6;: + 6; 6;) Val.
(A.1.86) 1
Every 2-form, being :in elcmcnt of M " , can now be represented as (P = ( P I E by its six coliiponcnts with rcspcct to t,lie basis (A.1.79). A 4-form w, i.e., a form of the liiaximal rank in four dimensions, is expanded with respect t o the wedge protlucts of the Gbnsis as w = i ~GI A~ GJ. .Tlie~coefficients WIJ form R, symmetric G x G matrix since the wedge product between 2-forms is evidently comn~ut,ativc. A 4-for111 has only one component. This simplc observation enables us t o iritrotlucc a natural metric 011 the 6-diniensional spacc Ad%s the syrnmctric bilincar form E(W,(P) := ( W A ( ~ ) ( e o , e l , e z , e s ) ,
w,cp
M6,
(A.1.87)
where e, is a vector basis. Although the metric (A.1.87) apparently depends on the choicc of basis, the linear transformation e,! -t L,! e, induces the pure rescaling E -t det(L,,") E . Using the expansion of the 2-forms with respect to tlic hivcctor bitsis G', tlic bilinear form (A.1.87) turns out t o be (1
whore
E(W,p ) = wI p j E'.',
&IJ =
(G1 A E1)(eo,e l , ez, e3).
(A.1.88)
A rlilrct inspection by using the clefinition (A.1.79) and the identity (A.1.85) shows tllnt the 6-l~lctriccomponents rei~tlexplicitly With tlie GI LS basis (say "cyrillic B" or "Bch"), we can set up a 6-dimensional A 2 V*. Tliis vector spacc will play an important role in our vector spacc A[%= consiticrations in Parts D and E. The extra dcco~npositionwith rcspcct t o P" ant1 ibis convcnicnt for recognizing where tlic clcctric and where the magnetic pieces of tlic field arc locatctl. WP denote the clcmcntary volumc 3-form by i = t9' ~ 6A@. ' Thcn i,, = r , J i is t8hc basis 2-form in t,hc spacr spanned by the 3-cofraliic 79", see (A.1.65). Tliis notat,ion has bcen used in (A.1.79). Moreover, as usrlal, the 1-form basis Citn then be tlcscribccl by in/) = rb J 2,,. Sonir useful algebraic relations can be immedi;~tclydcrivcd: (A.1.80) ('4.1.81) (A.1.82)
Here I3 :=
(
0
: :) 1 0
is tlie 3 x 3 111iitmatrix. Thus we see that the metric
(A.1.87) is ;~lwaysnondegeneratc. Its signature is (+, +,+, -, -, -). Indeed, the cigenvnlucs X of tlic niatrix (A.1.89) are defined by the characteristic equation d c t ( ~ ' . ]- AdrJ) = (A2 = 0. The synimetry group that preserves the 6-metric (A. 1.87) is isomorphic t o 0 ( 3 , 3 ) . By construction, t,he clcmcnts of (A.1.89) numerically coincide witli the components of t,lic Lcvi-Civit,a sylnbol F ' J ~ ' , see (A.1.62):
A.1. Algebra
42
Similarly, the covariant Levi-Civita symbol P,,, sentcd in 6D notation by the matrix
, see
(A.1.69), can be repre-
43
2.1.11 Almost complex structure on M6
&ere the determinant of the transformation matrix (A.1.92) reads det L := det L,,o = [Lo0 - L , O ( L - ~ ) ~ ~ L ~ ~ det ] ~,d.
(A.1.99)
One can write an arbitrary linear transformation L E GL(4, R) as a product One can inlmecliately provc by multiplying the matrices (A.1.89) and (A.1.91) that their product is equal to 6D-unity, in complete agreement with (A.1.71). Thus, the Levi-Civita symbols can be consistently used for raising and lowering indices in M G .
(A.1.100)
L=L1L2L3 of three matrices of the form
@Transformationof the M6-basis What happens in M h h c n the basis in V is changed, i.e., e, --+ eat? As wc know, such a change is descril~edby the linear transformation (A.1.5). Thcn the cobasis transforms in accordance with (A.1.6): dQ = Lo," d") .
In the (1
(A.1.92)
+ 3)-matrix form, this can be written as
+ + +
3ere V n Ub,Roo,A ha , with a , b = 1,2,3, describe 3 3 1 9 = 16 elements )f an arbitrary linear transformation. The matrices {L3) form the group R 8 TL(3, R) which is a subgroup of GL(4,R), whereas the sets of unimodular natrices {L1) and {Lz) evidently form two Abelian subgroups in GL(4, R). In the study of the covariance properties of various objects in M 6 , it is thus sufficient t o consider the three separate cases (A. 1.101)-(A.1.103). Using (A.1.96) and (A.1.97), we find for L = L1 l
Correspondingly, the 2-form basis (A.1.79) transforms into a new bivector b x i s similarly, for L = L2 we have
and for L = L3 Substituting (A.1.92) into (A.1.79), we find that the new and old 2-form bases are related by an induced linear transformation
pah= Roo h b n ,
Qba = (det A ) ( A - ' ) ~ ~ ,
wab= Zab = 0.
(A.1.106)
A.1.11 Almost complex structure on M~
pnb
=
wab=
LoOLha- LoaLbO, Qoa = det L,"L-')~", LcoLdaFbcdl Znb =
The 3 x 3 matrix (L-')ha is inverse to the 3 x 3 subbloclc L,"n invcrsc transformation is easily computed:
(A.1.96) (A.1.97) (A.1.92). The
A n almost complex structure o n the space of 2-forms determines a splitting of the complexification of M6 into two invariant 3-dimensional subspaces. Let us introduce an almost complex structure J on My We recall that every tensor of type [i] represents a linear operator on a vector space. Accordingly, if cp E M G ,it is of type and J(cp) can be defined as a contraction. The result will also be an element of M6. By definition, J(J(cp)) = -I6 cp or
I:[
412
A.1. Algebra
see (A.1.19). the opcrator J can be represented as a G x G matrix. As a tensor of type Since the basis in M G is naturally split into 3 3 parts in (A.1.79), we can write it in terms of the set of four 3 x 3 matrices,
[;I,
+
45
A . l . I 2 Computer algebra
Let us denote the 3-dimensional subspaces of MG(@)that correspond to the eigenvalues +i and -2 by (8)
M
:= {w
E M ~ C I )J(w) = i w ) ,
-
Because of (A.1.107), the 3 x 3 bloclcs A, B , C, D arc constrained by
+
AncBcb C'"CCh
-d;, c ~ ~ A ~ ~ = ~ 0,+ A ~ ~ D ~ ~ Ba,Ccb DoCBc~) = 0, t~ D,,A"" D,"D," = -dn, (A.1.109) =
+ +
The significant tliffcrcncc between these two metrics is that E' assigils a real length to any complex vector, whereas E defines complex vector lengths. We will nssulne that the J operator is defined in MG(@)by the same formula linear operator in MG(@), i.e. as J(w) in Ad? 111other words, J remains a rral for every coml~lex2-form w E A/lG(@)one has J(w) = J(a).The eigenvaluc problem for the operator J(wx) = Xwx is meaningful only in the cornplcxificd space A/lG(Q1) because, in view of the property (A. 1.107), the eigcnvalues are X = fz. Each of these two cigenvalucs has multiplicity three, which follows from the reality of J. Note that the G x G matrix of the J operator has six eigenvcctors, but the numl~crof eigcnvectors wit11 cigenvaluc +z is equal to the number of cigenvcctors with cigenvalue -z beca~lsethey arc complex conjugate to each other. Indccd, if J(w) = zw, tllcn tlie conjugation yields J(LJ)= J ( Z ) = -155.
(a)
-J
) ],
with
2 ) 1-2 [ An allnost complex structure on M G motivates a complex generalization of M G to the complezified linear space Ad G(@).The elements of My(@ are the complex 2-forms w E My(@, i.e. their component,^ WI in a decomposition w = WIG' are conlplex. Alt,ernat,ively,one call consider Ad"(@ as a real 12-dimensional lincar space spanned by the basis (c', i ~ ' ) ,where i is tlie imaginary unit. We denote by M"(@)t,hc complex co~ijugatcspace. The same sylnnlct,ric bilinear form as in (A.1.87) also defines a natural metric in Ad"(@). Note howcvcr, that NOW an orthogonal (complex) basis call always be introtluccd in Ad"(@ so that E'.' = 6I.I in that basis. Incidentally, one can tlefinc nnotllcr scalar product, on a complex spacc MG(C) I)Y
(8)
respectively. Evidently, M = Ad. Therefore, we can restrict our attention only to the self-dual subspace. This will be assumed in our derivatiolls from now on. Accordingly, every form w call be tlecomposed into a self-dual and an antiself-dual piece4,
(*)
1 w = - [w 2
(a)
(3)
+ iJ(w)] .
(a)
It can be checked that J ( w ) = +i w and J ( w ) = -z
. (a) W.
A.1.12 Computer algebra Also in electrodynamics, research usually requires the application of computrrs. Besides numerical methotls and visualization techniques, the manipulation of formulas by means of "computer algchra" systelns is nearly a must. By no means are these nicthods confined to pure algebra. Differentiations ant1 int,egrations, for example, can also he exccutecl with the help of computer algebra tools. "If wc do work on the foundations of clnssical electrodynanlics, wc can dispense wit11 computer algebra," some true fundamentalists will claim. Is this really true? Well later, in Chap. D.2, we analyze the Fresnel equation; we couldn't have done it to the extent we did without using an efficient computer algebra system. Thus, our fundamentalist is well advised to learn some computer algebra. Accordingly, in addition to introducillg some mathemat,ical tools in cxterior calculus, we mention computer algebra systems like Reduce" Maple" and Math4 A discussion of the use of self dual and nnti-self dual 2-forms in general relntivity cnn be found in Kopczyriski and Trnutman [13],e.g. "earn (81 crented this Lisp-based system. For introductions t o Iieduce, see Toussaint 1361, Groain [GI, MncCnllum and Wright [Is],or Winkelmann and I-Iehl [38];in the lnttcr text you can learn how to get hold of n Reduce system for your computer. Reduce as npplied t o general-relntivistic fleld theories is described, for example, by McCren [lG] and by Socorro ct al. [28]. In our presentntion, we pnrtly follow the lectures of Toussaint [36]. ' ~ a p l e , written in C, was created by a group nt the University of Waterloo, Canada. A good introduction is given by Chnr et nl. [3].
A.1. Algchra
46
4*
A.1.12 Computer nlgcbra
brary of McCrcal' and GRG12, in Maple G R T e n ~ o r l I ' ~and , in Mathematica, besides MathTensor, the Cnrtan package14. Computer algcbra systems are almost cxclusively interactive systems nowadays. If onc is inst;~llcdon your computer, you can usually call the system b:r typing its llalne or an abhrcviation thereof, i.c., 'reduce', 'maple', or 'math', and then hit.t,ing the return key, or clicking on the corresponding icon. In the cas? of 'reduce', thc systcm introcluccs itsclf and issues a '1 :'. It waits for your first command. A command is a statcment, usually somc sort of expression, a part cf a formllla or a. formula, followed by a tcrminator15. Thc latter, in Reduce, is i semicolon ; if you want to see the answer of the system, and othcrwisc a dollar sign $. Reduce is case inscnsitivc, i.c., thc lowcrcasc letter a is not clistinguishe~l from the uppercase letter A.
Formulating Reduce input
1
Figure A.1.5: "Hcrc is thc new Rcclucc-updatc
011
a hart1 disk."
emntlcr~~ and specifically explain how to apply thc Rcducc package E : ~ c n l c ~ o fclrms that occur in clcctrotlynal~lics. tllc work in solvilig problems by mcalls of colllpllt,er algcbra, it is our In cxpcricllcc thct it is best to have access to diffcrcnt computer algebra systems. Even tllougll 111 thc coursc of tirrlc good fenturcs of one system "migrated1' to other systems still, for a certain spccificd purpose one systcln may lbc brtter suited tllall alother one - and for different purposes thesc may bc different systems. Thelt' does not cxist as yct, tire optimal system for all purposes. Thcrcfore, it is llot a rare occasion that wc have to feed the results of a calculation by means of d1c systcnl as input into another system. For ~ ~ ~ ~ ~ ~ ~inu clcct,rotlynamics, tztions rcl~tivit~y, and gravitation, we keep the three general-11urpose computer algebra systems: Reduce, Maple, ant1 Mathematica. Otllcr systelils arc available? Our workhorse for corresponding calcu lati on s ill cxtc2rior calculus is the Rcducc package Excalc, but also in tlie Maf/rTcnsor rackage'' of Mnthematica exterior calculrls is implemented. For the m ~ n i ~ u l a i i oofn tensors wc usc tAic followillg packagcs: In Rcducc the li7 W ~ l f r a n r( s c [ N ] ) created the C-based Mathematica software packagc whicll is in very widespreacl use. R S ~ l l r ~[25 ~ f26) ~ ris the creator of that packagr (cf. also [27]). J3xcalc is applied t o Maxwell's theory by P u ~ ~ t i , ' a et m al. [22]. "111 the rcvicv of IIartley [7] possiblc alternative systcnis arc discusscd (scc also IIeinickc ct al. [ l l ] ) . "I'arkrr and Zhristenscn [21] created this package; for a simplc application scc Tsantilis r t nl. 1371.
As a11 illplit statrlncnt to Retlucc, we type in a certain legitimately formcl expression. This means that, with the help of somc opcrators, we compose formulas accortlillg to well-defined rulcs. Most of thc built-in opcrators of Reclucr, like the arithmetic opcrators + (plus), - (minus), * (times), / (divided by), ** (to tlic power of)'%re self-explanatory. They are so-called znfix operators since they arc positiolled zn between their arguments. By means of thcm we can construct combined cxprcssions of the typc (r or x3 sins, which in Redutc read (x+y)**2 ant1 x**3*sin(x), respectively. If thr comlnalltl
+
+
+
is cxccl~t,cd,you will get the expanded for111 r2 2sy y 2 . Therc is a so-callcd switch exp in Rrducc that is usually switched on. You can switch it off by the commalitl I
--
o f f exp; I
See, McCrca's lecturrs [IG]. l21'he GRG s y s t ~ r n creatctl , by Zhytnikov [do], ant1 the GRCrZc system of Tertychniv (35, 34, 201 grew frorn the same root; for an application of GRGEc t o t h e Einstein-Maxwell equations, see [33]. '%ec t h e docrl~nentatiollof Musgrave et al. (191. Maple applications t o t h e EinstcinMaxwcll system arc covered in thc lcctrrres of McLenaghan [17]. '"Solcng (301 is t h c creator of 'Cartan'. l51~1asnothing t o rlo with Arnold Schwarzcncggcr! '%sually one takcs t h c circumflex for exponentiation. I-Iowcvcr, in t h e Excalc package this operator is rcdcfinctl a l ~ dused as tlre wedge syrnbol for exterior multiplication.
A.1. Algebra
48
* * * *
* *
*
description if switch is on factorize simple factors divide by the denominator expand all expressions make (common) denominator cancel least common multiples cancel greatest common divisor display as polynomial in f a c t o r display rationals as fraction dominates a l l f a c ,d i v , r a t , r e v p r i display polynom. in opposite order calculate with floats simplify complex expressions don't display zero results display in Reduce input format suppress messages display in Fortran format display in TeX format
Switch allfac div exp mcd lcm gcd rat ratpri pri revpri rounded complex nero nat msg fort tex
The operator neq means not equal. The assignment operator := assigns the value of the expression on its right-hand side (its second argument) t o the identifier on its left-hand side (its first argument). In Reduce, logical (or Boolean) expressions have only theI truth values t (true) or n i l (false). They are only allowed within certain statements (namely in i f , while, r e p e a t , and l e t statements) and in so-called rule lists. A prefix operator stands in front of its argument(s). The arguments are enclosed by parentheses and separated by commas:
examnle
Table A.1.2: Switches for Reduce's reforlnulation rules. Those marked with are turned on by default; the other ones are off.
cos (XI i n t (cos ( X I ,x) f actorial(8)
In ordinary notation, the second statement reads S cos x dx. The following mathematical functions are built-in as prefix operators:
*
Type in again
Now you will filid that Reduce doesn't do anything and gives the expression back as it received it. With on exp; you can go back to the original status. Using the switches is a typical way to influence Reduce's way of how to evaluate an expression. A partial list of switches is presented in a table on the next page. Let us give some more examples of expressions with infix operators: (u+v) * (y-x)/8 (a>b) and (c
Here we have the logical and relational operators and, > (greater than), < (less than). Widely used are also the infix operators: neq
>=
<=
sin cot a si n acot sinh asinh sqrt dilog factorial
s ind cotd a s ind acotd cosh acosh
exp erf
or
not
:=
C08
sec acos asec tanh atanh In expint
cosd secd acosd asecd coth acoth log cbrt
tan csc atan acsc sech asech log10 abs
tand cscd a t and acscd csch acsch logb hypot
Identifiers ending with d indicate that this operator expects its argument t o be expressed in degrees, l o g stands for the natural logarithm and is equivalent to In. logb is the logarithm in base n; accordingly n must be specified as a second argument of logb, hypot calculates the hypotenuse according to hypot(x, y) = csc is the cosecant, d i l o g the Euler dilogarithm with dilog(z) = - S : log(1 - <)I< dC, e r f the Gaussian error function with erf(x) = 2 / f i :S e-t2dt, abs the absolute value function, expint the exponential integral with expint(x) =, :S et/t dt, and finally, c b r t the operator for the cubic root. Reduce only knows a few elementary rules for these operators. In addition to these built-in rules and operators, the Reduce user may want to define her or his own rules and operators (i.e., functions) by calling the command operator. No arguments are specified in the declaration statement. After the declaration, the specified operators may be used with arguments, just like s i n , cos, etc.:
d m .
I
I --
49
A.1.12 Computer algebra
50
A.1. Algebra
4.1 -12 Computer algebra
51
Integer expressions evaluate to whole numbers, as in:
I
clear f,k,m,n$ operator f$ f (m) ; f(n):=n**4+h**3+~**2+u; f (4,k):=g;
If an opcrator is given witli a certain argument, say f ( n ) , and a11 expression (here n**4+h**3+p**2+u, which contains the argument n of the operator) is assigned to the operator f (n), this is a specific assignment only. There is no general functional relationship establisliecl between the argument of thc operator antl thc same identifier that appcars in tlie assigned expression. Such a relationship can only be created by self-defined rules. Let us demonstrate this solnewllat difficult point as follows: f(n):=n**4+h**3+p**2+u$ f (k); f (n) ;
% does not evaluate to the value % k**4+h**3+p**2+u, but only to f ( k ) % again yields n**4+h**3+p**2+u
A newly crrntctl ol)c.rator, which has no previously assigned valuc, carries as its value its ow11 nnmc with the arguments (in contrast, to tlrc clemcnts of an arrtiy which arc initialized witli value zcro antl which can nevcr lravc ,zs valncs thr array 11:ilnc with t,licil indices!). Tl~cscopcrtitors 1i:ivc no propcrties unlcss l e t rulcs arc spccifictl. All opcrtitors niay a
provided the variables j , k , h evaluate to integers. Scalar expressions consist of (syntactically correct) sequences of numbers, variables, operators, lck and right parentheses, and commas and arc the 11sua1 representation of mathematical expressions in Reduce:
havc values assigned to them, as in --
-
log(u) : =12$ cos (2*k*pi) : =1$
11me properties tleclnrctl for some collcctions of arguments (for example, t,hc valuc of sin(intrger*pi) is always 0), a 1)c. frilly defined, cithcr by the uscr or by Reducc, as is the case for the operator df for differentiation. Witli the operators definctl so far, we can construct Reduce cxpressions by combining variablcs antl operators in such a way that thcy represent our mathenlatical formulas. Rccluce distinguishes between three kinds of expressions: iiitcgcr, scalar, ttnd Boolcan.
The minimal scalar expressions that are known t o Reducc are variablcs or numbers. The following rules are applied on evaluation of scalar expressions: Variablcs and operators with a number of arguments liave thc algebraic value they were last assigned or, if ncvcr assigned, stand for tliemselvcs. a Nevertheless, sornc special cxpressions, such as elcmcnts of arrays (indexctl variables), initially havc the valuc 0. a Operators act according t o tlie rulcs that are dcfined for tlicln. If tllrre is no matching rule, the operator with its argument stands for itself (cos 0, for cxample, evaluates to 1, but cos x won't get evaluated as long as x is an rinbound variablc). Note that an (inappropriate) assignment such as cos (0) : =7 will have thc samc effect, as a rule that defines cos (0) to bc 7. a Procedures of expressiolls are evaluated with tlie values of tlreir actnal parameters used in tlie procedure call. a The algebraic evaluation of expressions (also called simplification) is controlled by the switches, which lnay be turned on or off by the Rcdlicc user. In any case, the standard rules of algebra apply. Parentheses are allowed. Exprcssions may be combined with legal operators to build new expressions. Those new expressions take on tlie new valuc built from the v:dncs of the subexpressions via the operators, taking into account the control switches.
A.1. Algebra
52
Examples: clear a,b$ a*b; pol ; pol:=(a+b)**3$ pol ; on gcd$ off exp$ pol; f:=g*m*m/r**2; on div$
I
% still not assigned % now assigned % greatest common divisor switch on % expansion switch off
% removes identical factors in % numerator and denominator % reset switches
We tlidri't give the output. You should try to get this yourself on your computer. Roolran expressions use the well-known Boolean algebra. and have truth values t for true and n i l for false. For handling Boolean expressions we havc alrcatly mentioned tlir Boolean infix operators. Boolean exprcssiolis arc only allowed within i f - , while-, or repeat-statements. Examples of typical Boolean cxprcssiolls are j neq 2 a=b and (d or g) (a+7) > 18
clear g,x$ a:=sin(g:=(~+7)**6) ; cos (n:=2)*df (x**lO ,x,n) ;
% a and b are declared to be unbound
f;
off gcd,div$ on exp$
are always evaluatcd first. I11 the first case, the value of (x+7)**6 is assigned to g, and then s i n ( ( x + 7 ) **6) is assigned to a. Note that the value of a whole assignmellt statement is always the value of its right-hand side. In the second case, Reduce assigns 2 to n, then computes df (x**lO,x,2), and eventually returns 90*x**8*cos(2) as the value of the whole statement. Note that both of these examples represents bad programming style, which shoulti be avoitfed. One exception to the process of evaluation exists for the assignment opcrator := . Usually, the arguments of an operator are evaluated before the operator is applied to its arguments. In a11 assignment statement, the left side of the assignment operator is not evaluated. Hence 1
clear b, c$ a: =b$ a: =c$ a;
will not assign c to b, but rather c to a. The process of evaluation in an assigllment statement can be studied in the following examples: % if a evaluates to an integer
I
clear h$ g:=l$ a: =(g+h) **3$ a; g:=7$ a;
If you want to display tlie truth valuc of a. B O O ~ Cc~x L ~ )~rIe ~ ~ illse o n ,th(: i f statcmcnt, as ill t l l ~following example: if
53
A.1.12 Computer algebra
2**28 < 10**7 then write "less" else write "greater or equal";
Rudiments of evaluation A Reduce program is a follow-up of commands. And the evaluation of the commands may be conditioned by switches that we switch on or off (also by a com1naiid). Let us look into t l ~ ccvalr~ationproccss a bit closer. Aftcr a command has been sent to the computer by hitting the return key, the whole command is evaluated. Each expression is evaluatcd from left to right, and the values obtained arc combiricd with the operators specified. Substatements or snbexpressions (.xisting within other expressions, as in
% yields: (l+h) **3 % yields: (l+h)**3
1
After the second statement, the variable a hasn't the value (g+h)**3 but rather (i+h) **3. This doesn't change by the fifth statement either where a new valuc is assignctl to g. As onc will recognize, a still has the value of ( l + h ) **3. If we want a to dcpcntl on g, then we must assign (g+h)**3 to a as lollg as g is still unbound: (
clear g,h$ a:=(g+h) **3$ g:=l$ g:=7$ a; I
% all variables are still unbound
% yields: (7+h)**3
54
A.1. Algebra
Now a has the value of (7+h)**3 rather than (g+h)**3. Sometimes it is necessary to remove the assigned value from a variable or an expression. This can be achieved by using the operator clear as in I
clear g,h$ a:=(g+h)**3$ g:=I$ a; clear g$ a;
55
A.1.12 Computer algebra
with denoting the exterior and -I (underline followed by a vertical bar) the interior product sign. Note that before the interior product sign -I (spoken in) there must be a blank; the other blanks are optional. However, before Excalc can understand our intentions, we better declare u to be a (tangential) vector r
tvector u; I
f to be a scalar (i.e., a 0-form), and x, y, z to be 1-forms: 7
pform f=O, x=l, y=l, z=l;
or by overwriting the old value by means of a new ~ssignmentstatement: I
clear b,u,v$ a:=(u+v)**2$ a:=a-v**2$ a; b: =b+l$ b;
A variable that is not declared to be a vector or a form is treated as a constant; thus 0-forms must also be declared. After our declarations, we can input our command I
f*xAy+u - 1 (y-2-x); I
The evaluation of a; results in the value u* (u+2*v) since (u+v) **2 had been assigned to a and a-v**2 (i.e., (u+v) **2-v**2) was reassigned to a. The assignment b: =b+i; will, however, lcad to a difficulty: Since no value was previously assigned to b, the assignment replaces b literally with b+l (wherehs the prcvious a:=a-v**2 statement produces the evaluation a:=(u+v) **2-v**2). The last evaluation b; will lcad to an error or will evcn hang up tho system because b+l is assigncd to b. As soon as b is evaluatccl, Rcducc returns b+l, whereby b still has the value b+l, and so on. Therefore the evaluation process leads to an infinite loop. Hence we should avoid such recursions. Incidentally, if you want to finish a Rcduce session, just type in bye; After these glimpses of Reduce, we will turn to thc real object of our interest.
Loading Excalc We load the Excalc package by I
load-package excalc$
Of course, the system cannot do much with this expression, but it expands the interior product. It also knows, of course, that I
u -If; I
vanishes, that y A z = -x A y, or that x A x = 0. of an expression, we can use
If we want to check the rank
I
exdegree (x-y);
This yields 2 for our example. Quite generally, Excalc can handle scalar-valued exterior forms, vectors and operations between them, as well as nonscalar valued forms (indexed forms). Simple examples of indexed forms are the Kronecker delta 6 i or the connection 1-form I',P of Sec. C.1.2. Their declaration reads
I
pform delta(a,b)=O, gammal(a,b)=l;
The system will tell us that the operator A is redefined since it becamc the new wedge operator. Excalc is designed such that the input to the computer is the same as what would have been written down for a hand calculation. For example, the statement f *xAy + u - I (yAzAx) would be a lcgitimatcly built Excalc expression
The names of the indices are arbitrary. Subsequently, in the program a lower index is marked by a minus sign and an upper index with a plus (or with nothing), i.e., 6: translates into delta(-I , I ) , and so on.
A.1. Algebra
56
A.2 Exterior calculus
Figure A.1.G: "Catastrophic crror," a Reduce crror message. Excalc is a good tool for studying differential equations, for making calcnlntiolls in field t,lieory and general relativity, and for such simple things as calc~~lating the Laplacian of a tensor field for an arbitrarily given frame. Excalc is complotcly cmbccldccl in Rcducc. Thus, a11 fcnturcs and facilities of R c d ~ ~ c c arc available in a calculation. If we tlrclarc the dimension of tlie underlying spacc by spacedim 4 ;
then pform a = 2 , b = 3 ;
a"b;
yicltls 0. Thrsc arc the funtlamcntal commands of Excalc for exterior algebra. As soon as we liavc introduced exterior calculus with frames and coframcs, vector fields and ficltls of forms, not to forgct exterior ant1 Lie differentiation, we will come back to Excalc ant1 brttcr appreciitte its real power.
Having developed the concepts involved in tlie exterior algebra associated witli an 11-dimensional linear vector space V, we now look a t how tliis structr~rccan be "liftetl" ont,o a11 n-dimensional differentiable manifold X,, or, for short, onto X. The procedure for doing tliis is the same as for tlie transition from tensor algebra to t,cllsor calcul~ls.At each point .7: of X t,l~creis an 17,-climcnsional vector spacc X:,, the tangent vector spacc at 2 . Wc identify tlle space X, witli the vector spacc V collsidcrcd in the previous chapt,cr. Tlicn, a t each point 2, tho exterior nlgcbrii of forms is dctcnninctl on V = X,:. Howcvcr, in differential geometry, one is concerned not so mucli witli objects defined at isolatccl points as with fields over tlic lnanifold X or over open sets U C X , A field w of pforms on X is defined l,y assigning a pform t,o each point n: of X and, if this assigllrnclit is pc?rformed in a sinooth manner, wc shall call tlie res~iltiligficltl of pforms a n e3:terioi. differential p-form. For simp1icit.y we sh;\ll take "smootli" to 1ner211 Cwl altliougli in physical applications tlic dcgrcc of diff~rcntiabilit,y may be lcss.
A.2.1 @Differentiable manifolds A topolo!jical space becomes a differen,ti& ~rlnn,.foldrulren 0.71, atlas of coordincl.te clrnrts is introd~~cerl i n it. Coorrlinate trclmsform.ations are smootll i n the intersection,s of the clrnrts. T11.r: a,tlna i.9 o.r.ien.tcr1 lulren irr n.ll in~ter.sectio71.s the ,JncoOian.s of tlre coorrlirl.ate tmn,sfornrc~t~ion,s o,re positive. In ortler to dcscril~cmore rigorously how ficltls arc introducccl on X , we have to recall some basic facts about manifolds. At tllc start, one ~lccds:I topological
-4.2. Exterior calculus
58
~ ~ 2@Differentiable .1 manifolds
59
Figure A.2.1: Nan,-Hausdorff m:~nifold: Take two copies of the line segment {0,1) and identify (paste together) their left halves excluding the points (112) and (112)'. In the resulting manifold, the Hausdorff axiom is violated for the pair of points (112) and (112)'. stm~cture.To be specific, we will normally assume that X is a connected, Hausdorff, and paracompact topological space. A topology on X is introduced by the collection of open sets 'T = {U, c X l n E I) which, by definition, satisfy three conditions: (i) both the empty set Q) and the manifold itself X belong t o that collection, Q),X E 'T, (ii) any un.ion of open sets is again open, i.e., UaEJ U, E T for any subset J E I, and (iii) any intersection of a finite number of open sets is open, i.e., U, E 'T for any finite subset I( E I. A topological space X is con,n,ectedif one cannot represent it by the sum X = X1 U X2 with open Xl,2 and X I X2 = Q).Ust~allyfor a spacetime manifold, one further requires a linwar connectedness, which means that any two points of X can be connected by a continuous path. A topological space X is Hausdorff when for any two points p1 # p2 E X onc can find open sets U1, U2 c X wit,h pl E U1 and 212 E U2 such that Ul U2 = Q). Hausdorff's axiom forbids the "brancl~ecl"manifolds of the sort dcpictcd in Fig. A.2.1. A connected Hausdorff manifold is paracompact when X can be covered by a countable number of open sets, i.c., X = UnEI( U, for a countable subset I< E I. Finally, a manifold X is compact when it can be coveretl by a finite number of open sets selected from itasarbitrary covering. A collection of filnctions {p, : X + R) is called a partition of unity subordinate to a covering {U,) if support of p, C U, and 0 p,(p) 5 1, C , p,(p) = 1 for all points p E X . The partition of unity always exists for paracompact manifolds, and it is a standard tool that helps to derive global constructions from the local ones. A differentiable manifold is a topological spacc X plus a differentiable structure on it. The latter is defined as follows: A coordinate chart on X is a pair (U, $) where U E 'T is an open set and the map $ : U --t Rn is a homeomorphisn~ (i.e., continuous with a continuous inverse map) of U onto an open subset of the arithmetic space of n-tuples Rn. This map aqsigns n labels or coordinates d)(p) = {xl(p),. . . ,xn(p)} to any point p E U c X. Given any two intersecting Up # Q),the map charts, (U,, 4,) and (Uo, $o) with U,
n,,,
n
n
<
n
Figure A.2.2: Rectangle A R C D in R2.
is C". The latter gives a coordinate transformation in the intersection of the charts. The whole collection of the charts {(U,,$,)J a E I) is called an atlas for every open covering of X = UaEI U,. The two atlases {(U,, $,)) and {(V,, $,)) arc said to be compatible if their union is again an atlas. Finally, the differentiable structure on X is a maximal atlas A ( X ) in the sense that its union with any atlw gives again A(X). The atlas {(U,, 4,)) on X is said to be oriented if all the transition functions (A.2.1) arc orientation preserving, i.e., the corresponding Jacobian determinants are everywhere positive, J(fap) = det
(g)
> 0,
where $, = {x i ) and $o = {yi), with i , j = 1 , . . . , n. Then faP = (xl(y', . . . , y n ), . . . , x71(y1,. . . , yn)). The differentiable manifold X is orientable if it has an oriented atlas. The notions of orientability and orientation on a manifold will prove to be very important in the theory of integration of differential forms. It is straightforward to provide examples of orientable and nonorientable manifolds. The following 2-dimensional manifolds can be easily constructed with the help of the cut and paste techniques. Consider an R2 and cut out a rectangle ABCD, as shown in Fig. A.2.2. As such, this is a compact two-dimensional manifold with boundary that is topologically equivalent to a disc.
12.2. Exterior calculus
GO
61
,,2.2 Vector fields
Figure A.2.3: Torus T 2
Figure A.2.5: Klein bottle K2. dialnetrically opposite points (a more spectacular way is to say that tlic hole is closed by a Mol~iusstrip). This 2-dimensional manifold is also compact, but it, is nononentnble. Finally, we can glue the rectangle ABCD by turzstznq one pazr of opposite sides while matching the two otlicr sidcs untwisted, as shown in Fig, A.2.5. The resulting compact 2-dimensional manifold is n famous K1rm bottle K 2 . The Klein bottle carinot be drawn in R%itliout self-intersections. However, it is possible to undeistand it as a sphere with t u ~ odiscs removed ant1 the holes closcd up with two "cross-caps" (the Mobius strips). The Ier of small discs r e ~ i i o v ~and d a finite liulnbcr of either handles or "cross-caps" attached in order to close the holes.
Figure A.2.4: Rcnl projective plane P 2
I-Iowever, after glr~ingtogether the sidcs of this rectangle, one call construct a nirml~erof compact manifolds without a boundary. The first example is obtained when we identify the opposite sides without twisting them, as shown in Fig. A.2.3. The resulting manifold is a 2-dimensional torus T2 that is topologically equivalent to a sphere with one handle. This 2-dimensional compact manifold is orientable. Another possibility is to glue the rcctallgle ABCD together after twisting both pairs of opposite sides. This is shown in Fig. A.2.4. As a result, one obtains a real projective plane P2, which is rcprescntetl by a sphere witli a disc removed and the resulting hole is closcd up by a '"cross-cap", i.e., by identifying its
A.2.2 Vector fields Vector fields s~nootl~ly assign to each p0in.t of ( I manifold an element of the tan,gent space. Tile com.muta.tor of two vector Jielrls i s a new vector field. Let us denote by C ( X ) the algebra of differentiablc functions on X . A tc~ngent vector u a t R point 2 E X is defined as an ooperator which maps C ( X ) into R and satisfies the conditioil
+
1,
(A.2.3) V f ,g E C ( X ) . A physical motivation comes from the. not,ion of vclocity. Indced, lct us consider a smoot,h curve .c(t) such that 0 5 t 5 1 and n(0) = 2 . Tlieli the dircctiollal derivative of a function f E C ( X ) along n(t) a t n., l ~ (9f)
= f (z) ~ l ( y )
!?(.T) 1 1 ( f
A.2. Exterior calculus
62
is a linear mapping v : C(X) -+ R satisfying (A.2.3). Choose a local coordinate system {r2) (z = 1 , . . . , IL) on a coordinate neighborhood U 3 x. Then the differential operator 3, := d / 3 x Z ,for each i, satisfies (A.2.3). It can be demonstrated that the sc.t of vectors {a,}, z = 1 , .. . , n , provides a basis for the tangent space X, a t x E U . Wc will rlse Latin letters to label coordinate indices. A mapping 11 that assigns a. tangent vector u, E X, to each point x is called a vector field on the manifold X . If we consider a smooth function f ( s ) on X, then v ( f ) := ~ r (, f ) is a function on X . A vcctor field is called diffcrentiable when a. filnction ~ ( f is) differentiable for any f E C ( X ) . In local coordinates {rZ},a vector field is described IL = ua(x)8% by its components u2(x),which are smooth functions of coordinates. For every two vector fields ILand v a comm7~tator[u, v]is naturally defined by
(a) arbitrary vector v at a point P
(b) arbitrary 1-form o at a point P
Figure A.2.G: (a) Picture of a vector ("contravariant vector") at a point P. (h) Picture of a 1-forni ("covcct,or" or "covariixnt vcctor") at a point P. This is again a vcctor ficlcl. Please check that the condition (A.2.3) is satisfied. In local coordinates, ~ I C C ~ I I S Cof u = ua(x)3, and v = va(x)a,, we find the components of the commutator [11,41]= [u,vIa(x)8, as
A.2.4 Pictures of vectors and one-forms
A. 2.3 One-form fields, differential p-forms One-form fields assign to each point of a manifold an element of tlrc dvnl tangent space. Differential p-forms are then defined pointwise as the exterior products of 1-form fields. Thc dual vcctor spacc X: is callctl cotangent space a t x. The elements of X: are 1-forms w that 1Tlilp X, into R, An extenor differentzal I-form w is defined on X if a 1-for111w,. E X: is assignet1 to each point x. A natural example is provitlcd by the tliffcrential df of a function f , which is defined by (df,(l~)):= 76,(f),
V~E L X,,
are diffcrrntinblc functions of the. coordinates. where w , ~, , , ( r ) = w[,, ,,,](r) e by All(X). I11 this context The space of tliffcrential p-forms on X will l ~ tlrnotctl we may write Ao(X) = C ( X ) for tllc set of diffelcntiablc filnctiorls on X .
(A.2.7)
or in local coordinates { r 7 ) ,
Obviously, the coordinate differentials dx2dcscribc fields of 1-forms that provide a basis of X: at each point. The basrs (3,) and {dx" are dual to each other, i.c., &'(a,) = 63.Repeating pointwisc tlic constructions of Sec. A.2.2, we find that, a differential p-form w on U C X is expressible in the form
The p ~ : c t ~of~ r(I.e vector field (n,r~,nrrow) arises fiorn tlre velocity of n, poin,t ~n,ovin.gcrdong an arbitrurg cvnte. The prototljpe of n picture of a 1-form ((1 pair of ordered llljperplancs) emerges frorn tlrc diflerentic~lof n fi~n.ction,. It seems wort.liwliilo to ~)rovidcsiniplc pictures for difft?rcntia.l 1-form fic?ltlsant1 vcctor ficltls. The physical prototype? of a tangent vcct,or v is the ~ ~ e l o c ioft ? ~ a particlo nioving along n given curve. Tlierefore a vector can pictorially be reprcscnt,c:tl by a n mrolu (see Fig. A.2.G). The prototype of a 1-form w can be representzed by tlie differential (if of a function f ; its componcnt,~represent tlie gradient. Therefore, a suitable pict,urc for a, 1-form is givcn by two pnmllel 1rlyperplan.e~(scc Fig. A.2.G), that dcscribc surfaces of constant value of f . An arrowhead indicates in wllicli direction tlic value o f f is increasing. The "stronger" the 1-form is, the closer tliose two planes arc. In physics, a gcncric 1-form is tlic 1l1a.11e1-form, dcfincd as the gradient of the ph;~sc(tliinl< of a dc Broglic wave!), or tlie 7nomentl~m1-form, tlefincd as the grxtlicnt of the r~ct~ion function. To give a specific example, Ict 11scollsidcr a 3-t1in1cnsion;~llnanifoltl with local coordinixtcs (n:', x2, 2 ) .A 1lyl)erplanc is then simply a 2-dimensional plane. A local b:tsis of vectors is givcn by ei = a , , with i = 1 , 2 , 3 , wliicli arc tangent to the coort1in:ltc lines 2:" rcspcctivcly. Siniilarly, it local basis of 1-forins 79' = dz'
A.2. Exterior calculus
64
Figure A.2.7: Local coordinates ( r ' , z 2 , z 3 ) a t a point P of a 3-dimensional manifold and the basis vectors ( e l le2, e 3 ) .The basis 1-forms 19' = d z l, 2 = 1, 2,3, arc also supposed to be at P. Note that d l ( e l ) = 1, 19'(e2) = 0, d1(e3) = 0, . . . , i.c., 19"s dual to e, according to dZ(e,) = 6,". is rel>rcsentetl Ily the local planes, which depict surfaces of constant value of the relevant coordinatc r2(see Fig. A.2.7). A 1-form is clcfined in sl~clia way that, if appliccl to a vector, a number (scalar) pops out,. The pictorial representation of such a numher is straightforward (see Fig. A.2.8): A straight line starting a t P in the tlircction of the vector v dissects the second liypersurfacc a t P'. The n~imberw(v) is then the size of the vector v ~ilcasureclin terms of the segment PP' uscd as a unit.'
65
A.2.5 @Volumeforms and orientabilit,y
Figure A.2.8: Two-dimensional pictures of 1-forms w, n,p of different strengths at P. They are applied to a certain vector v a t the same point P and, in the case of p, also to the vector u.
A manifold X is orientable (for the definition, see Sec. A.2.1) if and only i f it has a global volume form. Indeed, given a volume form w which, by definition, docs not vanish for any x , the function wl,.,,(x) is either everywhere positive or everywhere negative in U,. If it is negative, we can simply replace the local coordinate a' by xl' = -zl. Then wl,2.,,,(z) > 0. Thus, without any restriction, we have positive coefficient functions wl,,,,(x) in all charts of the atlas {(U,, 4,)). In an intersection of any two charts U, Up with local coordinates 4, = { z Z )and 4 p = {y Z), we have
n
A.2.5 @Volumeforms and orient ability A n eve7y~uller.e n,on,vnn.ishing n-form on an it-dimen.sion.al m.an2fold is called a volun~,efonn.
As in thc c u e of a vector space V, differential forms of maximal rank on a manifold X are closely relatctl to volume and orientation. Any everywhere nonvanislling n-form w is callcd a volume form on X . Obviously, it is determined Ily a single co~nponentin every local coordinate chart (U,, 4,): I
I
I
I
Thus wl,.,,,(y) = wl~,,,(z)J(faB). Since both wl,,,, (z) and wl,,.,(y) are positive, we conclude that the Jacobian determinant J(f,p) is also positive for all intersectzingcharts, cf. (A.2.2). Hence the atlas {(U,, 4,)) is oriented. Conversely, let the atlas {(U,, &)) be oriented; i.e., (A.2.2) holds true. Then in each chart (U,, 4,) we have an evidently nonvanishing n-form w(") := dx l A A dz". In the intersections U , Up one finds w(,) = J ( fa@)~ ( 0 )Let . {p,) be a partition of unity subordinate to the covering {U,) of X.Then we define a global n-form w = C , p, w("). Since in the overlapping charts all nontrivial forms w(") are positive multiples of each other and p,(p) 2 0, C , p,(p) = 1 (i.e., all pa cannot vanish at any point p), we conclude that w is a volume form. a
'Pictures of vectors artd 1-forms, and of rnarty other gcomrtrical quantities, can be found in Schortten [24];see, e.g., p. 55. Also easily accessible is Misner et al. [18], where in Chapter 4 a nutnbcr of corrrsponding worked out examples and nice pictures are displayed. More recent I~ookswith bea~itiftilpict~rresof forms and their mnnipulntion include those of Burke 12) and Jancewicz [12].
n
A.2. Exterior calculus
GG
i.2.7 Exterior dcrivat,ive
67
In atlas {(U,, do)), a twisted p-form is represcntcd by a faniily of tliffercntitl >-forms{ w ( ~ )such ) that in the inter~ect~ions U,, Up,
n
Figure A.2.9: Mobius strip.
A.2.6 @Twistedforms Th.e diflerential forms that can be defined on a n.onorien,table man,zfold are called twisted differential forms. Tlre?~ arc: orien,tation-7ial1r.edin terms of the con,uen.tionnl differential forms of Scc. A.2.3.
1 I
I I
"Sincc tlic integral of a diff(~rcntia1form on RT"s not invariant under the whole group of diffconiorphisms of Rn, but only unclcr the subgroup of oricntationprcsc~rvi~ig diffcomorphislns, a diffcrclitial form cannot bc integri~t~cd over a nonoriclntablr mnnifolcl. Howcver, by modifying a tliffcrcntial for111 wc obtain so~nc>thingcalled a rlrnszt?/, which can he int,cgratcd over any manifoltl, oricntablc or ~ i o t , . " ~ Besides the co~~vcntionnl tliffcrcntial forms, olic can define slightly different objccts calld t ~ m s t e dforms 3 (or, ~omct~imrs, "odd fornis", "impair forms", "tlcnsiti~s'~, or L 1 ~ ) s e t ~ d o f o r ~Tlic i ~ ~twisted "). forms are ncccSsiuy for a11 nppropriate rrprcscntation of certain physical quantities, sticli as thc electric current tlrnsity. Moreover, they arc indispensal>lc wllc11 one consi~lersthe integration tlicory on manifoltls and, in part,icular, on nonorientablc manifolds. I11 Sec. A.1.3, scc Examplcs 4) nntl ti), a twisted form was clcfincd on a vector spacc V as a geornctric quantity. Intuitively, a twistcd forni on the manifolcl X can be dcfinetl as an "orientation-valtiecll' conventional exterior form. Given 2Bott k TI] [I], p. 79. ""l'iui.ristcd tensors were introd~rcedby I l e n n n n W e y l . . . and de R h a m . . . called t h e m tcnsors of odd kind . . . . W e coirld make out a good case tliat the usiinl diflercntiol forms are actualE~/the twisted ones, hut the language is forced o n us by history. Twisted differential for.ms are the natural representations for densities, and sometimes are actually called densities, which u ~ ~ i t lhed a n ideal n a m e were i t n o t already i n iise in tensor analysis. 1 agonized over a notation for twisted tensors, say, a different typeface. I n the end I decided against i t . . . ," Willin~nI,. Burkc [2], p. 183.
"here, as in tlic previous section, J ( f a p ) is the Jacobian of tlie transition fun(ion fao := &% 0 4;'. Example: Consitler the Mobius strip, a nonorientablc 2-dimensional compact nanifold wit,h boundary (sce Fig. A.2.9). I t can be easily rcalized by taking rl ~ectanglc( ( 8 , [) E R2)0< 0 < 27r, -1 < [ < 1) ant1 gluing it together with ore ,wist along vcrtical sides. The simplest atlas for the resulting manifold consisis )f two charts (U1, and (U2,42). Tlie open domains UlY2arc rectangles aril ,hey can be chosen as shown in Fig. A.2.9 with the evidently dcfined loc~l :oordinat,e maps dl = ( x l , x2) and 4 2 = ( y l , y2), where the first coordinaie u n s along tlie rcctanglcs ailcl the second one across them. The intcrsectim ;TI U2 is comprised of two open sets, (Ul U2)left and (Ul U2)rig~lt.Tle ,ransition funct,ions f12 = 41 o qizl are f 1 2 = {x l = yl, x2 = y2) in (UI fl U2)lflt ~ n df12 = {xl = ?,11z2= -y2) in (Ul U2)riKll+,, SO tliat J ( f l z ) = r t l in t,he!e lomains, respectively. Tlie 1-form w = {w(') = dx2,w(2) = dy2) is a twistcd orln on tlie Mobius strip. In general, given a chart (U,, $,,), both a usual a.ntl a twist,ccl ??-form is givm )y its colnponeiits wi,,,,i,,(x),see (A.2.9). With a change of coorclinates, tle :omponcnt,s of a twist,etl form, via (A.2.12), arc transformctl as
n
n
n
n
For a conventional pform, tlic first factor on t,lic right-liantl side, the sign ~f the Jacobian, is abscnt. Normally, in gravity and in fieltl theory one worlts on orientable ninnifolls with an orientcd atlas chosen. Then tlic clifferencc bet,wcen ordinary and twistrtl objects tlisappenrs bccausc of (A.2.2). However, twisted forms are vcry impcrtant 011 ~ionoricnt:tbleli~aliifoltls011 which t,lic ~isualforms ca~inotbc integrat,cI.
A.2.7 Exterior derivative The exterior derivative nrn.ps a p-fornl in.to a ( p ciu.cial propelsty is n.ilpntency, r12 = 0.
+ 1)-form. 'ts
Denotc t,lic set of vector fields on X by Xd. For 0-forms f E Ao(X), tlie ctiffercntial 1-form df is defined by (A.2.7), (A.2.8), i.e., by df = f,, dxZ.We wkh t o extend this map d : Ao(X) A1(X) to a map d : Ap(X) -+ A"+'(J). Ideally this should be performed in a coortlin;~tc-frcc way and we shall glvc such ;I. definition a t tlie cntl of this section. Howrvcr, thc drfinition of cxtcror
-
A.2. Exterior calculus
68
dcrivativc of a pform in terms of a coordinate basis is very transparent. Furt l i e r ~ ~ ~ oitr cis, a simplc matter t o prove that it is, in fact, independent of the local coordinate syste~rithat is used. Starting with the expression (A.2.9) for a w E A Y X ) , wc tlefinc rlw E A"+] ( X ) by
G9
A.2.7 Exterior dcrivative
Thus
Then d ( w ~ 4 )=
d(flt)~dx'~~...~dxJ~l = (f dl!. h rlf) A dz" A . . A dxjq = (df A dzil A . . . A r~n:~l') A (1%dxjl A . . . A dn:j")
+
By (A.2.7), (A.2.8), the riglit-hand side of (A.2.14) is (llp!) wi, ,,,,
,,,,d x j A dxil A . . . A dxip .
(A.2.15)
A
+(-1)l' (f dx"' A . . . A dsi1') A (dh A dxjl A . . . A dzi4)
Hcncc, bccausc of the antisymmetry of the exterior product, we may write
=
dw~d,+(-l)"w~dqi.
(A.2.22)
To prove 4), wc? first of a11 notc that, for a function f E AO(X), Under a coordinate transformation {xi) -+ {n:i'), it is found that sincc partial dcrivat,ivcs comninte. For a pforrn it is sufficient t o consider a monoinial Hence, Then so tliat thc exterior dcrivative, as defined by (A.2.16), is independent of the coordinate systclil clioscn.
-
Proposition 1: T h r exterior tlcrivativc, d : AYX)
,?s
tfcfiricd by (A.2.1G), is a map A"+'(x)
(A.2.19)
with the following properties:
+ A) = dw + dX 2) d(w A 4) = dw A 4 + (-1)"
1) d(w
3) df (11) = d f )
and rcpcntetl application of propt.rty 2) ;ultl (A.2.23) yields tlle drsirctl result d(dw) = 0. By linearity, this m:~ybe extcndctl t o a general pform wliicli is il linear coinbination of tcrlris liltc (A.2.24). Proposztion 2: h v n ~ i a i z tes:pression for tire exterior derivr~tive. For w E A"(X), we can express c(w in a coordinatc-free nlanller as follows:
[linearity], A db
[(anti)Leibniz rule], [partial derivative for f~inctions],
4) d(dw) = 0
[nilpotency].
Here, w, X E Al'(X),4 E A"(X), f E A O ( X ) , uE XA(X). Proof. 1) and 3) are obvious from the definition. Because of 1) and the distributive propcrty of the exterior multiplication, it is sufficient t o prove 2) for w and 4 of the 'monomial' form:
wherc ?LO, u l , . . . ,lip arc arl)it,rnry vcctor ficltls ant1 ii indicates tlint t h r field 11 is omztted as an i~rgunient.It is a straightforwartl m;itter t o verify tliat (A.2.26) is consistent with (A.2.16). Wc shall nlnlte particular use of the casc i11 wliich w is a 1-form nlld (A.2.9) becorncs
A.2. Exterior calculus
70
A.2.8 Frame and coframe A natural frame and natural cofram,e are defined at every local coorrlinate patch by aian.d d z i , respectively. A n arbitranj frame e , and coframe 6" are con.structed by a 1in.ear transformation th.erefrom. T h e object of an,ll.olonom,ity measures how m u c h a cofra,m.e differs from a natural one. A local frame on an n-tlimcnsional differentiable n~anifolclX is a set e,, a = 0 , 1 , . . . , I ? , of n vcctor fields that arc l ? n e a r l ~i n d ~ p e n d ~ an tt each point of an open subset U of X. They thus for111 a basis of tlic tangent (vector) space X, at every point .I' E U . There exist quite ordinary manifolds, the 2-dimensional splicrr for cxamplc, where no continuous frame ficld can be introduced globally, i.r., at racll point of tlle manifold X . Therefore, speaking of frames on X , we will always have in mind local frames. If P , is a framc, then the corresponding coframe is the set 6" of n different 1-forms such that
is valid a t each point of X . In other words 1 9 ~ 1 , a t each point x E X is the dual basis of 1-forms for X:. We note in particular t l ~ a t as , a consequence of (A.2.28), every vcctor ficld u E X,' can be dccomposcd according to
A local coordinate system defines a coortiinate frame aion the open neighborhood U . Tllus an arbitrary framc e , may be expressed on U in terms of 8, in tl1c for111 of
wllcrc e",, arc tliffercntiable functions of the coordinates. For the corresponding coframc 6" wc have
e t a e2p= 6; If
iL
~ofrill~lc do
l i i the ~
p r ~ p ~that r t ~
it is said to be natural or holonornic. In tliis case, in the ncighborliood of each point, thcrc exists a coordinate systcln {.r2) such that
..2.9@Mapsof manifolds: push-forward and pull-back
71
lnder these circumstances, the frame e , is also natural or holonomic with = 6: 8,.The 2-form
--
rith C(p,)" 0, is the object of anholonomity with its 24 independent compoents. It measures how much a given coframe 29" fails to be holonomic. There ; also a version of (A.2.35) in terms of the frame e,. With the help of (A.2.27), ; can be rewritten as
'he object of anholonomity has a nontensorial transformation behavior. Example: On a 2-dimensional manifold with local coordinates {x, y ) , the 1)rms { f i i = x d y , 79% = y dx) are linearly independent. Such a coframe is anolonomic with d ~ = 9 d~ z A d y and d d 2 = - dz A d y , i.e., Ci = - C' = d z A d y .
1.2.9
@Mapsof manifolds: push-forward and pull- back Pull-back cp* and push-forward cp, maps are the companions of every diiffeomorphism cp of the manzfold X . T h e y relate th,e corresponding cotangent and tangent spaces at points x and cp(x). T h e m a p cp* com.mutes with the exterior diflerential.
If a differcnt,iable map cp : X -+ Y is given, various geometric objects can be transported either from X to Y (pushed forward) or from Y to X (pulled back). A push-forward is denoted by cp, and a pull-back by cp*. Given a tangent vector u at a point x E X, we can define its push-forward cp,u E Y,(,) (which is also called the diflerential) by determining its action on a function f E C(Y) as
However, if is not merely a tangent vector, but a vector field over X, it is in general not possible to define its push-forward to Y. There might be two reasons for that. Firstly, if cp is not injective and cp(x1) = cp(z2) for X I # x2, then the vectors pushed from X,, and X,, are different in general. Secondly, if cp is not surjective, the pushed forward vector field is not, in general, be determined all over Y. It is always possible to define cp,v of a vector field if cp is a diffeomorphism (which can only be considered when d i m X = dimY). Using the rule
A . 2 . Exterior calculus
72
wc can tlefinc the push-forward of an arbitrary contravariant tcnsor a t z E X t o the space of tensors of the same typc a t cp(z) E Y. So cp, bccomes a l~omornorphismof the algebras of contraviiriant tensors a t z E X ant1 cp(x) E Y. In a diagram we can depict the push-forward map p, of tangent vectors 11 a i ~ dtlie pull-back map p* for 1-forms w:
.2.10 @Liederivative
73
If cp is a difleomorphism, or a t least a loct~ldiffeornorphism, we shall use the pull-back cp* of arbitrary tensor fields. For contravariant tcnsors, it can be defined as p* = ( e l * = (p*)-* .
(.!.2.43)
b define it for an arbitrary tensor of type [jj], we liavc to rccluirc only that * is an dgcbra isomorphism. Technically, in local coordinates, this alnounts ) thc invc.itil)ility of the square matrices 6'?j1/6'r2, Wheri p is a (local) tliffcon~orpl~ism, wc can also pull-baclc (or pnsli-fcrwwd) ?omctlic quantit,ics constructed on tlie tangent space. Let [(w, e)] he a gcoletric quantity; llcrc e = ( e l , .. . , e,,) is a frame in the tangcnt space Yy and bclo~igsto the set W in which there is the left action p of GL(n, R). As for vectors, wc clcfinc p*r: = ( p * e l , . .. , p*e,) and I
Let { r Z )be local coordinates in X and {yJ) local coordinates in Y (with thc rnngcs of indices i and J dcfinetl by the dimellsiollality of X and Y , respectively). Thcn thc lnap p is clescribed by a set of smooth functions y3(zZ),and the pushforward map for tcnsors of type ]:[ in components reads
Comparing with (A.2.8) for tlic case when Y = R, it becomes clcar why cp, is also cnllcd a differential map. i c pull-back p'w E A?(X) For a p-form LJ E A;=(P(z)(Y), we can d c t c r ~ n i ~its
I)Y This definition can be straiglitforwardly cxtcntlcd t o a liomomorpllism of tlic algebras of covariant t,ype tcnsors. In local coordinates it reads, analogously to (A.2.39),
I:[
Let w be a11 exterior p-form (i.c., a, pform field) on Y . I11 order t o determine its pull-l~aclcp*w t,o X by (A.2.40), it is sufficient to liavc cp,ul,. . . , p,v, on t,hc right hand sidc of (A.2.40) tiefincd as vectors (i.e., not lieressarily as vcctor fields). Therefore, thc pull-back of cxtcrior forms (and, in general, of covariant tcvsors) is determined for an arbitrary map cp. In cxtcrior calcl~lus,an import,allt property is the corlllnutativity of pull-back and exterior differentiation for any p-form LJ:
i t . , tlic transportcd object has the same components n s the initial objctt with respect t o t,lic transported frame. Certainly, this definition of thc pull-3ack is consistent with that given earlicr for tensors.
A.2.10 @Liederivativc A t~ectorfield generutes a group of diffeornorphisms on z mnnifold. Making use of this group action,, the Lie derivative enables us to com.pare tensors and geometric q.uan,tities at differen.t poin.ts. The main rcsr~ltof the present section will I)(? cquatio~i(A.2.51), thc Lie rlerivntive of a tliffcront,ial for~n.Howovcs, wc slinll first explain tllc conccpt of tho Lit: derivativc of n geiicral gcomctrict~lqr~ant~ity. Note that for Lic tleriva:ivc n,o metric un.d no connection. is required; it can bc tlcfii~cd011 eacl~diffcrcut,iahlc manifold. For cach point 21 E X , a vccf,os field .u witli ~ ( p #) 0 dcternlincs a rlriique curve o,,(t), t 2 0, srlcli that a,(()) = p with IL as tllc tangcnt vector field t o thc curvt:. Tlic family of curvcs dcfincd in this way is called thc congmlen.ce of curvcs generat,ctl by the vcctor field rr. Lct {x i ) be a local coortliliate system with xi,as thc coort1in:itcs of p ant1 decompose 71 according t o 11 = ui(n:', . . . , xTL) (3,.Then the clrrvc o,)(t) is found by solving tlic systcnl of ordinary differential eql~at,iolls
witli init,ial values xl(0) = xi,.Tlic congrllence of curvcs obtained in this way tlcfines (at least locally) a 1-parameter group of diffcomorpliisms cp, on X given by
with thP propcrtics that (a) cp,' = cp-,, (11) cp, o cp, = cp,+,, and (c) cpo is thc idcntity mal). The integral curves of the congruence arc cnlletl tlie frajectones
A.2. Exterior calculus
'igurc A.2.11: Tlie clcfinition of thc Lie derivat,ive L,,v with rcspcct to a vcct,or A: The 1-paramctcr group pt, gcncratcd by the vector field u, is 11scd in ordcr ,o transfer thc vector v(pt(p)) back to the initial point and to compare it with 4~).
Figure A.2.10: Translations ( n ) and circular motion (b) gcnerated on R2. of thc group. F~lrthcrmorc,the equations (A.2.45) are equivalent to
for all p E X alitl all diffcrclltiable functiolls f . Exam.ples in R 2 :
+
1) Thc vector field ?L = 3 / d x generates translations cp,(x, IJ) = (x t,y), -oo < t < 4-oo.Thc tr;~j~ctories arc the lints y = const. See Fig. A.2.10a. 2) The vector ficld IL = (n.a/a?j - y d l a x ) gcneratcs thc circular motion cpt (x, 11) = ( 2 %cos t - y sin t , x sill t y cos t), 0 5 t < 27~.The trajectories are concc~ntriccloscd curvcs around thc origin, see Fig. A.2.10b.
~t is suficirnt to have cxplicit cxprcssions for tllc Lie derivatives of f ~ n c t ~ i o n s , vectors, n~ltl1-forms in order be in a position to do tjhc sallie for a gcnrrnl tensor. Thc two most important c:tsc.s nro ns follows. For vectors 1, E Xd,
+
In g c ~ i ~ r aifl ,we take a coordinate patch U of a differentiable manifold with coordinates { x l ) , tlicll cp, is defined in terms of s q y
where f t ( t ; x1) are diffcrentiablc functions of ( t , x J ) . Property (a) states t,hat r2= x 7 ( t ;y 7 ) = f 2 (- t; yJ). By p r o p ~ r t y(h) we have f 2 (t; f3(s;x k )) = f 2 ( t s;.xJ), while (c) lncans that f "0; r J )= r 2 . For every vnluc of t in a ccrtain interval, tlie diffeomorphism cpt induces corrcsponding pull-backs cp? 011 functions, vcctors, cxterior forms, ancl general tensor firltls of type Accordingly, the Lze denvatzve of a tensor T with respect to a vector ficld IL is defined by
+
[;I.
sec (A.2.6) for a component vcrsion. For p-forlns w E AYX) and p thc ?nuin theorem for t,lic. Lic tlcrivntivc of an cxtcrior form:
1 f ,,w
=
11 J
(dw)
2 0, wc find
+ (I(u J w ) .I
An altcrntitivc coordinnte-free general formllla for this Lie dcrivttt,ive reads: ( w ) ( ,. . )
=L
(
w
(
(, u p-
u
, , [u,vi],. . . , 4 ) (A.2.52)
7=1 Tllc Lie tlerivativc for tlie fullctiolls f E C ( X ) is obtained as a of (A.2.51) for p = 0: Lllf =
1~(f= ) 2LJ
(If.
articular case (A.2.53)
A.2. Exterior calculus
7G
nd we casily find for u = t i ( x ) di,
The last, formula is straightforwardly chccl
=
lim
f (vt(P)) - f(p)
= IL(f)(p),
(A.2.54)
t
t-to
77
-2.10 @Liederivative
by use of (A.2.47). Tlic proof of (A.2.50) and (A.2.51) is left as an exercise to thc readers. As a hint, we mcntion that the formula (A.2.51) follows from the Lie derivative of a 1-form w,
For conlpleteness, let us consider the Lie derivative of a geometric quantity. br this purpose we note the following: If e,(x) is a frame taken a t a given oint p E X , then cpt (e,(cpt(x))) can be decomposed with respect to this framc rith some t-dependent coefficients:
)ifferentiating this equation with respect t o t at t = 0, we get where IL,v are arbitrary vcctor fields. The most important propertics of the Lie deiivati71es of exterior forms may be sum~narizedas follows: 1) L,,dw
=
[L,, and d commute],
dL,,w
nd thus formally we find the matrix
[Leibniz rule], [rescaled vector], [noncommutativity],
4) LvLllw - L,,LT1w= f[,,,v]w
5) L,,(ILJw)
I
-
UJ
L,,w =
[V,IL]J
[L, ,
w
IL J
do not commute].
Let us consider a field of a geometric quantity [(w(x),e(x))] of type p, for hort, 7u(x). According to the definition of the pull-back of such objects, we .ave
The formulas above contain all the necessary information about the Lie tlcrivativc for arbitrary tcnsors of type In particular, by construction we have tliat L?, is type prcscrving, i.c., if T E T:(X) then (L,T) E T:(X). Moreover, it is clear that for any two tensor ficlds T and S of the same typc,
.I:[
I
I
I
IVe diffcrcntiate a t t
= 0 and find:
=
I'
Finally, for T E T l ( X ) , S E T r ( X ) ,
I
The proof follows straightforwardly from
Thcsc propertics cnablc us to express thc Lie derivative of a general tensor in t,er~ns of a local coordinate bmis. Consitlcr a tensor field of type for example. In terms of a local coordinatc system {xZ),
[:I,
1
(UW)
+
(z) p+$(x) W ( X ) .
(A.2.65)
n all practical applications, a geometric quantity is described as a smooth field In X that takes values in the vector space W = R N of a prepresentation of he group GL(n, R) of local linear frame transformations. In simple terms, the :eometric quantity w = w A en is represented by its components w A with respect o the basis en of the vcctor space W = R N . Hereafter A, B, . = 1,. . . , N. rhus, recalling (A.1.16), p(L) = pAB(L)E GL(N, R), and p, maps the tangent pace Btd(n,,R) of the group GL(n, R) into the tangent space &d(N, R) of the youp GL(N, R) by means of the matrix
.2.11 Excalc, a Reduco packngc
79
Tlicrcforc, talting into accoullt (A.1.17), Eq.(A.2.65) reads for the components of the gcolnctric quantity:
where 41,,0 is defined by (A.2.64). For a tensor field T o p of typc
[:I,
for instance, we have
As an interesting exercise, we propose to the reader to calculate tlic Lie tlerivative of a scalar density S of weight w, scc (A.1.56):
It is a simple matter to generalize tlic relation for t,hc Lie derivative of a p-form of typc p. If w is sucll a form, then
For a vcctor-valr~cdp- form, for cxample, wt. have Figurc A.2.12: Tlic perennial computcr algebra problem.
A.2.11 Excalc, a Reduce package 111 Chap. A.2, after specifying an n-dimensional manifold, we introduced succcssivcly ficltls of l-forms, p-forms, and vectors. Their dcclarntions by n~eallsof pform ant1 t v e c t o r arc already ltnown to us. Thrn the cxtcrior derivative was specified. In Excalc, not surprizingly, the lcttcr d is reserved for this opc3rator. Partial differentiation is tlcnotcd by thr operator 0. Thus, @ ( s i n x , x ) ; yicltls c o s ( x ) . We collect tlic? different Excalc operators in Table A.2.1. Math.
Excalc
A
.
J
-1
(3 d d:
@
*
d
I_ #
Operator
Operator Typc
exterior protluct interior product partial derivative exterior dcrivativc Lie derivative
nary infix l~inz~ry infix nary prefix unary prefix binary infix ~uiaryprefix
Hodgc star operator
Tahlc A.2.1: Translation of mathemnt,ical symbols into Excalc. The Horlgc st,nr opcrator will not. bc tlcfinetl before Scc. '2.2.8. Unary means tliat tlierc is one, binary tliat tlierc arc two, tint1 "nary" that thcrc is any number of arguments.
Let us load again Excalc by l o a d j a c k a g e excalc$. By means of a declaration with fdomain, a11 itlcntificr can l)c dcclarcd to be a function of ccrtnin variaI>les. With f domain f =f ( X, y) , h=h ( x ) ; Q(x*f,x); Q(h,y);
0 1
i.e., f +.rD, f ant1 0. T h r partial tlcrivativc symbol can also be 8111 operator with a single arguiiient, :LS in @(z). T~IPIIit rrprescnts the leg 8, of a natural frame. Coming back t,o the cxtcrior derivative, t,lle following cxan~plcis now sclfcxplanatory: pform x=O,y=O,z=O,f=O; f domain f =f ( x , y) ; d f;
A.2. Exterior calculus
80
i.e., clf eva111;~tcst,o (3,j)dn: o l ~ t ix., ,
+ (a,f)dy. Products are normally differentiated +
pf orm x=0, y=p,z=q; d(x*yAz) ;
i.e., ~ ( U JZ ) U J dr. The operator of the Lie derivatives fulfills the rules displayed after (A.2.55). We will check the rule for the rescalcd vector as an example. Already above, the form w has been declared to be a p-form, f to be a scalar, and u to be a vector. Hence we call type in directly
In an ordinary formula, we have d(xyAz) = ( - l ) ~ x y ~ d z + z c l y ~ ~ + d ~ ~ y ~ ~ . This expansion can be suppressed by tlie command noxpnd d ; . Expansion is xpnd d ; is executed. performed again when the coin~na~lcl i.e., The Excalc operator d knows all the rules for the exterior derivative as speciI fied in Proposition 1 in the context of (A.2.19). Let us declare the corresponding ranks of the forms in order to check the first two rules (note that lambda is a I reserved identifier in Reduce ant1 cannot be used):
Then we give the commands I
d(omega+lam) ;
d(omegaaphi) ;
i.c., d(w 4-A) ant1 rl(w A $), a ~ ~find, t l respectively, I
d lam + d omega
(
-
P 1) *omegaad phi + d omega-phi
+
i.e., dw + dX nnd ( - 1 ) " ~A d$ dw A 4. The sccond to last entry in our table is the Lie derivative L . In Excalc, it can be applirrl to an rxtcrior form with respect to a vector or to a vector again with rcspcct to a vector. It is represented by the infix operator I - (vertical bar followed ~y an nnderline). If the Lie derivative is applied to a form, Excalc remembers tl~emain thcorcm of Lie tlerivativcs, namely (A.2.51). Thus, pform z=k; tvector u;
81
,.2.11 Excalc, a Reduce package
ul-z;
I
Lf,, w, and d(u
-I
find omega)*f + u
-I
d omega*f + d f ^u
-I
omega
i.e., d ( u w) ~ f + ( U J clw)f +df A U J W.The rule is verified, but Excalc substituted (A.2.51) immediately. Anyway, we also see that 1- does exactly what we expect from it. In Sec. A.2.8, we introduced the frame e , and the coframe 19" as bases of the tangent and the cotangent space, respectively. In Excalc we use the symbols e(- a) and o(a), respectively. In Excalc a coframe can only be specified protl~drd a metric is given at the same time. This feature of Excalc is not ideal for our purposes. Nevertheless, even if we introduce the metric only in Part C, we have to use it in the Excalc program already here in order to make the programs of Part B executable. As we saw already in Scc. A.1.12, wc can introduce Excalc to the tlimcnsion of a manifold via spacedim 4 ; . This can also be done with the cof rame statementl, since we specify thereby the uildcrlying four 1-forms of the coframc and, if the coframc is orthonormal, the signature of the metric. For a. Minlcowslci spacctimc' with time coordinate t and s~hcricalspatial coordinates r, 0, cp, we state o(t) o(r) theta) = r * = r * sin(theta) o(phi) with signature (1,-1,-1,-1); frame e; coframe
*
d d d d
t, r, theta, phi
With frame e ; , we assigned the identifier e to tlie name of the frame. 111ortlinary mathe~naticallanguage, the coframe statement would read dt
= dt, d r = d r , dO=rdO, d4=~-sin0cl4, g = dt@~9t-dr@dr-de@d0-d1b@d4.
(A.2.72)
,.2.12 @Closed and exact forms, de Rham cohornology groups
A.2. Exterior calculus
82
Of colirss, the frame e ( - a ) ant1 the coframe o ( a ) are inverse to each 0the1., i.e., the conln~nntle ( - a ) -lo(b) ; will yisltl the Kroneckcr drlta (if you switch o n n e r o ; thcn only the c o m p o n ~ i ~which ts nonvanishing values will be printed out). The cofranlc statenlent is very fi~lldamclltalfor Excalc. All quantities will be evalr~ntctlwith resprct to this coframe. This yicltls the anliolonomic (or physical) components of an object. T h s coframe statement of a corrrsponding spherically symmetric Riclnallniall metric with unknown function $(r) reads:
A.2.12 @Closedand exact forms, de Rham cohomology groups Closed f0rm.s are n.ot exact in gen,eral. Two closed f0rm.s belonsgto the same col~omologyclass when they differ by an exact form. Groups of cohomologies are topological invarian,ts. L
1 I
I
load- package e x c a l c $ pform psi=O$ fdomain p s i = p s i ( r ) $ coframe o ( t ) = psi * o(r) = (l/psi) * theta) = r * o(phi) = r * sin(theta) with signature (1,- 1,- 1,- I)$
*
t, r, theta, phi
1 I
r
p = 1 , . . . , n,
(A.2.74)
is also a (real) vector space, ant1 evidently BYX) c Zp(X) (each exact form is closed, since dd r 0). One puts B O(X) = (4. Obviously the rxterior derivative defines an cquivalcncc relation in the space of closed forms: two fornls w, w' E Zl'(X) are said to be cohomologically cquivalent if t,licy differ by an cxact form, i.r., (w - w') E Bp(X). The quotient. space
consists of cohomolog?jclasses of p-forms. Each Hl'(X; IW) is a vector space and, moreover, a n Abclinn group with an evident group action. Tlie spaces H"(X; R) are namcrl as dc Rllam cohomology groups. Unlilte the AYX) which :ire infinite dilnc~~sional functional spaces, the dc Rhnm groups, for compact mnriifolds X , are finite dimensional. Tlie dinlension
d r =
forms a (real) vector s ~ ~ b s p a of c e AP(X). A I)-form w is call~clexact if a (p - 1)form p exists sucli that w = dp. The spacc of all cxact p-forms Bl'(X) := {w E A1'(X)Jw= dp} ,
If a commantl is tcrn~inatcdby a tlollar sign $, the output will be suppressed. Consecluently, if we input the program seglnent into Reduce, only the coframc 49" will be tlisplnycd:
0
7L
let us consider the ext,crior algcbra A*(X) = @ Ap(X) together with the p=o exterior derivative defined in (A.2.19). A pforln w is called closed if dw = 0. The space of all closed p-forms
! d d d d
% d i s p l a y s t h e coframe o ( a ) , i n p u t c a n be checked
displayframe; frame e $
83
----psi
theta
o
is called the p-th Bett~,number of the lnanifold X. Locall?/, an exterior derivative does not yield a difference between closed and xact forms. This fact is usually formulated as the Poincnre' lemma: Locally, in given chart (U,q5) of XI every closed p f o r n ~w with dw = 0 is cxact, i.e., a n - 1)-form cp exists in U C X stlcl~that w = dp. Let us illustrate this by an rxplicit cxamplc. Suppose we have a closet! one-form w. In local coordinates,
= d theta*^
phi o
= d phi*sin(theta)*r
Pcrliaps wc should remind ourselves that $' = 1- 2 m l r rcprescnts tlle Schwarzschild solution of general relativity.
*
12.2. Exterior calculus
84
Then this form is locally exact, w = dq, where the 0-form cp is given explicitly 4) by in the chart (U, q(z) =
I
wi(tx) x i dt.
0
Indeed, let us check directly by differentiation: dq
=
]
.
.
dt [(ajwi(tx))t x' dx2
wi (x) dxi = W.
and exact forms, de Rham cohomology groups
85
With the help of this map onc can prove a fundamental fact: If X and Y are homotopically equivalent manifolds, their de Rham cohomology groups are isomorphic. As a consequence, their Betti numbers are equal, bP(X) = bP(Y). Homotopical equivalence essentially means that the manifolds X and Y can be "continuously deformed into one another". An n-dimensional Euclidean spacc IEn is homotopically equivalent to an n-dimensional disk Dn = { ( x ,. , x n) E l E n l J m 5 I}, for example, and both are homotopically equivalent to a point. Another example: A Euclidean plane lE2 with one point (say, the origin) removed is l~omotopicallyequivalent to a circle S1.More rigorously, manifolds X and Y are homotopically equivalent, if there are two differcntiablc maps f l : X 4 Y and f 2 : Y 4 X such that f 2 o f l : X 4 X and f l o f 2 : Y Y are homotopic to identity maps idx and idy, respectively. Two maps are homotopic if they can be related by a smooth family of maps. Thc alternating sum
+ wi (tx) dxi]
0
=
A.2.12'Closed
(A.2.79)
We used (A.2.77) when moving from the first line to the second one. The explicit construction (A.2.78) is certainly not unique but it is sufficient for demonstrating how the proof works. One can easily generalize (A.2.78) for the casc when w is a pform, p > 1,
is a topological invariant called the Euler characteristic of a manifold X. In two dimensions, every orientable closed (compact without a boundary) manifold is diffeomorphic t o a sphere with a finite number of handles, Mz := S2 " h handles", where 11 = 0 , 1 , 2 , .. . (for h = 1, we find a torus Mf = 'If2from Fig. A.2.3). Euler characteristics of these manifolds is X ( M i )= 2 - 2h. Analogously, for the nonorientable 2-dimensional manifolds N: := S2 "k cross-caps" (Figs. A.2.4, A.2.5 show Nf = P2 and Ni = K 2 , respectively), the Euler characteristic is equal X(N:) = 2 - k.
+
where the vector field u is locally defined by u = x'a,. Its integral lines evidently form a "star-like" structure with the centcr a t the origin of the local coordinatc system. Globally, i.c., on the whole manifold X, however, not every closed form is exact: One usually states that topological obstructions exist. The importance of the de Rham groups is directly related to the fact that they present an example of topological invariants of a smooth manifold. Of course, the Betti numbers then also encode information about, the topology of X . The zeroth number bO(X),for instance, simply counts the connected components of any manifold X. This follows from the fact that 0-forms are just functions of X, and hence, a closed form cp with d q = 0 is a constant on every connected component. Since there are no exact 0-forms, B O ( X ) = 9), the clemcnts of the group H O ( X ; R) are thus N-tuples of constants, with N equal to thc number of connected components. Hence b O (X) = dim H O ( X ; R) = N. Moreover, recall that in Sec. A.2.9 for any differentiable map f : X -+ Y we have described a pull-back map of exterior forms on a manifold Y to the forms on X. Since the pull-back commutes with the extcrior derivative, see (A.2.42), wc immediately find that any such map determines a map between the relevant cohomology groups:
+
Integration on a manifold
[n this chapter we will describc thc integration of exterior forms on a nia.nifold. rhe calculus of differential forms providcs us with a powerful tcchniql~c.This oc:urs because one theorem, known as the Stokes or the Stokes-Poincarb throrem, "eplaccs a ntimbcr of diffcrent theorems known from 3-dimensional vector cal:ulus. Both types of pforms, ordinary and twistcd ones, can be integrated over +dimensional submanifolds, and in botl1 cases onc necds an additional struc;ure, thc oricntation, in order to define them. For ordinary forms onc ~ieedsthe inner and for twisted forms the ovter oricntation. Tllcre are two cxceptio~is:To ntegrate an ordinary 0-form or a twistcd n-form, no oricntation is necessary.
A.3.1 Integration of 0-forms and orientability of a manifold Tlie integral of a 0-form f over a 0-dim.ensiono.1 subrnnn~ifoln! (set of points i n X) is j ~ ~ as sum, t of vn11l.e~o f f at tlrese poin,ts. Let f be a function on X , i.e., f E AO(X),and let Z be a finite collcctio~iof points, Z = ( p l y . .. , p k ) . We can then define the integral of f over Z by
If f is, instead of being an ordinary function, a twisted function, then this definition is not satisfactory. Then thc f (pi)'s change their signs together with
4.3.3 @Integrationof pforrns with 0 < p
tlie change of tlic orieut.ntion of tlic rcfcrcnco frames a t each point p,. If we fix one of the orientations at,, say, tlie point p l , then we call try to propagate this oricntatioil by ~ont~inuity to a11 other points pz, . . . ,pk. If tliis call be don(. unainl~iguously,then we say that tlie manifold is orientable, ant1 we have just chosen an oricntation for X . I11 such a case, the values of the function f , i.e., f (PI), . . . , f (pk), can be t,akcn with respect to any frame wit,h positive orientation, and f o r n ~ ~ l (A.3.1) la defines unambiguously the integral f f of a twisted
85
'A.3.3). One has thus to fin. on omentatfon in X in order t o have a definitt lotion of the intcgrnl w. X
If Supp(w) is not contained in tlic domain of a single coordinate chart, thc ;ituation is inore complicated. Tlicn, for any atlas {(U,, d , , ) ) , one has t o use : sartition of unity {p,) subordinate to tlic covering {U,) of X . In particular ;ince C , p, = 1, we can represent the n-form as a suin w = C, w,, wliert >achW, := wpn vanishes outsidc U,, i.e., Supp(w,) c U,. For every w, we car ,lien construct the integral via (A.3.3), and finally, an integral for an arbitrarj %-formis defined by
A.3.2 Integration of n-forms The integral of an n.-fom luill be defined over an orientable n-dirnen.siona1 manifold. In the case of a tu~istedn-form tlrc 07ien.tation is not needed for tll.at plrrposc. A support for a pform cp on X is defined as a set S~ipp(cp):= {p E Xlv(p) f 0). Let w he an n-forni on X , i.c.., w E A7'(X).At first, let 11s consider the case wheil its support Supp(w) C U is containcd in one coordinate chart (U, qi), & = {xZ). Tlien ill U c X we have
where f,,(r", = p, wl ,,(z;). The intcgral of an ordinary n-form can be definec unambiguously only in the case of a11 orientable manifold. Moreover, one car prove that it is ~ i n ~ q ~ defined ~ e l y over X if the orientation is prescribed. I1 particular, tliis tlcfinition is invariant under the change of an oriented nt,la~ {(U,, 4,)) and/or the partitioil of unity {p,). The situation is quite different if, instead of an ordinary n-form, we considel a t ~ u ~ s t 11-form. rd If (A.3.2) holds for a twisted form w in one coordinate system then in ;~notlicrone we have
wlicre we denoted the only co~nponentof the n-form as f ( r ) = wl,,,,,(n:), cf. (A.2.10). Wc can try t,o define with
wlicre 4(U) is tlic iiilage of U in R" for the coordiriate map 4, and on the righthand sitlc of (A.3.3) we have a usual Ricmann intrgral. The "definition" above is, howevcr, nnibiguous if one clianges the coorclinatcs in (A.3.2). Without touching U, one can consider, for cxamplc, an arbitrary diffeoii~orphisnlA : R7' + RfL that will introd~icea new local coordinate map 4' = Aod, wit,l~coordinates {?j7) in U. Untlcr tlie change of variables z' = rZ(y7),the riglit-liand sidc of (A.3.3) transforlns into
Here the ;\tltlitional sign factor of (A.2.12) lras been takrii into accoullt. W6 don't nccci any orientation in order to fix uniquely the ineaning of (A.3.3). I' many U,,'s cover Sripp(w), then we &ill iiectl tlie partition of unity, but the consistency in tlir intersections U, n Uo is automatically provided by (A.3.7) Sumniing up: Any tu~zstedn-form w with compact support call be integr:ttct ovcr an la-tlimensional manifold, regardless of whether the latter is orientable or not.
A.3.3 "Integration of p-forms with 0 < p < n where J(A-') = dct drL/3!jJ is the Jacobian determinant of the variable change, cf. (A.2.2). But from (A.3.2), we have w{ ,,(?j) = f ( y ) J ( A - I ) . Thus the traiisforinccl lcft-liancl sitlc of (A.3.3) is equal h of the right-hand sidc, depending on thc sign of t l ~ cJacobian tlcterininant. A diffeomorpliisn~A that changes the orientation of the coordinatc system leads t o a change of sign of tlic integral
Thr: integml of n p-form is defined for a sivlgular p-simplez. In ortlcr t o tlcfinc ;in integral of a p-form with 0 < p < n soriie prelirnillary constr~rctionsare needed that introduce suitable p-dimensional dolllains of X ovcr wliicli onc can int~grat~e.
90
A.3.
Integration on a manifold
A.3.3
@Integrationof pforms with 0 < p < n
91
+
Figure A.3.2: A 1-chain el = - u ( ~ ) u ( ~ )With . U(O) = (PI, P2), u ( ~ = ) (Po,P2), u p ) = (Po, P I ) , the resulting chain is a boundamj of a 2-simplex: cl = B(P0, PI,P2). Arrows show the ordering of the vertices.
Figure A.3.1: Simplices uo, a', u2, and u3. It is callcd the boundary of up. More explicitly, n
+
As a first step, one considers psimplices in R . Take a set of p 1 ordered points Po, PI,.. . , P, E Rn that are independent in the sense that the p vectors (P, - Po) with i = 1 , . . . , p are linearly independent (recall that R n is a linear vector space). One calls a p-dimensional simplex, or simply a p-simplex, the closed convex hull spanned by this set of points: ul' := (Po,P I , .. . , P,).
(A.3.8)
Geometrically, it is represented by
The boundary of the 2-simplex (Po, PI,Pz), for example, reads
(see Fig. A.3.2). The boundary of a boundary is zero: for any psimplex we have BB(P0, PI,.. . , P,) = 0.
(A.3.13)
Let us check this for the 2-simplex (Po,PI,Pz).From (A.3.12) and the definition (A.3.11), we find: 1,
where t o ,. . . , tP are real numbers. Accordingly, every 0-simplex is simply one point (Po);a 1-simplex is a directed line segment (Po, P I ) ; a 2-simplex is a closed triangle with ordered verticcs (Po,P I , P2); a 3-simplex is a closed tetrahedron (Po, PI,P2,$3), and so on (see Fig. A.3.1). Each psimplex u p 11n3 a natural (p - 1)-dimensional boundary that is composed of faces. An i-th face ort:' with 0 i p of a simplex (PO,P I , . . . , PI,)
> >
is defined a5 a (p - 1)-simplex u ~ := ~ (Po,. ~ '. . , P,,. . . , P,) obtained from (PO,P I , . . . , P,) by removing the vertex P, (as usual, the hat denotes that an element is omitted from the list). Defined in this way, a face lies oppositc to the vertex P,. For any psimplex a" (PO,P I , .. . , P,), with the help of the faces, one can define the formal sum of (p - 1)-simplices by
From simplices one is able to construct chains. An arbitrary p-chain is a formal sum
whore the a, are real coefficients and the ori) psimplices. The boundary of a pchain is a (p - 1)-chain defined by Be, =
C ai &Ti).
(A.3.16)
In Fig. A.3.2, we demonstrate the construction of a chain from simplices. Ih this particular case the chain turns out to be a boundary of a 2-simplex.
92
A.3. Integration on a manifold
Figure A.3.3: Singular simplices on a smooth manifold X. We are now in a position to define the integral of a pform on the manifold X . For this we need to extend the concept of a p-simplex from Rn to the manifold X . A suitable domain of integration is given by singular simplex in X which is defined as follows. Given a psimplex a" R y a singular p-simplex in the rriailifolcl X is defined as a differentiable map s : op -+ X . Every point p E X can evidently be treated as singular 0-simplex, and any smooth curve on X is just a singular 1-simplex. For a 2-simplex see, for example, Fig. A.3.3. Now consider a pform w on tlie manifold X . Given a singular psimplex s : a" + X, the pull-back s* maps w to IWP and we define the integral of the form over the singular simplex by
where t' = ( t l , . . . , tl') arc the standard coordinates in Cr' and f ( t l , .. . , t") := ( s * ~ ,(t2) ) ~ is the single component of the form s*w on a" CP. The righthand side of (A.3.17) is understood in the usual spnse of a Riemann integral. As in the case of the integral for n-forms, the questions related to the orientation should be carefully studied separately first for ordinary and then for twisted pforms. Let w now be an ordinary p-form on X. No orientation should be specified for X in the definition above. Instead, the preferred (and standard) orientation in RP is used. This orientation, by means of the map s , is transported t o s ( a ) . In otlicr words, the push-forward s, maps the standard frame (&,. . . , to the frame (s,&, . . . , s,&) tangent to s(a) c X. This frame determines an inner orientation all over s ( o ) . The value of the integral (A.3.17) is not changed if we change s and U P in such a way (keeping w untouched) that the orientation induced on s(a) is not changed. Such changes of s to s' = (s o A) can be inducccl by diffeomorphisms A : RP + IWP that have positive determinant J(A). The value of the integral (A.3.17) does not depend on s; it depends, however, on the choice of inner orientation of the simplex a". Therefore it must be assumed that UP is inner oricntable.
6)
A.3.4 Stokes' theorem
93
Let us now turn to twisted pforms. In this case, equation (A.3.17) is ambiguous since the choice between +w and -w depends on the oricntation in X,and this fact is not taken care of properly. In order to overcome this ambiguity, we have to determine the outer orientation first of the tangent space 5 , and subsequently of the whole %. Hereafter we denote a singular p-simplex by E := s(up) C X and an arbitrary point by x E %. The tangent space of the singular simplex Z, is a subspace of the tangent space X, a t this point. An outer orientation in the tangent space %, is an orientation in the complementary space X,/Z,. As usual, it is given by an equivalence class of frames in X,/%, that are related by a matrix with positive determinant. In other words, the outer orientation is given by a sequence (e,+l,. . . , e,,) of linearly independent vectors in X, tliat are transversal to E. The submanifold is outcr orientable if the outer orientations in the tangent spaces Z, can be chosen continuously on the whole %. Since % is connected, there are only two orientations allowccl. If the submanifold is outer oriented, we can require tliat the sign in front of w on the right-hand side of (A.3.17) is consistcnt witli the choice of tlie orientation for the frame
all over the singular simplex % = s ( a ) . This finally removes, for twisted forms, the ambiguity inherent in (A.3.17). To put it differently, by fixing the outcr orientation, the only allowed orientation-reversing coordinate transformations are induced by the orientation-changing diffeomorphisms of Rll, ant1 (A.3.17) is obviously invariant under such transformations. A good example of the notions introduced so far is tlie Mobius strip, Fig. A.2.9, consiclcrcd as a submersed submanifold (witli boundary) of IW? In this case, neither an inner nor an outer orientation can be attached to the Mobius strip in a continuorls way. Therefore neither ordinary nor twisted 2-forms, given on R3, can be integrated over the Mobius strip. Howcvcr, one can einbcd the Miil)ius strip into a nonorientable 3-dimensional manifold that is dcfincd :IS the direct product X3 = "IW x Mobius strip". In this Xg, the Mobius strip is a two-sidctl submanifold, and one can thus introduce the outer oricntation on it. After fixing the outer orientation, any twisted 2-form on Xg can be iiltcgratcd 011 the Mobius strip.
A.3.4 Stokes' theorem Stokes' tlieorem provides an n-dimensional gen,eralization of the familiar 3-dimensional Gauss and 2-dimensional Stokes theorems. The Stokes theorem is a far-reaching generalization of the fundamental integration theorem of calculus. Its importance for geometry and physics cannot be
4.3.4 Stokes' t,hcorem
95
overcstiniatcd. Tlicrc arc sovc?ralformulations of Stokcs' theorem. Usually, that basic result is prescntcd for an n-dimensional manifold X with a houndary d X . Let w be an ( n - 1)-form on X . Then
This theorem is true for an ordinary forni on orientablc manifolds as wcll as for a twisted forni on nonorientable manifolds. One can show that, in the former case, a natural iniicr oricntation, and in the latter casc, a natural outer orientation is induced on the boundary d X . We will not give a rigorous proof here.' Another version of Stokcs' thcorem, thc so-called cornbinatorial onc, is relatcd to tlie singular homology of a manifold (sce Sec. A.3.5). For any singular p-simplcx s : up -+ X and a (p - 1)-form w on the manifold X, the (combinat,orial) Stokes theorem states that
Figure A.3.4: Standard 2-siniplcx witli canonical coordinates on it. with the two indcpc.lldent components f,(t1,f2)= (s*w),, z = 1,2. Accordingly, the exterior derivative rends
Hcre, in accordance witli tlic definition (A.3.10), the bounda.ry of a. singular simplcx s : a" X is defined by
~' I
Now we have to apply the definition (A.3.17). For the lcft-hand side of Stoltes' tlleorem wc find, using the conventional rules for thc multiple integrals, Lct us demonstrate this tlieorem for a 2-sirnplm-. It is clcar that, without loss of generality, one can always clioosc coordini~tcs(t1,f2) in R2 > a 2 in s11ch a way that the vcrticcs of the 2-simplcx iuc the points Po = (0,0), Pl = (1, 0), ant1 Pz = ( 0 , l ) . The simplex is then callctl starldnrd witli thr canonical clioicc of cooidinutcs. The standard 2-simplex is depicted on Fig. A.3.4. Incidentally, tlic generalizat,ion to lligller-dimcnsiolial sirnpliccs in IW" is straightforward: A standard ysimplex aP = (Po, P I , .. . , P,) is dcfined by the points Po = (0,. . . ,0), Pl = ( 1 , 0 1. . . ,O), . . . , P p = ( 0, . . . ,0 , l ) . Given the parametrization of thc standard 2-simplcx, cf. (A.3.9), 02={(1-t
1
- t 2 ) ~ O + t 1 ~ l + t 2 ~ 2 } O, < t l < l ,
(A.3.22)
its bo~lndaryis described by tlic thrcc 1-simplices (its faccs): o [ ~ )=
{t1Pl+t2P2},
ail
=
{t
at2)
=
{(I - t l ) p0 t1 P I ) ,
For a. 1-for111 w
011
2
P2
+ (1
-
f1
t 2 ) PO),
+
+ t 2 = 1,
0 5 t 2 5 1,
(A.3.23)
o < t1 5 1.
Tlic right-hand sitlr of Stokcs' tlicorem consists of the three integrals ovcr thc faces (A.3.23). Direct calculation of tlic corresponding line integrals yiclds:
X tlie pull-hack on a 2 c R2 is givcn by
+
(s*w) = f l ( t l l t 2 ) d t 1 f 2 ( t 1 , t 2 ) d t 2 , ' A rigorous proof can bc found in Clioquct-Urul~atct aI. [4].
(A.3.24)
96
A.3. Integration on a manifold
97
A.3.5 @DeItham's theorems
Taking into account that
and recalling (A.3.21), we compare (A.3.26) and (A.3.27) to verify that (A.3.20) holds true for any I-form and any singular 2-simplex on X .
Figure A.3.5: Simplicia1 decomposition (triangulation) of (a) the torus T 2 , (b) the rcal projective plane P 2 , and (c) the Klein bottle R2. is introtluced, in analogy to (A.3.16) and (A.3.21), by defining for every singular pchain c, a singular (p - 1)-chain:
A.3.5 @DeRham's theorems The first theorem of de Rham states that a closed form if and only if all of its periods vanish.
1:s exact
Recall that the de Rham cohomology groups, which wcro defined in Sec. A.2.12 with the help of the exterior derivative
in the algebra of differential forms A*(X), "feel" the topology of the manifold
X. Likewise, singular simplices can also be used to study the topological properties of X . The relevant mathematical structure is represcntecl by the singular homolo.qy groups. They are defined as follows: Like a chain constructed from simplices, see (A.3.15), a singularp-chain on a manifold X is defined a formal sum
with real coefficients ai and singular p-simplices s f . In the space C , ( X ) of all singular p-chains on X, a sum of chains and multiplication by a rcal constant are defined in an obvious way. The bounday map
In complete analogy wit11 the de Rham complex (A*(X),d), a singular psimplex r is callccl a cycle if a z = 0. The set of all p-cycles, Zp(X):={z~CP(X)Idz=0),
p = O , . . . ,n,
is a real vector subspace, ZP(X) C CP(X). A singular pchain b is called a boundarg if a (p b = a c . Tllc space of pboundaries
(A.3.33)
+ 1)-chain c exists such that
B , ( X ) : = { ~ E C , ( X ) I ~ = ~ C ) ,p = 1 ,
... , n ,
(A.3.34)
also forms a. (real) vector space and B , ( X ) C Zp(X), since 813 r 0. One sets B,, ( X ) = Q). Finally, the singular homology groups are defined as the quotient spaces H,(X;R):=Z,(X)/B,(X),
p = O ,...,n.
(A.3.35)
As an instructive example, let us briefly analyze the homological structure of the siinplest compact 2-dimensional manifolds: The sphere S 2 , the torus T 2 (these two are orientable), the real projective plain P 2 , and the Klein bottle R2 (tllese are nonoricntable). The three last n~anifoldsare seen in Fig. A.2.3, Fig. A.2.4, and Fig. A.2.5, respectively. A standard approach to the calculation of lion~ologicsfor a manifold X is to triangulate it, that is, to subdivide X into simplices in such a way that the resulting totality of simplices (called a simplicia1 con~plex)contains, together with each simplex, all of its faces. Every two sinlplices either do not have common points or they intersect over a common
A.3. Integration on a manifold
98
face of lower diincnsion. The triangulation of a sphere obviously reduces just to a collection of four 2-simplices that form the boundary of a 3-sin~plex,that is, tlie surface of a tetrahedron (see Fig. A.3.1). The triangulations of the torus, tlie projective plane, and the Klein bottle arc depicted in Fig. A.3.5.
A.3.5 @DeRham's theorems
dcf,, = (B)- (A) and act2) = (A) - (B),thus proving that z1 = ct,) +ct2) is a l-cycle. Moreover, it is a boundary because of 2z1 = dP2. No other l-cycles exists in P2. Thus we conclude tliat tlic 1st homology group is also trivial. Because of connectedness, the final list reads:
1) S2 lins as the only %cycle the manifold itself, z 2 = S 2. Direct inspection shows that tliere arc no nontrivial l-cycles (they are all boundaries of 2-diniensional chains). Finally, each vertex of the t,etrahedron is trivially a O-cycle, and tliey are all homological to each other because of the connectedness of S 2 . These facts are summarized by displaying the homology groups explicitly:
+
2) T 2 is "cornpo~cd~~ of two 2-simplices, T 2 = Stl) SE), namely Stl) = (A, C, D) ant1 St2)= (A, B, C) with the corresponding identifications (gluing) of sides and points as shown in Fig. A.3.5(a). The direct calculation of the boundaries yields = (A, B ) (B, C ) - (A, C ) and as,",,= (C, D ) - (A, D ) (A, C). Taking tlie identifications into account, we then find dT2 = 0; hence the torus itself is a 2-cycle. There are no other non-trivial 2-cycles. As for the l-cycles, we find two: zfl) = (A, B)lo=n and = ( B ,C)lc=B.A (end points are identified). Geometrically, these cycles are just closed curves, one of which goes along ancl another across the handle. There are no other independent l-cycles (z&) = (C, A)Jc=n, for cxamplc, is l~omologicalto tlie sum of zfl) ancl z(:)). Thus we havc verified t h t the 1st homology group is two-dimensional. For O-cycles the situation is exactly the same as for the sphere. In summary, wc havc for the torus:
+
+
99
+
4) R 2 = Stl) S&, wliere St,) =.(A, C, D ) and St2)= (A, B, C ) witli sides and points glued as shown in Fig. A.3.5~.By an analogous calculation, we find dK2 = 2(B, C ) . There are no nontrivial 2-cycles on the Kleiri bottle. As for the torus, tliere are two independent l-cycles, zfl) = (A, B)IR=A and z(:) = (B, C)Ic=B=n However, tlie second one is a l~ounclary2zt2) = dK2. Hence zll) generates tlie only nontrivial hornology class for t,llc. Klein bottle. T ~ U Sfinally, tlic 11omologygroups are:
+
2 3) P2 = Sfl) S&),where Sfl)= (A,C, D ) and S#) - (A, B, C) with sides and polnts identified a5. shown in Fig. A.3.5b. epeating the calculation for tlie torus, we find dP 2 = 2cf1) t 2cf2),wliere the l-chains are cf,) = (A, B) and cf2)= (B, C ) . Tlius, the projective plane itself is not a 2-cycle. Since there are no other homologically inequivalent 2-cycles, we conclude tliat the 2nd homology group is trivial. Moreover, we immediately verify
Like the dc Rham coliomology groups II"(X; R) (see Scc. A.2.12) tlic singular homology groups Hp(X;R) are topological invarian,ts of a manifoltl. In partic11lar, tliey do not change under a 'smooth deformation' of a manifoltl, i.c.., tliey are hoinotopically invariant. Cohornologies ancl homologics arc tlccply related. In order to demonstrate this (altliough without rigorous proofs), we ncod the central notion of a period. For any closc(1 pform w E P ( X ) and eacli p-c?lclr: z E Z,(X), n period of the form w is t,llc number 1 ' ~ ' (w) ~ :=
[
w.
This real number is not merely a function of w and z; it rather depends on tlie whole coliomology class of the form [w] E Hp(X; R) and on tlie whole llolnology class of the cycle [z] E H,(X;R). Stokes' theorem underlies the proof: For any coliornologically ccluivalcnt p-form w + dv and for any homologically cquivnlcrit pcycle z dc, we find
+
A . 3 . Intcgrntion on a manifold
100
sincc d t = dw = 0. Therefore, in a strict sense, onc has to write a periotl as per[,]([w]).Wc recall thc definition of a form as a linear map from a vcctor space V to thc reals (see Scc. A.1.1). Accordingly, one can treat the pcriod (A.3.40) as a 1-form on the space of cohomologies with V = H"(X; R), i.e., as an elcmcnt of the dual spacc per[,] E H,,(X; R)* ,
I
I
for all [w] E N1'(X; R).
In siniple tcrms, the first theorein tells that DR.([w])= 0 H [w] = 0. A 1-form on a vcctor spacc V is dctcrnlinccl by its components wliicll give the values of of tlic that for111 with respect to a basis of V. Suppose wc have chosen a. basis [ti] p t l i homology group Hr,(X;R), i.e., a complete set of hon~ologicallyinequivalent singular p-cycles ri. (For a compact manifolds this set is finite.) Denotc as Oi E V* = IIp(X;R)" the dual basis to [ri]. Each 1-form on V = II),(X;R) is then an elelllent (ti Oi spccificd Ily a set of real nutnh~rs{ o , ~with } i rrinliing ovcr the wliolc range of t,hc basis t i . Tlic second de Rll,nnr. t11,eoremstatcs that, tlic cle Rhani 111;q) is ~:n,vertible, that is, for every set of rcnl num11c:rs { a i } tlicrc cxist,s a closcd p-form w on X such that i.c.,
[w] = DR-'(a, 0 % ) .
101
n particular, dim Hl'(X; R) = dim H,(X; R). Then one can, for examplc, cal:ulatc the Euler charactcrist~ics(A.2.82) easily. Returning again to the 2-dimen,ional examples, we find: X(S2) = 1 - 0+ 1 = 2, see (A.3.36); X(T2)= 1 - 2 + 1 = 0, scc (A.3.37); X(P2)= 0 - 0 1 = 1, see (A.3.38); and X(JK2)= 0 - 1 1 = 0, see (A.3.39).
+
+
(A.3.41)
Thc linear 111a11DR : HI'(X; R) + Hr,(X;R)*! defincd via the equations (A.3.41) and (A.3.40) as DR([w])([t]):= pcrlz1([w]), is callctl the de R h a m map. A fundamental tllcorem o f de R h a m statcs that this map is an isomorphism. Sometimes, the proof of the cle Rham theorem is subdividetl into tlic two separate propositions known as the first ant1 second de Rham theorems. The first de R h a m tl~eoremrcads: A closet1 f o m zs exact zf and only if all of zts penods vanzsh:
DR([w])= a; O',
i.3.6 @DoItham's theorems
(A.3.43)
In con~binationwith the second theorem, thc first dc Rham tllcorem clcarly guarantees that the dc Rham map is one-t80-onc:Suppose that for a given set {a,) onc can fincl two 1-forms w ancl w' that both satisfy (A.3.43). Then we get DR([w - w']) = 0 ancl (A.3.42) yields [w] = [w'], i.c., w ancl w' differ by an exact, form. Incidentally, our earlier study of the Iiomologica1 st,ructurc of the 2-dimensiona1 manifoltls S2,T 2 , P2,K2 gave explicit constructions of tlic basrs [r,] of the homology groups. Onc can show that for an arbitrary compact lnanifold X both thr cohonlology ttnd homology groups are finite-dimensional vcctor spacrs. Thcn the d r Rliai~lnlap establishes tlic canollical isomorphism
Problem Problem A.1 Show that properties 1)-4) of Proposition 1 in Sec. A.2.7 lead uniquely t o the forlnula (A.2.1G), i.e., they provide also a definition of the exterior derivative.
References
[I] R. Bott and L.W. Tu, Differential Forms i n Algebraic Topology. Corr. 3rd printing (Springer: Berlin, 1995). [2] W.L. Burke, Applied Differential Geometry (Cambridge University Press: Cambridge, 1985). [3] B.W. Char, K.O. Gecldes, G.H. Gonnet, B.L. Leong, M.B. Monagan, and S.M. Watt, First Leaves: A Tutorial Introduction to Maple V (Springer, New York, 1992). [4] Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick, Analysis, Manifolds and Ph?jsics, revised ed. (North-Holland: Amsterdam, 1982). [5] J. Grabmeier, E. Kaltofen, and V. Weispfenning, eds. Computer Algebra Handhook: Foundations, Applications, Systems (Springer: Berlin, 2003). [GI A.G. Grozin, Ussing REDUCE i n High Energy Physics (Cambridge University Press: Cambridge, 1997).
[7] D. I-Iartley, Overview of computer algebra i n relativity. In [lo] pp. 173-191. [8] A.C. Hearn, REDUCE User's Man.ua1, Version 3.6, RAND Publication CP78 (Rev. 7/95). The RAND Corporation, Santa Monica, CA (1995). [9] F.W. Held, J.D. McCrea, E.W. Mielke, and Y. Ne'eman, Metric-ABne Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance, Phys. Rep. 258 (1995) 1171.
104
Part A. Mathematics: Some Exterior Calculus
1101 F.W. Hchl, R.A. Puntigam, and H, Ruder, eds., Relativity and Scientific Com.p~~tin.g: Computer Algebra, Num.erics, Visualization (Springer: Berlin, 1996).
[25] E. Schriifcr, EXCALC: A S?jstem for Doin,g Calculo.tion..s in the Co.lc~~lus of Modern Diflerential Geornetr?j. GMD-SCAI, St. Augustin, Gcrriialiy (1994).
[ l l ] C. Hcinickc and F.W. Hehl, Computer algebra in gravity, 5 pages in 151.
[26] E. Schriifcr, Differen.tin.1Geom.etr?jand Applico,tions, 3 pages in [5].
[12] B. Jancewicz, Wielkos'ci skierowane w elektrodynam,ice (in Polish). Directed Quantities in Electrodynamics. (Univcrsity of Wroclaw Press: Wroclaw, 2000); an English version is under preparation.
[27] E. Schriifcr, F.W. Hchl, nntl J.D. McCrea, Exterior calculus on the corn,puter: Tlre REDUCE-package EXCALC applied to gen.eral relativit?/ and tlre Poincart !lauge thcory. Gen. Rcl. Gr8.v. 1 9 (1987) 197-218.
[13] W. I
1281 J . Socorro, A. Macias, and F.W. Helil, Com,p~~ter algebra in gra~lity: Reduce-Excalc Programs for (non-)Riem,a.nnic~nspacetimes. I. Con~put. 1711,~s. COI~IIILIII. 115 (1998) 264-283.
[14] B. Laurent, Introduction to Spacetime, a first course on relativity (World Scientific: Singapore, 1994).
[29] I.S. Soltolnikoff, Tensor An,alysis (Wiley: New York, 1951).
[15] M. MacCallum and F. Wright, Algebraic computing with REDUCE. Vol.1 of [23]. [16] J.D. McCrea, REDUCE in general relativity and Poincark gauge theo y . In Vo1.2 of [23] pp. 173-263. McCrea's library can be found via ftp://ftp.maths.qmw.ac.uk/pub/grlib. [17] R.G. McLenaghan, MAPLE applications to general relativity. In Vo1.2 of 1231 pp. 265-354. [18] C.W. Misner, I<.S. Thornc, and J.A. Wheeler, Gravitation (Freeman: San Francisco, 1973). 1191 P. Musgrave, D. Pollney, and I<. Lake, GRTensorII, Release 1.50. For Maple V releases 3 and 4 (Quccn's Univcrsity a t Kingston, Ontario, 199G); see also h t t p : / / g r t e n s o r . phy . queensu. ca/. [20] Yu.N. Obukhov, Computations ~uitlzGRG in gravity th,eoy, 3 pages in (51. I
[21] L. Parker, S.M. Cliristcnscn, Matl~Tensor:A System for Doing Tensor A11,alysis by Computer (Addison-Wesley: Redwood City, 1994). 1221 R. Printigam, E. Schriifcr, and F.W. Hehl, The use of computer algebra in. Ma,xwell'.s theory. In Computer Algebra in Science and Engineering, J . Fleischer et al., eds. (World Scientific: Singapore, 1995) pp. 195-211. Eprint Archive gr-qc/9503023.
[30] H.H. Solcng, Tlre Mnthematica Packogc.~CARTAN and Mntlr.Tensor for Tensor Analysis. In R.cf.[lO],pp. 210-230. [31] D. Stanffcr,F.W. Hc1.11, N. Ito, V. Winkclnialln, iilltl .J.G. Znbolit,zky, Cornputer Sim.1~1ationan.d C o m p ~ ~ t eAlgebra: r Lectrrres for. Be,qinncrs. 3rd ed. (Springer: Berlin, 1993). [32] J.L. Syiigc ttnd A. Schilcl, Tensor C ~ ~ l c u lRc~~rintcd ~~s. froin thr corrcctctl cdition publishctl in 1969 (Dover: Ncw York, 1978). 1331 S.I. Tcrtychniy, Searching for electrouac soh~tion.~ to B~:~e.stein,-M~rnn:~ucIl e q ~ ~ a t i o~uitl~, n , ~ t l ~ chelp of conrpzrte~algeb~v,s?y.stem GRGI-,c, Los Alnllios Eprint Archive gr-qc/!)810057 (1998). 1341 S.1. Tortychniy, Cornpt~tcralgebra s ~ s t ~ GRGEc, 7r~ 3 pages in [5]. [35] S.I. Tc>rt;ychniyaiitl I.G. Obukhova, GRGrsc: Com.puter Algebm S?jsterrtfor Applicntion.~in Gro,vitation. Tlt,ro17j. SIGSAM Brlllctin 31, $11, issue 119 (1997) G-14. [36] M. Toassaint, Lect,r~reson. Rerluce and Mr~plrnt UAM-I, Mcn:ico, G1 p;~.gcs, Los Alnmos Eprint Arcliivc h t t p : //xxx. l a n l .gov/abs/cs. SC/0105033 (2001). 137)
E.Tsantilis,
R.A. Puntigain, and F.W. Held, A (pnd7ntic C I L ~ V ( L ~ I LLa~C grangic~nof Paurlowski and Rqczka: A finger exel-cise zuitlr A4atlrTen.s01-.In [lo] pp. 231-240.
[23] M.J. Rcbouqas, W.L. Roque, eds., Algebraic Computing in General Relati~lit?j, Lecture Notes from the First Brazilian School on Computer Algebra, Vol. 1 ant1 Vol. 2 (Clarendon Press: Oxford, 1991 and 1994).
[38] V. Winkclmnnn ant1 F.W. Hclil, REDUCE for Begi~an,ers:Seven L e c t ~ ~ r c . ~ on tlre Application of Con~putcr-Algebm.In [31] 1111. 145-280.
1241 J.A. Schout,cn, Tensor Analysis for Ph?jsicists. 2nd ed, reprinted (Dover: Mincola, New York 1989).
1391 S. Wolfrnm, Tll,e A4atlrerr1,nticc~ BnoX:, 4th ctl. (Ca.lnbriclgc Univcrsit,y Press: C i ~ ~ ~ i l I r i1999). dg~,
106
Part A . Mathematics: Some Exterior Calculus
[40] V.V. Zhytnikov, GRG: Compu.ter Algebra System for Diflerential Geometry, Gravitation an,d Field Theory. Version 3.2 (Moscow, 1997). GRG3.2 is nvailablc via anoliymous ftp from f tp . maths . qmw . ac . uk in thc directory pub/grg3.2.
Part B
Axioms of Classical Electrodynamics
108
Part B. Axioms or Classical Electrorlynamics
I11 Part B we put phcnornenological classical eltctrodynamics into such a form that t,hc unclcrlyirig physical facts arc clcarly visible. Wc recognize that the conscrvation of electl-rc cltarqe and of magnetic flux arc the two main expcrimcntally wcll-founded axioms of clectrodynan~ics.To formnlatc them we usc cxtcrior calculus because it is the appropriatc m;tthcmatical framrwork for handling fields thc integrals of which-here charge and flux-posscss an invariant meaning.
B.1 Slectric charge conservation
T h e densitics of the clcctric charge and electric current arc ,2ssumctl to be ~~l~enomenologically spccificd. Tliese quantities will not be resolved any fi~rther fundamental for classical elcctrodynnmics. and will bc consitlered
. . . it is now discovered and demonstrated, both here and i n Europe, that th,e Electrical Fire i s a real Elemcn,t, or Species of Matter, not created by ttte Friction, 611t collcctccl only. Benjamin F'rnnklin (1747)'
B.1.1 Counting charges. Absolute and relative dimension Progress i n semicon.ctucto~~ tecltnology lras en,abled tlre fabiication of str~icturesso srnnll that they can contain just on,e mobile electron. By varying controllably the n.umber of electrons in these 'artificial atoms' and measl~ringth,e en.ergy required to ad$ s ~ ~ c c e s s electrons, i~~e one can conduct atoin ptrysics exper-
' ~ c cIIeilbron [lc], p. 330. On the sarnc page Heilbrorr statcs: "Althor~glrR a n k l i r ~did not. 'discover' corrservation, he was 2~11iquestiotiablythe first t o exploit the concept fruitfully. Its full utility appeared in his clmsic annlysis of the conclenser." Note that the discovery of cliargc conservation prccedcd the discovery of tlre Coulomb law (1785) by more than 40 years. This historical scquencc is rcflccted in our axiomatics. IIowever, our reason is not at1 Iristoricnl but a conceptual one: Charge conservatiotl should come first ant1 the Maxwell equations shoultl be forniulatctl so a.q to bc compatible with that law.
B.1. Electric chnrge conservation
110
B.1.1 Counting charges. Absolute and rclntivr dinlension
111
imcn.ts in. a re,qim,e that is in.acce.s.sible to experiments on real atoms. R. C. Ashoori (1996)~
i
1
P11enomenologic;tlly speaking, electromagnetism has two types of sources: The electric charge density p ant1 thc electric current density j. The electric current dcnsity can be understood, with respect to some reference system, as moving elcctric charge dcnsity. I11 this scction we give a heuristic tliscussion of cliargc conservation that will bc used, in t,hc next scct,ion, as a motivation for the formulation of a first axiom for electrodynamics. Imagine a 3-tlimensionnl simply conncctcd region Q3 that is ellclosed by the 2-dimensional boundary d o s (see Fig. B.1.1). In tlie region 03, there are elenientnry pa.rticles wit,h charge zte (e = elementmy electric charge) and quarks with charges *$e ancl *+e, respectively, which all move with some velocity. It is assumctl in clcctrodynamics that we can attribute to the 3-region 523 a t any t,inlc in a certain rofcrence system a well-defined net charge Q with tlie dim.ension~bfcharge q (in SI unit,s, coulomb, abbreviated C):
Q=
I
IP, IQI
=B,
[PI
=(r.
(B.l.l)
3
I
I
Figure B.1.1: Charge conservation in 3-dimensional space. The electric current lensity 2-form j flows out of the volume Q3.
Here [Q] should be rcnd as "dimension of Q" and, analogously, [p] as "dimension of p." We call [Q] and [p] also absolute tlimension of Q and p, respectively. T h r intcgrand p is callcd tlie electric charge density 3-form. It assigns to the volumc 3-fonn in an arbitrary coordinate system (o, 6,. . . = 1, 2, 3) 0.r." A dzt' A d.rC, (B.1.2)
Tlicreforc .. we* introduce an arbitrary 3-tlimcnsional local cofi.iunc
anhololiomic 3D intliccs. A physicist neetls such a (co)framc to ;~ncliorhis or her nl)pnrntus. Wc assign to each of tlie tlircc 1-fornis 17' the rlirnenslon of a length C alld to tllc corresponding vrctors cb the tlimcnsion of P-': [d"
1
*
t,
3
[ej] =
2See his article [I]. For a rcvicw on the counting of single electrons, see Devoret and Grabert (51 and, for tnorc recent dcvcloptn~nts,Fujiwara and Takahimhi 112). 3We base our dimensional analysis on the work of Dorgelo and Schouten (see Schouten [45], p p 126-138 ant1 Post [37], pp. 23-46). A guide of how t o use the International System of Units (SI) ha=becn ~>rovitledby Nelson [34]; see also the references therein.
with
h
a scalar-val~ictlcliargc
which, as a scalar, can bc adtlcd 11p in any coortlinatc system to yield Q. Thus, as alrrady noted, cvcn t.he charge density p carries the same absolute dinielisioll as the net clinrgc Q. In spatial spherical coordinates z" = (r, 0,4), for instance, the coordinates carry different dimcnsions: The .rl or r-coordinate has the dimension of a length, wliercns .r2 = 0 ant1 "r 4 a1c dimensionless. Accordingly, the different coordiliatc coniponcnts pa,,c of p have different dimcnsions that cannot be lneasurecl straiglitforwartlly in an cxpcrimcnt. Thus these components of p are unsuitable for a. gcweral definition of the relative (also called physical) dimension of a quantity.
19"
6 , . , . = 1 , 2 , 3 , atid its d u d frmne r i wit11 r j J 19' = 6;. We mrwk tlie aumbrrs i, 2, by a circumflex in order to bc able to identify thcm as being rclatrd to
*
A
for 6,= 1,2,:3,
P-' ,
A
A
*
,
(13.1.4)
.
for b = 1 , 2 , 3 .
(B.1.5)
Length is here understood :IS n spgmrnt, that is, as part of n straiglit line between two points. Accordingly, lciigtl~is syl~ol~yn~ous with a 1-dimensional 2xtcnsiou i11 ztffinr geometry: it is not, however, the distance in the spnsc of Euclidean geometry. In other words, length as tlinicnsion tlocs not prcs~lpposc :he existence of a metric; it is a, prc-metric conccptg. Now we call dccomposc the cliargc density 3-form with respect to the coframc 19', 1
p=
% PiLCB%
19"
\9?
rcl. dim. := [phi,] = P - ~.
(B.I.G)
The din~cnsionsof all n.nholonomic components /);,je of the charge tlcnsity p arc the samc. We cnll [pibe] the relative dim.ension,of p. Accordingly, t,l~oabsolute dimension of p is charge, i.c., [p] = q, whereas the rc1ativc dilnc1ision of p is c/~.ar~e/(len.~tl~,)', i.c., = q t-'.
112
I3.1. Electric charge conservatiori
In the h~jpotlres~s of local~t?/"t is assumcd that the mcasuring apparatus in the coframta 6" evcn if thc lattcr is accelcratctl, mca-ures thc anholonomic components of a physical quantity, such as the components pAb2,cxactly as in a momentarily comoving inertial frame of rcfercnce. In the special case of the mcasulmcnt of time, Einstein spoke about thc clock hypothcsis. If we assume, as suggested by cxpcricncc, that the electric charge Q has no zntnnszc scrcw-sense, then thc sign of Q docs not tlepend on thc orientation in space. Accordingly, thc charge tlellsity p is reprcsentcd by a twzsted 3-form; fol the definition of twisted quantitics, sec the cnd of Scc. A.1.3. Providcd the coordinates T" arc givcn in Rj, we can dctcrmine dzn and tht. volume 3-form (B.1.2). Thcrc is no nccd to use a lnctric nor a connection, the propertics of a 'bare' differcntial ~nanifoltl(continuum) are sufficient for the definition of ( B . l . l ) . This can also be lccognized as follows: Tlic nct chargc Q in (0.1.1) curl be tlctermined by counting thc chargetl elclnentary particles insidc t3R 3 and adding up their clcmentary electric charges. Nowadays one catches single electrons in traps. Thus the counting of electrons is a11 cxpcriment,ally fcnsihlc procedure, not only a. thought experiment devisetl by a theoretician. This consideration shows that it is not necessary to use a clistancc. concept nor a. length standard in 0 3 for the determination of Q. Only 'counting proccdurcs' arc rcquirctl ant1 a way to delimit an arbitrary finite volume R j of 3-dimensional space by a boundary (303 and to know what is inside aR3 and what outside. Accordingly, p is the prototype of a charge density with nbsolutc tlilncnsioll [p] = q and relative tlinicnsio11 [pabC]= rl P - 3 . It becomes thc convcntional charge densit,y, that is chargc per scnlrd unit volume, if a unit of distance (111 in SI-units) is introduced adclitionally. Then, in SI-units, we have -9 = C 1Tl . Out of the rcgion Rj, crossing its bounding surface i3R,3,thcrc will, in general, flow ;L net elcctric cuncnt, see Fig.B.l.l,
with absolutc dimcnsion qt-I (t = tilnc), whicli must not depend on the orientation of spacc either. Thc int,cgrand 3 , thc twisted electric current density CS-' = ampcre = A, here s tc qt-' 2-form wit,li the same a b s o l ~ ~dimcnsion = sccond, assigns to the area clclncnt 2-for111
4T11c formulation of Mashlloon (301 rcads: " . . . thc hypotliesis of locality-i.e., the p r o sullietl c?quivalenceof an accelerated observer with a ~iiorncntarilyconloving inertial observerunclcrlies the stitndard relativistic f o r ~ n a l i s ~byn relating tlre menqurcrnents of a n accelerntecl observer to tllosc of an inertial observer." T h c two observers are assumed to be otherwise identical. That is, the two obscrvers are copics of one observer: One copy is inertial and the othcr nccelcrated; this is the only difference betwccn them. T h e limitations of the hypothesis of locality are rliscussccl in [31].
3.1.1 Counting charges. Absolute and relative dimension
113
'igure B.1.2: Charge conservation in 4-dimensional spacetime. The 3limensional "cross section" 0 3 sweeps out a 4-.dimensional volume f14 On its Yay from t = t l t o t = t 2 . 1 scalar-valued
charge current
The postulate of electric charge conscrvation requires
xovidcd the area 2-form dxa Adzb is directed in such a way that the outflow is :ounted ~ositivelyin (B.1.7). The time variable t, provisionally i n trodu c ed here, loes not nced to possess scale or a unit. It can be called a "smooth causal time" 11 the sensc of parameterizing a future-directed curve in the spacetime manifold ~ i t tha9. a monotone increasing and sufficiently smooth variable. Substit u ti on of :B.1.1) and (B.1.7) into (B.1.lo) yields an integral form of charge conservation:
By applying the 3-dimensional Stokes theorern, the differential version turns out to be
B. 1. Electric chargc conservation
114
115
3.1.2 Spacetime and the first axiom
Let, us put ( B . l . l l ) into a 4-dimensional form. For this purpose wc integrate (B.1.11) over a certain time interval from tl to t2 (see Fig. B.1.2). Note that this figure depicts the same physical situation as in Fig. B . l . l :
Obviously we are integrating over a 3-dimensional boundary of a compact piece of the 4-dimensional spacetime. If wo int,roclucc in four tliincnsions tlic twisted 3-form J:=-jAdt+p,
with
[J]=q,
(B.1.14)
Figure B.1.3: Spacetime and its (1
+ 3)-foliation.
then the integral can be written as a 4-dimensional boulidary integral,
wlierc Q4 = [ t l lt2] x Q3. The twisted 3-form J of the electric charge-current, tlensity witli absolute dimension q plays the central role as source of the elcctroniagnetic ficltl.
lefinc twisted and ordinary untwisted tensor-valued differential forms. In order o avoid possible violations of causality, we will, as usual, consider oilly non:ompact spacetime manifolds X4. The X4 with the described topological properties is known tjo possess a (1+3)oliation (see Fig. B.1.3); i.e., there exists a set of nonintcrsect,ing 3-di~nensional iypersnrfaces It, that call be parameterized by a monotone increasing (would)e timc) variable a with the clirncnsion of time: [a]= t. Althongh at this stage vc do not introduce any metric on X 4 , it is well known that tlic cxistcncc )f a (1 3)-foliat,ion is closely rclatcd t,o the cxistcncc of pseudo-Rien~:innian tructures. Among the vector fields transverse to the foliation, we choose a vector field z (not to be confiiscd with the dimension n of a manifold discussed in Part A) iormalizcd by tlic condition
+
B.1.2 Spacctime and the first axiom Motivatetl by the integral for111 of charge conservation (B.1.15), we can now turn to an axiolnatic approacli of electlodynamics. First we will forniulatc a set of minimal assumptions tliat we sliall need for clefining an appropriate spacctimc manifold. Let spacetime be given as a 4-dimensional ronnected, Hausdorff, oomentnble, and pnmcompact t1iffcrenti;tblc manifold X 4 . This manifold is "bare," that is, it carries neitlicr a. connection nor a nictric so far. We assume, however, tlic conventional continuity and differelltiability requirements of physics. To recall, a topological space X is Har~stlorffwhcn for any two points $. p2 E X one can fintl ope11 sets pi E U1 c X , pa E U2 c X, such tliat U1 n U2 = Q).An X is connected when any two points can be connected by a continuous curve. Finally, a connected Hausdorff manifold is paracompact whcn X can be covered by a countable number of coordinate cliarts. The (smooth) coorclinatcs in arbitrary cliarts will be called r t ,witli i, 3, k,. . . = 0,1,2,3. The vector basis (frame) of the tangent space will be called e, ailti the 1-form basis (cofrainc) of tlie cotangent space 6" with (anholonomic) illdices a, P, y,. . . = 0,1,2,$. On tlie X4 we call *
.
.
A
Physically, the folia h, of collstalit a represent a simple model of a "3-space," vhile the function a serves as a "time" variable. The vector ficltl n is usually ntcrprctetl as a congruence of observer worldlines. In Sec. E.4.1, this rather ormal mathematical construction becomes a full-fledged physical tool whcn the metric is introduced. Now we arc in a position to formulate our first axiom. We require the existence of a twisted charge-c~~~-rent density 3-form J with the absolutc diincnsioll of SI :hargc q, i.e., [ J ] = q = C which, if integrated over an arbitrary closed 3iimensional submanifold C3 c X4, obeys
f
J=O,
c 3
ac3=0
(first axiom) .
(B.1.17)
116
B.1. Electric charge conservation
117
B.1.4 Timc-space decomposition of tllr inhomogoncolls Maxwcllrquation
We recall, a rnanifold is closed if it is conlpact and has no boundary. In particular, tlic 3-dimcnsional boundary C3 = 804 of an arbitrary Cdimensional region 0 4 is a closcd manifold. However, in general, not every closed 3-manifold is a 3-boundary of some spacctime region. This is the first axiom of elcctrotiynamics. It has a firm phenomenological basis.
9.1.4 Time-space decomposition of the
B.1.3 Electromagnetic excitation H
ziven tlie spacetime foliz~tion,wc can decolnposc any exterior form in "timc" tnd "space" pieces. With respect to the fixctl vector ficld n, normalized hy :B.l.lG), we tlefi~lc,for a p-form {I!, thr part longitudinal to tlie vector n by
Since (B.1.17) holds for an arbitrary 3-dimensional boundary C3, we can choose C3 = dCl4. Then, by Stol
inhomogeneous Maxwell equation T i m e i s n.nturels luny of keel~in,gevenjthzng from. /rnppen,in,g nt oncc. Anonyn~ous
~dthe part transversal tlo tlie vector Since R4 can I,c chosen arbitrarily, the electric current turns out to be a closed forni:
This is, in four dimcnsions, thc differential version of charge conservation. Having proved tliat J is a closetl 3-form, we now recognize (B.1.17) as the statement tliat all periods of tlic current J vanish. Then, by de Rham's first theorem (see (h.3.42)), the current is also an exact form:
This is tlie i n h o m o g ~ ~ i c oM,zxwcll ~~s equation. Thc twisted clcctromagnetic excitation 2-form H has the absolute dimension of charge q, i.e., [HI = q. Thc cxcitatiori H in this sctup appears as a potential of the electric current. It is dctermiiictl olily up to a closcd 2-form
111 Sec. B.3.4, however, we tliscuss how a unrque cxcitation field H is selected from tlie multitude of H's occurring in (B.1.21) by a very weak assumption: If the ficld strength F , to be d~finedbelow via the second axiom in (B.2.8), vanishes, then tlie excitation H vanishes too. In this context we recognize tliat tlic electric and maglletic pieces of H call be measured by means of an ideal electric conductor and a s ~ ~ p ~ r c o l i d ~ofc ttype o r 11, respectively. Chargc conservation, a s forinul~tcdin the first axiom, is experinlentally verified in all lnicroscopic experiments, in particular in those of high-energy elementary particle physics. Therefore the excitation H represents a microscopic field as well. Charge conservation is not only valid as a macroscopic average. Similarly, the excitation is not only a quantity tliat shows up in macrophysics, it rather is a microscopic field, too, analogous to the electromagnetic field strength F t o be introduced below.
q~:= (1 -
- I)*=
TLJ(
TL
by
d~Agl),
n~ q = 0 .
(B.1.23)
Then, we finally have
with the absolute dimensions
Thus t,]ie projection operators " I " alitl "-"for111 i t complrt,c set. Frlrthcrmorc, every 0-form is tr:tnsvcrsal, whereas every 4-fonii is longitutlinal, that is, for p = 0, wc fintl 9 = 9 antl, for p = 4, 9 = I * . In ortlcr to apply tliis dccomposit,ion t.o firltl theory, tllc following rules for exterior niultiplication can be dcrivctl fro111 (B.1.22) and (B.1.23),
if Q is p-form. We iritrotlucc the 3-tliiiicnsionnl exterior tlcrivativc as the trallsvcrsnl part of rl, that is, nccortling to (13.1.231, as d := 71 J (du A r f ) . Then the cxtcrioi tlerivativr of n p-for111tlccolnposcs as follows:
According to (A.2.51), the Lre den vat?^^^ of a p-form along a vector fic.1~1 be writtcn as
< can
B.1. Electric charge conservation
118
Notice that the Lie tlcrivativc along tllc foliation vcctor field n commutes with the projection operators as well as with tlie exterior derivative, i s . ,
3.1.4 Time-space decomposition of the inhomogeneous Maxwell equation
119
In three dimensions, we recover the twisted current density 2-form j (see 'B.1.9)) and the twisted charge density 3-form p (see (B.1.3)). Now charge :onservation (B.1.19) can easily be decomposed, too. We substitute (B.1.36) nto (B.1.28) for = J. Then, with the abbreviation (B.1.32), we recover 'B.1.12):
+
These rules also imply that
L,, dl = d &,4. The Lie derivative of the transversal piece vcctor 11, is abbreviated by a dot,
9 of
(B.1.31) a form with respect to the
since this turns out to be the time drrivative of the corresponding quantity. I11 order to make this dccomposit,ion formalism more transparent, it is instructive to consider natural (co)framcs on a 3-dimensional hypersurface k, of constant a (see Fig. B.1.3). Let rn be local coordinates on 11, with a = 1,2,3. The differentials dx" arc not transversal to n in general. Indeed, in terms of the local spacctimc. coortlintitcs (a, r"),the nor~nalization(B.1.16) allows for a vector field of the structure n = 8, +nn & wit11 some (in general, no~lvanishing) functions n". We use (B.1.22) and (B.1.23) and fincl a nontrivial longitudinal piece '(dn.") = n" da, whereas the transversal piece rcads & = dzn - n" do. Obviously d a is purely longitudinal. I-Iencc it is collvenient t o choose, a t an arbitrary point of spacetime, a basis of the cotangent space
rhis, a t thc same time, gives an exact meaning t o the time derivative, which we treated somewhat sloppily in Sec. B.1.1. Note that d p = 0, as a 4-form in ;hrec dimensions, represents an identity with no additional information. Before we turn to the right-hand side of (B.1.20), we decompose the excita;ions according to 5
with the twisted magnetic excitation 1-form 7-l and the twisted electric excitation 2-form 'D: ?i:=HI
'D:=H.
(B .1.40)
The signs in these two definitions are discussed in section B.4.2. The absolute and relative dimensions of 3.1 and V are, respectively,
[x]= qt- 1 SI= A , [xi]= 1x1 SI= A m-l, (-1
which is compatible with thc foliation given. The coframe (13.1.33) manifestly spans the longituclinal and transversal subspaccs of the cotangcnt space. Thc corrcspondi~~g basis of the tangent space reacls
and
['Dl = 9 SI = c,
(B.1.41)
[ V ~=J[VI e-2 2 c mV2.
Everything is now prepared: The longitudinal part of (B.1.20) reads
and the transversal part This coframc and this frame are a1111010110niic, in general. The gcnernl decolnpositioll scheme can I,e applied to the inhomogeneous Maxwell eq~lation(B.1.20). First, wc decompose its left-lii~ndside. Then the current reads
By means of (B.1.35), (B.1.36), and (B.1.39), the last two equations can be rewritten as
and j := -Jl
and
p := J .
(B.1.36)
The minus sign is chosen in conformity with (B.1.14). Since [ J ]= q , we havc for thc absolutr ant1 tlic relative dimensions, respectively, [j] =
[j,,;l
=
q t - 1 SI =Cs-' = A , 2 SI
SI
[p] = q = ~ , 3 SI
[ j ][--= A mP2, [phLs]= [p] l- = c nlp3.
(B.1.37)
"I'lle historical name of 7-1 is "magnetic field" and that of 'D "electric displacement." T h e I S 0 names (International Organization for Standardization) are "magnetic field" for li and "electric flux density" for V.As we can see, the I S 0 names are in conflict with a reasonable 4D interpretation; that is, the 4D quantity H remains without a name. The I S 0 names are based on the difference between a 1-form, here 7-1 (field), and a 2-form, here 'D (flux density).
120
R.1. Electric charge conservation
respectively. We find it remarkable that all we need to recover the inhomogeneous set of the Maxwell equations (B.1.44), (R.1.45) (see also their vector versions in (1.6) of the introtluction) is electric chargc consel.vation in the form (13.1.17) for arbitrary periods - and nothing more. Incidentally, the boundary ~ondit~ions for H ' and 23, if required, can also be derived from (B.1.17). Bccausc of the (lot,, forniula (B.1.44) represents an evolution equation, whereas (B.1.45) constitrltes a constraint on the init,ial distributions of V and p.
B.2 Lorentz force density
B.2.1 Electromagnetic field strength F y now we have exhausted the information contained in the axioin of charge )nservation. We have t,o introduce new concepts in order to complete the fun,.amcntal str~~ctrrrc of electrotlynamics. Whereas the excitation H = (X, 2)) is linl
-
Thereby wc can define the elcctric field strength E in 3-dimensional space. The electric field strength E has three independent components cxactly as the electric excitation 2). In order to link up mechanics with the rudimentary elcctrotlynamics of the first axiom, we consider a delta-function-like test charge current J = (3, p) centercd around some point with coordinates zZ.Generalizing (B.2.1), the simplest 4-dimensional aasatz for defining the electromagnetic field strength r e d s : force
-
field strength x charge current.
(B.2.2)
R.2. Lorentz force density
122
Also in four dimensions, the force N dL/3za is represented by the components f, of a, covector. Accortlingly, the ansatz (B.2.2) can be made more precise:
The componcnts fa represent a twisted form since they change sign under spatial reflections. Tlic Lagrttngian 4-form lias the absolute dimension of an action, h. Correspondingly the forcc, which is understood as the (variational) derivative of the action with respect to tlie spacetime coordinates, has the dimension [fa] =
11 [do]-' = (h t-', 11P-l) = (energy, momentum) .
(B.2.4)
A forcc cannot possess more tlian four indepentlcnt components. Since the current J in (B.2.3) is a twisted 3-form, tlie only possibility seems to be that f, is a covector-valuetl 4-form (rc.call the general defii~itiongiven in Sec. A.1.3, example 3). In other words, t,lic covector index transforms under the 4-dimensional linear group as the frame does, namely f,, = La!" f, (see (A.1.5)). Then f, has four components, indeed, and assigns t o ally 4-volume element the components of a covcctor:
3.2.2 Sccond axiom rclatir~gmechanics a n d elcct,rodynamics
n SI, we havc the abbreviations J = joule (not to be mixed up with the current ;-form J ) , V = volt, ant1 Wb = wober. Finally, thc force equation (B.2.3) for a t,ost chargc current rca.tls f, = , J F ) A J . Accordingly, wc tlefincd tlie fic?ld streligt,h F as the incorpora\"C tio n of the possible forces acting in an electroma.gnct,ic ficld on test ch:trgcs, thc?reby relating tlie mechanical notion of a force to tlle electromagnetic state s . the first axiom, we consider the nc-- lund elcctric charges and c ~ ~ r r c n tIn tive role of cllargc that creates the excitation ficlrl; here we study its pc~s.sive rol e, namely wliich forces nct on it in an clectroniagnctic environment. This ,,ncludcs orir hetvistic consitlcrations and we now turn to our new axiom. nn,
3.2.2 Second axiom relating mechanics and elcctrodynamics Ne havc then
f, = (e,, J F ) A J The anliolonornic componcnts of f,, namely f,,,,,,, determine the relative di~ilensionof f,. Sincc fi,,,,,, = f[,l,pal , one of the indices p , v , p, a has t o be a time index and the remaining t l ~ r e espace indices. Tlierefore, we find (W = watt, N = newton), =
(powcr density, force density)
(B.2.6)
Thus we have identified tlic f, in (B.2.3) indeed as 4-force density, the spatial This shows once more components of wliicli carry the relativc dimension o f f t3. that the assignment of the climensions in (B.2.4) was correct. As a consequence, tlie field F, in (13.2.3) turns out to be an untwisted covector-valued 1-form PC,= Fop190 with tlie coframc 6p. Tlic covector index of F, transforms a9. F,, = L,f"F,, whereas, as a 1-form, we have F, = Fa,dxz = Fap194. As sucli, Fa@ would Iiavc 1 G intlependcnt components in general. However, we know already tliat the elcctric field strength E has 3 independent components. If we expect the analogous to be true for the magnetic field strength B, then Fopshoultl carry G intlcpendent componcnts a t most. Then, it is near a t hand to nssume tliat FnPis antisymmet,ric: F-0 = -Fp, Accordingly, the electromagnetic ficld strength turns out, to be a 2-form F = Fap6, A 6 0 and F, = P , J F = Pep1 9 with ~ t,hc frame r , . For the absolute dimension of F this yields, if we use (B.2.3), (B.2.4), and [J]= q,
IF]
=
[F,] [ep]-'
=
[f,] [J]-'[ep]-'
= Itq-'
[da]-' [ea]-'
123
(second axiom) .
(13.2.8)
The untwzsted rlectromngnet~cfield strenqtli %-form F carries the absolutc di$1 mension of a magnet>icflux: [ F ]= hq-' Wl,. This equation for the Lorcntz forcc density yields an operntloncd definition of the elcctromngnctic field sticllgtll F and represents our secontl axiom of clccti otlyn:imics. Obsci vc that (B.2.8), like (B.1.17), is a11 equation that is free fiorn tlic met,ric and the connectioii; it is defined on any 4-dimensional tliffcrcntiable mnnifoltl. Since F A .J is a 5-form, (B.2.8) can alternatively be wrlttcn :is f, = -F A ((,, J J). A tlecoiiiposition of tlic 2-forin F into "timc" and "space" picccs accortliiig to
'
fields, in t,liree dinlensions, the ulltwistcd electric field strength 1-form E :tnd ,he untwisted magnc.tic ficltl st,rc>ngth12-form B,
r h e choices of the signs ill tliesc tlcfinitions art. rclatctl to the Lcnx rulc, inter tlia. Tllcy arc tliscusscd in section B.4.2. Tlic. absolute ant1 relative dimensiolis ' ~ h historical c names arc "electric field" for E ant1 "magnetic induction" for 13. T h e I S 0 namcs "electric ficld" and "tnagnctic flux density," respectively, again don't merge to a 4D nnmc. Note also that tile I S 0 narnes don't recognize the difrerence between n twisted and an untwisted form. 'rile 1-forms 'H ant1 E arc botll called "fielti" and 'D ant1 TI "flux dcnsity," if thcy were quantities of the same type. As a ~iiatterof fact,, thc four qrrantities 'H, 'D, E , B , constituting the electromagnetic field, obey four distinctly diflcrent transforn~ationlaws under tlifleomorphisrns (coordinate transformat,ions)..
B.2. Lorentz force density
124
of E and B arc, respectively,
[El
=
[E&]= [Elt-
[ B ]= hq-
=V,
1 SI
= V m-
',
1 SI
= Wb,
(B.2.11)
[B,[,]= [B]e- = Wh m-2 =: T , 2 SI
where T abbrcviatcs tesla. We can decompose the 1-form E and the 2-form B as follows: E = E, dz" and B = BUbdru A dz" = BCi,. For the 2-form i, in three dimensions, one sl~ouldcolnparc with (A.1.65). The compollents E, and = eabCBbc corres~ondto the conventional Cartesian components of thc clcctric and the magnetic field vectors and respcctively. E Clearly then, the electric line tension (elcctromotivc force or voltage)
a
an
a
125
We substitute this into the second axiom (B.2.8),
1 -1 SI
hq- t
B.2.2 Second axiom relating mechanics and electrodynamics
B,
Iince i is a 4-form, the expression in the square bracket is a 5-form and vanishes. rhen the antisymmetry of the interior product yields:
=onsequently, for the components 3, of the 1-form 3, we have
a1
and the magn,etic flux
J B must play a decisive role in Maxwell's theory. Hence, 0 2
vhereas the 1-form itself reads
as building blocks for laws governing the elcctric and magnetic field strengths,
we have only the elcctric linc tension ant1 the magnctic flux a t our disposal. Thc Lorentz forcc density f,, as a 4-form, that is, as a form of ~naximal rank ("top dimensional form"), is purely longitutlinal with respect to the normal vector n. Thus only its longitudinal piece (f,)l, a covector-valued 3-form, survives. It turns out to be
g=.F,19~= - q u ~F ,
with
u ~ F = 0 .
(B.2.20)
Phis is the point particle version of the Lorentz force, which is in accord with )ur starting point in (B.2.1). In contrast, we chose as a second axiom (B.2.8) ,he field-theoretical version of the Lorentz force, namely its density, since only n that version does there appear the current 3-form 3 of the first axiom. If there is an electromagnetic field configuration such that the Lorentz force lensity vanishes (f, = 0), we call it a force-free electromagnetic field:
If we now display its time and space components, we find 3ere we have substituted the inhomogeneous Maxwell equation. In plasma physics such configurations play a decisive role if restricted purely ,o tlic magnetic field. We (1 3)-decompose (B.2.21). A look at (B.2.13) and :B.2.14) shows that for the magnetic field only the space components (B.2.14) ~f the Lorentz force density matter:
+
The zero in kg carries a circumflex since it is an anholonomic index. The relative dirnensions pick U D a11 additional factor of P-3 m-3. The time component kg represents the electric power denszfly, the space components k, the 3-dimensional Lorentz force density. Let us come baclt to the Lorcntz forcc density fa. We consider now a point portrcle with net electric charge q and thc velocity vector IL = uNe,. Its electric currcnt J is a 3-form. Hence we define the flow 3-form of the particle as
with the volumc 4-for111i and the 3-form inof (A.1.74). Clearly, the flow 3-form U lias the same number of indcpcndcnt co~nponcntsas the velocity vector 11. Thcn the clcct,ric current of the point particle reads
[f the electric excitation and its time derivative vanish (V = 0, fi = 0) :learly a frame dependent, i.e., noncovariant statement - then we find for the force-free magnetic field2
or, in components,
2 ~ e Lundquist e (261 and Liist & Schfiiter 1271: However, they as well as later authors put €3 = po *1-I right away and start with 3-1 x V x 3-1 = 0.
13.2. Lorentz force density
126
B.2.3 @Thefirst three invariants of the electromagnetic field The first axiom supplied us with the fields J (twistcd 3-form) and H (twistcd 2-form) ant1 the secolld axiom, additionally, with F (untwisted 2-form). Alge\,raically we can construct therefrom the so-called first invariant of the clcctromagnetic field,
Il := F A H ,
[II]= It.
127
t.2.3 @Thefirst thrcc invariants of the electromagnetic ficlcl
This is as far as we can go by algebraic manipulations. If we differentiate, wc an const,rllct d.J, d H , clF. The first two expressions arc known, d J = 0 and !H = J , whcrcm the last one, d F , is left open so far. We turn to it in the next haptcr. Lct us collect our results in a table:
Table B.2.1: Invariants of tlie electromagnetic field in 4D.
(B.2.25)
I t is a twistcd 4-form or, equivdently, a. scalar density of weight +1 with one ill~le~endellt component. Clearly, I 1 could qualify as a Lagrange 4-form: It is a twisted form and it has tllc appropriate dimension. firtherrnore, a second invariant can be asscn~blcd,
Invariant
dimension
un-/twisted
Il=Fr\H
h
twistcd
I2= F A F
(l~/q)~
name N
Chern 4-form
untwisted
I,=HAH
q2
untwisted
14=AAJ
11
twisted
Lagrangian
... coupling term in Lagr.
a11 uiltwisted .?-form with the somewhat strangc dimension of the square of n magnetic flnx, and n third onc,
I3:= H A H ,
[I:3]= q 2 ,
(B.2.27)
equally an untwisted form. 111 order to get some insight illt,o tlie incailillg of these 4-forms, we s ~ ~ h s t i t ~ l t c the (1 + 3)-decompositions (B.1.39) and (B.2.9):
If for an clect,romagilctic field configuration the first invariant vanishes, then we find,
As wc will see in (B.5.61), this means that the magnetic energy dclisity equals the electric energy density. Similarly, for the second invariant, we have
Ttletl tllc electric field strength 1-form can he called "parallel" tlo tlic magnctic field strcngt,h 2-form. Analogously, for the vanishing t,liird invarinnt,
tlie excitations arc "parallel" to cach other. I t should bc underst,ood that the characterizatio~lsII = 0 and so on arc (diffeomorphism ant1 frame) iilvariallt, statclnent,~about electromaglletic field configurations.
rhc fourth invariant I4 and tlic names are explained in Sec. B.3.3. The I k l s Ire 4-forms, respectively, with one independent component each. We call them nvariants. With the tliainond operator 0 , tlic dual with respect t o the Lcvi3 v i t a epsilon (see (A.1.78) a t the end of Sec. A.1.9), we can attach t o each I-form Ika (metric-free) scalar density O I ~ .
Magnetic flux conservation
B.3.1 Third axiom The spacetime manifold, which underlies our consideration and which has been defined at the beginning of Sec. B.1.2, is equipped with the property of an orientation. Then we can integrate the untwisted 2-form F in four dimensions over a 2-dimensional surface. Since F is a 2-form, the simplest invariant statement that comes to mind reads
f
F=O,
w2=0
(third axiom),
(B.3.1)
For any closed 2-dimensional submanifold C;, c Xq. Indeed, this is tlic axiom we are looking for. It is straightforward to find supporting evidence for (B.3.1) by using the dccomposition (B.2.9). Faraday's induction law results from (B.3.1) if one cliooscs the 2-dimensional surface nq Ci = 80'3 with 52'3 = [ao,o]x Rb, whcrc 0; is represented in Fig. B.3.1 by a line in h,,; i.c., it is transversal to the vector field n:
What is usually called the law of the absence of magnetic charge we find Ily again choosing the Zdimensional surface C2 in (B.3.1) as the boundary of a 3-dimensional submanifold 0 3 that lies in one of the folia h, (see Fig. B.3.1):
130
R.3. Mngnctic flux conservat,ion
0.3.1 Third axiom
131
Figurc B.3.1: Different 2-tlimensional periods of tlic magnetic flux integral. Tlie 3-dimensional integration region 0; lies transversal t o tlie hypersurface 11, whereas 0 3 lies in it.
g
B=O
on:, c IL,,
T h c proofs are nnnlogons t,o thosc givcn in (13.1.13). They arc givcn bclow on a differelitid level. Magnetic flux B and its cons~rvatiollare of central importance t,o clcctrodynamics, At low tempcrat,nrcs, certain materials call become superconducting; i.c., they lose their electrical resistance, At the same time, if t,licy arc ~ X I I O S C ~ to an external (sufficiently weak) magnctic field, the maglietic field is expelled from tlieir interior except for a thin layer a t tlieir surface (Meissncr-Oclisciifeltl effect). In t,he case of a supercond~ictorof type I1 (niobium, for example), provided the external ficld is higher than a certain critical value, qt~nntizrdmagsI nctic flux lines carrying a flux quantum of1 die := li/(2r) = 2.068 x lo-'% can penetrate the surface of the supcrcontl~~ctor and build up a triangular lattice, an Abriltosov lattice. A cross section of such a flux-line lattice is tlcpicted in Fig. 13.3.2; a schcmat,ic view is provided in Fig. B.3.3. In high-tcniperaturc su~>crconductors half-integer flux quanta have also been ~ b s c r v c t l What .~ is important for us is that, at least under certain circumstances, single quantized d tliat tliey behave like a conserurd quantity; i.c., flux lines can be c o ~ ~ n t rand they migrate but arc not sl~ontancouslycreated nor destroyed. The counting argumcnt supports tlic view that magnctic flux is determined in a metric-frcc way, t,he migration argument, even if this is somewhat intlirctct, tliat the flux is conscrvcd. Tlit. computer sim~ilationof tlic magnetic ficld of the Earth in Fig. B.3.4 may give one an iiltuitivc feeling that tlie magnctic lines of force,
' I I r r r 11 is Planck's collstant and e t he elelncntary cllargc. 2 ~ e tlrr e discr~ssionof Tsuei alld Kirtlcy 1501.
Figurc B.3.2: Direct observation of individual flux lines in type I1 supcrconductors according to Essmaiirl 6c Tr1.iiublc [7, 81. Tlie image shown lierc belongs t o a small supcrcontluctin~Niobium disc (diameter 4 inm, thickness 1 inm) wliicli, s t a tcmpcrz~tureof 1.2 I<, was exposctl to an external magnetic excitatioii of H = 78 ltA/m. At the surface of the disc t,he flux lines were dccoratctl by small Ferromagnetic pnrt,icles that were fixed by a replica technique. Eve~ituallythe replica was 01,scrvctl by means of an electroll microscope. Tlie parameter of the flux-line lattice was 170 nm (courtesy of U. Essmann).
R.3. Magnetic flux conservation
132
R.3.2 Electromagnetic potential
133
Figure B.3.3: Sketch of an Abrikosov lattice in a type I1 superco~ldtictorin 3-dimensional space. which can be understood as unquantized magnetic flux lines, arc controlled by the induction law and, at the same timc, are close to our visual perception of magnetic field configurations.
B.3.2 Electromagnetic potential In (B.3.1) we specialize to the case when C2 is a 2-boundary C2 = 8 R 3 of a11 arbitrary 3-dimensional domain f13. Then, by Stokes' theorem, we find /dF'=0.
(B.3.4)
Since RJ can be cllosell arbitrarily, the electromagnetic field strength trlrns out to be a closed form:
This is the 4-dimensional version of the set of homogeneous Maxwell cq~~ations. The axiom (B.3.1) now shows that all periods of F arc zero. Consequently the field strength is an exact form F=dA.
(B.3.6)
Equation (B.3.5) is implied by (B.3.6) because of dd = 0. However, the inverse statement that (B.3.5) implies (B.3.6) does not hold in a global lnanncr unless the conditions for the first de Rham theorem arc met.
Figr~rcB.3.4: Figure of G. A. Glatzmaier: Snapshot of magnetic lines of force in the core of our computer-simulated Earth. Lines are in gold (blue) where they are inside (outside) the inner core. The axis of rotation is vertical in this image. Thc ficld is directed inward at the inner core north pole (top) and outward at the sotit11 pole (bottom); the maximum magnetic field strength is about 30 mT (scc [14] and (431). In color you can look at such pictures at the website htt~://www.cs.ucsc.cc11~/~glatz/geodynamo.htn~l. The untwisted electromagnetic potential 1-form A has the dimension of Dccomposcd in "timc" and "space" pieces, it reads
11 q - l .
B.3. Magnetic flux conservation
134
with cp := -AL
and
A := 4 .
(B.3.8)
Here we recover the familiar 3-dimensional scalar and covector potentials cp and
B.3.3 @AbelianChern-Sirnons and Kiehn 3-forms
135
which has a topological meaning in that it is related to the winding number of the so-called U(1) vector bundle of electromagnetism. Also in all other dimensions with n 3, the Abelian Chern-Simons form is represented by a 3-form. If we (1 3)-decompose C A ,we find
+
>
A, rcspectively. The potcntial is only determined up t o a closed 1-form
a fact that has far-reaching consequences for the quantization of the electromagnetic field. We decompose (B.3.6) and find straightforwardly
which includes the so-called magnetic helicity density3 A A B, which is also a 3-form (however in 3D) with onc independent component. Obviously,
Consequently, even though CA is not gauge invariant, its differential ~ C isA gauge invariant. What we just did t o the second invariant, we can now implement in an analogous way for the first invariant:
and
A decomposition of (B.3.5) by means of (B.1.28) and (B.2.10) yields the homogeneous set of Maxwell's equations,
or in 3-dimensional notation,
~ -
E + B = O
We define the twisted Kiehn. 3-form"
It carrics the dimension of an action. In n dimensions, it would bc an (n - 1)form - in sharp contrast t o the Abelian Chern-Simons 3-form. If we decon~pose I( into 1 3, we find
+
and
dB=O, respectively. If we integrate these cquations over a 2- or 3-dimensional volulnr and apply Stokcs' theorem, we find (B.3.2) and (B.3.3). Again, by analogy with the inhomogeneous equations, (B.3.13) and (B.3.14) represent an evolution equation and a constraint, respectively.
The 2-form A A 7-t is the purely magnetic picce of I( in 3D. By differentiating K, or by rearranging (B.3.19), we find
Evcn though thc I
+
B.3.3 @AbelianChern-Simons and Kiehn 3-forms For the electromagnetic theory, we now have the 1-form A as a new building block. This has an immediate consequence for the second invariant, namely
It decomposes according to since d F = 0. In other words, 12,the so-called Abelian Chern 4-form, is an exact for111 and, accordingly, cannot be uscd as a nontrivial Lagrangian. We read off fro111 (B.3.15) the untwisted Abelian Chem-Simons 3-form
3 ~ h magnetic e helicity is given by the integral and Raliada ancl Dueba[40, 41, 49)). 4See Kiehn and Pierce [24, 22, 231.
1" A A B (see Moffatt 1331, Marsh (28, 293,
13.3. Magnetic flux conservatiorl
136
R.3.4 Measuring the excitation
137
Because d J = 0, it is gauge invariant up t o an exact form: I4
+
I4+(d?j,)AJ=I4+d(?,bAJ).
(B.3.25)
Accordiligly I,,, like 11, also qualifies as a piece of a11 clcctrodynanlic Lagrangian 4-form since it is twistccl, of dimension h , and gauge invariant (up to an irrelcvant exact form). Note, however, that d K in (B.3.22), as an cxact form, cannot feature as a Lagrangiali in 4D even though botli pieces on t,hc right-hand sitle of (B.3.22) indcpel~tlent~ly have the correct behavior. In other words, tlie relative factor bcwccn II and I4is inappropriate for a Lagrange 4-form. Let us put our results together. If we start with the 2-form H and differentiatc~ and multiply, then we find the sequence ( H , d l J , H A H ) . We cannot go ally further since forms witli a rank p > 4 will vanish identically. Similarly, for thc 1-form A, we have (A, dA, A A dA, dA A dA) and, if we mix botli, tlic sequence (A A H , dA A H, A A d H ) . Note that A A A =. 0 since A is a 1-form. I11 tliis way, we create the new 3-forms I( (see (B.3.20)) and CA (see (13.3.16)): Table B.3.1: Three-forms of tlie electroinagnetic field in 4D.
3-form
dimension
un-/twisted
name
I(=AAH
i~
twisted
I
untwisted
Cliern-Simons
ulitwisted
electric clirrent,
CA=AAF
J
(i~/q)~ (1
By tlic snme tol<en, tlic four different illvarialits .,I of the c1cctrom;ignetic fieltl arise as 4-form (srr Table B.2.1, p. 127).
+++ + +++
external charges
+
++++
Figure B.3.5: Measurement of D ' a t a point P. Technically, one can realize the limiting process by constructing the plates in the form of parallclograrns with tlie sides a d l and a 6'2. Then (B.3.26) is replaccd by
&
'Dl, = lim - . n-o a2
(B.3.27)
Repent tliis lneaslirement witli the t,wo other possible orientations of the plates (be careful t o choose one of the two plates in accordance with the orientation prescription above). Then, similarly, one finds 'D23 and '031. Thus finally tlic elcctric cxcitation is measured:
13.3.4 Measuring the cxcitation The electric excitation 2) can be operationally defined using the Gauss law (B.1.45). P u t a t some point P in free space two small, tllin, and electrically conducting (metal) plates of arbitrary shape with insulating handles ( "Maxwelliaii double plates") (see Fig. B.3.5). Suppose we clloose local coordinates ( x l , .r2, .r3) in the ncigl~l~orl~ood of P s11c11that P = (0,0,0).The vectors a,, witli a = 1 , 2 , 3 , span tlir tangcnt space a t P. Press the pltites together a t P and oricnt them in such a way tliat tlicir common bountlary is given by tlic cqllatioii .r3 = 0 (and thus ( a ' ,x 2 ) arc the local coordinates 011 each plate's surface). Separate the plates ant1 measure the net charge & and the area S of that plate for which the vector points ol~twardsfrom its boundary surface. Determine expcrimel~tally liins+o & / S as well as possible. Then
Tlir tlimcnsion of D ' is tliat of a charge, i.e. ['Dl = q; for its components we have [Dab]= q pP2. Let us prove the correctness of this prescription. The double plates are assullied to Ile ~ d r acondc~ctors. l If an external field is applicd t o an ideal conductor, a redistribution of the carriers of tlie electric charges (usually electrons) takes place in its interior by means of flowing electric currents such that the Lorentz force acting on each L L ~ l e ~ t rfinally ~ l i l l vanishes. This was found in experiments alitl later nsed to define an ideal condl~ctor.I11 the end, no interlial currents are flowing any more and, relative to tlic conductor, the charge distribution is a t rest. Tlicreforr the vanishing 3D Lorentz force density (B.2.14) iniplies that the electric fieltl E inside tlie conductor has to vanish as well:
q
inside
= 0.
R.3. Magnetic flux conservation
138
R.3.4 Measuring the excitation
139
The vanishing of E defines a unique clcctrody~~amical state in the conductor. In particular, a certain elcctric charge distribution is specified. If we asslime that (B.3.29) implies
7
inside
= 0'
(B.3.30)
then this charge distribution, as we will see in a moment, determines 2) uniquely: We can read off from the charge distribution the electric excitation."hat (B.3.29) has (B.3.30) as a consequence is a rudiment of the spacetime relation that links F and H and that makes the excitation field V unique. For that, reason we have postponed this cliscussion until now, since knowledge of the notion of the field strength E is necessary for it. By (B.3.30), we selected from the allowed class (B.1.21) of the excitations that 2) that is measured by thc double plates. In a more general setting, let us consider the electromagnetic excitation 2) near a 2-dimensional boundary surface S c h, that separates two parts of space filled with two different types of matter. Choose a point P E S . Introduce local coordinates ( x l , x2,x" in the neighborhood of P in such a way that P = (0,0,0) while ( z l , x2) are the coordinates on S (see Fig. B.3.6). Let V be a 3-dimensional domain that is half (denoted Vl) in one medium and half (V2) in another, with the two halves Vlf2 spanned by the triples of vectors (dl, a2,ad3) and (dl,d2, -a&), respectively. For definiteness, we assume that 83 points from medium 1 to medium 2. Now we consider the Gauss law (R.1.45) in this domain. Integrate (B.1.45) over V and take the limit a -+ 0. The result is
where V(1) and V(2)denote the values of the electric excitations in the medium 1 and 2, respcctjvcly, while ASl2 is a piece of S spanned by (dl, 82). Despite the fact that the volume V clearly goes to zero, the right-hand side of (B.3.31) is nontrivial when there is a surface charge exactly on the boundary S. Mathematically, in local coordinates ( z l , r2,x", this can he described by the 6-function structure of the charge density:
Figure B.3.6: Electric excitation on the boundary between two media.
Returning to the measurelnent process with plates, we have = 0 inside an ideal conductor, and (B.3.34) justifies the definition of the elcctric excitation as the charge density on the surface of the plate (B.3.26). Thereby we recognize that the electric excitation V is, by its very definition, the ability to separate charges on (ideally conducting) double plates. A similar result holds for the magnetic excitation 3-1. Analogously to the small 3-diinensional domain V, let us consider a 2-dimensional domain C that is half ( E l ) in one medium and half (C2) in another one, with the two halves ClP2 spanned by the vectors (1, ad,) and (1, -ad3), respectively. Here 1 = I'dl 12d2 is an arbitrary vector tangent to S at P. Integrating the Maxwell equation (B.1.44) over C, and taking the limit a -+ 0, we find
+
Substituting (13.3.32) into (B.3.31), we then find
where F = Flzdxl A dx2 is the ,?-form of the electric surface charge density. Since P and ASl2 are arbitrary, we conclude that on the separating surface S between two media, the electric excitation satisfies 5 ~ h only e really given by Pohl [36].
appropriate discussiot~of the definitions of V and E thnt we know of
is
The second term on the left-hand side of (B.1.44) has a zero limit because fi is continuous and finite in the domain of a loop that contracts to zero. However, the right-hand side of (B.3.35) produces a nontrivial result when there are surface currents flowing exactly on the boundary surface S. Analogously to (B.3.32), this is described by
13.3. Magnetic flux conservation
140
R.3.4 Measuring t h e excitation
141
+
they generate an ?find t h a t compensates the 'H t o be measured: Xi,,,! 3-1 = 0. We have t o change the angular orientation such that we eventually find the maximal current Smnx. Then, if the x 3 -coordinate axis is chosen tangentially t o the "maximal orientation," we have 3-13
= lim L-0
3mnx
L '
where L is the length parallel t o the wire axis trallsverse t o which the currcnt has been measured. Accordingly, we find
SI
I
Figure B.3.7: Measurenlent of 3-1 a t a point P. We denote the surface currents by j. Substituting (B.3.36) into (B.3.35), we find
and, since 1 is arbitra.ry, eventually:
Thus we see that on the boundary surface between the two media the magnetic excitation is clircctly related to the 1-form of the electric surface cumn,t den,sity j . To measure 3-1, followillg a suggestion by Zirnbauer,%e can use the Meissner effect. Because of this effect, the magnetic field B is driven out of the superconductor. Take a thin superconducting wire (since the wire is pretty cold, pcrhaps around 10 K , "taking" is not to be understood too literally) and put it a t the point P where you want to measure 3-1 (see Fig.B.3.7(a)). Because of tllc Meissner effect, the magnetic field B and, if we assume analogously t o the electric case that B = 0 implies 3-1 = 0, the excitation 3-1 is expelled from the supercontlucting region (apart from a thin surface layer of some 10 nm, where 'H can penetrate). Accordillg to the Oersted-AmpBre law, (B.1.44) (t3-1= j (we assume quasi-stationarity in order t o be permitted to forget about v ) .T h e compcnsation of 3-1 a t P can be achieved by surface currents j flowing around the superconducting wire. These induced surface currents are of such a type that
-
3
'See his lecture notes [55];compare also Ingarden and Jamiolkowski [17].
Clearly the dimension of 3-1 is that of an electric current: [HI = qt-' = A and sI ['Ha] = q t-' t-' = A/m ("ampere-turn per meter"). This is admittedly a thought experiment so far. However, the surface currents are experimentally known t o exist.' Furthermore, with nanotechnological tools i t should be possible t o realize this experiment. It is conccivablr that the surface currents can be measured by means of the Coulomb drag.8 Alternatively, one can take a real small test coil a t P , orient it suitably, anrl read off thc corresponding maxima1 current a t a galvanometer zs soon as t h r effective 3-1 vanishes (sce Fig. B.3.7(b)). Multiply this current with the winding number of the coil and find j,,,. Divide by the length of the test coil and you are back t o (B.3.39). Whether 3-1 is really compensated for, you can check with a magnetic needle which, in the field-free region, should be in an indifferent equilibrium state. Note that today one has magnetic "ncedlrs" of nanometer size. In this way, we can built1 up the excitation H = 2) - 3-1 A da as a rncasurablc electromagnctic quantity in its own right. The ~xcit~ation If, together with the field strength F , we call the "elcctromagnetic field."
'see the corresponding recent magneto-optical studies of Jooss et al. [21]. BThe idea is the following (M. Zirnbnuer, private communication): If in a first t h ~ nfilm a currcnt flows (this represents the surface of the sr~perconductor),the11 via the Coulornh interaction, it can drag along with it in a separate second concl~~cting thin film the c l ~ a r g r carriers which can then be measured (outside the superconductor). This Coulomb drag, which is known to exist (see [44]), requires, however, the breaking of trnnslational invariauce.
Basic classical electrodvnamics summarized, example d
B.4.1 Integral version and Maxwell's equations We arc now in a position to sunitllarixc the fulidalncntal structure of electrodynalnics in a fcw lirics. According to (R.1.17), (B.2.8), ancl (B.3.1), the tlirce axiorns on a collnected, Hausdorff, paracompact, and oriented spacetimc read, for any C3 :iiic1 C2 wit11 aC3 = 0 ant1 3C2 = 0:
Tlic first xiom om governs matter and its conserved electric charge, the sccond axioln linlts tlic notion of that charge and the concept of a mechanical forcc t o an operational definition of the elcctrolnagnetic field strength. The third axioin dcterrilincs the flux of the ficld strength as sourcefree. 111 Part C we learn that a mctric of spacctiinc brings in tllc temporal and spatial distance concepts and a linear connection the inertial guidance ficld (and thclcl~yparallel disp1:tccment). Since tlic mctric of spacctinie represellts Einstein's gmv1tnt7onnl potential (and thc lincar conncction is also rclatcd t o gravitatiollal properties), the three axioins (B.4.1) of clcctrody~~amics arc not colltnminatcd by gravitational properties, in contrast to what happens in the usual tcxtboolt approacli t o electrodynamics. A curved mctric or a non-flat linear conllcction do not affect (E.4.1), sincc these geolnetric objccts don't enter tlic axioms. U p to now, we could do without a mctric and without a connection. Yet we tlo 11:tvc the basic Maxwcllian strlicturc already a t our tlisposal. Conseclucntly, the strncture of clcctrodynamics that has cmergcd so far has nothing
144
13.4. Basic cla3sical electrodynamics summarized, examplr
B.4.1 Integral version and Maxwell's equations
145
t o do with Poincarb or Lorentz covariancc. The transformations involved arc diffeomorpliisms and frame transformations alone. If one desires to generalize special relativity to gencral relativity theory or to the Einstein-Cartan theory of gravity (a viable alternative to Einstein's theory formulated in a non-Riemannian spacetime; see again Part C), then thc Maxwellian structure in (B.4.1) is untouched by it. As long as a 4-dimensional connected, Hausdorff, paracompact and oriented cliffcrcntiablc manifold is used as the spacetime, the axioms in (B.4.1) stay covariant and remain the same. In particular, arbitrary frames, holonomic and anliolonomic ones, can be used for the evaluation of (B.4.1). For what reasons are charge conservation (B.4.1)1 and flux conservation (B.4.1)3such "stable" natural laws? Apparently sincc the clementary "portions" of charge e/3 and of flux .rrti/e (with e = clemcntary charge and h = (reduccd) Planck constant) are additive 4-dirncnsional scalars that are constant in timc and space. In other words,' e = const,
e, ti are 4D-scalars ,
ti = const,
(B.4.2)
guarantees that elcctric chargc and magnetic flux arc conserved quantities in classical electrodynamics. Thereforc (B.4.2) secures the validity of the first and third axioms. A time dependence of e and/or ti, for examplc, would dismantle the Maxwellian framework. According to (B.4.1), the more specialized differential version of electrodynamics (skipping the boundary conditions) reads as follows:
dJ = 0, J = dH,
f,=(e,_rF)~J,
dF=0, F = dA.
(B.4.3) I
This is thc structure that we defined by means of our
with
dx=O,
(B.4.4)
very seriously, since measurable quantitics must be gauge invariant. The system (13.4.3) can be straightforwardly translated into the Excalc language: We denote the electromagnctic potential A by p o t i . The "1" we wrote in order to remember better that the potential is a I-form. The field strength F is written as f arad2, i.e., as the Faraday 2-form. Thc excitation H is named e x c i t 2 , and the left-hand sides of the homogeneous and the inhomogeneous Maxwell equations are called maxhom3 and maxinh3, respectively. Then we necd the elcctric current density c u r r 3 and thc left-hand side of the continuity equation cont4. For the first ,axiom, we have 'See the corresponding discussion of Peres 1351.
Figure B.4.1: Faraday-Schouten pictograms of the electromagnetic field in 3dimensional space. The images of 1-forms are represented by two neighboring planes. The nearer the planes, the stronger the 1-form is. The 2-forms are pictured as flux tubes. The thinner the tubes, the stronger the flow. Thc difference between a twisted and an untwisted form accounts for the two different types of 1- and 2-forms, respectively.
pform cont4=4, curr3=3, maxinh3~3, excit2=2$
ancl for the sccond and third axiom, I
pf orm f orce4 ( a ) = 4 , maxhom3=3, f a r a d 2 ~ 2 ,pot l = l $ /', has t o be preceded by a coframe statement frame e$ f arad2 := d potl; f orce4(-a) : = ( e ( - a ) -If arad2)^curr3; maxhom3 : = d farad2;
Tlicse program bits and picces, which look almost trivial, will be integrated into a complete and executable Maxwell sample program after we have learned about the energy-momentum distribution of the electromagnetic field and about its act,ion.
R.4. R,wic classical electrodynamics summarized, example
l4G
The physical interpretation of equations (B.4.3) can be found via the (1 dccolnposition that we had derived earlier as
+ 3)-
(sce (B.1.35), (B.1.39), (B.2.9), and (B.3.7), respectively). The signs in (B.4.5) to (B.4.8) will bc discussed in the next section. A 3-dimensional vis~~alization of the clcctron~agncticfield is given in Fig. B.4.1. Wc first concentratc on equations that colltain only measurable quantities, na~nelytlic Maxw(~11equations. For thcir (1 3)-decomposition we found (scc (B.1.44), (B.1.45) and (B.3.13), (B.3.14)),
+
v =dX-j B
=
-d E
(1 constraint eq.) , (3timecvol.eqs.),
(1constraint eq.) , (3 time evol. eqs.) .
Accordingly, we have 2 x 3 = 6 time cvolution equations for the 2 x 6 = 12 variables (V, B , X , E) of t,hc electromagnetic field. Thus the Maxwellian structurc in (13.4.3) is underdetermined. We need, in addition, an electromagnctic spacctimc rclation that expresses the excitation H = ('FI,V) in terms of the ficld strength F = (E,B ) , i.c., H = H [ F ] .For classical electrodynamics, this functional bccomcs the Maxwcll-Lorcntz spacetime relation that we discuss in Part D. Whcrcas, on thc unquantizctl levcl, tlic Maxwellian structure in (B.4.1) or (B.4.3) is bciicved to be of universal validity, the spacetime relation is more of an ad lloc nature ant1 alnenablc to corrections. The "vacuum" can havc different spacetimc rclations depending on whether we take it with or without vacuum polarization. Wc will come back to this question in Chapters D.6 and E.2. An overview of thc clectron~agneticficld and its source in (1+3)-decornpositio~~ is given in Table I, the corresponding SI units are displayed in Tablc 11.
B.4.2 @Lenzand anti-Lenz rule In gcncrally covariant form, tlic Maxwell cquations read dH = J and dF = 0.
+
However, their physical interpretation requires, via (B.4.5)-(B.4.7), an (1 3)-decomposition. We discussed the operational interpretation of the different quantities in ordinary 3-space, and now we turn to the signs involveda2Generally, we could write thc 4-current as 2 W follow ~ lierc the paper of Itin et nl. [ l n ] , scc also the references given tlicre.
B.4.2 @Len2and anti-Lenz rule
147
148
13.4 Basic classical electrodynamics summarized, cxamplc
Table 11. SI units of the electromagnetic field and its source: C = coulomb, A = ampere, Wb = webcr, V = volt, T = tcsla, m = meter, s = sccontl. Thc units oersted and gauss arc phased out ant1 do not exist any longer in SI. In the second coll~mnwe fintl tlie units related to the absolute and in tlie third colulnll those related to tlic relative dimensions.
B.4.2 @Lenzand anti-Lenz rulc
149
where we introduccd again the sign factors hT anti hs with values frorn { + I , -1). We tliffcrcnt,iate (B.4.14)
Accordingly, the inhomogeneous Maxwell equation d H = J (see (B.4.9)) with the source (B.4.13) decomposes as
The convcntiollal definition of the sign of the clectric excitation is that the (positive) electric charge p is its source: d V = p. Consequently, hs = -iT and (B.4.1G)1, tlw Ocrstecl-Amp&rc-Maxwelllaw, reads
The cont,rih~~tions of j and
v adtl up correctly to the the Maxwell current j +v.
In analogy to V, tlie magnetic excitation 31 is defined such as to have j
+ V as
its source:
Accordingly, where we introduced tlic factors ZT and is with values from { + l , -1). Any factor the absolute value of which differs from 1 is assumed t o be absorbetl in the corresponding form. Htwcc (B.4.11) is the most general drcornposit,ion rclntivc t80n given foliation. We differelitiatc (B.4.11) and use d p = 0. Tlicn the first xiom om yicltls
We know that p is t,hc charge dcnsity and J tlic current density. With tlic usual conventions on orientation, namely tliiit tlic outward direction fro111 an cwcloscd 3D volu~ilris positive ctc., both quantitirs fulfill the continuity cquntion (B.1.12). Thus, zs = -zT and
We liavc not yct made n co~ivcntionfor t l ~ sign c of the charge. If wc lnaliipulatc wit11 ;I cnt's skin ;ultl a rod of amber, we can take tlie conventional choice. This amounts to iT = -1. Then t,lie decompositions (B.4.13) anci (B.4.19) coincide wit11 (B.4.5) and (B.4.6)) respectively. Finally, we atldrcss the field strength:
The Lcnz fnct,or fA can be either +1 or -1. The overall sign of F is again chosen by a convcnt,ion in connection with the sign of the Lorcntz force relative to a positive charge. Conventionally, f T = 1. Tli? liomogencous Maxwell equation dF = 0 (see (B.4.10)) decomposes accordil~gto
Let us now turn to the inlioniogcncous Maxwell equation. For the cxcitatioll we liavc Thus, the only sign left open is the one in Farnday's induction law (B.4.21),. Thc clioicc f A = +l is cq~iivnlentto the Len,z rule: The induced voltage S E is
150
B.4. Basic classical electrodynamics summarized, example
directed such that it acts opposite to the B that creates it. The (unphysical) choice f A = -1 we call the anti-Lenz rule. Our cl~oicein (B.4.7) corresponds to the Lenz rule. We will justify this later. Therefore, we have derived all signs in (B.4.5)-(B.4.7) with the exception of that of the Lenz factor f A . Since the Lenz rule follows from energy considerations, we have to postpone further discussion until we have an axiom about the electromagnetic energy distribution. At the end of section B.5.3, we will resumc our discussion.
B.4.3 @Jumpconditions for electromagnetic excitation and field strength
I
ll
I
I
Equations (B.3.34) and (B .3.38) represent the so-called jump (or continuity) conditions for the co~nponentsof the electromagnetic cxcitation. In this section we give a more convenient fomulation of these conditions. Namely, let us consider on a 3-dimensional slice h, of spacetime (see Fig. 13.1.3) an arbitrary 2-dimensional surface S , the points of which are defined by the parametric equations
Here [" = ([i,[2) arc the two parameters specifying the position on S. We denote the corresponding indices A, 13,.. . = i , 2 by a dot in order to distinguish them from the other indices. We assume that this surfacc is not moving; i.c., its form and position are the same in every a = it const hypersurface. We introduce the 1-form density u normal to the surface S ,
R.4.4 Arbitrary local noninertial frame: Maxwell's equations in components
151
are the excitation forms in the first half and 2)(2) and 'Fl(2) Here, D(l) and in the second half of the 3D space h,. One can immediately verify that the analysis in Sec. B.3.4 of the operational determination of the elcctromagnetic excitation was carried out for the special x2 = t', x3 = 0 (then case of a surface S defined by the equations z1 = v = dx%~nd T A = a,, A = 1,2). It should be noted though that the formulas (B.3.34) and (B.3.38) are not less general than (B.4.25) and (B.4.26) since the local coordinates can always be chosen in such a way that (B.4.25), (B.4.26) reduces to (B.3.34), (B.3.38). However, in practical applications, the use of (B.4.25), (B.4.26) turns out to be more convenient. In an analogous way, one can derive the jump conditions for the components of the elcctromagnetic field strength. Starting from the homogeneous Maxwell equations (B.3.13) and (B.3.14) and considering their integral form near S, we obtain
ci,
As above, B ( l ) and E(l)are the field strength forms in the first half and E(2) and E(2)in the second half of the 3D space h,. The homogeneous Maxwell equation does not contain charge and current sources. Thus, equations (B.4.27), (B.4.28) describe the continuity of the tangential piece of the magnetic field and of the tangential part of the electric field across the boundary between the two domains of space. In (B.4.25) and (B.4.26), the components of the electromagnetic cxcitation are not continuous in general, defining the corresponding with t,hc surface charge and current densities discont,inuitics.
F, 7
13.4.4 Arbitrary local noninertial frame: Maxwell's equations in components and the two vectors tangential to S,
Accordingly, we have the two conditions T A J v = 0. The surface S divides the whole slice h, into two halves. We denote them by the subscripts (1) arid ( 2 ) , respectively. Then, by repeating the limiting process above for the integrals near S , , we find, instead of (B.3.34) and (B.3.38), the jump conditions
In (B.4.9) and (B.4.10), we displayed the Maxwell equations in terms of geometrical objects in a coordinate and frame invariant way. Sometimes it is necessary, however, to introduce locally arbitrary (co)frames of reference that arc noninertial, that is, accelerated in general. Then the components of the electromagnct~icfield with respect to a coframe go, the physical components emerge and the Maxwell equations can be expressed in terms of these physical components. Let the current, the excitation, and the field strength be decomposed according to
I
152
1 H=-H,p6"~6~, 2
1 F=-F,p6"~6~, 2
153
R.4.5 @Elcct.rodynamicsin flatland: 2DEG and QIIE
B.4. Basic claqsical electrodynamics summarized, example (B.4.30)
respectively. We substitute these expressions into the Maxwell equations. Then the coframc needs t o be differentiated. As a shorthand notation, we introduce the anholonomicity 2-form C" := d6" = &C,," 61' A 6" (see (A.2.35)). Then we straightforwardly find:
~ , the cxcitaIf we use the (metric-free) Levi-Civita tensor density E " ~ Ythen tion and the current,
are rcprescntcd as densities. Accordingly, Maxwell's equations (B.4.3) in components read ~ltcrnat~ively
+
Figure B.4.2: Electrodynamics in 1 2 dimensions: Tlic arsenal of clcctromagnetic quantities in flntlnnd. Tlic sources arc tlie charge tlcnsity p,, and the current dcnsity (1, ,jy)(see (B.4.40)). The excitations (Vz, V , ) and 7-l (twisted lc for ~ n c a s ~ ~ r i n p scalar) nrc so~newhat11nusua1(scc (B.4.41)). T h e t l o ~ ~ b p1:ttes V , c.g., become double wires. The magnetic fic.1~1 has only o ~ i cindrpentlent colnponcnt B,,/, whereas tlic clcctric ficld has two, na~nely(E,, ETl) (sce (B.4.42)).
T h e tcrlns with thc C's emerge if Maxwell's equations are referred t o an arbitrary local (co)frame. Only if we restrict ourselves t o natural (or coordinate) frames is C = 0, and Maxwell's equations display their conventional form.3 This rcprcscntation of electrodynamics can be used in special or in general relativity. If one desires t o employ a laboratory frame of reference, then this is the way t o tlo it: T h e object of anholono~nicityin the lab frame has t o be calculated. By s ~ ~ b s t i t u t i nitg into (B.4.31) or (B.4.33), we find the Maxwell cqr~ationsin terms of tlic components FaBand so on of the electromagnetic ficld quantities witli respect to the lab frame - and these arc the quantities one observes in the laboratory. Therefore the F,p and so on are called physzcal components. As soon as one starts from (B.4.3), the derivation of the sets (B.4.31) or (B.4.33) is an elementary exercise. Many discussions of the Maxwell equations within special relativity in nonincrtial frames could be appreciably shortened by using t>liisformalism.
first axiom, then the rank of t,lic clcctric current lnust 1)c 17.- 1. The force tlcnsity in mcclianics, in accordance witli its tlcfinition aLli3.x' within the Lilgriillg~ forn~alisn~, slioultl rcnlain a covoctor-va111od11-form. Hcncc wc kccp t,lic? sccontl axiom in it,s origin;tl forln. Accordingly, thc ficltl strc?ngt,h F is again n 2-forni: N
.f
fO=(r(?~F)AJ,
C., 1
.f
F
=
0.
(B.4.34)
((2
This nlay sccnl liltc an academic exercise. However, a t lcmt for 71 = 3, tlicrc cxists an application: Sincc the niidtlle of tlic lSGOs, cxpcrimcntnlists w ~ r ablc c to create [t ,?-d7rnrns1onol rlrctron gas (2DEG) in suitable transisto~sa t sufficiently low tcinl~c.raturcs ant1 to position the 2DEG in a strong cxtc'rn:tl transversal niagnctic field. Undcr such circ~lmstanccs,the electrons can only move in a planc transvcrsc t o B and one spacc tlimension can bc supprcssctl.
B.4.5 @Electrodynamicsin flatland: 2-dimensional electron gas and quantum Hall effect
Electrodynamics in 1
Our formulation of elcctrodynan~icscan be generalized straightforwardly t o arbitrary di~nensionsn. If we assume again the charge conservation law a5 a "o formulate electrodynamics in acceleratcd systems by means of tensor analysis, sce J. Van Blatlel [51]. New experiments in rotating frames (with ring lavers) can be found in Stcclman [47].
J=0,
+ 2 dimensions
In clcct,rotlyna~nicswit,li onc time ant1 t,wo spacc dimensions, wo have from thc first and third axiolns (in arbitrary coordinates),
1
I
and
R.1. I3asic classical electrodynamics summarizcd, examplr
154
respectively, wliere we indicated the rank of thc forms explicitly for better transparency. Here, thc remarkable featurc is tliat field strength F and current J carry thc same rank; this is only possible for n = 3 spacctime dimensions. Moreovcr, the c~irrentJ, the excitation IT, and the field strength F all liavc thc snmc number of indcpcndent components, namely threc. Now wc (1 2)-dccomposc the current and the electromagnetic field:
+
twisted 2-form: twisted 1-form:
(2)
J
=
-j A
d o + p,
H = -'FI d a + V , (1)
(B.4.37) (B.4.38)
Accordingly, in the space of the 2DEG, wc 11:tvc (again in arbitrary coordinates)
We recognize thc rather degcncrate nature of such a system. The magnetic field B , for example, 11213 only one intleprndent colnponcnt B I Z .Such a configuration is visualized in Fig. B.4.2 by using rcctilincar coordinates; they need not bc Cartesian coordinatcs. Chargr conservation (B.4.35) in dccomposrd form and in components rcads,
+
The (1 2)-dcconiposcd Maxwcll cquations look exactly as in (B.4.9) and (13.4.10). Wr also C X P ~ C S S them in componcnt,~:
R.4.5 @Electrodynamics in flatland: 2DEG and QHE
155
Maxwell equations (B.4.44) to (B.4.48) would still be valid since they had been derived for arbitrary (curvilinear) coordinates and arbitrary frames. Before we can apply this formalism to the quantum Hall effect (QHE), we irst remind ourselves of the classical Hall effect (of 1879).
Hall effect (excerpt from the l i t e r a t ~ r e conventional ,~ formalism) rhis subsection on the Hall effect and the next one on QHE are excerpts from ;he literature. Thus equations (B.4.49) to (B.4.56) are written in the conven;ional notation, and they refer to Cartesian coordinates and may contain the metric of 3-dimensional space. However, our subsequent presentation of QHE, which starts with (B.4.57) and which contains all relevant features of a phenomenological description of QHE, is strictly metricfree. We connect the two yz-faces of a (semi)conducting plate of volume I, x 1, x 1, with a battery (see Fig. B.4.3). A current I will flow and in the plate the current density is j,. Transverse to the current, between the contacts P and Q, there exists no voltage. However, if we apply a constant magnetic field B along the z-axis, then the current j is deflected by tlie Lorentz force and the Hall voltage Uw occurs which, according to experiment, turns out to be
BI
i3
(B.4.49) with [AH]= - . UII = R H I = AH - , 1, (I RIj is called the Hall resistance and All the Hall constant. We divide UH by 1,. Because E, = UH/ly, we find
Lct us stress that the classical Hall effect is a volume (or bulk) effect. It is to be dcscribetl in the frameworl< of ordinary (1 3)-dimensional Maxwellian electrodynamics.
+
Quantum Hall effect (excerpt from the literat~re,~ conventional formalism) and
Wc assume an infinitr cxtcnsion of flatland. If tliat cannot be assumed as a valid npproximat,ion, onc has to allow for linc currents a t tlie boundary of flatlancl ("cdgc curr~nts")in ordcr to fulfill t,he Maxwell cquations. In our formulation thc Maxwell cquations don't tlepcnd on the metric. Thus, instead of thc planc, as in Fig. B.4.2, we coultl havc drawn an arbztranj 2din~cnsionnlmanifold, a surfacc of a cylinder or of a sphere, for example; the
A prerequisite for the discovery in 1980 of QHE were the advances in transistor technology. Since the 1960s one was able to assemble Zdimensional electron gas laycrs in certain types of transistors, such as in a metal-oxide-semiconductor (see a schematic view of a Mosfet in Fig. B.4.4). field effcct transistor"Mosfet) Thc electron laycr is only about 50 nanometers thick, whereas its lateral extension may go up to the millimeter region. 4See Landau & Lifshitz (251, pp. 96-98 or Raith [39],p. 502. "ee, for example, von Klitzing [52],Braun [3], Chakraborty and Pietilainen [4], Janssen et al. [20],Yoshioka [54],and references given there. ' A fairly detailed description can be found in Raith [39],pp. 579-582.
156
B.4. Basic classical elcctroclynamics summarized, example
B.4.5 @Electrod y namicsin flatland: 2DEG and QME
157
Figure B.4.3: Hall effcct (schematic): The current density j in tlic collductilig platc is affected by tlic exterlial constant magnctic field l3 (in the figure only sy~nbolizcdby one arrow) such as t o crcatc the Hall voltage UH.
Figure. B.4.5: Sclicmatic view of a quanttun Hall cxpcriment witli a 2dimcnsionnl electron gas (2DEG). Tlie current density gT in tlic 2DEG is cxposed to a strong trarisversc ~iiagncticfield B,,,. Tlic Hall voltage Url can I)c mcasr~rcdin the tralisvrrse direction t,o j , betwccli P ant1 Q. In t l ~ cinset wc clcnoted the longitudinal voltage witli U, and tlie transversal onc witli U,(= UIl).
In the quantum Hall reainic, wc liavc very low temperatrires (between 25 mK ant1 500 mK) and vcry high magtffnetic fields (between 5 T and 15 T ) . Then thc conducting clcctrous of tlie spccirncn, because of a (quantum mechanical) excitation gap, cannot lnovc in the z-direction, thcy are confined t o the zyplanc. Thus an almost idcal %dimensional electron gas (2DEG) is constituted. Tlic Hall coliductance (= llresistance) exhibits very well-defined plateaus a t integral (and, in tlic fractional QHE, a t rational) multiples of tlic funtlamcntal SI cond~~ctance' of e2/11 1/(25 812.807 R), where e is the clemelitary charge and 11 Plallck's constant. Thcrcforc this effcct is instrumental in precision expcrimcnts for measuring, in conjuliction with the Josephson effect, e and 11 vcry accurately. We concentrate here on the integer QHE. nlrliing to Fig. B.4.5, wc colisider thc rectangle in thc ~ y - p l a n ewitli side lcngtlis 1, and l,, rcspcctively. The qualltulli Hall effect is observed in such a
'
Figure 13.4.4: A Mosfct witli a 2-dimclisional clcctron gas (2DEG) laycr between a semiconductor (Si) ancl an insulator (Si02). Adaptcd from Brauli [3].111 1980, with such a transistor, vo11 I
'Rcccntly, Lhc conductance of a single Hz-molcc~~le has been measured (see Srnit ct a1. [46]). T h c vall~e2e2/h way round, wl~icllis consistc~ltwith thc fundamcntnl conductance of QHE.
13.4. 13asic classical clectrociynamics summarized, example
158
159
R.4.5 @Elcct.rodynnrnicsin flatland: 2IlEG nntl QIIE
2-dimensional system of electrons subject to a strong uniform transverse magnetic ficld @. The configuration is similar to that of the classical Hall effect (scc Fig. B.4.3), but the system is cooled down to a uniform temperature of about of 0.1 I(. The IIall res~stanceRI1is defined by the ratio of the Hall voltage Uy and tlie clectric current I, in tlie x-direction: RII = U,/Ix. Longitudinally, we havr the ordinary dissipative Ohm resistance RL = U,/Ix. For fixed values of the lnagnetic field B and area charge density of electrons n, e, the Hall resistance RrI is a constant. Phenomenologically, QHE can be described by means of a lznear tensorial Ohm-Hall law as a constitutive relation. Thus, for an zsotropw material,
If we introduce the electric field Ex = Uz/l,, E, = U,/l, densities j, = Iz/l,, j, = I,/l,, we find +
l? = p j
-
or j =
and the 2D current
-#
oE,
with
(B.4.53)
where p , t,he specific resistivity, and a,the specific conductivity, are represented by second-rank tensors. Witli a classical electron model for the conductivity, the Hall resistance can be calcr~latcdto be RH = B/(n, e), where B is the only component of the magnetic field in 2-space. Tlic rnagnetic flux quantum for an electron is h/e = 2a0, witli
Figure B.4.G: The Hall resistance RIrat a temperature of 8 mI< as the magnetic ficld B (adapted from Ebcrt ct al. [GI).
;t
function of
Tliis yicltls
SI
Q)o = 2.07 x 10-'"1. Witli the fundamental resistance we can rewrite the Hall resistance wit11 tlic dimensionless filling factor v as a plicno~ncnologicallaw vnlitl at low flt~qucncicsant1 Inrgt. tlistances. We will S constitt~tivol i l ~ill :I g ~ n see in tlic n ~ x tsubsection tliirt we c:\n C X ~ ~ C Stliis crally covarinnt forln. Tliis slio\vs t1i;tt the laws govrrning QI-IE iirc cntilely indcpcndcnt of tlie pa~ticulnrgeometry (lnctric) under consitlei'a t'ion.
Observe tliat v-I lncasrlrcs tlic amount of magnetic flux (in flux units) per electron. Thus classically, if we increase tlie magnetic ficld, keeping n, (i.e., the gate voltagc) fixed, we woultl rxpcct a strictly linear increase of the Hall resistance. Sr~rpl.isiligly,however, we find the following (see Fig. B.4.6):
Covariant description of the phenomenology at the plateaus
1. The Hall RH has ploteous at rational heights. The plateaus at integcr height occur with a high accuracy: RH = (li/e2)/i, for i = 1 , 2 , .. . (for holes we would get a negative sign).
+
2. When (v, R H )belongs to a plateau, the longitudinal resistance, RL, vnnishes to a good approxirntrtion. I11 accortlancc witli these results, we choose such tliat RI, is zero and constant at a plateau for one particular system, namely
-
For a (1 2)-tliniclisionnl covi~riilntf o r ~ i i ~ ~ l n t i lct o n ,us ~ turn back to the first, ant1 tlie tliirtl irxiolns in (B.4.35) and (B.4.36), rcspcctivcly. In (B.4.56), 1 is linearly related to l?. Accordingly, we will also assume lineant?, in 1 + 2 dimensions bctwccn tllc 2-forms J and F,
OH
I
X ~ l ~call i s be fou~itlin t,lic work of 1;'lohlicli ct a!. [11, 9, 101 (see also Avron [2], Ricllt,cr [42], and rcfcrcnccs giver1 thcrc). An rarly article on QHIS and topology is l.llat, of Post [38], whicli sccms t,o Itave bee11 largely ovcrlooketl.
and Sciler
I (i0
Basic classical clrct~rodyni~mics summnrizcd, example
4
161
B.4.5 @Electrodynamicsin flat,land: 2DEG and &HE
witli tlie Hall resistance a,,". The minus sign is chosen bccause we want to recover (I3.4.56) cvcntually. Silicc J,, = -J,, ant1 Fkl= -Flk, we havc
+
Therefore tlie simplest (1 2)-covariant ansatz is found by assuming additionally isotropy in the 1 2 climcnsions, that is,
+
witli tlic generalized I
el. current (twisted)
P
j
2 + 1 decomposition
Substitutctl into (B.4.57), this yields
The Hall rcsistancr a11is a twistctl scalar (or ~ ~ s c u d o ~ c a l Of a r )course, . we can only hope that such a simple ansatz is valid provided the Mosfet is isotropic witli respect to its Hall conductance in the ~>laiicof tlie Mosfet. Because of (B.4.35) 1 and (B.4.36)1, wc find quite generally that tlie Hall conductance must be constant in space and time:
-
We slioultl be awartl that somctliing rcinarkablc happenecl 11cre.We have not introducecl a spacctinir rrlation A la H F, as is done in (1 3)-dimensional clcctrodynamics (scc Part D). Since for n = 3, H and F have still tlie same n ~ ~ m b of c r independent components, this woul(l I)c possible. However, it is the ansatz .I F that lrixds to a S L I C C C S S ~ U ~description of the phenomenology of QHE. One can consiclcr (B.4.60) as a spacctimc relation for thc (1 2)dinicnsional quantum Hall regime (see Fig. B.4.7). Of coursc, the integer or fractional numbers of aHcannot bc derived from a classical tlicory. Nevcrtlicless, classical (1 2)-climensional e1ectrotlyri;~micsimmediately suggcsts a relation of the type (B.4.60), a relation tliat is free of tiny metric. I11 other words, thc (1 -t 2)-dimensional clcctrodynamics of QHE, as a classical theory, is mctricfrec. Equation (B.4.60) transforms the Maxwcll equat,ions (R.4.35)2 and (B.4.36)2 into a complete systcni of partial tliffcrential cqrlations that can be integrated. At t,hc same time, it also yields an explicit relation between N and F , namely
+
N
+
+
rlH =
-011
F
or
dH =
dA .
Figure B.4.7: Interrelationship between current and field strength in the ChernSimons electrodynamics of QHE: In the horizontal direction, the 2-dimensional space part of a quantity is linked with its 1-dimensional time part t o a 3dimensional spacetime quantity. Lines in the vertical direction connect a pair of quantities that contributc to the Lorentz force density. And a diagonal line represents a spacetime relation. In (B.4.56), the current is represented as a vector. Thercfore we introducc the vector density 3 := O j by means of the diamond operator of (A.1.78) or, in components, 3" = cabj n / 2 Then ( a ,b = 1,2),
(B.4.62)
Tlic last equation can bc integrated. We find
since tlie potential 1-for111A is only clcfincd up to a gauge transformation anyway. The mctricfrcc differentla1 relation (B.4.62)1 between excitation H and fieltl strrngtli F is certainly not what we would havc cxpcctccl from classical I \ / I R x w c ~eloctrodyilalnics. ~~R~~ It rc])rcscnts, for 11 = 3, a totally new type of (Clicrii-Siliiolis) clcctrotlynnnlics. We colnpnrc (B.4.60) witli (B.4.37) and (B.4.39) to fintl
tliat is, we recover (B.4.56), thereby verifying the ansatz (B.4.60). For the chargc dcnsity P := O P = cabpnb/2, we find
As mentioned, (B.4.57) and the subsequent equations in this subsection are all mctricfrec. In particular, this is true for (B.4.65). Nevertheless, we recover (B.4.56) from it. In our derivation the crystal structure does not enter. Tlic assumptions of linearity and isotropy were sufficient. The (crystalline) material, in which QHE takes place, does define orthogonality by suitable crystal axes. However, this information is not needed for the derivation of (B.4.64).
R.4. Basic classical electrodynamics summarized, examplc
Moreover, we really wo111cl nccd a theory of rlcctrodynamics inside matter for the discussion of QHE. Such questions are trcated in Part E. In spitc of our nai'vc ansatz (13.4.60) in the realm of (1 +2)-dimensional v a c l i ~ ~electrodynamics, m wc get tlie (long wavclcngth limit of tlic) phcnon~cnologyof QHE correct and wr can post-dict tlic topological nature of QHE. This is not a minor achievement, and we consider it as a definite proof of the relevance of our way of formulating clcctrodynamics. Our guess is that the isotropy assumption in (B.4.59) is n sort of cffcctivc rnacroscopic average already. And probably for that rcason thr ansatz (B.4.60) works.
Slectromagnetic energy-momentum :urrent and action
Preview: 3D Lagrangians We introduce Lagrangians in Sec. B.5.4. Neverthclcss, let us collcludc this scction on clcctrodyna~iiicsin flatlantl with a few remarks on the appropriate Lagrangian for &HE. The Lrtgrangian in (1 2)-dimensional clectrodynamics has to bc a twistrtl 3form with the dilnension 11 of an action. Ol~viously,the first invariant of (I3.2.25) qualific.~,
+
But also the Cliern-Simons 3-form of (B.3.16), if multiplied by thc twistcd scalar 011, has thc right cliwactcristics,
with the I
In other wor(ls, they yield the sarnc Lagrangian since they only differ by all irrelevant exact form. Lct 11s define t,hen the Lngrangc 3-form
By means of its Eulcr-Lngrangc cqrlation,
2 A similar computation as in (13.4.71) yields tlirectly (B.4.60).
Min,kom.ski's "greatest discoverg was that at any point 1:n the electromagnetic field in vacuo there exists a tensor of rnn,k 2 of outstan,din,g physical importance. . . . each component of the tensor E,Q has a physical interpretation, which in every case had been discovered many years before Minkoluski s11.01oedthnt tl~ese16 components con.stitute n tenwor of rm.nk 2. Tlre tensor E,q i s called the energy tensor of the electromagnetic field." Edmund Wliit,t,al<er(1953) et us consitlcr tlie Lorcntz force cl(~nsityf,, = (e, J F ) A J in (B.4.3). If we want to derive tlie energy-momentum law for clectrodynarnics, we lii~vcto try to express f, as an e m c t form. Then energy-momentum is a kind of generalized potent,id for the Lorentz force dciisit,y, namely f, dC,. For that pIIrpose, wc start from fa. We substitute J = d H (inhoniogeneous Maxwcll equation) nntl subtract out a tcrni with H and F excliangcd and mnltiplied by a constant factor a: N
we can rrcovcr the 3D spacrtimc relation (B.4.60) t,hat wc started from. Altcrnatively, we can consider J as the external field in tlie Lagranginn
L3D' := -OH A A ~ A - A A J .
g.5.1 Fourth axiom: localization of energy-momentum
(B.4.72)
f , = ( e , J F ) A d H - a.(e,
J
H)AdF.
Bccausc of d F = 0 (homogeneous Maxwcll equation), the subtracted term vanishes. Tlic factor n will be left open for the moment. Note that we llecd a nonvanishing currcnt J # 0 for our derivation to bc sensible.
R.5, Elcctromagnctic energy-momcntum current and action
164
Wc pa.rtially integrate bot,h terms in (13.5.1):
B.5.l Fourth axiom: locnliznt,ion of energy-momcnti~m
165
and the remaining forcc tlcnsity 4-form t,urns out to be
X,, The first term already has the desired form. We recall the main formula for the Lie derivative of an arbitrary form Q), namely LC,,@= d(e, J @) e, J (d?)) (see (A.2.51)). This allows 11s to transform the second part of (B.5.1):
+
1 2
:= - - ( F r \ L e , , H -
IJAL,,F).
(B.5.8)
he absolute tlimcnsion of kC,, as well as of X, is lzle. Our tlerivation of (B.5.G) tiocsn't lead to a. unique dcfil~itio~l of kC,,. The idition of any closcd 3-form worlltl bc possible, kC:, :=
k
~
+N Y,
,
with
dY, = 0 ,
(B.5.9)
~ c l that i
f,,= ti k ~ : , + X,,.
The last line can be rewritten as
As 5-forms, the expressions in the square brackets vanish. Two terms remain, arid we find
Bccausc d F = 0, the third line adds up t o -af,.
Accordingly,
Now we have to make up our rninds about the choice of the factor a . With a = -1, tlic lcft-hand sitlc vanishes and we find a mathematical identity. A real conservation law is only obtained when, eventually, the second line vanishes. 111 other wortls, herc we need an a posteriori argument, i.e., we have to take some inforlnation fro111 experience. For a = 0, tlie second line does not vanish. However, for a = l , we can hope that the first term in the second line compensates the second tcrm if soli~cllowH N F. In fact, under "ordinary circumstances," to be explored below, the two terms in tlie second line do compensate cacli other for a = 1. Thcreforc we postulate this choice and find
Here the kin ern at^:^ energy-morncntum 3-form of tlie electromagnetic field, a central result of this section, reads k ~ := ,
1
- [ F A (e, 2
J
H) - H
A (e, J
F)]
(fourth axiom),
(B.5.7)
(B.5.10)
In particular, Y, could bc exact: Y, = dZ,. The 2-form Z,, htts the same dimension as k C n . It serms i~nl~ossiI)lo to build up 2, exclusively in ternis of the quantities e,, H, F in an n1gchr:tic way. Thcreforc, Y,, = 0 appears to bc the most na.t,ural choice. Thus, by the forlrtli axiom wc postulate that kC, in, (13.5.7) represents the ene~:q?j-momentu~r~ current that corrcct.ly localizes the energy-momentum distribution of the electro~naglieticfield in spa.cctimc. We call it the kinem.atic energy-momcntrlm current since we didn't find it by a dynamic principle, which we will not formulate bcforc (B.5.89), but rat,lier by means of some sort of ltincmatic argr~mcnts. The current kC, can also be rewritten by applying the :~nt,i-Lcibnixrldc for e, J either ill the first or sccolitl t c r ~ n011 tlic right-liand side of (13.5.7). With the 4-form
we fintl
Let 11s coliic bacl< to (13.5.G). Of course, rl kC, and kC, cannot Ilc integrntcd since they 1~1.~ covrctor-valtl~dforms. HOWCVC~, if we have a suitablc vector firltl ( = ("c~, ilvitilal>lc,wr citn transvrct its coniponcnts wit11 kCl,. Accordingly, we dcfinc t hr "cliargc" clcnsity
whicli should not be 11iixcd I I with ~ tlic notion of an elrct~xcchargt~.111 fiiit spacctinie, fol example, with t as tilnc coordinate, a suitablc (Killing) vcctol worild 1)c ( = 8, and Q = ["'(C, = k C t would signify tlie enrrgy tlrrisity. Generally, thc integrals J d Q ant1 Q arc wc.11 dcfinc.d.
r3.5. Electromagnetic energy-moment~imcurrent and action
1 GG
B.5.2 Energy-momentum current, electric/magnetic reciprocity
Lct us now comput,c d Q . Wc transvect (B.5.6) with the vector components I n . Then,
167
which amounts to one equation. This property - the vanishing of the "trace" of kC, - is connected with the fact that the electromagnetic field (the "photon") carries no mass and the theory is thus invariant under dilations. Why we call it the trace of the energy-momentum will become clear below (see (B.5.38)).
kC, is electriclmagnetic reciprocal F'urthermore, we can observe another property of kC,. It is remarkable how symmctric H and F enter (B.5.7). This was achieved by our choice of a = 1. The cnergy-momcntum current is electric/magnetic reciprocal; i.e., it remains invariant under the transformation
If wc evalrlatc the last term by subst,itr~ting(B.5.8), we have to remember the rule for the multiplication of a vcctor by a scalar in a Lie derivative (scc Sec. A.2.10): LftLw= f L l L w + d f A ( u - 1 ~ ) .
(B.5.16) with the twisted 0-form (pseudoscalar function) C = ((x) of dimension [C] = [H]/[F]= q 2 / l ~= llresistance. It should be stressed that in spite of kC, being electriclmagnetic reciprocal, Maxwell's equations are not,
WC sr~bstitutef = t", u = em,w = H and reorder: <"L,<, H = LCH - clt"
(e,
J
H).
(B.5.17)
An analogo~~s formula is valid for tlic 2-form F. Then (B.5.15) bccomes
or, in holonomic components, with f i i j := C i j k l ~ k l / 2 ,
$2
:= ~ ~ j ~ ~ j i Q := ~ ~c i j k~l ~/ 6 ~, anti ~
~
Under which conditiolls is Q conscrvcd, i.c., d & = O? We have to assume vacrlllml i.c., J = 0. Moreover, the conditions L c H = a(.r)If and L c F = a ( x ) F , with all arbitrary function ru(m), are sufficient for thc vanishing of the expression within thc last two ~ ~ a r c n t l ~ r sW e sC . will come back to this type of question a t the cnd of this section under Preview.
Thus, thc products 0 I10 I1and 0 I20 1 3 are invariant under reciprocity transform a t 1011s. ' Equation (B.5.21) expresses a ccrtain reciprocity between electric and magnrtic effects with regard to their respective contributions to the energy-momentum current of the field. We call it elcctric/magnetic reciprocity.' That this naming is appropriate can bc seen from a (1 3)-decomposition. We recall the decompositions of H and F in (B.4.6) and (B.4.7), respectively. We substitutc them into (B.5.21):
B.5.2 Properties of the energy-momentum current, electric/magnetic reciprocity k ~ , is
+
trace free
The (.nergy-momentumlonetu current kC, is a 3-form. Wc can blow it up to a 4-for111 according to 7YP A kC,. Since it still has sixteen components, wc haven't lost any iriforniation. If we recall that for any p-form Q) we have 29, A (e, J @) = p4', we imlnccliatcly rccognizc from (13.5.7) that
/ not 6even for ~ d C = 0, since we don't want to restrict ourselves t o the free-field case with vanishing source J = 0. For thc invariants of the electromagnetic field, we find under a reciprocity transformation the following responses:
'
I
. . . following Toupin [48] even if he introduced this notion in a somewhat more restricted context. Maxwell spoke of the mutual embrace of electricity and magnetism (see Wise [53]). In thc case of a prescribed metric, discussions of the corresponding Rainicli "duality rotation" were given by Gnillard & Zurnino 1131 nnd by Mielke [32], amongst others. Note, however, that our transformation (B.5.21) is metricfree and thus of a different type.
1I
1
Figure B.5.1: Different aspects of the clrctrolnagnctic field: In thc horizontal dilection, tlic 3-clirnc~nsionalspace part of a quantity is linked with its 1-dimcwsionitl timc part to a 4-dimensional spacetime ql~antit~y. The clic'rgy~nomcntinnC L I ~ ~ rCe I~ ~n ~a ~ ninvariant s untlcr the rxcliangr of tl~oseqnantitich that are connected by a d7aqonnl linc. And a, vrrt7cal linc rcprcsents a spacetime relation between quantities that arc canonically related like morncntl~n~ arid velocity (see (B.5.84)1).
H -+(I:
'li -CE, V - t CB, -+
-
Here it 1s clearly visiblr that n. magnetic quantity is replaced by an electric magnctlc. In this one and an elcctric quantity by a magnetic one: rlectr7c fol sense, we can speak of an electric/niagnetic reciprocity in the cx~~rrssion tlic rncrgy-morncntum current C,. Alternatively we can say that C, fulfills elcctric/n~agneticrcciprocity; it is clcctric/magnctic rc~i~)rocal. Let us pause for a ~liomentand wonder of how the notions "elcctric" ant1 "n~agnctic"arc attaclietl to certain ficlds and whether t l i c ~ cis a conventional clcinent involvcd. By lnaking experi~nentswith a cat's skill and a rot1 of amber, we call "libe~ate"~ l l i t we t call rlectnc charges. In three dimensions, they arc
described by the charge density p. Set, in motion, they produce an electric j . The electric charge is conserved (first axiom) ant1 is linked, via the Gauss law dV = p, to the electric excitation V . Returning to the Oersted experiment, it is clear that moving charges j induce magnetic effects in accordance with tlie Oerst,ecl-Ampkre law h3-1- v = j, also a consequence of the first axiom. Hence we can unanimously attribute t,he tcrrn magnetic to the excitation 3-1. There is no room left for doubt about that. The second axiom links the electric charge density p to the field strength E according t,o ( e , J p) A E and the clectric currelit j t,o tlie field 13 accorcling to (e, J j ) A B . Consequently, for the field strength F also, there can be no other way than to label E as electric and B as m.agnetic field strength. These arguments imply that tlic substitutions H + ( I: as well as F + -HI< both substitute a11 elcctric by a magnetic field and a magnet,ic by an elcctric one (see (B.5.27) and (B.5.28)). Because of the minus sign (that is, because cc = 1) tliat we found in (B.5.21) in analyzing the electrolnagllctic energy-momentum cuucnt C,, we cannot spcak of an equivalence of electric and magnet,ic fields; the expression reciprocity is much more appropriate. Fundamentally, electricity and magnetism enter into classical electrodynamics in an c~symmetricway. Let us try to explain the elect,ric/maglletic reciprocity by means of a simple example. In (B.5.Gl) wc? will show tliat the electric energy density reads ?L,] = E A V . If one wants t,o try to guess the corresponding expression for the magnetic energy density u,,,,, one substitutes for an electric a. corresponding magnetic quantity. However, t,he electric field strength is a 1-form. One cannot replace it by the magnetic field strength B since that is a 2-form. Therefore one has to switch to the magnetic excitation according to E -+ 'li,with tlie 1-form 3-1. The function { is needed because of the different dimensions of E and 3-1 and since E is an unt,wistccl and 7-1 it twistetl form. Analogously, one substit,ut,cs V -+ (h', t,her&y fincling u,,,, = 7-1 A 13. This is the corrcct result, i.c., u = (E V + B A X),and we can be happy. Naively, one woliltl t,lien postulate the invariance of . t ~1111(lertile substitution E - t 53.1, V - t CB, B -t $ V , 3 - 1 - + C E But, . a s a l o o k a t (B.4.6) ant1 (B.4.7) will show, t.his cannot bc implcmellted in a covariant way. More generally, if we comparc (13.4.14) with (B.4.20), then the mentioned invariance wo111d require ItT = f T and = f A fT. C011scq1ierit,ly,f A = /lS/hT. Iiowevcr, according t,o (B.4.18), wc have /zs/hT = -1. Hence only the unpliysical anti-Lcnz rulc with f A = -1 could save the sit,uation, an option t,lrat wc tlon't take; we rather stick to the Lenz rulc. How are we going t,o save our rulc of thumb for ext,ractirlg the magnetic energy from the electric one? Well, if we turn to the substit~~t,ions (B.5.27) and (B.5.28), i.c., if we i n t r o d ~ ~ ctwo e minus signs according to E 1-1, V + CB , D -t -$ V ,Fl -+ -{E, then u still remains invariant and we recover the covariant rule (B.5.21). In other words, the naive approach works u p to two minus signs. Those we can supply by having insight into the covariant versio~l of electrodynamics. Accortlingly, the electric/magnctic reciprocity is the one that we knew all the time; we just have to be careful wit11 the sign.
5
4
-+
5
@
kC, expressed in terms of the complex electromagnetic
field We can understand the clcctric/magnetic reciprocity transformation as acting on the colulnn vector consisting of N and CF:
Now, accortling to (B.5.31), clcctric/magnctic reciprocity of the energy-lnornentum current is manifest. If we execut,c successively two el~ctric/magncticreci1)rocity transformat,ions, namely U -r U1 -r U", tlien as call bc seen from (B5.31) or (B.5.21), wc. find a reflect,ion (a rotnt,ion of -n), nalrlely U1' = -Ul i.e., U-r-U
I11 order to compact,ify this formula, we introduce the com,plex electromagnetic field 2-form2 U:=H+iCF
,!
and
U*=H-iCF,
(B.5.30)
wit11 * denoting the complcx conjugate. Now the electric/magnetic reciprocity (B.5.29) translates into
This corresponds, in the colnplex plane, where U lives, to a rotation by an angle of -n/2. We call rcsolvc (B.5.30) with respect to excitation and field strength:
We differentiate (D.5.30)1. Then the M~xwcllequation for the complex field turns out to be
or
(H+-H,F---F).
Only four elcct~ric/111aglleticreciprocity t,ransforn~ationslead I~ackt o the idcntity. It should be stressctl, howcvcr, that already onr electric/magnctic reciprocity transformation lcnvcs kC, ilivari:\nt. I t is now straigl~t~forward to fornliilly cxtcnd the clcctric/magnetic reciprocity transformation (B.5.31) t o ,-J/
= c+7cQ
~
*
=l e - 7 d ~
k
~
i 4c
=n - [U* A (c"
J
U) - U A (e,
J
U*)]
2Eve~ tl~otrgli ~ we introduce the concept of a lnctric only in Part C, it is necessary to point, out that the complex clcctrornagnetic Jeld U , srrbsr~rningexitation and field strength (see l d carefully tlistinguished from the con~plexelectromagnetic field strength Fig. 0.5.1), s h o ~ ~ be introduced convrtltionally: F" := i P b c I;b,, with F ~ ":= g b l g " ~ , J .This can only be defined a f t e r a, metric hw been i~!troduced. Similarly, for the excitatiol~we would then have fi" := - lIb" z c " " ~ Ifb,, with If0" := g n ' g a ~I f , , .
-
+
I."" +
U* ,
(B.5.36)
1
with 4 = d)(x)as i\n i~rl)it,ri\ry"rotation" angle. Tlie energy-momentum current kC, is still invariant r~nderthis extended transformation, but in later applirntions only the s1111cascof 4 = -n/2, trcatetl above, will be of interest.
@Energ y-momentum tensor density k7a@ Since kC,, is a 3-form, we call decompose it :ithe!. co~~yenti?nally or with respect t o the I);isis 3-for111 ilj= PO J ?, with i = 19'' A d1 A 7Y2 A d 3 (sec (A.1.74)):
Thc sccontl-rank tcnsor density of wcight 1, k7nBl is the Mtnkotu.~k.lcncrgy tensor tlensity. We can resolve this cqllation with respect t o k7nP by exterior mult,iplicntion with 80.Wv rccall 211' A i y = St i al~tlfind k7m0i =1 9 A ~ k
Clearly, if we choose a constant C, i.c., dC = 0, the second term on the lefthand side vanishes. T l ~ casyn~mctrybetween electric and magnetic fields finds its expression in the fact that the source term on the right-hand side of (B.5.33) is a real quantity. If we substitute (B.5.32) into the energy-momentum current (B.5.7), we find, after some algebra,
(B.5.35)
or with tlic new diamond opcrator
(13.5.38)
~ , ,
* of (A.1.78),
Thereby we recognize that d" A kC, = 0 (scc (B.5.20)) is equivalent to tho vanishing of the trace of the cncrgy-lnolncntuln tcnsor dcnsit,y k7,"= 0. Thus k7,P as we11 as k C n liave fifteen indcprlidelit componcnt;~a t this stago. Both qunnt,itics arc c?quiva.lent. If wc sukst,it,~~tc (13.5.7) into (B.5.39), thc.11 wc citn cxprcss t l ~ cncrgy-niolncnc tutri tensor tlcllsit,y ill the componcnt,~of H :u~tlF as follows:"
we
lcavc it, t.o the rcatlers to prove t.lle formula first by Minkowski in 1007.
k7,y
k77L3 =
,I
k72 which wm derived
172 @
R.5. Electromagnetic energy-momentum current and action
kT,P alternatively derived by means of tensor calculus
0 . 5 . 2 Energy-momentum current, electric/magnetic reciprocity
173
and the force density
We start with the Maxwcll eqt~;ltions(B.4.33) in holonon~iccoordinates, i.e., in the nat,ural frame e, = 6Ld,: we finally have the desired result, Here ?? is defined accortling to ??" = $ fkklmnHm,(see (B.4.32)1). We substitute the inhomogenco~lsMaxwell equat,ion into the Lorentz force density and integrate partially:
The last term can be rewritten by means of the lion~ogcncousMaxwell equation, 1.C.,
+
fi = ajkzj Xi,
(B.5.51)
which is the component version of (B.5.6). By means of (B.4.32)1,it is possible to transform (B.5.49) into (B.5.40).
-
@Preview:Covariant conservation law and vanishing extra force density X, The Lorentz force density f, in (B.5.6) and the energy-momentum current kC, in (B.5.7) are covariant, with respect to frame and coordinate transformations. Nevert,heless, both of the two terms on the right-hand side of (B.5.6), namely d k C, and X,, are not covariant by themselves. What can we do? For the first three axioms of electrodynamics, the spacetime arena is only required to be a (1 3)-decomposable 4-dimensional manifold. We cannot be as economical as this in general. Ordinarily a linear connection raPon that manifold is needed. The linear connection is the guiding field that transports a vector, for example, from one point of spacetime t o a neighboring one. The connection is only introduced in Part C. There, the covariant exterior I',Pp(Lpa) (see (C.1.63)). With the help of differential is defined as D = d this operator, a generally covariant expression D kC, can be constructed. Then (B.5.6) can be rewritten as
+
Again, we integrate pnrtially. This time the two last terms arc:
+
We collcct the first three terms 011 the right-himd sitlc and substitute the lefthand side of the inliomogenco~rsMaxwcll equation into the l i ~ term: ~t
The last term represents thc negative of a Lorentz force density (see (B.5.42)). Thus we find
with the new supplementary force density
+
The first and the third term
011
the right-hand side are of a related structure.
We split the first term into two equal pieces and differentiate one piece:
I
Introtlucing the kinematic energy-monlcrtun tensor density
which contains the covariant Lie derivative LC = D J J J J D (see (C.1.71)). , the same; only the force Note that the energy-moment_um current k ~ remains density X, gets replaced by X,. It is remarkable that in (B.5.52) or in (B.5.6) the energy-momentum current can be defined even if (B.5.52) (as long as X, # 0) doesn't represent a genuine conservation law. In this subsection only, and not in the rest of Part B, wc use the linear connection and the covariant exterior derivative. Thus we are able to show that the fourth axiom is exactly what is needed for an appropriate and consistent derivation of the conservation law for energy-momentum. Let us exploit then, as far as possible, the arbitrary linear connection r a p introduced above. As auxiliary quantities attached to roo,we need the torsion 2-form Ta and the A
n . 5 . Elrctromagnctic energy-momentum current anti action
174
-
+
t~nnsposrdconn,ection 1-form l?,P := I',P e , J TP, both t o be introduced i l l Part C in (C.1.42) and (C.1.43), rcspectivcly. Let us now go back to thc extra force dcnsity 2, of (B.5.53). What we ncwl is the gauge covariant Lie derivative of an arbitrary 2-form 9 = 9,,,19/% 19'"/2 in terms of its components. Using the general formula ((3.2.128) we have
-
-
B.5.3 Time-space decomposition of the energy-momentum and the Lenz rule
175
+
If we 1 3 clecomposc the Lorentz force and thc cnergy-momentum current, we arrive a t t,hc 3-dimensional version of the energy-momentum law of electrodynamics in a rather direct way, Recall that wc work with a foliation-compatible framc e, as specified in (B.1.34), i.e., with eg = n, e, = d,, together witli thc transversality condition e, J d a = 0. Consider tllc definition (B.5.7) of kC,. Substit,ute into it the (1 3)-decompositions (B.4.6) and (B.4.7) of the excitation H and tlic field strength F , respectively. Then, we obtain
+
where D , := e,, J D , witli D as the exterior covarinnt differential with respect. to the transposed connection. Thus, wllcrcl we introduced the en,ergy density 3-form or since 19'' A 6" A 191' A 8" =
P"iL"
2 , wc finel as an alternative t o (B.5.53),
1 u:=-(EAV+BA~~), 2 the enc?lg?/fl1~3:density (or Poynting) 2-form
This is ns far as we can go with an arbitrary linear ronnectjon. Now it bccolnes obvious how to achieve thc vanishing of X,. Our four axioms don't malte electrodyna~nicsa completc theory. What is missing is the rlrcfromagnetzc spacetzme rrlatzon between excitation H and ficld strength F . Sucll a fifth axiom is introduced in Chapter D.4. The starting point for arriving at such an axiom is tlic linear ansatz fill,
=1 - XiLUP"F
2
P"
Witll 3
fii~1'
=1 - EILI/PO
2
Hpv .
thc m.omcntum density 3-form
pa := - B A ( e , J V ) ,
(R.5.63)
ant1 the Maxwell stress (or momentum flux density) 2-form of the clectromagnetic ficld
(B.5.57)
Substit,ut,ing into (B.5.56), wc have Accortlingly, we can represent the scheme (73.5.59)-(B.5.60) in the form of a 4 x 4 matrix (for dcnsity wc usc tlic ahbrcviation tl.): Thus, the extra, force dcnsity X, vanishes providcd x P " i l " is covariantly collstallt with respect to the transposetl connectiori of thc underlying spacrtimc. Wc will come back t o this discussion in Sec. E.1.4.
B .5.3 Time-space decomposition of the energy-momentum current and the Lenz rule An.other tlreory of electricity, v l ~ i c hI prefer, denies action at n distance and attributes e1ectr.l:~action to tensions and pressures in, an all-pe~varlingmedium, the.se .stresses being the su.me in. kind with those familiar to engineers, and the medium being identical wit11 that i n which light i s supposed to be propagated. James Clerk Maxwell (1870)
) (,
-mo111. (I. energy d. encrgy flux d, morn. flux d.
=
(B.5.65)
The entries of the first row are 3-forms and those of the scconcl row 2-forms. The absolute clirncnsions of the quantities emerging in tllc 4 x 4 matrix can be determined from their respect,ive definitions and the clccompositions (B.4.G) ancl (B.4.7):
The relat,ivc tlirnensions, that is, those of their respcctivc components, read (herc z , j , k = 1 , 2 , 3 ) ,
176
B.6. Electromagnetic energy-momentum current and action
This coincides with the results from mechanics. A momeritum flux density, e.g., should have the dimension pule3 = mv2/l" f /t2 = stress, in agreement with SI (Sij,]= h/(te3) = energ?y/t3 = stress = Pascal. Note that the dimension of the energy flux dcnsity s i j is the same as that of the momentum density p i j k , times the square of n velocity ([It)'. Transvecting t,hc Mnxwcll stress S , , "familiar to engineers," with da, we find straightforwardly
which is thc 3-dirncnsional version of (B.5.20). As soon as an electromagnetic spacetimc relation is available, we can relate the energy flux density sAda, which has the sarnc number of independent coinponcnts a?the 2-form s , namely three, to tllc lilomeiituln density pa. In Sec. E.1.4, by means of the Maxwell-Lorentz spacctimc relation, we prove t,hc symmctry of the energy-momentum current in this way (src (E.1.30)). The Lorrntz forcc density is longitudinal wit11 respect to 72, i.e., fa = fa, wlicrcas the forms 11, s, pa, and S, arc pi~rclytransversal. Equations (B.5.59)(B.5.60) provide the tlccomposition of the energy-n~omentum3-form into its "tinie" and L ' ~ ~ ~~ a~ ~C eC "If C SWP. apply (B.1.28) to it, wc find for the exterior differentials: dkxi,
= r l o (~ u
d k ~ ,=
tic7 A
+ fds),
(B.5.69) (B.5.70)
(-Pa + d S , ) .
Combining all t,hc results, we evcntually obtain for the (1 + 3)-decon~position of (13.5.G) t,lic balance cqr~ationsfor the e1cctroni;~gncticfield encrgy and mon~ent~um:
Let us now cornc back to our discussion of thc Lenz rule. We take kC6from t,he fourth axiom (B.5.7) and substitute thc decompositions for excitation and field strength (B.4.14) and (B.4.20), respectively. After some nlgcbra we find
In section l3.4.2 we have sliown that
/lT
= -1, ItS = $1
, and
fT
= $1. Thus,
Thc elcct,ric encrgy density u,l = $ E A 'D = 42)A E and thc magnetic u , , ~ = $13 3.1 = $3.1 A B add up for f A = +1 (Lenz rule). In the opposite case,
for fA = -1 (anti-Lenz rule), they subtract. It is true, at the present stage we cannot make any statements about the signs of either uol or u,,,~.However, from the similarity of both expressions one would conclude that if uei is the electric has to be the magnetic one and similarly for -u,l and -urn,. energy then urn@; In any case, pre-metric electrodynamics clearly bifurcates into the MaxwellLenz and the Maxwell-anti-Lenz equations. Already in (B.4.7), we had opted for the Maxwell-Lenz case in anticipation of the adding up of electric and magnctic energy. For the rest of the book, we put f A= $1. Obscrve finally that all the formulas displayed in this section are independent of any metric and/or connection.
Why have we postponed the discussion of the Lagrange formalism for so long even though we know that this formalism helps so much in thc effective organization of field-theoretical structures? We chose to basc our axiomatics on the conservation laws of charge and flux, intcr alia, which arc amenable to direct experimental verification. And in the second axiom we uscd the concept of forcc from mechanics that also has the appeal of being able t o be grasped directly. Accordingly, the proximity to experiment was one of our guiding principles in selecting the axioms. Already via the second axiom the notion of a force density came in. We know that this concept, accorcling to f, 6'L/6'xE,also has a place in the Lagrangc formalism. When we "derived" the fourth axiom by trying to express the Lorentz force density fa as an exact form fa d C,, we obviously had already movcd towards the Lagrange formalism. It becamc apparent in (B.5.11): the 4-form A is a possible Lagrangian. Still, we proposed thc energy-momcntnm currentl without appealing t o a Lagrangian. That seemed to be more sccurc becausc we could avoid all the fallacies related to a not directly obscrvablc quantity like L. We were led, practically in a unique fashion, to the fourth axiom (B.5.7). In any case, having formulated the integral and the diffcrential versions of electroclynamics including its energy-momentum distribution, wc have enough understanding of its inner working to be able to rcforrnulate it in a Lagrangian form in a very straightforward way. As wc discussed in Sec. B.4.1, for the completion of elcct,rodynarnics wc nrctl an electromagnetic spacetime relation H = H [ F ] .This could be a nonlocnl and nonlinear functiollal in general, as we will discuss in Chapter E.2. Tlir field variables in Maxwell's equations arc H and F . Therefore, the Lagrange 4-form of the elcctromagnctic field should depend on both of them: N
From a dimensional point of view, it is quite obvious what type of action we would expect for the Maxwell field. For the excitation we have [HI = q and for
178
R.5. Elcctromagnetic energy-momentum current and action
the field strength [F]= il q-', Accordingly, J H A F would qualify as action. And this is, indeed, what we will find out later. We hasten to add that in Part D, when we introduce the hypotlictical skcwon field, a Lagrangian description turns out to be too narrow. We instcad usc tlic spacetime relation H = H [ F ] directly without taking recourse to a Lagrangian. Maxwell's equations are first order in N = H[F] and F, respcctively. Since F = dA, the field strength F itself is first ordcr in A. We take A to be n field variablc. The Er~lcr-Lagrange equations of a variational principle witli a Lagrangian of diffcrcntial order m are of differential order 277%. The field equations arc assumcd at most of second differential ordcr in the field variables A, 111. Tliereforc, tlic Lagrangian is of first ordcr in thcse fields. Consider an electrically charged matter ficld 11, which, for the time being, is assumcd to be a p-form. The total Lagrangian of the system, a twisted 4-forni, should consist of a free field part V of thc electronlagnetic field and a mattcr part L,,,t, tlie latter of which describes the matter field and its coupling to A: N
$J
In Lmat,tlie potential A is needed for tlic coupling of matter t o the elcctromagnetic ficld. However, dA is not assumed to arisc: dLIllat/ddA = 0. This is dubbetl minimal coupling since this is thc most economical way of bringing a coupling about. Furthermore, ST are what we call tlie structural fields. Their presence, at this stage, is rnotivatcd by a tcclinical concern: In order to construct a viable Lagrangian other t,han thc trivial "topological" F A F Lagrangian, onc slioultl have supplcment,ary fields, namely ST, that allow one to achieve this goal. We won't specify tlicir nature now. As soon as we have introduced a metric (and ;I connection) on spacctimc, those fields will play thc part of ST. Wc require V to be gauge znvanant, that is SV = V(A -t SA, d[A
+ 6/11, ST) - V(A, dA, ST) = 0 ,
(B.5.77)
The ficld cquations for A are given by the stationary points of W under a variation S of A, which comnlutes with the extcrior derivative by definition, that is, Scl = d6, and vanishes at the bountlary, i.e., SAlsn4 = 0. Varying A yields
where the variational clcrivativc of the 1-form A is defined according to
Stationarity of W leads to the gauge Jield eq~~ation.
Kccl~ingin mind thc inliomogencous Maxwell field equation in (B.4.3), we definc tlic excitation ("ficld rnoment,um") conjugated to A and the matter current by
I{
av
= --
ddA
dv dF
= --
and
SL,l,,t (FA
-
dLn1at dA
J=- --
respectively. Then we recover, indeed:
wlicre 6A = dw represents an infinitesimal gauge transformation of tlic type (B.3.9) for x x dw. Wc obtain
V = V(dA, ST) = V(F, ST).
(B.5.79)
Hence the frce ficld Maxwcll or ga.uge Lagrangian can depend on the potential A only via the ficltl strength F = dA. The mattcr Lagrangian should also bc gaugeinvariant. The action reads
Tlir liomogcneous Maxwcll cquation is a consequence of working witli tlic potential A, since F = dA and d F = ddA = 0. In (B.5.85), we were also able to arrive a t the inliomogcneous equation. The excitation H and the current J are, howcvcr, only implicitly givcn. As we can see from (B.5.84), only an explicit form of the Lagrangians V and LInat promotes H and J to more than sheer placelioldcrs. On thc other hand, it is vcry sat,isfying to rccover the structure of Maxwell's thcory already a t such a n implicit level. Equation (B.5.84)) represents the as yet unknown spacetzme relntzon of Maxwell's theory. Let us t,urn to the variational of the matter ficld $. Its variational clcrivative reads
180
R.5. Electromagnetic energy-momentum current and action
Since V does not depend on $, we find for the matter field equation simply
All that is left to do now in this context is to specify the spacetime relation (B.5.84)1 and thereby to transform the Maxwell and tlie matter Lagrangian into an explicit form. At this stage, the structural fields ST are considered nondynamical ("background"), so we do not have equations of motion for them.
B.5.5 @Couplingof the energy-momentum current to the coframe
R.5.5 @Couplingof the energy-momentum current to the coframe
+
6 = LC = 5 J d dE J . Substituting this into the left-hand and right-hand sides of (B.5.91), we find, after some rearrangements, the identity
<
Since is arbitrary, the first and second lines vanish separately. Now, the final step is to put = e,. Then (B.5.92) yields two identities. F'rom the first line of (B.5.92) we find that the dynamic current dC,, defined in (B.5.89), can be identified with the canonical (or Noether) energy-momentum current, i.e.,
<
d ~ , ~ ~ ,with , In this section we show that the canonical definition of the energy-momenturn current (as a Noether current corresponding to spacetime translations) coincides with its dynamic definition as a source of the gravitational field that is represented by the coframe, and both are closely related t o the kinematic current of our fourth axiom. Let us assume that the interaction of the electromagnetic field with gravity is "switched on." On the Lagrangian level, it means that (B.5.79) should be replaced by the Lagrangian V = V(dnlF ) .
181
C, : = - ~ , J V - ( ~ , J F ) A H .
(B.5.93)
Thus we don't need to distinguish any longer between the dynamic and the canonical energy-momentum current. This fact matches very well with the structure of the kinematic energy-momentum current kC, of (B.5.7) or, better, of (B.5.12). We compare (B.5.93) with (B.5.12). If we choose as our Lagrangian V = -F A H/2, then kC, = C,, and we can drop the k and the d from kC, and dC,, respectively. However, this choice of Lagrangian is only legitimized by our fifth axiom in Chap. D.6. The second line of (B.5.92) yields the consentation law of energy-momentum:
(B.5.88)
F'rom now on, the coframe assumes the role of tlie structural field ST. In a standard way, the dynamic (or Hilbert) energy-momcntum current for tlie coframe field is defined by
Tlie last term vanishes for the Lagrangian (B.5.88) under consideration. As before (cf. (B.5.84)), the electromagnetic field momentum is defined by
Tlie crucial point is the condition of general coordinate or diffeomorphism invariance of the Lagrangian (B.5.88) of the interacting electromagnetic and coframe fields. The general variation of the Lagrangian reads
where we have used the inhomogeneous Maxwell equation d H = J. The presence of the first term on the right-hand side guarantees the covariant character of that equation. It is easy to see that we can rewrite (B.5.94) in the equivalent form
by making use of the Riemannian covariant exterior derivative in Part C.
5 t o be defined
Our discussion can even be made more general14going beyond a pure electrodynamical theory. Namely, let us consider a theory of the coframe da coupled to a generalized matter field @. Note that the latter may not be just one function or an exterior form but an arbitrary collection of forms of all possible ranks (0)
(PI
+
and/or exterior forms of type p, i.e., 9 = ( $ ", . . . , *, . . . ). The ranges of indices for forms of different ranks are, in general, also different; that is, U and A If [ is a vector field generating an arbitrary 1-parameter group of diffeomorphisms on X, the variation in (B.5.91) is described by the Lie derivative, i.e.,
4See, for
example, reference
[15] and also [la].
182
0 . 5 . Electromagnetic energy-momentum current and action
run over different ranges, for example. We assume that the Lagrangian of such a theory depends very generally on frame, matter field, and its derivatives:
B.5.5@Couplingof the energy-momentum current to the coframe some "partial integrations," we find
Normally, the matter Lagrangian does not depend on the derivatives of the coframc, but we include d79" for greater generality. (It is important to realize that the Lagrangian (B .5.96) describes the most general theory: For example, the set 3 can include not only the true matter fields described, say, by a p-form (P)
$ " but formally also other virtual gravitational potentials such as the metric 0-form gap and t,hc connection 1-form I',p as soon as they are defined 011 X and are interacting with each other.) The basic assulnption about the Lagrangian (B.5.96) is that L is a scalarvalued twisted n-form th,at is invariant under the spacetime diffeomorphisms. This simple input has amazingly general consequences. The dynamic energy-momentum current of matter is defined as in (B.5.89):
Now we rearrange the equation above by collecting terms under the cxterior derivative separately. Then (B.5.100) can be written a3.
where
+ (E J d*) Similarly, the variational derivative of the generalized matter reads
B
:=
EJ
L
6L
+ (-~)"(FJ
6L 9 ) A d6@ ,
a~ aL ( E ~ 2 9 " ) -~ ( c ~ d . 1 9 ~ddt9a )~-
-
-((J*)A-
where the sign factor (-1)P correlates with the relevant rank of a particular component in the set of fields @. This is a simple generalization of (B.5.86). According to the Noether tllcorenl, the conservation identities of the matter system result from the postulated invariance of L under a local symmetry group. Actually, this is only true "weakly," i.e., provided the Euler-Lagrange equation (B.5.98) for the mat,tcr fields is satisfied. Here wc consider the consequences of the illvariance of L under the group diffeomorpllisms on the spacetime manifold. Let ( be a vector field generating an arbitrary 1-parameter group ;Tt of diffcon~orphismson X. In order to obtain a corresponding Noether identity from the invariance of L under a 1-parameter group of local translations c 7 x Dqf (4, R), it is important to recall that infinitesimally the action of a 1-parameter group '& on X is described by the conventional Lie derivative (A.2.49) with respect to a vector field (. Since we work with fields that are cxterior forms of various ranks, the most appropriate formula for the Lie derivative is (A.2.51), i.e., LC = J d dE J . The general variation of the Lagrangian (B.5.96) reads:
A
a~ a@-
a~
(B.5.103)
( ( J ~ * ) A -a, d 3
The functions A and B have the form
B=("B,.
A = (" A,,
(B.5.104)
Hence, by (B.5.101), ("(A, - dB,) - d(" A B, = 0,
(B.5.105)
where both (" and d(" are pointwise arbitrary. Hence we can conclude that B, as well as A, vanish:
Since the vector field ( is arbitrary, it is sufficient to replace it via the frame field. Then, for B = 0, we obtain from (B.5.103)
E
-+ e,
by
+
- d-
aL 8d29"
aL + ( c , J d19O) A add@ '
For A = 0, equation (B.5.102) yields For the variations generated by a 1-parameter group of the vcctor field E we have to substitute 6 = LC in (B.5.99). This is straightforward and, after performing
ddE, E (e,
J
d@) A
+ 3, ,
d ~ l j
(B.5.108)
184
R.5. Electromagnetic energy-momentum current and action
B.5.6 Maxwell's equations and the energy-momentum current in Excalc
o(3) = r * sin(theta1 with signature (1,-1,-1,-I)$ frame e$ In fact, when the mat,t,cr Lagranginn is indcpcndent of the derivatives of tlic coframe ficld, i.c., aLlddI9fl = 0, cquation (B.5.107) clcmonst,ratcs the equality of the dynamic energy-momentum current that is coupled t o the coframc with the canonical one, t,hr Norther current of the translations.
13.5.6 Maxwell's equations and the energy-momentum current in Excalc It is a merit of exterior calculus that electrodynamics and, in particular, Maxwell's equations can be formulated in a very succinct form. This translates into an eq11allyconcise form of the corresponding computer programs in Excalc. Tlir goal of better understanding the structure of clectrodynamics leads us, hand in hand, to a morc transparent and morc effectivc way of computer programming. In Excalc, as we mentioneti in Sec. A.2.11, tlle electrodynamical quantities are evaluatcd with respect t o the cofrnme that is sprcified in the program. If we put in an accelerating and rotating coframe w e , for example, then the electromagnetic ficld strength F in the program will be evaluated with rcspect 2 . is, of course, exactly what we discussed to this frainc: F = Fapwo ~ w ~ IThis in Sec. B.4.4 when wc introduced arbitrary noninertial coframes. We cautioned our readers already in SCC.A.2.11 that wc need to spccify a coframe togcthcr with t,hc metric. Thus, our Maxwell sninple program proper, to be displaycycd I>elow, is prcccded by coframc and frame commands. Wc pick the splicrical coordinate system of Sec. A.2.11 and rcqriirc tlic spacctimc t,o bc Minkowskian, i.e., $ ( I -= ) 1. Afterwards wc spccify tllc elcctrolnagnctic potential A = A, d o , llamcly poti. Since we haven't defined a specific problcln so far, we lcavc its components aaO, aai, aa2, aa3 open for tlie morncnt. Then we put in thc pieces discussctl after (B.4.3). I11 order to relate N and F , we havc t o make use of the fifth axiom, only t o be pinned down in (D.6.13):
*
185
d phi % coframe defined
% flat spacetime assumed
psi:=i;
% start of Maxwell proper: unknown functions aaO, aal,
..
pform (aa0,aal ,aa2,aa3)=0, pot i=i, {f arad2,excit2)=2, {maxhom3,maxinh3)=3$ fdomain aaO=aaO(t ,r,theta,phi),aal=aai(t ,r,theta ,phi), aa2=aa2(t ,r,theta,phi),aa3=aa3(t ,r,theta,phi)$ pot1 f arad2 maxhom3 excit2 maxinh3
aaO*o(O) + aaI*o(i) + aa2*0(2) + aa3*0(3)$ d potl; : = d farad2; : = lam * # farad2; % spacetime relation, see : = d excit2; % 5th axiom, Eq.(D.6.13)
:= :=
% Maxwell Lagrangian and energy-momentum current assigned pform lmax4=4, maxenergy3(a)=3$
% Use of a blank before the interior product sign is obligatory! end ;
If this salnplc program is written onto a file with name mustermaz.exi (cxi stands for excalc-input; the corresponding output filc has the extension .exo), t,hen this vcry filc call bc read into a Reduce session by the command inumustermax.exi" ; As a trivial test, you can spccify tlie potential of a point charge sitting a t the origin of our coordinate system:
% file mustermax.exi, 2002-11-15 aaO : = -q/r;
load-package excalc$ pform psi=O$ fdomain psi=psi(r)$ coframe o(0) = psi = (l/psi) 0(1> 0(2) = r
* * *
d t, d r, d theta,
aai := aa2
:=
aa3
:=
0;
Dcterminc its ficld strength by f arad2: =f arad2; its excitation by excit2: = excit2; and it,s cnergy-momentum distribution by maxenergy3(-a) :=maxenergy3(-a) ; You will find the results you are familiar with. And, of course, you
186
R.5. Electromagnetic energy-momentum current and action
want to convince yourself that Maxwell's equations are fulfilled by releasing the commands maxhom3: =maxhom3; and maxinh3: =maxinh3; This sample program can be edited according to the needs one has. Prescribe a noninertial coframe, i.e., an accelerating and rotating coframe. Then you just have to edit the coframe command and can subsequently compute the corrcsponding physical components of an electromagnetic quantity with respect to that frame. Applications of this program include the Reissner-Nordstrom and the Kerr-Newman solutions of general relativity; they represent the electromagnetic and the gravitational fields of an electrically charged mass of spherical or. axial symmetry, respectively. They are discussed in the outlook in the last part, of the book.
References
[I] R.C. Ashoori, Electrons in artificial atoms, Nature 379 (1996) 413-419. (2) J .E. Avron, Adiabatic quantum transport. In: Mesoscopic Quantum Pt~ysics,Les Houches Session LXI 1994, E. Akkermans, G. Montambaux, J.L. Pichard, and J. Zinn-Justin, eds. (Elsevier: Amsterdam 1995) pp. 741791. [3] E. Braun, The quantum Hall effect. In: Metrology a t the frontiers of Pl~ysicsand Teclinology, Proc. Intern. School of Physics "Enrico Fermi" Course CX (1989)' L. Crovini and T.J. Quinn, eds. (North Holland: Amsterdam, 1992) pp. 211-257. [4] T . Chakraborty and P. Pietilainen, The Quantum Hall Effects, fractional and Integral, 2nd ed. (Springer: Berlin, 1995). [5] M.H. Devoret and H. Grabert, Introduction to single charge tunneling. In: Single Charge Tunneling: Coulomb Blockade Phenomena in Nanostructures, H. Grabert and M.H. Devoret, eds. (Plenum Press: New York, 1992) pp. 1-19.
[GI G. Ebert, K. v. Klitzing, C. Probst, and K. Ploog, Magneto-quantumtransport on GaAs-Al, Gal-, as heterostructures at very low temperatures. Solid State Comm. 44 (1982) 95-98. [7] U. Essmann and H. Trauble, The direct observation of individual flux lines in type II superconductors, Ph-ys. Lett. 24A (1967) 526-527.
Part B. Axioms of Classical Electrodynamics
188
References
189
[8] U. Essmanrl and H. Trauble, Th,e magnetic structure of superconductors, Sci. American 224 (March 1971) 74-84.
(221 R.M. Kiehn, Periods on manifolds, quantization, and gauge, J. Math. Phys. 18 (1977) 614-624.
[9] J. F'rohlich and B. Pedrini, New applications of the chiral anomaly, in: "Mathematical Physics 20001', Eds. A. Fokas, A. Grigoryan, T. Kibble, and B. Zegarlinski (Imperial College Press: London, 2000), pp. 9-47; Eprint Archive heI)-th/0002195.
[23] R.M. Kiehn, The photon spin and other topologicalfeatures of classical electromagnetism. In Gravitation and Cosmology: From the Hubble Radius to the Planck Scale. R. Amoroso et al., cds. (Kluwcr: Dorclrecht, Netherlands, 2002) pp. 197-206.
[lo] J . F'I.ohlich, B. Pedrini, C. Schweigcrt, and J . Walcher, Universality in quantum Hall systems: Coset constmjction of incompressible states, J. Statist. Ph.ys. 103 (2001) 527-567; Eprint Archive cond-mat/OO02330 (2000).
[24] R.M. Kiehn and J.F. Pierce, Intrinsic transport theorem, Phys. Fluids 12 (1969) 1941-1943.
[Ill J . F'rohlich and U.M. Studer, Gauge invariance and current algebra in non.relntivistic m.any-body theoy, Rev. Mod. Pl~ys.65 (1993) 733-802.
[25] L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media. Volulne 8 of Course of Theor. Physics. Transl. from the Russian (Pergamon Press: Oxford, 1960).
[12] A. Fujiwara and Y. Takahashi, Manipulation of elementary charge in a silicon charge-coupled device, Nature 410 (2001) 560-562 [see M. Devoret, Physics World June 2001, pp. 27-28].
[26] S. Lundquist, Magneto-hydrostatic fields, Arkiv for Fysik 2 (1950) 361365; On the stabilit?~of magn.eto-hydrostatic fields, Ph.ys. Rev. 83 (1951) 307-311.
[13] M.K. G ~ i l l i ~and r d B.Zumino, Duality rotations for interacting fields, Nucl. Pl~~ys. B193 (1981) 221-244.
[27] R. Lust and A. Schliiter, Kraftfreie Magnetfelder, 2. Astroplysik 34 (1954) 263-282.
1141 G.A. G1at:nlaier and P.H. Roberts, Rotation and magnetism of Earth's inner core, Science 274 (1996) 1887-1891.
[28] G.E. Marsh, Force-Free Magnetic Fields: Solutions, topology and applications (World Scientific: Singapore, 1996).
[Is] F.W. Hehl, J.D. McCrea, E.W. Mielke, and Y. Ne'eman, Metric-afine gauge t h e o ~of gravit?~:Field equations, Noether identities, world spinors, and hreakil,g of dilation, invariance. Pl~ys.Rep. 258 (1995) 1-171.
1291 G.E. Marsh, Topology in electromagnetics. Chapter 6 of Frontiers in Electromagnetic~.D.H. Werner, R. Mittra, eds. (IEEE Press: Ncw York, 2000) pp. 258-288.
1161 J.L. Heilbr011, Electricity in the 17th and 18th Centuries. A Study of Early Modcrn Pl~ysics(University of California Prcss: Bcrkeley, 1979).
130) B. Mashhoon, The hypothesis of locality in relativistic physics, Ph~ls.Lctt. A145 (1990) 147-153.
[IT] R. Ingarden and A. Jamiolkowski, Classical Electrodynamics (Elsevier: Amsterdam, 1985).
[31] B. Mashhoon, The Irypothesis of locality and its limitations, Eprint Archive: gr-qc/0303029.
[18] Y. Itin, Energy momentum current for coframe gravity, Class. Qnant. Grav. 19 (2002) 173-189. Eprint Archive: gr-qc/0111036
[32] E.W. Miclke, Geometrodynamics of Gauge Fields: On the geometry of Yang-Mills and gravitational gauge theories (Akadernie-Verlag: Berlin 1987) Sec.V.1.
[19] Y. Itin and F.W. Held, Is the Lorentz signature of the metric of spacetime electroma!netic in origin? To be published (2003). (201
M.Janssrl, 0. Viehweger, U.
Fastenrath, and J. Hajdu, Introduction to tlre Tl1.eo7.1of the Integer Quantum Hall Effect (VCH: Weinheim, Germany, 1994).
[21] Ch. Jooss, J . Albrecht, H. Kuhn, S. Leonhardt, and H. Kronmuller, Magneto-rptical studies of c u m n t distributions in high-T, superconductors, Rep.Progr. Phys. 65 (2002) 651-788.
(331 H.K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids (Cambridge University Press: Cambridge, England, 1978). [34] R.A. Nelson, Guide for m.etric practice, Internationally recognized collvcntions have been established for standard usage of SI units, Ph.ysysics Totlay 55,No.8, Buyer's Guidc (Aug. 2002) BG15-BGlG. [35] A. Percs, The speed of ligh,t need not be constant, Eprint Archivc: grqc/0210066, 3 pages (Oct 2002).
190
Part B. Axioms of Classical Electrodynamics
References
191
1361 R.W. Pohl, Elektrizitatsleh,re, 21st ed. (Springer: Berlin, 1975) pp. 27/28, see also earlier editions.
1511 J . Van Blntlel, Relativity and Engineer-in,g, Springer Series in Electrophysics Vo1.15 (Springer: Berlin, 1984).
[37] E..J. Post, Formal Structure of Electromagnetics: General Covaria,nce and Electromagnetics (North Holland: Amsterdam, 1962, and Dover: Mineola, New York, 1997).
1521 I<. von I
[38] E.J. Post, O n the quantization of the Hall im,pedance, Pl1.y~.Lett. A94 (1983) 343-345. [39] W. Raith, Beyqman.n-Sch,aefer, Lehrbuch der Ezperimentall)h?~s~:k, Vol. 2, Elektromagnetismus, 8th ed. (de Gruyter: Berlin, 1999). [40] A.F. Raiiada, Topolog,ica.l electromagnetism, J. Phys. A25 (1992) 16211641. [41] A.F. Raiiada, O n the m.agn.etic Aelicity, Eur. J . Phys. 13 (1992) 70-76. [42] T. Richter and R. Scilcr, Geometric properties of transport i n quantum Ha11 systems. In: Geometry and Qrianturn Pllysics. Proc. 38th Schladming Conference, H. Gausterer et al., cds. Lecture Notes in Physics 543 (2000) 275-310. [43] P.H. Roberts ant1 G.A. Glatzmaier, Geodyna,mo theory and simulations, Rev. Mod. P11,ys. 72 (2000) 1081-1123. [44] A.G. R.ojo, Electron-drag effects i n coupled electron, systems, .7. Phys., Condens. Matter, 11 (1999) R31-R52. [45] J.A. Schoutcn, Ten.sor Anahjsis for Pl~,ysicist.s.2nd ed, reprinted (Dover: Mineola, Ncw York 1989). [46] R.H.M. Smit, Y. Noat, C. Untiedt, N.D. Lang, M.C. van Hcmert, J.M. of the conductance of a hydrogen nlolecule, van Ruitenbeck, Mea.ct~~m.en,t Nntrlro 419 (2002) 906-909. [47] G.E. Stcdman, Ring-laser tests of fun,damental physics anri geophysics, Rcpt. Prog. P1i.y~.60 (1997) 615-688. [48] R.A. Toupin, Elasticity and electro-magnetics, in: Non-Linear Continuum Tl~eories,C.I.M.E. Conference, Bressanone, Itcl1.y 1965. C. Truesdcll and G. Grioli, coorclinators, pp. 203-342. 1491 J.L. Tmcba and A.F. Raiiada, The electromagnetic helicity, Eur. J. Phys. 17 (199G) 141-144. [SO] C.C. Tsuci and J.R. Kirtley, Pairing symm.etry i n cuprate superconductors, Rev. Mod. Pl~ys.72 (2000) 969-1016.
153) M.N. Wise, The mutual embraaceof electricity and magnetism, Science 203 (1979) 1310-1318. [54] D. Yoshioka, The Qua,ntum Hall Effect (Springer: Berlin 2002). [55] M.R. Zirnbauer, Elektrodynamik. Tex-script July 1998 (Springer: Berlin, to bc published).
Part C
More Mathematics
194
Part C. More Mathematics
In Part A we outlinctl exterior algebra and exterior calculus on differential manifolds that are "bare" in the sense that no other geometrical structures arc assumed on them. The corresponding mathematical machinery proved to be sufficient for the formulation in Part B of the general framework of classical elcctroc1ynamics. In Part C we provide additional mathematical tools needed to complete electromagnetic theory: the connection and the metric. Quite generically, the connection and the mctric can be introduced on a smooth manifold as two completely independent fields. The (linear) connection defincs parallel transport, which enables us to compare values of quantities a t different points of the manifold. The existence of a connection in physical theories of spacetime is indispensable in that it naturally incorporates the inertial properties of spacetime. Chapter C.1 deals with the connection and its properties in detail. The mctric brings in the notions of distance and angles on a spacetime manifold, which arc crucial for the operational definition of most physical measurements in space and time, namely those that are beyond mere counting procedures as in charge conservation and simple (Lorentz) force measurements. Clocks and rulers come into play, forming a firm basis for the description and an understanding of most physical observations. An important characteristic of the metric is its deep relation t o certain duality operators. The relevant matliematical material, which is later heavily used in Part Dl is displayed in Chapter C.2.
Linear connection
" . . . the essential achievement of general relativity, namely to overcome LLrigid" space (i.e., the inertial frame), is only indirectly connected with the introciuction of a Riemannian metric. The directly relevant conceptual element is the "displacement field" (l?fk), which expresses the infinitesimal displacclnent of vectors. It is this which replaces thc parallelism of spatially arbitrarily separated vectors , equality of corresponding components) by fixed by the inertial frame ( i . ~ .thc an infinitesimal operation. This makes it possible to construct tensors by differentiation and hence to dispense with the introductioll of "rigid" space (the inertial frame). In the face of this, it seems to be of secondary importance in some scnse that some particular r field can be deduced from a Riemannian metric . . . "
A. Einstein (1955 April 04)'
C.1.1 Covariant differentiation of tensor fields The change of scalar functions f along a vector 71, i s rlescrihed by the directional derivative a,, f . The generalization of thi.7 notion from scalars f to tensors T is provided 011 the co7)nriant differentiation V,,T.
'Transl~tion by F. Gronwald, D. Hartley, and F.W. Ilehl from the German original: See the preface in [14].
C.1. Linear connect,ion
196
When calculating a d~rectzonalderivative of a function f (n.) dong a vector field u, one has to know the values o f f (x) a t different points on the integral lines of 71. With the standard definition, which involves taking a limit of the separation between points, the dircctional derivative reads
=
1Li(2)
afax i -
i
I
I:[
+
+
+
I
C.1.2 Linear connection 1-forms
Consider the chart U1 with tlie 1oc:il coordinates x2 that contains a point n. E U1 c X . Take the natural basis 8, at x. Tllc covariant differentiation V,, of the vectors 8, with respect to an arbitrary vector field u = ukdk reads
[t]
1) C(X)-linearit,y with rcspcct to
for all vcctor fields u, v, all functions f , g, all tensor fields T, S, and all forms w . The Lie derivative L,,, defined in Sec. A.2.10, is also a map T!(X) x T ; ( X ) -+ T t ( X ) . The properties of the covariant differentiation 2)-5) are the same as those of the Lie derivative, cf. (C.1.4)-(C.1.7)with (A.2.56), (A.2.53), (A.2.58) and (A.2.55),respectively. The property ((2.1.3) is, however, somewhat different from the corresponding propcrt,y of the Lie derivative (A.2.57), which can be vis~ializetl,for example, by substituting u -+ f u in (A.2.57). This difference reflcct!s the fact that the definition of .CUTa t a given point x E X makes use of u in tllc neigliborhood of this point, whereas in order to define V,T a t x one ha? to know only the value of u at s.
The difference between covariant and partial differentiation of a tensor field is determined by the linear connection 1-forms r i j . Tl~.e?j .show .up in the action of V,, on a frame ai and supply a constructive realization of the covariant differentiation V,, of an arbitrary tensor field.
which to any vector field 11 E T;(X) and to any tensor field T E T:(X) of typc assigns a tensor ficld V,,T E qy(X) of typc that satisfies the following properties:
I':[
197
5) the Leibniz rule with respect to the interior product,
in local coordinates {xi}.
Obviously, (C.1.1) describes a map: T t ( X ) x Ti(X) -+ T ~ ( x of ) scalar fields into scalar fields, i.c., tensors of type are mapped again into T t ( X ) = C ( X ) . This map has simple properties: 1) W-linearity, 2) C(X)-linearity with respcct to ?L, i.e., aq,,+lL,, f = gduf ha,, f , 3) additivity a,,(f 9 ) = a,,f a,,g. In order to generalize thc directional derivative to arbitrary tensor fields of typc [jF], one needs a recipe for comparing tensor quantities at two different points of a manifold. This is providcd by an additional structure on X called the llnear connect~on.The linear connection or, ecluivalently, the covariant differentiation is necessary in order to formulate differential equations for various ~~hysical fielcls such as the Einstein equation for the gravitational field or the Navicr-Stokes equation of hydrodynamics. In line with the directional derivative, a covanant dzff~rentiatzonV is defined as a smootli R-linear map
C.1.2 Linear connection 1-forms
7 ~ , I
Hcre wo used the property (C.1.3). These n vector fields can be tlecomposed with rcspcct to the coordinate frame ai:
2) additivit,y with rcspcct to T,
can be read off with their components rk2J. Suppose a different chart U2 with the local coordinates x 2' intersects with U1, and the point x E Ul U2 belongs to an intcrscction of the two charts. Then the connection 1-forms satisfy the consistency conclition
n
3) for a scalar ficld f a tlirectional tlcrivativc is rccovcrcd, (C.1.5)
4) the Lcibniz rule with respect to the tensor product, V,,(T@S) = V 1 , T 8 S + T @ V , , S ,
(C.1.6)
I
everywhere in the intersection of the local charts Ul
n U2.
198
C.1. Linear connection
Now, making usc of properties 1)-5), one can describe a covariant differentiation of an arbitrary tensor field
61
in terms of the connection 1-forms. Namely, we have explicitly the field
61 tensor
where the covariant differential of the natural tensor components is introduced by
C.1.3 @Covariantdifferentiation of a general geometric quantity the connection 1-form changes as
Although the transformation law (C.1.20) is inhomogeneous we cannot, on an open set U , achieve r,,P' = 0 in general; it can only be done if the curvature (to be introduced later) vanishes in U.At a given point xo, however, we can always choose the first derivatives of LetQ contained in dL,,, in such a way that I',tP1(xo) = 0. A frame e, such that r,B(xo) = 0
at a given point xo
(C.1.23)
is called normal a t xo. A normal frame is given up to transformations L,la whose first derivatives vanish at xo. This freedom may be used, and we can always choose a coordinate system {x i } such that the frame e,(x) in (C.1.23) is also "trivialized" : e, = &Lai
The step in (C.1.9) can be generalized to an arbitrary frame e,. Its covariant differentiation with respect to a vector field u reads
199
a t a given point xo .
((2.1.24)
Summing u p 2, for a trivialized frame we have (e,, r a p ) f
(&Lai, 0)
a t a given point xo .
(C.1.25)
with the corresponding linear connection 1-forms r,P. In four dimensions, we have sixteen 1-forms r,P a t our disposal. The components of the connection 1-forms with respect to the coframe 6 , are given by
Despite the fact that the normal frame looks like a coordinate frame (in the sense, e.g., that (C.1.24) shows apparently that the tangent vectors di of the coordinate frame coincide with the vectors of e, basis at xo), one cannot, in general, introduce new local coordinates in the neighborhood of xo in which (C.1.23) is fulfilled.
In terms of a local coordinate system {xi},
C.1.3 @Covariantdifferentiation of a general
= ri,pdxi
where
riaP= I'ap(ai).
(C.1.17)
The 1-forms I',P are not a new independent object: since an arbitrary frame may be decomposed with respcct to thc coordinate frame, we find, with the help of (A.2.30) and (A.2.31), the simple relation
Under a change of frame that is described by a linear transformation
the connection 1-forms transform in a nontensorial way,
For an infinitesimal linear transformation, La@= 6:
+ E,P,
geometric quantity What is the covariant differential of a tensor density, for example ? Let us now consider the covariant differentiation of a general geometric quantity that was introduced in Sec. A.1.3. As in Sec. A.2.10, we treat a geometric quantity w as a set of smooth fields w A on X. These fields are the components of w = wAeA with respect to a frame en E W = R N in the space of a prepresentation of the group GL(n, R) of local linear frame transformations (C.1.19). The transformation (C.1.19) of a frame of spacetime acts on the geometric quantity of type p by means of the local generalization of (A.1.17),
or in the infinitesimal case (C.1.21), 2See von der Heyde (101, Hartley [9],and Iliev (111.
C.1.5 "l'orsion
C.1. Linear connection
200
The generator matrix pBA a p was introduced in (A.2.66) when we discussed the Lie derivative of geometric quantities of type p. A covariant differentiation for geometric quantities of type p is introduced as a ilatural generalization of the map (C.1.2) with all the properties 1)-5) preserved: The covariant differentiation for a W-frame reads
and curvsturc
where ( a ,O) is an interval in IW. In local coordinates {z2},the tangent vector to the curve a = {x2(t)) is uZ= dx7/dt. A t e ~ ~ s ofield r T is said to be pamllell?/ tmn,.~portedalong a if
along a . Taking into account (C.1.14), we get for T whereas for an arbitrary geometric quantity 7u ((3.1.14) are replaced by Vt,711 = I L J ( D T U eA, ~)
with
D W ~ :=
= wA
dwA
eA of type p, eqs. (C.1.13),
+ p B A a p rap?uB.
(C.1.29)
The general formula (C.1.29) is consistent with the covariant derivative of usual tensor fields when the latter are treated as a geometric quantity of a special kind; one call compare this with examples 2), 3) in Sec. A.1.3. Two simple applications of this general technique are in order. First, wc recall that the Lcvi-Civita symbols have the same values with respect t o all frames, see (A.1.64). This means that they are geometric quantities of the type p = id (identity transformation) or, plainly speaking, 6~"'..."- = 0. Comparing this with (C.1.27) and (C.1.28), we conclude that
- ( ( t ) )T
~
~
,
,
C.1.5 @Torsionand curvat urc
(C.1.31)
Torsion and c u r ~ ~ ~110th t ~ rrr.eas~rre ~re the deviation from tlre p a t spacetime geometq (of spec,in.l relativit?/). When both of tirem are zero, on,e can globally define the trivialized fro,me (C.1.25) all over the spacetim.e manifold.
a
Comparing with (C.1.27), we find p p = -ru 6;, and hence (C.1.29) yields
If we generalize this to a tensor density , Ta...Pv', we find
We now want to associate with a connection two tensor fields: torsion and curvature. As we llave seen above, the connection form can always bc transformccl to zero a t onc given point. However, torsion :~11tlcurvat,ure will in general give nonvanishing characterization of the conncctioll at this point. The torsion of a connection V is a map T that assigns to cacll pair of vector v ) by fields u and v a vector field T(IL,
C.1.4 Parallel transport By means of the covariant differentiation V,, a tensor can be parallell?/ transported along a curve on a manifold. This provides a convenient tool for comparing values of the tensor field at different points of tlre manifold. A connection enables us to define parallel transport of a tensor along a curve. A differentiable curve a on X is a smooth map
~ ( (~ t ) ~ )
If 0 E ( a ,1)) and T(u(0)) is given, then there exists (at least locally) a. unique solution T ( a ( t ) ) of this linear orclillnry differential equation for t E ( a ,b ) . Tl1erc.fore, we have a linear map of trnsors of type [y] at tile point o(0) to tensors of type a t the point a @ ) . Taking T = IL, we obtain a differential equation for aufopnralle1,s:
for an arbitrary connection. Second, let us take a scalar density S of weight w. This geometric quantity is described by the transformation law (A.1.5G) or, in the infinitesimal form, by
6s= WE,"^.
~
I
The coinmutator [u, v] has bccn definctl in (A.2.5). One can straightforwartfly verify the tensor character of T(IL,v) so that its value at any given point is dctcrmincd by the values of 11 and v at that point. Fig. C.1.1 illustratcs scliclnatically the gcomctrical mcaning of torsion: Clloosc two vectors v and u at a point P E X. Transport 71 parallelly along 1) to the point R and likewise 11 along u to the point Q. If the resulting parallelogram
,
C.1.5 @Torsion and curvature
(3.1. Linear connection
Figure (3.1.2: On the geometrical interpretation of curvature: parallel transport of a vector around a closed loop. Figure C.1.1: On the geometrical intcrpretation of torsion: a closure failure of infinitesimal displacements. This is a schematic view. Note that R and Q are infinitesimally near to P.
anholonomity, see (A.2.36). Accordingly, we find for the components of the torsion
T ~ ? "= rpyn - ryPn -t- cpya.
is broken, i.e., if it has a gap or a closure failure, then the connection carries a torsion. Such situations occur in the continuum theory of di~locations.~ Since the torsion T ( u , v) is a vector field, one can expand it with respect to a local frame,
The torsion 2-form T" allows one to define the 1-form e, J TP. If added to tlrc connection 1-form ",?l it is again a connection. We call it the tmnsposed connection,
By construction, tllc coefficicnts T " ( ~ Lv), of this expansion arc 2-forms, i.c., functions that assign real numbers to every pair of thc vector fields u, v. Taking the vectors of R frame, T ( ~ Pe,), = T"(ep, e,) e, = T'," e, ,
1
Ta = - T,,," 8" A 8"= 2
5 Tij0dr' 2
A d2j .
e,lne,.
(C.1.41)
The first t,wo tcrlns on the right-hand side bring in the connection coefficients (C.l.lG), whereas the conlmutator can be rewritten in terms of the object of 3 ~ e Kroner e [13] ~ r l dreferences give11 there.
?-.
are transposed with respect to that of r a p , The curvature of a conncction V is a map that assigns to each pair of vector fields u and v a linear transformation R(u, v) : X, -+ X, of the tangent space a t an arbitrary point x by
I
Thc explicit form of these coefficic~lts is obtained directly from the definition ('2.1.37) which we evaluate with respect to a frame e,, Tn(eg,e,)e, =V,,e,-V,,ep-[eg,
sincc in a natural frame, i.e, for Cn,P = 0, the indices of the components of
(C.1.39)
we can read off the coeficicnts of this vector-valued torsio~t2-form:
((2.1.42)
R(U,V)UI := V,V,w - V , ~ , , W- V[,, ] w .
(C.1.44)
One can verify the tensor character of the curvature so that the value of R(u, v)tu at any given point depends only on the values of vector fields u, v , and w at that point. Analogously to torsion, we can expand the four vector fields R(u, v)e, with respect to a local vector frame, R(u, v) e, = R , ~ ( U v) , ep .
(C.1.45)
Thereby the curvature 2-form R,P is defined. Similarly to (C.1.40), we may express these 2-forms with respect t o a coframe 19" or a coordinate frame dx i as follows:
C.l .G @'cartan's
Taking (C.1.45) witli rospcct to vature 2-form,
;I
frame, one finds the componc~ntsof the cur-
205
C.1.6 @Cartan'sgeometric interpretation of torsion and curvature
R(ep,e,)e, = R ,,, Oep
and
R(d,,a,)e, = ~ , , , ~ e p .
((2.1.47)
On a manifold with a linear connection, th,e notion. of a position vector can be defined along a curve. If we transport the position vector around an infinitesimal closed loop, it is subject to a translation and a linear transformation. The translation reveals the torsion and the linear transformation the curvature of the manifold.
Tl~us,by (C. 1.44),
Rij,k
geometric interprctation of torsion and curvature
air,," 8.r.0 + riaBrj,IT - rjIT/31'itrb. 2,
(C.1.49)
Here we introcluccd thc abbreviation a,, := e2/,8,. The curvature 2-form Roo can be contractcd by Incans of the frame ep. In this way wwc find tlie Ricci 1-form Ric, := ep J Rap = RicQB1 9 ~ .
(C.1.50)
Using (C.1.46), wc immediately find
Let us start,, following Cartan, with thc flat affinc space in which il connection V has zero torsion and zero curvaturc. In such a manifold, we may define an affinc position vector field (or radius vector field) r as one tliat satisfies the eqilation V1,r =
for all vector fields v. With respect to the local coordinates {xi), r = ri ai, ant1 ((2.1.53) is a syst,em of sixteen part,ial diffwcrcntial equations for tlie four functions ri (z), namely
D r' Thc geometrical mcani~igof the curvaturc is revealed when wc consider n parallel transport of a vector along a closed curvc in X (see Fig. C.1.2). Let, a : {x 2(t)}, 0 t 5 1, be a smooth curvc that starts and ends a t a point P = x"0) = r 7 ( l ) (in other words, a is a 1-cycle). Taking a vector 11(0) at P and transporting it parallclly dong u (which tcchnically reduces to tllc solution of a differential equation (C.1.35)),one finds a t the return point r 7 ( l )a. vcctor I L ( ~ that ) differs fro111 ~ ( 0 )Tlie . tliff(.rcncc is determined by tlie curvaturc,
<
where A S is an infinitesimal 2-tlimcnsional surface element that is bound by a , i.e., a = 8AS. Wlicn the curvaturc is zero everywhere in X, we call such a manifold a flat afine space wzth torsion (or a tcleparallelism spacetime). If the torsion vanishes aclditionally, we speak of a flat afine spacc. Affinc means that the connection V is still thcrc and it allows onc to comparr tensors at tlifferent points. Clearly, the curvature is vanishing for an cvcr~ywhcre trivial connectioli form FOP = 0. However, as wc know, the components r a p depend on the frame ficlcl and, in gencral, if we have curvature, it is impossible to clioose frames e, in such a way that reP= 0 on the whole X. On a flat affine space, the components of a vector do not change after a parallel transport around a closcd loop; in other words, parallel transport is integrable. Curvature is a measure of thc deviation from thc flat case.
((2.1.53)
II
= dxi,
or
di r j ( z )
+ rikj(5) r k ( x ) = 6;.
(C.1.54)
In flat affine space, a coordinate basis can always be chosen, a t least within one chart, in such a way tliat rtk3 = 0, and then tlie cquatiori (C.1.53) or (C.1.54) is simply d,r" 66;. The sollltion is r' = x" An"wherc A7 a constant vector, so that r7is, indeed, the position (or radius) vcctor of x b i t h respect to an origin x7 = -A 2. In an affine spacetime, thc integrability condition for (C.1.53) is V,,V,,,r - V,,V,, r - V[7,,w1r= V,,IL)- VZUv - [v, w] ,
((2.1.55)
for all vcctor fields v, w. Hcncc a sufficient condition for the existence of global radius vcctor ficlds r is R(V,PU)=O
and
T(v,w)=O
('2.1.57)
for arbitrary v, w ,i.e., vanisliing curvaturc and torsion. In a general manifold, when torsion and curvature are nonzero, thc posit,ion vector field docs not exist on X. Nevertheless, it is possible to define a position . objcct turns out to bc cxtremcly useful for revealing vcctor along a c u r v ~This the geometrical meaning of t,orsion and curvaturc. Let p E X bc an arbitrary point and u = { r 2 ( t ) ) , t 0, be a sniootli curve that starts at p, i.e., x"0) = r;,. We now attach at p an affine tangent spacc or, to put it diffcrcntly, wc consider the tangent space X, a t p as an n-~lime~~sional afine vector space. Rccall that in an sffine vector spacc, each element (vcctor) is given by its origin and its components witli respect to some fixed basis. We
>
C. 1. Linear connection
206
I-
C.1.7 @Covariantexterior derivative where A S is the 2-tlimensional infinitesimal surface enclosed by a . Thus, in going around the infinitesimal closcd loop a , the position vector r in the affine tangent space at p undergoes a translation and a linear transformation of the same order of lnagnitutlc as the area of AS. The translation is determined by the torsion
whereas the lintxu transformation is determined by the curvature (It is instructive to compare this with thc changc of a vector under the parallel transport (C.1.52).).
C.1.7 @Covariantcxtcrior derivativc Figure C.1.3: Cartan's position vector on manifold with curvature and torsion: Affine transport of a vector r around an infinitesimal loop. Here p and p' denote the initial and find positions of the position vector in the affine tangent space. construct Cartan's position vector as a map that assigns to each point of a curve a a vector in the affinc tangent space X p . Geometrically such a construction can be conveniently unclerstood as a generalized development map that is defined by "rolling" the tangent space along a given curve. The defining equation of the position vector is again (C.1.53), but this time v is not an arbitrary vector field but tangent to the curve under consideration, i.e. v = (dx'/dt)a,. Sub~tit~uting this into (C.1.53), we get dz i dr" (t) dt - [e:(x(t)) - rip"(z(t)) r ~ ( t ) ]
--
If o,n exterior form is generalized to n tensor-vnluctl exterior form, tll,cn the ~ r s ~ irLefin,ition al of the exterior d e ~ ? v a t i ~can ~ e he nat~rra.llyexten.ded to the covariant exterior deriantive. Covnriant exterior derivatives of torsion. and c~~rvature o.re in,volved in the two Bianchi icientities. Thc torsion 2-form (C.1.38),(C.1.40) and the curvature 2-form (C.1.45), (C.1.46) are examples of t,ensor-valucd p-fonns, that is, of generalized geometric quantitics. For such ohjccts we llccd to introduce thc notion of covarinnt cxtcrior derivative which shares the properties of a covariant derivative of a. geometric quantity itnd of an exterior derivative of a, scalar-valuetl form. be an arbitrary pforni of type p. It can be written as a sun1 of Let clccomposnblc p-forms of type p, namely P" = lo" w whcrc 111" is n scalar of type p ant1 w i L llsual exterior p-for111. For S I I C ~ Ia for111 we dcfinc
-ij;
As t increases, one "moves" along the curve a and the functions r a ( t ) always describe the colnponcnts of the position vector in the affine tangent space a t the fixed original point p. Thus, for example, a displacement along the curve from z 2 0 xZ d t Zyields a change of the position vector
+
Following Cartan, we may interpret this equation as telling us that the position vector map consists of a translation e:dti and a linear transformation dt' rp in the affine tangent space a t p. Let us now consider a closcd curve a such that x2(1) = zb (see Fig. C.1.3). Then, upon integrating arountl a , it is found that the total change in r" is given by
ant1 cxtcntl t,liis definition by IW-linearity to arbitrary pforlns of type p. Using (C.1.29), it is st,rniglltforward to obt,ain t,he gcncr;tl for~llula
and t80prove th:it D satisfies t,hc Lcibniz rulc
-riaa
I I I
Ar" = A S (T" - Rpa TO)
(C.1.60)
whcrc p is the degree of cp". Unlike the usual exterior tlcrivativc, which satisfies dd exterior derivative is no longer nilpotcnt: DDcpA = p ~ j A " / jR , h
y7".
=
0, the covarialit (C.1.65)
208
'2.1. Linear connection
The simplest proof makes use of the normal framc (C.1.23) in which DvA 5 dvA, R a p A drop. We choose a normal frame and differentiate (C.1.63). Since the resulting formula is an equality of two (p 2)-forms of type p, it holds in an arbitrary framc. The relation (C.1.65) is called the Ricci identity. Now we can appreciably simplify d l calculations involving frame, connection, curvature, and torsion. At first, noticing that the coframc 6" is a 1-form of vector type, we recover the torsion 2-form (C.1.38), (C.1.40) as a covariant exterior derivative
C.1.8 @Theforms o(a), connl (a,b),torsion2(a), curv2(a,b) 7 -
-
pf orm torsion2(a)=2, curv2(a,b)=2;
+
209
% preceded by coframe command
I11 Excalc, the trace of the torsion T := e, J T" reads e(- a) -1 t o r s i o n 2 ( a ) , and the corresponding trace part of the torsion T a = 6" A T becomes pf orm trator2(a) ; trator2(a) : =o(a)^(e(-b)
Tliis equation is often called the first (Cartan) structure equation. Applying (C.1.65), we obtain the 1st Biancl~iidentity:
- 1 torsion2(b)) ;
The Ricci 1-form Ric, := epA n a p , see (C.1.50), is encoded as Analogously, after recognizing the curvature 2-form as a generalized 2-form of we immediately rewrite (C.1.45), ((3.1.46) as tensor type
r
[:I,
which is called the second (Cartan) structure equation. Using again the trick with the normal framc, we obtain the 2nd Binnchi identity:
IDR,~-o
1.
(C.1.69)
Finally, we call link u p the notions of Lie derivative and covariant derivative. For dccon~posablcp-forms of typo p, vA= ?uAw, where wA is a scalar of type p and w a p-form, we define the covnriant Lie derivative as
and extend this definition by R-linearity to arbitrary pforms of type p. It is an interesting exercise to show that
C.1.8 @Theforms o(a), connl (a,b), torsion2(a), curv2 (a,b) We come back to our Excalc programming. We put n = 4. On each differential manifold, wc can specify an arbitxary coframc field t9", in Excalc o ( a ) . Excalc is made familiar with o ( a ) by means of the coframe statement as described in See. A.2.11. Moreover, since we introduced a linear connection 1-form r a p , we tlo the same in Excalc with pf orm connl ( a , b) = l ; Then it is straightforward to implement the torsion and curvature 2-forms T" and R,D by means of the structure equations (C.1.66) and (C.1.68), respectively:
--
pform riccil(a)=l; riccil (-a) : =e(-b) - I curv2(-a,b) ;
Weyl's (purely non-Riemannian) segmental (or dilational) curvature 2-form $6,p RyY is the other generally covariant contraction of the curvature. We have I
pf orm delta(a,b)=O, dilcurv2(a,b)=2;
These are the quantities that play a role in a 4-dimensional differential manifold with a prescribed connection. The corresponding Excalc expressions defined here can be put into an executable Excalc program. However, first we want to get access to a possible metric of this manifold.
c.2 Metric
I
Although in our axiomatic discussion of electrodynamics in Part B we adhered to tlic connection-free and metric-free point of view, the notions of connection and metric arc unavoidable in the end. In the previous Chapter C . l , we gave thc fundamentals of the gcolnetry of manifolds equipped witli a linear connection. Herc we tliscuss the metric. In Special Relativity theory (SR) and in the corrcsponding classical ficld theory in Hat spacetime, the Lorentzian mctric enters as a fundamental absolute element. In particular, all physical ~~articlcs are defined in terms of representations of the Poincard (or inhomogcncous Lorcntz) group wllicli has a mctric built in froni tlie very beginning. 111General Relativity theory (GR), the metric field is upgraded to tlie status of a gravitattonal potential. In particular, the Einstein field cquation is forrnulatecl in terms of a Riemannian niet,ric witli Lorentz signature carrying on its right-hand side thc symmetric (I-Iilbert or mctric) energy-momentum tensor as a. material source. The physical significance of the spacetime lnetric lies in the fact that it determines irltervals ds 2 between events in spacetime and, furtliermorc, establishes the causal structure of spacetime. It is important to realize that tlie two geonietrical structures - the conricction ancl thc metric - are a priori absolutely independent from each other. Modern data convincingly demonstrate the vdidity of Riemannian gcolnctry and Einstein's GR on inacroscopic scales where Inass (energy-momcntt~n~) of matter alonc deterinines the structure of spacctime. However, a t high energies, the properties of matter are significantly different, with additional spacetimerelated characteristics, such as spin and scale charge coming into play. Corne-
212
C.2. Metric
spondingly, one can expect that the geometric structure of spacetime on small distances may deviate frorn Riemannian geometry. "In the dilemma whether one should ascribe t o the world primarily a metric or an affine structure, the best point of view may be the neutral one which treats the g's as well as the r ' s as independent state quantities. Then the two sets of equations, which link them together, become laws of nature without attributing a prefcrcntinl status as definitions t o one or the other half."'
C.2.1 Mctricvcctor spaces A metric tensor introduces the length of a vector and an angle between e11ery two vectors. The component.s of the metric are defin.ed by the valr~esof the scalar products of the basis vectors. Let us consider a linear vector space V. It is called a metric vector space if on V a scalar product is defined as I~ilinearsymlnctric and nondegenerate map
(3.2.2 @Orthonormal,half-null, ant1 null frames, the coframe statement
213
Thus tlic isonlorpllis~n(C.2.3) is tcclinically rcd~~cecl to tlic vertical motion of indices,
Accordingly, t,hc basis vectors define the 1-forms via
Under a change of the basis (A.1.5), the metric coefficients goo transform according to (A.1.11). Recall tliat a symmetric matrix can always be brought into a diagonal form by a linear transformation. A basis for which
is called orthonor~nal.We are mainly interested in 4-dimensional spacctimc. Its tangent vcct,or space a t each cvcnt is Minkowskian. Therefore, from now on, let 11s take V t o be a 4-dimensional Minkowskian vcctor space unless s1,ecificd othcrwisc. The components of tlic nletric tensor with respect to an ortl~onorlnal basis are the11 given by
In other words, a scalar product is introduced by a metric tensor g of type [i] tliat is symmetric, i.c., g ( u , v) = g(v, u) for all u, v E V, and nondegenerate in the sense that g ( u , v) = 0 holds for all v if and only if u = 0. The real number
is called a length of a vcctor IL. The metric g defines a canonical isomorphism of the vcctor spncc ant1 its dual,
where the 1-form g(u), if applied t o a vector v, yields ( u ) (v) : (
for all v E V .
II )
(C.2.4)
Alternatively, we nlay write g(u) = g(u, In ternls of a basis e, of V and tlie dual basis 19" of V*,
C.2.2 @Ortkionormal,half-null, and null frames, the coframe statement A n , ~ ~~jector ll has zrro length. A set of null vectors is in many cases a convenient tool for the con,stm~ctionof a special ba.sis in, n Min.kowski vector space.
a ) .
g = gcro19" @ ool with
gap := g(e,, ep) = go,.
(C.2.5)
'" In den] Dilemnin, ob Innn dcr Welt urspriinglich eine ~netrischeoder cine nffine Struktur zuschreiben soll, ist viclloicht der beste Standpunkt der neutrale, der sowohl die g wie die r als t~nabhiingigcXustnndsgroDcn behnndelt. Dnnn wcrden die beiden Satze von Gleichungen, welche sie verbinden, zu Natrlrgesctzen ohne daO die eine oder andere Halfte als Definitionen cine bevorzugte Stcllung beko~nnien."I-I. Weyl: 50 Jahre Relativitatstheorie (181, our trnnslntion.
The Minltowski (or Lorcntz) metric has many interesting "faces" which we lnention licrc only briefly. Traditionally, in relativity theory the vectors of an ortlronormnl basis arc labeled by 0 , 1 , 2 , 3 , thus underlining the funclanlcrital difference bct,wccn ro, which has a positive length go0 = g(eO,eo) = 1, and e,, a = 1 , 2 , 3 , which l l i ~ ~ ~ negative length g,, = g(c,, e,) = -1. In general, a vcctor I L V is ~ i ~ l l ~ d timelike if g(u, u ) > 0, spacelike if g(u, 11) < 0, and null if g ( u , u ) = 0. In Excalc one spccifies tlie coframc rts the primary quantity. If we use Cartcsian coordinates, an ort1ionorm;~lcoframc and frame in Minkowski space read, rcspectivcly,
C.2. Metric
214
C.2.2 @Orthonormal,half-null, and null frames, the coframe statement
215
I
coframe o(0) o(l) o(2) o(3) metric g frame e;
= = = = =
d t , d x , d y , d z with o(0)*o(0)-o(l)*o(l)-o(2)*0(2)-0(3)*0(3);
coframe h(0) h(1) h(2) h(3) metric hh
The blank betwcen d and t and so on is necessary! Note that the phrase with ; in the case of a dimetric g=o(O) *o(0)-o(1)*o(i)-o(2)*o(2)-o(3)*o(3) agonal metric can also be abbreviated by with signature 1,-1,-1,-1; We recall that, in Excalc, a specific spherically symmetric coframe in a 4dimensional Riemannian spacetime with Lorentzian signature has already been defined in Sec. B.5.6 in our Maxwcll sample program. In general relativity, for the gravitational field of a mass m witli an angular momentum per unit mass a , one has an axiall?~ symm,etric ~net~ric witli a coframe like pform rr=O, delsqrt=O, ffsqrt=O$ fdomain rr=rr(rho, theta), delsqrtzdelsqrt(rho)
, ff sqrt=ff sqrt (theta)$
cof rame o(0) = (delsqrt/rr)*(d t-(ao*sin(theta)**2)*d phi), o(1) = (rr/delsqrt)*d rho, o(2) = (rr/ff sqrt) *d theta, o(3) = (ffsqrt/rr)*sin(theta)*(-aO*d t+(rho**2+a0**2)*d signature 1,-1,-1,-1$
= h(o)*h(i)+h(l)*h(o)-h(2)*h(2)-h(3)*h(3);
You can convince yourself by displayf rame; and on nero; hh(-a, -b) ; that all has been understood correctly by Excalc. Following Newman & Penrose, we can further construct two more null vectors as the complex linear combinations of ez and es:
Herc i is the imaginary unit, and overbar means the complex conjugation. This transformation leads to the Lorentzian metric in a null (Newman-Penrose) basis eat# = (1, n , m , m ) :
hi) with
I
Here rr , delsqrt , f f sqrt are functions to be determined by the Einstein equation. This is an example of a coframe that is a bit more involved. Starting from an orthonormal basis e, witli rcspect to which thc metric has thc standard form (C.2.9), wc can build a new framc e,, = (I, n , e2, eg) by thc linear transformation:
arid e21 = ez, eg~= e3. The first two vectors of thc new frame are null: g(1,l) = g ( n , n) = 0. Correspontlingly, the mctric in this half-null basis reads
In Excalc, again with Cartesian coordinates, we find
= (d t+d x)/sqrt (21, = (d t-d x)/sqrt(2), = d y, = d z with
Such a basis is convenient for invcstigating the properties of gravitational and electromagnetic waves. In Excalc, we havc I
coframe n(0) n(1) n(2) n(3) metric nn
d x)/sqrt (2), = (d t d x)/sqrt(2), = (d y + i*d z)/sqrt (2), = (d y i*d z)/sqrt(2) with = n(o)*n(l)+n(l)*n(o)-n(2)*n(3)-n(3)*n(2) = (d t +
-
-
;
In the Newman & Penrose frame, we have two real null legs, namely 1 and n , and two complex oncs, m and m. It may be surprising to learn that it is also possible to define a null symmetric frame that consists of four real null vectors.' We start from an orthonormal basis 2 ~ u c hcon~pletely symmetric real null frames had first been used by R. Coll and J.A. Morales [4] (see also Finkelstein and Gibbs [7]) and the corresponding coordinates even earlier by G.1-I. llcrrick [5] and Coll 121. Coll [3] also recognized the role of such frames in the framework of the Global Positioning System (GPS), see also Rovelli [15] and Blagojevid et 81. [I].
C.2.3 Metric volume 4-form
C.2. Metric
e,, with g(e,, ep) = o , ~ ,and define the new basis
fi,according
to
I
Sirice g(f6, f6) = 0 for all &, the null symmetric frame consists solely of real nonorthogonal null vectors. The metric with respect t o this frame reads
The metric (C.2.15) looks completely symmetric in all its components: Seemingly tlie time coordinate is not preferred in any sense. Nevertheless (C.2.15) is a truly Lorcntzian metric. Its deternlinant is -3 and tlie eigcnvalues are readily cornplitcd to be which shows that tlie mctric (C.2.15) has, indeed, the correct signature. There is a beautiful geolnetrical interpretation of the four null legs of tlie null symmetric frame. In a Minkowski spacetime, let us considcr the 3-dimensional spacclikc hypcrsnrfacc t,llat is spanned by (el, en, eg). Tlle four points that arc defillcd by the spat,ial parts of the null symnletric basis vectors ((2.2.14) with coordinates A = (1,1, I ) , B = (1, -1, -I), C = (-1,1, - I ) , and D = (-1, - 1 , l ) form a perfect tetrahedron ill the 3-subspace (scc Figure C.2.1). Thc vertices A, B, C , ant1 D lie at tlic same tlistanccs of f i / 2 from the origin 0 = (0,0, 0), r~ntlc o r r c ~ ~ ~ o ~ i d i all n g lsides y of tlic tetiahetlron arc of equal length, namelv fi.If we now send, at t = 0, a liglit pulse from the origin 0, it rrachcs all four vcrticcs of t,hc trtralicdron a t t = &/2. Thus four light rays provide thc operatiollal definition for the n?~llsynlmetric CoH-Moralcs basis ((3.2.14). In a Riemannian spacetime, we can pick Riemannian nornlal coorclinates at an arbitrary point. Thcn, provitlcd tlie tctrahcdron is sufficiently small, we have ~ ~ analogous 1 1 interpretation : ~ the t point choscn.
C.2.3 Metric volume 4-form Given a metric, a co~respondingortllonormal coframe determin,es a metnc vol~~me 4-form on eve? vector space. Let I9" be a n ort,honormal coframe in the 4-dinlensional Minkowski vector space (V*, g ) . Let us tlefinc the product
If 19"' is another ortlionor~nalcoframe in (V', g), then
Figure C.2.1: A tetrahedroll that represents the real symmetric null frame in 3-dimensional space. At time t = 0, light is emitted in 0. It reaches the points A, B , C , and D a t t = f i / 2 . The cvcnt (0, O ) , together wit11 the events (&/2, A), (&/2, B), etc., determine the four real null vectors.
where the transformation matrix is (pseudo)orthogonal, i.e., o,,,p L,"! Lop' = oa0
and
L := det (L,"') = &1.
(C.2.19)
Therefore, under this chai~geof t,he basis, we havc do' A dl' A ?J2' A d3' = L d0 /\ 1 9 ~A d 2 A ~ 9 ~ .
(C.2.20)
I11 other words, tlic definition (C.2.17) gives us a unique (i.e., basis independent) twi-sted v o l ~ ~ m4-form e of the Minkowski vector space. Alternatively, one may consider two untwisted volume 4-forms separately for each orientation. If 19"' is an arbitray (not necessarily orthonormal) frame, then we have 19"' = ~ ~ " ' 6and, " since 77 is twisted, q = ILI do' A 011 the
ol' A d2' A g3' .
(c.2.21)
other hand e,, = Lala e,. Thus, from tlie tensor transformation g,lfll =
Letn Lljlfl o , ~ ,we obtain ILI2 = - dct (g,tpt). Hence the twisted volulne eleirlent
with respect to the frame e,, reads 77 = B -JO'
A d l i A d2' A d3' .
(c.2.22)
C.2. Metric
218
Dropping the primes in (C.2.22), we may write, for any basis d a , the metric volume 4-form as
, where tlic twisted antisy~nlnetrict,cllsor Vnpya of type
[:]
5'
~
I
9
LBv
y p Jo
9 9
V I L V ~= U
-
1
is defined by
caP-ls,
JW
for all w, cp E M 6 . The explicit action of the duality operator on tlie basis 2-forms reads
(C.2.23)
= + l . If we raise and iapyg is tlie Levi-Civita permutation symbol with i0123 the indices in the usual way, we find the contravariant components as
17nP~g =
219
C.2.4. 1)ualit.y opertttor for 2-forms
(C.2.25)
It follows from (C.2.25) that
wliere the elenients of tlie matrix #,,"P are, by definition, the componellts of the almost complex str~lcturcin M G .In the 6-dimensional notatlion introduced in Sec. A.1.10, the dliality opcrat,or is t1111sclescribcd by
Exprcssod in tcrms of components, the metric reads ~ ( wcp), = ~ I . ' w ~ c psee ~, (A.1.88). Therefore, tlic self-acljointncss can also be written as ~ ~ " # ~ " ~ ~ ; = c p , ~ # J 1 < ~cpl<, l or "IJ
X
.JI
"IJ
x
for
=X
:=EI"#~<~
By construction, tlic components of tlie duality niat,rix can be cxprcssctl in t,crms of the 3 x 3 bloclts imported from (A.1.108),
C.2.4 Duality operator for 2-forms as a symmetric almost complex structure on M G The dualitlj operator grows out of the almost complex structure J on M G uhen J is additionally .self-adjoint with respect to the natural metric on M G . Let us now turn again to the space M%f 2-forms in four dimensions, wl~ichwe discussed in Sec. A.l.lO. When an almost complex structure J is ilitroduced in M G ill such a way that it is additionally symmetric, i.c., self-adjoint, with respect to tlie natural metric E (A.1.87) on M G ,then a linear operator on 2-forms
defined by means of # = J , is called the duality operator on M6. In view of (A.1.107), the tluality operator satisfies
Here tllc con~poncntsare co~lstrailicdby tlic self-adjointncss (C.2.29),
as compared to the aln~ostcomplex structr~rc(in particular, the D-block is exprcsscd in tcrms of C ) . Besitlcs t,l~at,the algcl~raiccontlitioll (A.1.109) is replaced by the closure relation
Tlic existence of tliv tluality operator has all immediate consequence for the ~omplcxifi~d spacc AJC'(C)of 2-forlns. As we saw in Sec. A 1.11, the a11ilost complcx structure providcs for a splitting of tlic M6(C) illto the two 3-dilncnsional subspaces coircsponding to tlic hz eigcnvalues of J. Now we can say cvcll more: Tllcsc two sul~spaccsarc ortliogonnl to each other in the scnsc of the natural 6-metric (A.1.87): E(W,cp) =
This will bc called the closure relation of tlie duality operator. The self-adjointness with respect to the 6-metric (A.1.87) Incans
0,
for
(s)
1111
w E A[,
E
(i\)
(C.2.36)
The proof is straiglitforwi~rtl:~ ( wcp), = ie(w, #cp) = za(#w, cp) = - E ( u , cp), wliere we used the definitions ( A . l . l l l ) and the symmetry property (C.2.29). Tllc G-metric a induces t t metric on t,lic 3-tlimensioual sul,spaccs (A.1.111), turning them into complex Euclicteall 3-spaces. Tlle syinnictry group, which
C.2. Metric
220 (8)
(a)
preserves the induced 3-metric on M (and on M ) , is SO(3, C). This is a grouptheoretic origin of the reconstruction of tlie spacetime metric from the duality operator: The Lorentz group, being isomorphic to SO(3,C), is encoded in the structure of the self-dual (or, equivalently, anti-self-dual) complex 2-forms on X. The significance of the duality operator will become clearer in the next sections where we explicitly demonstrate that # enables us to construct a Lorentzian rnetric on spacetime.
C.2.5 From the duality operator to a triplet of complex 2-forms G
Every duality operator on M determines a triplet of complex %forms that satisfy certain completeness conditions. Suppose the duality opcrator (C.2.27) is defined in M G with the closure property (A.1.107) and self-adjointncss (C.2.29). Its action on the basis 2-forms is given by (C.2.31), with the matrix (C.2.33), (C.2.34). Using the natural 3 3 split of the 2-form basis (A.1.79), cq.(C.2.31) is rewritten, with tlie help of (C.2.33),
C.2.5 From the duality operator to a triplet of complex 2-forms
221
Thus the 6-dimensional space of complex 2-forms M6(C) decomposes into two 3-dimensional invariant subspaces that correspond to the two eigenvalues fi of the duality operator. In order to construct the bases of these subspaces, we have to inspect the (3+3)-representation (A.1.79). One has, using (A.1.79) in (C.2.40), the two sets of self-dual 2-forms,
and similarly for the anti-self-dual forms. With the help of (C.2.38), we find explicitly:
+
Since the invariant subspace of the self-dual forms is 3-dimensional, these two triplets of self-dual forms cannot be independent from each other. Indeed, let us multiply (C.2.45) with B,, and (C.2.46) with C a C .Then the sum of the resulting relation, with the help of the closure property (C.2.35), yields
or in terms of its matrix elements,
This shows that these two triplets are linearly dependent. For the nondegenerate By rncans of tlie duality operator # (as well as with J of the almost complex structure), one can decompose any 2-for111 into a self-dual and an anti-self-dunl part. In terms of the 2-form basis, this reads,
(sIa
/3 A
where we define
.
(3)
(8)
B-matrix, one can express p a in terms of 6 explicitly. Let us compute the exterior products of the triplets (C.2.45) and (C.2.46):
.
(8)
E
(s)~
P
A
=-
(N)
6 t,
12 ( ~ A R ' + C(,,A~)' Vol = - -2i AabVOI, 1 . i ( z B , ~ CC(,Bb),) V O= ~ - - BabVal. 2 2
=--
+
((2.2.48) ((2.2.49)
We used here the algebra (A.1.83)-(A.1.85) and the closure property (C.2.35). Furthermore, we have Here i is the imaginary unit. One can check that
Here the overbar denotes complex conjugate objects.
C.2.6 From the triplct of complex 2-forms to a duality opcrator
C.2.6 From the triplet of complcx 2-forms to a duality operator Ever?j triplet of com.plex 2-forms with the comp1etene.s~property determ,ine a duality operator on M 6 ,
Both (C.2.45) and (C.2.46) are particular represelltatiolls of the following gcnera1 structure: Given is a triplet of self-dual complex 2-fomns S(n)such that
223
Combining (C.2.58) and (C.2.60), we find
whereas thc sum of (C.2.59) and (C.2.61) gives
We can count the nuniber of independent degrees of freedom. Thc total number of variables is 36 (= 4 x 9 unknown con~ponentsof tlie matrices V, U, X, Y). They 6 + 9) imposed by the equations ((2.2.62) are subject to 21 constraints ( = 6 and (C.2.63). Thus, in general, 15 degrees are left over for the unknown matrices in (C.2.55). In order to see how the triplet is related to the duality operator, let us define a new basis in M G by means of the linear transformation
+
where tlic matrix Gab is real and nondegenwrate. Thc overbar denotes complex co~ljrigat~c objects. Equivalently, one can rewrite (C.2.52) as
whcrc Gab denotes the matrix invcrsr to G ab . Wc call (C.2.52), (C.2.53) tlic completeness conditions for a triplet of 2-forms. I11 the previous section we saw that every duality operator defines a triplct of self-dual complex 2-forms. Here we show that thc converse is also true: Let IF an arbitrary triplet of complex 2-for111s that satisfies thc completeness conditions (C.2.52), (C.2.53). Then they determine a dualit,y operator in M 6 . Expancling the arbitrary 2-forms with respect to tllc basis (A.1.79), we can write
wliere Uao := GncUcband Yo" := GncYc
In view of (A.1.83)-(A.1.85), we find that (C.2.52) imposes an algebraic constraint on tlie matrix componcnts,
Thc direct clicck again involves only the complcte~lessconditions (C.2.G2) and (C.2.63). The original t,riplet (C.2.55), with respect to the ncw basis (C.2.64), then reduces to
as those in (A.1.83)-(A.1.85). The proof follows directly from ((2.2.62) and
(C.2.63). Interestingly, the transformatioll (C.2.64) is always invertible, with t,he inverse given by
Introducing the rcal variables Now we arc prepared to introtluce the duality operator. We define it by simply postulating that its action on the triplct amounts to a mere multiplication by thc imaginary unit, i.e.,
one can decompose (C.2.56) into t,he two rcal equations: v ( n c XWC -
v ( n Cy h ) c
u(", y W c
+ ($ac
xh)c
=
0,
=
-~
a
b
(C.2.58) ((2.2.59) = ( s ( ~ ) + S ( ~ )and ) / ~;b = i ~ , ~ ( s ( " ) - S ( ~ ) ) Then, /2. Fro111 ((3.2.67) we have immcdiately, wc find the action of the duality operattor on the ncw 2-form basis:
Analogously, (C.2.53) yiclds another pair of real matrix equations:
v(",X W C + u ( a c y o v[",y"]" - u['" X ~ I "
=
0,
(C.2.60)
0.
(C.2.Gl)
224
C.2. Metric
C.2.8 Ilodge star and Excalc's
#
225
We can now reconstruct the original basis in ((3.2.37). We usc (C.2.66) and ((3.2.64) and find the 3 x 3 matriccs
metric would have a Riemannian (Euclidean) signature (+I, +I, +I, +1) or a mixed one ( + I , + I , -1, -1). As a comment to the Schonberg-Urbantke mechanism, we would likt, to mention the local isomorphism of the three following complex Lie groups (symplectic, special linear, and orthogonal):
As a consequence, every triplet of complex 2-forms, which satisfy the completeness conditions (C.2.52) and (C.2.53), defines a duality operator with the closure and symmetry properties.
The easiest way to see this is to analyze the corresponding Lie algebras. At first, recall that the orthogonal algebra so(3,C) consists of all skew-symmetric 3 x 3 matrices with complex elements:
C.2.7 From a triplet of complex 2-forms to the metric: Schonberg-Urbantke formulas
Here ql, q2, q3 is a triplet of complex numbers. Since the linear algebra sl(2, C) consists of all traceless 2 x 2 corqplcx matrices, its arbitrary elemcnt can be written as
The triplet of complex %forms is a building material for the metric of spacetime. The Lorentzian metric, up to a scale factor, can. be constructed from the triplet of 2-forms by means of the Schonberg--Urbanfieformulas.
Assuming that the three complex parameters ql, q2, qs in ((2.2.77) are the same as in (C.2.76), we obtain a map so(3,C) -+ s1(2,C), which is obviously an isomorphism. It is straightforward to check, for example, that the cornmlltator [a, b] of any two matrices of the form (C.2.76) is mapped into the col-nn~utator [z,;] of the corresponding matrices (C.2.77). The symplectic algebra sp(1,C) consists of all 2 x 2 matriccs ?i that, sat,isfy
The importance of tlie duality operator # and the corresponding triplet of 2forms lies in the fact that they dcterrninc a Lorentzian metric on 4-space. that satisfy (C.2.52)be Let us formulate this rcsult. Let a triplet of 2-forms s(") givcn on V with somc symmetric regular matrix G. Then the Lorentzian mctric of spacetimc is recovered with tlie help of the Schonberg-Urbantke formulas:
-a s + s i i T = 0, with s := ( -
).o n e can easily verify that ally lllatrix
(C.2.77) satisfies this relatioh, thus proving the isomorphism sp(1, C) == s1(2, c ) . Herc, rkl""' is the Levi-Civitn symbol, ancl thc
s:)
are the components of tlic
/
basis 2-forms with rcspcct to tlic local coordillates {T", i.e., S(")= ~ ~ ( ~A ) d z ~ 2 23 dx3. Despite the appearance of thc Levi-Civita symbols in ((2.2.73)-(C.2.74), thcsc cxprcssions are tensorial. A rigorous proof of the fact that on a 4-clin~ensionalvector space V every three complex 2-forms S("),which satisfy the completeness condition (C.2.52), dcfinc a (pse~~do)Riemannial~ metric will not bc presented here.3 The metric is, in general, complcx; however it is real when ((2.2.53) is fulfilled. Note that t l ~ esign in thc algcbraic equations (A.1.107) and (C.2.35) is important. If insteatl of the minus thcre were a plus, then the resulting spacetimc 3 ~ e ehowcvrr, , Schotlbcrg [16], Urbnntke 1171, or IIarnett [8].
C.2.8 Hodge star and Excalc's
#
In a metric vector space, the Hodge operator e.stablislles a map between p-forms and (n - p)-forms. Besides th,e 29-basis for exterior forms we can define an 71-basis that is the Hodge dual to the 29-basis. Consider an n-dimensional vector space with metric g. Usually (when an orientation is fixed), the Hodge star is defined as a linear map * : APV* + ArL-PV*, such that for an arbitrary pform w and for an arbitrary 1-form cp it satisfies
The {ormula ((3.2.78) reduces the definition of a Hodgc dual for an arbitrary
1
lqor ~n to the definition of a 4-form dual to a number * l . Usually, *1 is taken ;Ly t l l ~ordinary (untwistred) volurnc form ant1 such a procedure distingl~ishcsa rettjPinorientation in V. Wc change this convention and require, instead of the I,sUP] tlcfinit,ion, t h t
I I
*:
APV*
*:
twisted APV*
-+
twisted An-"*,
(C.2.79)
;llld vice versa A"-pV*.
(C.2.80)
~ ~ ~ ~ r t l i nwe g lput y , *1 equal to the twisted volurnc forrn:
*1 = q .
((2.2.81)
~ c us t now rcstxict our attention to thc 4-dimensional Minltowslti vcct,ol. space. We can use ((2.2.78) to tlcfine the Hodgc dual for an arbitrary p-form. Take at first w = 1 and cp = 6" as a coframc 1-form. Then, wit11 (C.2.4), ( ~ . 2 . 8 1 )ant1 , (C.2.7), we find
where we used (A.1.49) in ordcr to compute the interior product e,' J 7 for (C.2,23). We can now go on in a recursive way. Choosing in (C.2.78) w = I?" and again (P = P ~ Pwe , obtain:
I-lcrc en := ep. Substituti~ig(C.2.82) into (C.2.83) ant1 again using (A.1.49) to valuate t l ~ einterior procluct, we find s~~ccessivcly the forrnulss
* (I?" A 1 9 ~ )= * ( d m A 19' A 6')
(e"
_I
1 7) = - 7"876 67 A 19' =: vnp, 2!
= c7 J ( e b
(en J 7)) =
7rflys I)"=:
* (6" A 19' A d7 A 19') = 7"p7R.
7
(C.2.85) ~
,
~
7(C.2.86)
a new mrtm'c dependent basis for the exterior algebra over the Minkowski vector space. This construction can evidently be generalized to an n-dimensional casc. In the 12-dimensional Minltowslti space, **w
(-1)1+-~)+1
W,
for w E APV*.
(C.2.90)
TIIIIS,for n = 4, we have **w = w for exterior forms of odd degree, p = 1,3, and * * w = -w for forms of even degree, p = 0,2,4. F'rom the definition (C.2.78), we can read off the rules
* ( 6+ $) = *(b + *1/)
and
*(ad) = a * d ,
((2.2.91)
for n. E R and (b, II) E A"*. These linearity properties togethcr with (C.2.84)((2.2.87) enable one to calculate the Hodgc star of any exterior form. Tlio implementation of these structures in Excalc is simple. By means of the coframc statement, the metric is put in. Excalc provides the operator # as Hodge star. In clcct,rotlyllan~icsthe most prominent role of the Hodge star operator is that it maps, up to a dimensionful factor A, the field strength F into the excitation H, namely H = X *F,as we will scc in the fifth axiom (D.G.13). Therefore, in Excalc we simply have e x c i t 2 := lam * # f a r a d 2 ; This spacctime relation is all we neccl in order to make the Maxwcll equations to a complete system. We used this Excalc command alrcady in our Maxwcll sample program of Sec. B.5.6. As a fnrt,lier example, we study the electromagnetic energy-momentum current. Recall that we constructed in (B.5.39) the electromagnetic energy-momentuni tcnsor density in terms of the energy-momcntum 3-form. We can now find thc corrcsponding tensor if we take, inst,cacl of thc tliamond operator 0 ,the Hodgc star opcrnt,or * . Thus, % definitions of o(a) and maxenergyd(a) precede this declaration pform maxenergyO(a,b)=O; maxenergyO(-a, b) : = #(o (b) ^maxenergy3(-a) ) ;
Or, turning to the 7-system of (C.2.88), we can write the program as follows: pform etaO(a,b,c,d)=O,etal(a,b,c)=l,eta2(a,b)=2,eta3(a)=3,eta4=4$
((2.2.87) eta4 eta3(a) eta2(a,b) etal(a,b,c) etaO(a,b,c,d)
The ncwly tlcfincd 7-system of p-forms, p = 0, . . . , 4 ,
: = # 1$
: = e(a) -leta4$ : = e(b) -leta3(a)$ : = e(c) -(eta2(a,b)$ : = e(d) -letal(a,b,c)$
c~l'Rtitutc~s, i~longwith the usual I)-systc.m, {1,
I)",
6" A dp, 79" A d p A 6',
I)" A
6% A 7 A 19").
((2.2.89)
We col~lcldefine 7 alternatively as e t a 4 : =o(O) Ao(1) Ao(2) Ao(3)$ see (C.2.17). With these tools, the Einstein 3-form G, := 7 7 a p 7 ~ ~now P 7 emerges simply as
C.2. Metric
228
pform einstein3(a)=3; einstein3(-a) : =(1/2)*etal(-a,-b,-c)^curv2(b,
C.2.9 Manifold with a metric, Levi-Civita connection
Let X, be an n-dimensional differentiable manifold. We say that a metric is defined on X, if a metric tensor g is smoothly assigned to the tangent vector spaces X, a t each point x. In terms of the coframe field, c) ;
g = gaP(x) fia 8 dP,
Accordingly, exterior calculus and the Excalc package are rcally of equal power.
@Froma metric to the duality operator Let us assume that a metric is introduced in a 4-dimensional vcctor space V. Then, the Hodge star map (C.2.79), (C.2.80) is defined for p-forms. Restricting our attention to Zforms, we find that the Hodge star maps M G = A2V* into itself. The restriction
where gap(x) = g(e,, eo)
(C.2.98)
is a smooth tensor field in every local coordinate chart. The manifold with a metric structure defined on it is called a (pseudo)Riemannian manifold, denoted V, = (X,,g). Usually, a metric (and, correspondingly, a manifold) is called Riemannian if the gap(x) is positive definite for all x. However, in order to simplify formulations, we will omit the "pseudo" and call metrics with a Lorentzian signature Riemannian. The metric brings a whole bunch of related objects on a manifold V,. First of all, a metric volume n-form emerges on V,: For every coframe field da = (.lgi,.. . ,6'), it is defined by
is obviously a duality operator in M G. Recalling the definitions of Sec. A . l . l l , wc can straightforwardly verify that both the closure (A.1.107) and the symmetry (C.2.29) are fulfilled by (C.2.92). The matrix of the duality operator is given by #poUP
=
Kpo
nP
.-.-
The world metric tensor with components rl,tvpa g n l l ~ g v l P , gij (5)= ei (x) ejP (x) gap (z)
Alternatively, in 6-dimensional notation, it reads
where a straightforward calculation yields the 3 x 3 blocks
is defined in every local coordinate chart {x i ). The principal difference between gap and gij is that the former can always be "gauged away" by the suitable choice of the frame field e,. One can choose an orthonormal frame field, e.g., in which gap has the diagonal form (C.2.8) independent of local coordinates. However, it is impossible in general to choose the coordinates {x i ) in such a way that gij is constant everywhere on V,. Thc Levi-Civita tensor densities in (C.2.99), (C.2.100) are introduced by
~ i...lin(x)
C. 2.9 Manifold with a metric, Levi-Civita connection The metric o n a manifold is introduced pointwise as a smooth scalar product o n the tangent spaces. The Levi-Civita (or Riemannian) connection is a unique linear connection with vanishing torsion and covarinntly constant metric.
(C.2.101)
=
eila'
a
einan~a ,...a,, = J ~ E ~ ~ . , , ~ , , ( C . ~ . I O
with the numerical permutation symbol chosen as el..., = tl,Hats over numerical indices help to distinguish components with respect to local frames form components with respect to coordinate frames. The next relative of the metric is the Hodge star operator. It is naturally introduced on a V, pointwise with the help of formulas derived in Sec. (2.2.8. Finally, the most far-reaching and nontrivial consequence of the metric g is the existence on a V, of a special covariant differentiation which is usually called a Riemannian or Levi-Civita connection. We use tilde t o denote this connection and any objects or operators constructed from it. As was shown in
C.2. Metric
230
Sec. C.l.l, a covariant clifferentiation is defined on a manifold as soon a5 in evcry local chart the connection 1-forms r,Jare given that obey the consistcncy conditicn (C.1.11). The Riemannian connection is defined, in each local coordinntc syrtern by thc Christoffel symbols rkt3:
-
The Leibniz rule for d was used a t an intermediate stcp, and the boundary integral (for noncompact manifolds) is vanishing due to the proper behavior of the forms at infinity. Accordingly, in n = 4 we have dt = *d * for all forms. In local coordinates, the codifferential of a pform p = $ pil...i,,dxil A . - .A dxip reads:
-
The Lcvi-cvita (or Riemannian) connection V has some special propertics that distingui5hit from other covariant differentiations. It has vanishing torsion. This is trivially seen from ((2.1.42) and (C.1.18) if we notice that the Cl~ristoff~l symbols (C.2.103) are symmetric in its lower indices. Moreover, the covariant exterior derivhtive of thc metric with respect to the Levi-Civita connection vanishes identcally:
Here is the covariant differentiation for the Levi-Civita connection (C.2.103). The codifferential is nilpotent,
scc ((2.2.136).A conncction for which the covariant derivative of the metric is zcro is called inetric compatible.
which follows directly from (C.2.107), (C.2.90) and thc nilpotency property d2 = 0 of the exterior differential (A.2.19). The operator d (resp., d t ) increases (resp., decreases) the rank of a form by one. Hence, the combinatiolls ddt and dtd both map p-forms into p-forms. However, these operators arc not sclf-adjoint with respect to the scalar product (C.2.105). Thc symmctrized second-order differential operator
C.2.10 @Codifferentialand wave operator, also in Excalc By means of the Hodge star operator we can define the codiflerential, which is adjoint to the exterior d~fferen~tial d luitlr. respect to the scalar product o n exterior forms. This yields directly a lllnve opero.tor. Consider the tpacc A"(X) of all snlooth exterior pforms on X. Thc Hodgc operator m~k@+ it possible to define a natural scalar product on this functional space: (~,(a)
:= J w A 'p,
for all
w, p E h p ( x ) .
is called the wave operator (or d'Alembertian, also called Laplace-Bcltrami operator). It is, by construction, self-adjoint with respect to (C.2.105). On one occasion, we had to check whether the wave operator, if applied to 7 a certain coframe field, vanishes, i.e., O d n = 0. Excalc could hell>. Aftcr the coframe statement specifying the appropriate value for the coframe, we defincd a suitable 1-form: I
pform wavetocoframel(a)=l$
((3.2.105)
X
The11 the cof'lfferential operator clt can be introduced as an adjoint to thc cxtcrior diffcrc1lt,iald with rcspcct to the scalar product (C.2.105), ( w , d t p ) := (dw, 9 ) .
(C.2.106)
By constrructicnlthe codiffercntial maps p-forms into (p - 1)-forms (contrary to cxteriordifferenti,zlwhich increases the rank of a form by one). Using the properN (C.2.\0)of the Hodge operator, one can verify that in an n-dimensional ~ o r e n t z i "'Pace ~ the codifferential o n p-forms is given explicitly by
The emerging expression we had to treat with switchcs and suitable substitutions, but the quite messy computation of the wave operator was givcn to t,hc machine.
Let a connection an,d a metric be defined independently on the same spacetime manifold. Then the nonmetricity is a m.easure of the incompatibility between metric and connection.
C.2.12 l'ost-Riemannian pieces of the connection @
Lct us consitlcr the general case when on a rnanifoltl X,, tlie metric and connection are defined independently. Such a manifold is denotctl (X,, V, g) ancl called a metric-a@ne spacetime. Sincc the mctric is a tensor field of type its covariant tlifferentiation yields a typc tensor ficltl tliat is called the nonmetricity:
I![
233
The factor l / r z is conventional. Then the nonlnetricity is decomposed into its deviator and its trace according to
['!I,
Thc trace of the curvature, which is called tlie segmental curvature, can be exprcsscd in terms of the Weyl l-form: n R,Y = - dQ. (C.2.119) 2 Thc pliysical importance of thc Wcyl l-form is related to the fact that, during parallel transport, the contribution of the Weyl l-form does not affect the light cone, whereas lengths of non-null vectors are merely scaled with some (pathdependent) factor. A space with gap = 0 is called a Weyl-Cartan, manifold Y,,. In this latter case, tlic position vcctor (C.1.GO) changes according to
for all vcctor fields u, v , ~Nonmetricity . rncaslires the extent to which a conncction V is incomptltible witli the mctric g. Mct,ric conlpatibility (also called mctricity) Q(71,V , w) = 0 implies the conservation of lengths and angles under parallel transport. A manifold that is endowed witli a mctric and a metriccompatible connrction is said to bc a Riemann-Cartan manifold (or a U,). In general, tlie Riemann-Cartan manifold has a nonvanisliing torsion. When the latter is zero, we recover the Riemannian manifold clescribcd in Scc. (3.2.9. Lilte the 2-forms of torsion (C.1.38), (C.1.40) and curvatlire (C.1.45), (C.1.46), we define the nonmetricity l-form
1 r e - R[Pn]rp), (C.2.120) n if it is Cnrtan displaced ovcr a closet1 loop that encircles tlic area clcment AS. T h r first cllrvature t,crm intluces a tliltttion, while the second onc is a purr rotation. A r e = A S (T" -
-R,T
C.2.12 "Post-Riemannian pieces of the connection Since ?~(g,lj)= dgnP (IL), equation (C.2.111) is cquivalcnt to
An arbitrary linear connection can, always be split into the LeviCivita connection plus a post-Riemannian tensorial piece called the distortion. The latter depends on torsion and nonmetricity. Correspondingl?y, nll the geometric objects and operators can be system.ntically decomposecl in-to Riem~n~nian and postRiemannian parts.
wllcre I',p := gp,T',,Y. If t,hc g , ~a" constants, then it follows from (C.2.114) that Q,,p = 2r(,p). Hcncc, in a U,,, where Q,o = 0, we have ~iit~isymmet~ric connection olic-forms
" . . . the qllestion whethcr this [spacctimc]continuum is Euclidean or structured according t,o the Riemannian scheme or still otherwise is a genuine physical question which has to be answered by cxpcricnce rather than being a mere convention to be chosen on the I~asisof c x p ~ d i c n c y . " ~
provided tho g,p arc constants, for example, witli respect to orthonormal coframc fields. We sha11 refer to (C.2.114) as the 0th Cartan struct?~ral relation and shall call the expression obtained as its exterior dcrivntivc,
tlic 0th B~nnchz~dentzfy.The proof of (C.2.116) makes use of the Ricci identity (C.1.65). It is convenient to separate the trace part of the nonmctricity from its traceless piece. Let us define the Weyl I-form (or Weyl cov~ctor)by
I
Thc gcomet,rical properties of an arbitrary metric-affine spacetime arc described by tlie 2-foniis of curvature R e P and torsion T" and by the l-form of nonmctricity Qap. Particular valucs of these funclamental objects specify different gconietrics which may bc rcalizetl on a spacetime manifold. Physically, one can think of a number of "phase transitions" tliat the spacctilne geometry undergoes at different energies (or distance scales). Correspondingly, it is co~ivcnicntt,o study particular realizations of geonietrical structures within the frillncwork of several specific gravitational models. The overview of these models and of tlic relevant geolnetries is given in Fig. C.2.2. 4 A . Einstein: Geo~netrieund Erfahriing [6], our translation.
C.2. Metric
234
-- R=O -T=O )
C.2.12 @Post-Riemannian pieces o f the connection
235
Furthermore, switching off the nonmetricity completely, one recovers the Riemann-Cartan geometry Uq which is the arena of Poincark gauge (PG) gravity in which the spin current of matter, besides its energy-momentum current, is an additional source of the gravitational field. The Riemannian geometry Vq (with Q,p = 0 and Ta = 0) describes, via Einstein's General Relativity (GR), gravitational effects on a macroscopic scale when the energy-momentum current is the only source of gravity. Finally, when the curvature is zero, Rap = 0, one obtains the Weitzenbock space P4 and the teleparallelism theory of gravity (when Q,p = 0) or a generalized teleparallelism theory in the spacetime with The Minkowski spacetime M4 with nontrivial torsion and nonmetricity vanishing Qap = 0, Ta = 0, Rap = 0 underlies Special Relativity (SR) theory. The relations between the different theories and geometries are given in Fig. C.2.2 by means of arrows of different types that specify which object is switched off. In a metric-affine space, curvature, torsion, and nonmetricity satisfy the three Bianchi identities (C.2.116), (C.1.67), and ((2.1.69):
~4f.
DQap
r 2R(ap),
DT" r Rp" D R , ~ = 0,
A dP,
0th Bianchi,
(C.2.121)
1st Bianchi, 2nd Bianchi.
(C.2.122) (C.2.123)
In practical calculations, it is important to know exactly the number of geometrical and physical variables and their algebraic properties (e.g., symmetries, orthogonality relations, etc.). These aspects can be clarified with the help of two types of decompositions. A linear connection can always be decomposed into Riemannian and post-Riemannian parts, Figure C.2.2: MAGic cube: Classification of geo~net~ries and gravit,y thcories in tile threc "dimensions" (R, T, Q).
The most general gravitational model-metric-affine gravity (MAG)-employs tile (L,,g) geometry in which all three main objects, curvaturc, torsion, and nonmetricity are nontrivial. Such a geometry could be rcalized a t extremely snia11 distances (high energies) when the hypermomcntum current of matter fields plays a central role. Other gravitational models and the relevant geometries appear as special ceses when one or several main geometrical objects are complctcly or partly "switched off". The Z4 geometry is characterized by T" = 0 and was used in tlte unified field theory of Eddington and in so-called SKY-gravity (theories of Stephe~lson-Kilmister-Yang). Switching off the traceless nonmetricity, gap = 0, yields the Weyl-Cartan space Y4 (with torsion) or standard Wcyl theory W4 (with Ta = 0).
where the distortion 1-form Nap is expressed in terms of torsion and nonmetricity as follows:
The distortion "measures" a deviation of a particular geometry from the purely Riemannian one. As a by-product of the decomposition ((2.2.124) we verify that a metric-compatible connection without torsion is unique: It is the LeviCivita connection. Nonmetricity and torsion can easily be recovered from the distortion, namely
If we collect the information we have on the splitting of a connection into Riemannian and non-Riemannian pieces, then, in terms of the metric gap, the coframe d", the anholonomity C", the torsion T a , and the nonmetricity Q,p,
C.2. Metric
236
C.2.13 @Excalcagain
wc have the highly symmetric master formula
237
I
coframe o(O)= . . . ; % input 1 frame e; riemannconx chrisl; chrisl (a,b) :=chrisl (b,a) ; pform connl(a,b)=l, distorl(a,b)=l, nonmetl(a,b)=l$ connl(O,O):= . . . ; % input 2 distorl(a,b):=connl(a,b)-chrisl(a,b); nonmetl(a,b):=distorl(a,b)+distorl(b,a);
Thc first linc of this formula refers to the Riemannian part of the connection. Together with the second linc a Ricmann-Cartan geometry is encompassed. Only the third line makcs the connection a really independent quantity. Note that locally the torsion T, can be mimicked by the negative of the anholonomity -C, and the nonmetricity Qep by the exterior derivative dg,p of the metric. In a 4-dimensional metric-affine space, the curvature has 96 components, torsion 24, and nonmetricity 40. In order to make the work with all thcsc variables manageable, onc usually decomposes all geometrical quantities into irrcduciblc pirccs with respect to thc Lorentz group.
and one coulcl continue in this linc and compute torsion :inel curvature according to pf orm torsion2 (a)=2, curv2(a, b) =2;
All other relevant geometrical quantities can be derived therefrom.
C.2.13 @Excalcagain
Problems
r,P
Excalc has ;I cornmoclity: If a coframe o(a) and a metric g are prescribed, it cdculates thc Riemannian picce of the connection on demand. One just has to issue the command riemannconx chrisi; Then chrisl (one could take any ot,hcr namc) is, without further dcclaration, a 1-form with the index structure chrisi(a,b). Since we use Schouten's conventions in this book, we have to redefine Schriifcr's ricmannconx according to chrisl ( a , -b) : =chrisl (-b, a) ; If you distrust Excalc, you could also compute your own Riemannian connection according to (C.2.127), i.c.,
Problem C.1. I I
Problem C.2. Prove ((3.1.52) in thc infinitesimal case, approximating thc curve parallelogram.
r
pf orm anhol2(a)=2,christl(a,b)=l$
I anhol2(a) : = d o(a)$ christl(-a,-b) := (1/2)*d g(-a,+) +(1/2)*((e(-a)-1 ( d g(-b,-c)))-(e(-b)-l (d g(-a,-c))))^o(c) +(1/2) * ( e(-a)- 1 anhol2(-b) - e(-b)- 1 anhol%(-a)) -(1/2)*( e(-a)-1 (e(-b)-lanhol2(-c)))^o(c)$
Rut wc can asslirc you that Excalc does its job correctly. In any casc, with chrisi(a,b) or with christl(a,b) you can equally well compute the clistortion 1-form anel the nonmetricity 1-form in terms of coframc o(a), metric g, and conncction conni(a, b). The calculation would run a? follows:
Check the geometrical interpretation of torsion given in Fig. C.1.1 by clircct, calcrllation using thc definition of the parallel transport.
(7
by a small
Problem C.3. Prove the following relations involving the transposed connection:
i
-
,-.
1. T" = -To, that is, T" = -D29";
I
4. The covariant Lie derivative of an arbitrary pform 9 = @,,.,, . . . A dmr1lp!:
79"l
A
1. Find a transformnt,ion matrix from a11 orthonormal basis t o tlie half-null franic in which the mct,ric has the form ((2.2.11). 2. Fintl a transfornlatioll matrix from an orthonormal lxc~ist o a. NewnlnriPenrosc null framc in which the metric has the form ((3.2.13). 3. Find a linear transformation e, = Lap f p that brings tlie null symmetric Coll-XiIornles basis f p back t o an orthonormal frame e,. Solution,:
JZ
0
-JZ
0
1
1 1 4
1
Checl
T h c matrix S just, scalcs tlic timc coordillate and a short calculation witli R E D U C E shows that R is an clerncnt of SO(4). 5. I11 alinlogy to (C.2.14), dctcrmine a null synlllictric coframe q)" and show that it, is not tlual t o thc corrcspontli~ignull sylnlnctric frame. T h c situation is tlcpictrtl in Fig. C.2.3. Discuss the rclation betwccn thcsc t,wo different tetralicdra. Problem C.5.
Figurc (2.2.3: Tlic outer tctraliedron represents a real null symmetric coframe 4)". The inner tct,rahcdron is that of Fig. (3.2.1 and it visualizes t h c corres~mndingrc>nl nr~llsymmetric frnmc - - -f,_N,ote that the 3-vcctors OA, O B , etc. t o tllc planes B C D , C D A , etc. arc pcrpcntlic~ilr~r 4. Show that in 4-dimcnsional space (a) 79" A qp = 6; q , (1)) dl' A qp, = (5; ~1p- 6; q, , (c) 79" A rlprs = 6; r/pr 4- 6; qsp 6; 71-,6 , ('1) 7 9 " 7)/j'-y(i11 = 6;; 7 1 - ~6; r/pyl~ ~ ~ + 6; 11/36/~- 'J; 'ITS/' .
+
Hint:19"
A
71
= 0 because it is x 5-form in n 4-dilncnsional spttcc.
1. Prove that for 9 , $ E AI'V*, *J,A1//= * $ A $ , 1. Prove that Chri~t~offcl symbols dcfinc n covnriant diffcrcntiation by cliecl
-
r,p 3. Provo that for any w E AI'V* ant1 9 , .II, E A2V*:
-
eJp cZa 1 = g"(dgcy,,, =
+ etP d eh + (e, A d g , ~- e,
dgnr) 8"
(F,JW)A(~"J*W)=~, (en -1 (b) A (e" J SCC
1121.
*dl) =
* [(e,
-I
9) A (e@J $)] ; wlicrc C" = d79'* is thc nnllolo~iomity2-form, see (A.2.35)
((3.2.136)
References
[I] M. Blagojevik, J. Garecki, F.W. Hehl, Yu.N. Obukhov, Real null coframes in general relativity and GPS type coordinates, Phys. Rev. D65 (2002) 044018 (6 pages). (21 B. Coll, Coordenadas luz en relatividad. In Proc. 1985 Spanish Relativity Meeting, A. Molina (ed.) (Ser. Pub. ETSEIB: Barcelona, Spain, 1985) pp. 29-38. An English translation Light coordinates in relativity is available on the author's homepage h t t p : //toll. cc [The abstract of this article reads: "The construction of coordinate systems by intersection of four beams of light is analyzed."]
.
[3] B. Coll, Elements for a theory of relativistic coordinate systems. Formal and physical aspects, In Proc. 23th Spanish Relativity Meeting ERES 2000 (World Scientific: Singapore, 2001) pp. 53-65. 141 B. Coll and J.A. Morales, Symmetric frames on Lorentzian spaces, J. Math. Pl~ys.32 (1991) 2450-2455. (51 G.H. Derrick, On a completely symmetric choice ofspace-time coordinates, 3. Math. Phys. 22 (1981) 2896-2902. [6] A. Einstein, Geometrie und Erfahrung, Sitzungsber. Preuss. Akad. Wiss. (1921) pp. 123-130.
(71 D. Finkelstein and J.M. Gibbs, Quantum relativity, Int. J. Theor. Phys. 32 (1993) 1801-1813; D.R. Finkelstein, Quantum Relativity. A Synthesis
o f the Ideas o f Einstein and Heisenberg (Springer: Berlin, 1996) pp. 326 328. 181 G . Harnett, The bivector Cliflord algebra and the geometmj of Hodge dual operators, J . P l ~ y s A25 . (1992) 5649-5662. [9] D. Hartley, Normal frames for non-Riemann.ian con,n,ections, Class. Quailturn Grav. 12 (1995) L103-L105.
Part D
[ l o ] P. von der Hcyde, The equivalence principle i n the U4 theo y of gravitation,,
Lettre a1 Nuovo Cimento 14 (1975) 250-252.
The Maxwell-Lorentz Spacetime Relation
[ l l ] B.Z. Iliev, Normal frames an.d the validity of tlre equivalence principle. 3. Th,e case a1on.g smooth maps with separable points of selfintersection, J . p11.y~.A31 (1998) 1287-1296. [12]Y . Itin, Coframe enerqy-momentum current. Algebraic properties, Gen. Rclat. Grav. J. 34 (2002) No. 11. Eprint Archive gr-qc/0111087. [13] E. Kroner, Con,tin..uum tlleoy of defects, in: Physics o f Defects, Lcs Houches, Scssion X X X V , 1980, R. Balian et al., eds. (North-Holland: Amsterdam, 1981) pp. 215-315. 1141 M . Pantaleo, cd., Cinquant'anni di Relativith 1905-1955 (Edizioni Giuntinc and Sansoni Editorc: Firenzc, 1955). [15] C . R.ovelli, G P S obsem~ablesi n general relativity. Phys. R.ev. D65 (2002) 044017 (6 pages). [16] M . Schonbcrg, Electromagnetism and gravitation, Rivista Brasileira dc Fisica 1 (1971) 91-122. [17] H . Urbantke, A quasi-metric associated with S U ( 2 ) Yan,g-Mills field, Actn Pliys. Anstrinca S u p p l X I X (1978) 875-816. [18] H. W e y l , 50 Ja.hre Relativit~tstheorie,Die Naturwisscnscl~aften38 (1950) 73-83.
I
244
P a r t D.
The Maxwell-Lorentz Spacctimc Itclation
So fa, tilc Mtxwcll cquat,ions (B.4.9) ant1 (B.4.10) represent an unclerdetermine(] system )f ~lartialtliffrrcntial equations of first order for the excitation N and field Jreiigt,l~ F. I11 order t o reduce the number of independent variables, we llaVp to set up a universal rclation brtween H and F , which is assumed to be local:
We call this tile electromagnctzc spacrtzme relatzon. Tllereforc we can coinplctr elcctlot]yllan,i~~, foimulated in Part B up t o now metric- and connection-fire, by ii~tlot~llcillg a mitable spacctirnc relation as a fifth arzom. TIlr simples choice is, of course, a linear relation H K F with the constit,itivr tensor x. Eventually, this yields tlic conventional Maxwell-Lorentz elcctrodynnlnics, ~t is reinarkable that this linear rclation, if supplemented merely by a, reczproclt/ property (basically equivalent t o tlie closure relation rc. K = -1) , a lzght cone a t and by the sllmmc'tnJ of K (symbolically, K = K ~ ) introduces (>ac]lpoint of spacetime. In other words, we arc able t o derive the metric of s p a c r t i l ~ l 111) ~ , t o a conformal factor, from the existence of a linear K that is closc~tl:ind ~ ~ l ~ i l n e t r i c . " Alternntivcly, one colild simply assume the existence of a (pseudo)Ricmannia~~ metric g of signature ( + I , -1, -1, -1) 011 tlie spacctimc manifold. In both cases, the Hodge star operator * is available and ordiliary electroclynan~ic~ can be rccovered via the spacetilnc relation H * F . Anyone that is not interested in lrnrnilig nbo~itoui method of deriving t l ~ elight cone of spacetime can j111111) right ;LW;-LVto Chap. D.G where a specific linear spacetime relatioll is formulatetl as a fifth axio111. N
D.1 A linear relation between H and F
Tlre excitation H is related to the field strength F by means of a 1~n.il)ersallaw. W e call it the (electromagnetic) spacetime relation and assume it to be local and linear. Tlt.en,, tlre corresponding constitutive ten.sor rc. carries 36 indepen.den,t com.ponents. W e decompose K into th,ree parts, the principal part (20 corn.ponents), the skc111onpart (15 com.ponents), a.nd the azion part (1 component), and discus.^ tlreir respective properties.
D.1.1 Tlie constitutivc tensor of spacetime The electro.rnagnetic spacetime relation1 expresses the excitation H in terms of tlie field strength F . Both are elements of the space of 2-forms A 2X. However, I1 is twisted nncl F is unwistcd. T h l ~ sone call formulate the spacetime relation as a local law "rllis tY1'e
idea goes back 1.0 Toupin [54] and SchGnberg [48]; see also Urbantke [5(i]
anti ~ a d c z yl2l1. ~ Wang 1571 gave a revised presentation of Toupin's results. A forerunner was Perm [301; see in this context also the rnorc recent papers by IPron and Moorr 1421. For new and recent result\, see [3G]. It wnq recognized by Brans [3] that, within general relativity, it is ~ ~ o s s i l ~ l "line c a drrality operator in much the same way a n s we present below (see (D.4.6)) ntld that duality operator the rnetric can bc rccovercd. Subsequently nrlrnerorls nrltllors 'liscUs"~ ssl~cllstrr~cturcsin the framework of general relativity theory; see, e.g., C ~ I ) O VJacOb'~or~, ~ ~ ~ ~ ~and Dell [5], 't Ilooft [18], Harnett [Id, 151, Obr~khovand Tertychniy [37], ant' t'lle refcre~ccsgivcn t.hcrc. In Part D, we also use the resr~ltsof Rul,ilar and Gross [12, 451.
'Post, [43] named it the constittitive map incll~clingalso the constitutive relation for rnatter (see (E.3.58)). 'IYuesdell k Toupin 1551, Tonpin [54], aritl Kovetz (261 use the terrn nether relations.
D.1. A linear relation between I3 a n d F
246
is an invertible operator that maps an untwisted 2-form into a twisted 2-form and vice versa. Local means that the excitation H a t some point P of spacetime depends exclusively on the field strength F a t the very same point P. An important subcase is that of a h e a r law between the 2-forms H and F . Accordingly, the operator (D.1.2) is required to be linear, i.e., for all a, b E AOX and 4,11, E A2X we have
For physical applications, it may be useful to present our linear operator tc in a more explicit form. Because of its linearity, it is sufficient t o know the action of tc on the basis 2-forms. The corresponding mathematical preliminaries were outlined in Sec. A.1.10. A choice of the natural coframe d i = dx i yields the specific 2-form basis GI of (A.1.79). The operator tc acts on the 2-form basis dx k A dxl (= E' in the equivalent bivector language) and maps it in twisted 2-forms the latter of which we can again decompose: 1 (dxk A dx') = - rcijkl dx i A dxj or 2 Now, we decompose the 2-forms in (D.1.1): K.
tc
(G") =
rcIIi GI .
The choice of the local coordinates in the deduction of (D.1.6) is unimportant. In a different coordinate system, the linear law preserves its form due t o the tensorial transformation properties of rcijkl. Alternatively, instead of the local coordinates, one may choose an arbitrary (anholonomic) coframe 8" = eiadxi and may then decompose the 2-forms H and F with respect to it according to H = Hap da A 8012 and F = FaPda A 79012. Then, if we redo the calculations of above, we find 1 2
Hap = - rcap7' FT6 with rcaB7' = ei ejp ekYe16rcijkl
X
ijkl
kl .-. - 2- caJrnn .' rimn
[;I
This twisted tensor field of type is called the constitutive tensor of spacetime. This is supposed to remind us that the constitution of spacetime, or rather its constitutive properties relevant for electronlagentism, should be reflected in tc. In tht bivector n ~ t a t i o n these , ~ 36 independent components are arranged into a 6 x 6 matrix rcIs(. Naively, one could try n.," = f 6: as a simple constitutive tensor with a ~seu~loscalar function f . Then H = f F and Maxwell's equations would transform into (df) A F = J and d F = 0. For f = const and J # 0, the inhomogeneous equation would be inconsistent - clearly indicating that a spacetimc relatibn rciJk" f 6: is unphysical.
(D.1.8)
or
XIK= & I M rcMK,
or
r c I K = < l ~MxK .
(D.1.9)
and conversely, rcijkl =
Thus a linear spacetime relation postulates the existence of G x G functions nijk1(x)depending on time and space with
.
Here we used also the components of the frame e, = ek, dk. As we recall from Sec. A.1.10, the Levi-Civita symbols (A.1.89) and (A.1.91) can be used as a quasi-metric for raising and lowering pairs of indices. We define the constitutive tensor density of spacetime,
(D.1.4)
Substituting (D.1.5), together with a similar expansion for F, and making use of (D.1.4), we find
247
D.1 .I T h e constitutive tensor of spacetime
-1, 2
&jjmn
X
mnkl
(D.l.10)
Similar to (D.1.7), we have Xijkl =
-X
jikl
= - Xijlk
(D.l.ll)
The 3G functions Xijkl(x)are equivalent to rcijkf(x).Because of the corresponding properties of the Levi-Civita symbol, the Xijklrepresent an (untwisted) tensor density of weight +I. In terms of X, the spacetime relation reads
Excitation H and field strength F are measurable quantities. The functions (or X i j k l )are "quotients" of H and F. Thus they first of all carry the absolute dimension of [tc] = [XI = q2/h = q/@ = (q/t)/(@/t) = cumnt/voltage SI = llresistance = 1/R = S (for siemens), and moreover, they are measurable too. Two invariants of tc, a linear and a quadratic one, play a leading role: The twisted scalar K~~~~
1 12
"
kl K k f E' '3 =
--1
cy
..
:= - 6..%3=
1
[ijkl] Zijkl x 4!
and the true scalar 2Ff\user [8] argues that the field strength (E,B) and excitation ('H,2)) represent line coordinfil,es in Rg in the sense of projective geoemtry (with only points and lines but without metric'). Only a linear relation between (E, R) and (7-1,V)would then yield a metric geometry.
~2
:= -4!
Kij
96
.; . 2 23kl mnpq
xijmnxpqkl
(D.1.14)
D.1. A linear relation between H and F
248
In later applications we will see that it always fulfills X 2 > 0. Note that [a]= [A] = llresistance. It is as if spacetime carried an intrinsic resistance or the inverse, an intrinsic impedance (commonly called "wave resistance of the V ~ C U I I ~or " "vacuum impedance"). One could also build up invariants of order p according to the multiplicative pattern of K T , ' 2 ~ ~ . . . ~K I , ', ~ with ~3 , p = 1,2,3,4,.. . , but there doesn't seem to Ile a need to do so.
D.1.2 Decomposing the constitutive tcrisor
For
249
via its definition (D.1.17), wc fincl
Tensor tcijkl
D.1.2
Decomposing the constitutive tensor
The tensor K,," as well as the equivalent tensor density X 2 ~ k 1have 36 components. I11 GD representation, the latter can be represented by the G x G matrix which can be decomposed into its symmetric and antisymmetric parts, = x('") X [ I K ] . Because of its antisymmetry (or skewsymmetry), we call XII"] the skewon part of x . It is irreducible under the linear group. From the := E L M xLM,which we symmetric part X ( I I < ) we can still extract the trace call the clzion part of X . This yields three irrcducible parts,
+
Clearly, our irreduciblr dccornposition can also Ilc cxpressccl in terms of the operator tc and its components K , , ~ ' . The contracted tensor of type
[;I,
has 16 iridcpcndcnt components. Tlic sccontl contraction yields the pseudoscalar function
The traceless piccc
has 15 indcpendcnt components. These pieces can now be subtracted from thc original constitutive tensor. Then,
Tensor density
K,3h1 = ( l ) K t j k l
xijkl
It is simple to translate these results into a11 irreducible decomposition in the 4D ( 2 ) X , ( 3 ) X . The axion representation. We denote the irreducible pieces by and skcwon parts read, respectively,
The symmetric tracefree part, which we call the principal part, is tlie rest:
It is desirable to bring ( l ) X i j k l as well as Therefore, we define tlie analog of X ( r l < ) :
Tlien the axion part can be written as
( 3 ) X i j k 1 into
a more compact form.
-
By construction,
(')ti,,"'
'"til7"
+ (2)K,,kl + (:UK,lkl
(D.1.24)
+ 2 $[,[*6:; + 1-G ~ 6 ; 6 : ] .
(D.1.25)
is tlic totally trncclcss part of tlic constit~lt,iverllap:
+
Thus, we split n nccortling to 3G = 20 15 1- 1, and tlic [i] tensor ( ' ) K , , " ' is subject to the 1G const,raints (D.1.26) ancl carries 20 = 3G - 1G coin1,oncnts. We rccall the clefinition (D.1.9) of X ' ~ " . We substitute (D.1.25) into its righthand side. Then, with X l ~ k l= ( 1 ) I J ~ I+ ( 2 ) y l j k l
X
we find the translat,ion rules
+ (3)X13A1 ,
(D.1.27)
11.1. A linear relation between H and F
$50
21
( 1 ) ~, + . ( 2 ) ~. .+ ( 3 ) ~ .
-
23
23
23 7
1 V=--FAH. 2
with Substitution of (D.1.33) yields
T ~ i = T k F ,r =, 1 , 2 , 3
Operator
251
For a linear spacetime relation H = n ( F ) , the electromagnetic Lagrangian is quadratic in F and has, in accordance with (B.5.90), the form of
Finally, we call express t,he spacetimc relation as H..-
D.1.3 @Decomposingenergy-momentum and action
K
However, because of (D.1.35),for 4 = 4 = F , the skewon part of the Lagrangian vanishes:
In operator language, the spacetimc relation reads H = (')H
+ (')H + (
3 )= ~ (')K(F) ~
+ (')K(F) + ( 3 ) ~ ( ~ ) .
(D.1.33)
For every pair of 2-forms 4 and $, the symmetries of the irreducible operators manifest themselves as follows:
4A(
)=
4 A (')n(ltl/),
(D.1.34)
$ A (2)n(q5)=
-4 A (2)n($),
(D.1.35)
4, A ( ' 3 ) ~ ( = 4)
A ( 3 ) ~ ( $, )
(D.1.36)
For the axion part (see (D.1.25)),we have (3)n(4)= 0 4 ,
Thus,
v = (1)v+ ( 3 ) ~
The skewon part drops out of the Lagrangian V. Had we based our considerations exclusively on Lagrangians, we would never have come across the notion of a skewon part. In the light of (D.1.40) and (D.1.44), the principal part ('IX behaves "normally" whereas the skewon and axion parts are more elusive (see Table D.1.1).
(D.1.37)
with a suitable psel~doscalarfunction a. Synonymously, one also speaks of an axial scalar; this is where tlie name axion part comes from.
Irreducible part principal ('IX
D.1.3 @Decomposingenergy-momentum and action Let us go back to our fourth axiom (B.5.7). We substitute (D.1.33) and fincl that the energy-momenta of tlie three parts are additive: k x n = (l)kxn
+
( 2 ) k ~ ,+
(D.1.44)
(3)kx,.
(D.1.38)
The (3)n part obeys (D.1.37). Therefore, we fincl
'[
F A (e, J ( 3 ) n ( ~ ) ) 13)n(F)A (e, J F)] 2 1 = - [ F A ( a e n ~ F ) - a F A ( e , ~ F )=]O . 2
(31k~, = -
(D.1.39)
T\lhs, the axion part drops out from the energy-momentum current:
kx, = (l)kxn+ ( 2 ) k ~ , .
(D.1.40)
skewon axion
(2)X
(3)X
or $
or a
Colltributes to energy-momentum
Lagrangian
T R tensor density
yes
yes
Yes
yes
no
Yes
no
Yes
no
Table D.1.1: How each irreducible part ('IX of the constitutive tensor density of spacetimc contributes to the electronlagnetic energy-momentum 3-form kC, = $ [Fr\ (e, J H ) -H A (e, J F)],to the Lagrangian 4-form V = H A F, and ~ ' be defined in (D.2.22). We see to the Tamm-Rubilar tensor density ~ ' 3 to that is omnipresent and thus indispcnsible, in accordance with the metric that will finally be derived from it. The skewon piece (2)X drops out of the Lagrangian because of its dissipative nature and has thus bee11 overlooked in theoretical discussions in the past. The axion part (3)X is the most elusive one: It contributes to tlie Lagrnngian but neither affects light propagation locally nor carries electromagnetic energy-momentum. So far, there is no experimental evidence for either the axion nor the skewon part nor for more than ten compo~lcntsof the principal part.
-;
262
D.1. A linear rclation bctween 11 ant1 F
D.1.4 Abelian axion field a
D.1.5 Skcwon field $TiJ and dissipation
263
equation
Since ( q X11)" only one indcpentlcnt component, we introduce in accortlancc witli (D.1.13), the azion field,"
(D.1.45) It is a twisted 0-form, :ilso called pscutloscalar or axial scalar (see (D.1.37)). As such, it is T and P otltl (T stands for time, P for parity). For definiteness, wc call it tlic Abelznr, axion4 field, since in our axiomatics it is exclusively rclatrtl to electrodynamics, the gauge tlicory of tlie Abelian grorip5 U(1). It is remarkable that the pscutloscalar axion field cu enters here as a quant,ity that docs not interfere at all with tlic first four axioms of electrodynamics. Already a t thc prc-metric level, such a field emerges as a not unnatural companion of the electromagnetic field. Hence a possible axion field has a high clrgrce of universality. After all, it arisrs, in t.11~framework of our axiomatic approach, even before the mrtric firld (Einstein's gravitatioiial potential) conies into bcing. The axion field a ( z )is some kintl of universal permittivity/pcrineal~ilityficltl. One could add a kinetic and a mass term of tlie a-field to tlic purcly c~lcctrornagnctic Lagrangian. Tlicii ~ ( rbecomes ) really propagating tind onc can assocititc. wit,li it it liypotlirtical quantum particle, the Abelian axion, with spin = 0 and parity = -1. Let us collcct the principal and tlic skcwon part of K, in
Then tlie linear sl)acctime relation (D.1.1) witli (D.1.3) can be written as
The Maxwell equations in this shorthand notation read
We can also cxccutc~the differentiation i11 tlic inliomogcneous equat,ioii ant1 srtbstitritc the homogencor~sonc. Then we find for the inliomogr~icousMaxwell "ickc wns sec~ninglythe first t,o introduce axion terms in a Lagranginn (see [7], p. 51, I3q. (7)). 1,nter Ni [31, 32, 331 cliscnssccl in detail an Abclian axion ficlcl cr in thc context of tllc c o ~ ~ p lof i ~electromagnetism ~g to gravity (see also Wilczck [GO], tie Sabbata & Sivaram [Mi], and thc rcferenccs given therc). Fieltl and Carroll [S] discr~sscdpsendoscalnr cflects in thc carly universe, including an Abelian axion ancl its rclation to a possible pri~nordinlcosmolo~icnl hclicity; also magnctic hclicity, which we addrcsscd carlier in (B.3.17), plays a role therein. Onc should also compare Haugan & Liim~ncrzahl[lG]. 'In c o n t r a t (.o tlic axions related to nowAbelian gaugc theories, see Peccci nntl Quinn [38], Weinberg 1581, Wilczck and Mootly [5S, 301, and thc rcvicws in Kolb and 'I'urncr [24] and Sikivic [ A n ] . 'See I-Iuang [ln], Cllaptcr 111.
Only the gradient of the axion field enters; that is, an axion field that is constant in time and space tlocs not emerge in the Maxwell equations. Eqriation (D.1.49), for ( ' ) K = 0, can be derived from tlie Lagrangian
As a look a.t (B.2.29) shows, the axion part of the Lagrangian reads 1
( 3 ) ~ = - - ~ ~ ~ ~ = - a ~ ~ (D.1.51) ~ ~ d o , 2 with the electric field strength 1-form E and the magnetic 2-form B in 3D. This term can also be rewritten as an exact form plr~sa sul>plementaryterm:
For the special case of cu = const, we are left witli a pure surface term. Since a surface term doesn't contribute to the field equation, we recover the result mentioned above. We proved in (D.1.39) that the axioii doesn't occur in the energy-momentum current of the elcctronittgnetic field. The axion is, in tliis energetic sense, a "ghost": ("k~, = 0. As we mentioned already above, the first four axioms are not touched 1)y tlic possible existence of the axion field. I11 particular, charge remains conserved: d J = 0. Experimentally, the Abrlian axion has iiot been found so far. In particular, ts.~ as we shall see it couldn't bc traced in ring laser c ~ ~ r r i i ~ i e nNeverthclcss, I~clow,the axion does not interfere with the light cone structure of spacetime at all. Thcreforc, tliis chapter is iiot yct closed, the Abelian axion rcn~ainsa serious option for a particle search in expcrimciital high energy physics and in cosmology.
D.1.5
Skewon field
,Fij and
dissipation
Can we si~iiplify('IX in a similar way as (qX?Well, it has fifteen intlependent components. Hence a traceless scconcl rank tensor could fit. That this is correct can be seen froin (D.1.29). 'Sec Cooper & Stcdman [GI and S t e d ~ n a n[51] for a systematic and extended series of experiments.
D.1. A linear relation between H and F
254
Bcforc we start witli tlie actual construction, we remind ourselves of the I)ro1)erties of ( 2 ) X . Becallsc of its irretlucibility, it possesses the symmetries (2)Xijkl
,
= -(2)Xklij
(2)Xlijkll
=0,
(D.1.53)
Being a piccc of X , tlie skewon part inherits its antisymmetries in the first and "cond pair of intlices:
D . l . 5 Skewon field
$,j
255
and dissipation
In order to 111akc tli(? syninietry (D.1.53)1 manifest, wc rc.nnnlc the intliccs ( 2 ) kl17 = 2 e k l n t ( ~ X Y ; ~ ~ ~ (D.1.62)
J'
and sl~bt~ract (D.1.62) from (D.1.61). This yields the final result ( 2 ) X l l k 1 = F t ~ r n [ k$nt 11 - E k l l n [ z t f f n 1 J l
.
(D.1.63)
For (D.l.G3), all t,hc sylnlnct,ries (D.1.53) and (D.1.54) can 1x2 verified straightforwardly. I11 (D.1.55), we chose t,hc convclltional factor as 114 in order to find in (D.1.63) a formula frcc of inconvenient factors. Thus, (D.1.63) represents thc inverse of (D.1.55) and ( 2 ) x 1 ~ ' 1and are cquivalcnt, intlced. $,J
$i3(z)
:= - Eiklnl ( 2 ) X k l m j A
4
.
(D.1.55)
@Twoidentities and a master formula The two tcrlns on the right-hand sidc of (D.1.63) call bc better untlerstood by means of the. following itlcntity: In four dzrnenszons, evenj tracc.less tensor of typc. [t] ptl fillfills
Bccausc of (D.I .53)2, its trace vanishes,
$11111
E1117'['
111 order to indicate tlie vanishing trace, we addcd a slash to the S .
rinl'(l
(-A(n'[?
$,,,J ] .
(D.1.64)
Let us sketch tlic proof. In 4D, any ol~jcctwitli five completely antisyrnn~c:trizctI intliccs is zcro, z!:!~'""'10. W l i ~ nfour of tlicsc five iiidiccs belong to the LeviCivita symbol, wc lii~vvct,hc identity:
@Invert(D.1.55) We multiply (D.1.55) by
-
c i , l ~ki ~ 1 - I j n ~ k~7 z,.,=F.
and find
f i l n ~ k~
+_
,,.
Applying this to tlic case of
, + j Cijlk ..,
Z
ZITI
...
+
ci.j~~ll
z",.
(D.1.65)
wo find tlic identity
S~~l)posc that $ ' , l is n tracrlrss tcnsor, i.c., ~ l t t 7 1 ' = 0. T h c ~ itl sinlplc rczurnngement of t,hc ternis ill the al,ovc. itlcnt,it,yyicltls, We cxpnnd the bracket: (2)Xi3kl
+
( 2 ) g k i l + (2)Xkijl -
-2
€ijkni
&1 .
(D.1.59)
The second term on the left-hand side of this equation, by means of the syrnmetries (D.1.53) and (D.1.54), , can be rrwrittcn as ( 2 ) X 3 k ' 1 = - ( 2 ) X ' 1 ~ k = ( 2 ) X L z J k . Thus,
By mcnns of t,lic itlcntity (D.1.64), it is ol~vioust,hat (D.1.61) ant1 (D.l.G3) arc cq~~ivalcnt rcprcscntntions of ( 2 ) x 7 ~ " ' . A comparison of o w results wit,ll (D.1.29) a11d (D.1.30) shows that tllc sl\von ant1 tlic axion field call I,c cxprcsscd in tcrlns of n as follows:
For K~~~~ one can find the following itlcnt,it,y: 178 four di,rnen.si.ons, every tensor tcij" ttlrat is traceless, ~ ~ ~ ~ j = 0, fi~ljlls' of type
[i]
fi.jo~7~ [klJ - k l i i ~ i ~ [ij] K,,,,,, Klflll
.
A~ncidcntnlly,this ~,rol>crt,ynpplicu 7
ill
(D.1.69)
part,icr~lart,o t,hc Wcyl cllrvnt.~lrct.cnsor
~
i
,
of~ n~
1D li.ienlnnr~ianspace. 111 this cnsc, wc? rccovrr fro111(D.1.69) Lhr wcll kriown n11t.i-srlf-dol~l>lcThe skcworl field $ , J ( z ) WELS first introtluccd by Hehl, Obukhov, and Rubilar [17].
d~lnlityo f t h e Wcyl tcnsor: Cijk'=
c
~ i i J 'l lC,,l,,llq. q ~ ~
~
~
~
'
D.1. A linear relation between H and F
25G
D.1.5 Skewon field
with -
X
I
(l)X[ijkl] = 0
,
JTmnl=O.
and dissipation
257
and has indeed been introduced in the context of the discussion of k C , (see (B.5.11)). Accordingly, in pre-metric electrodynamics, even when linearity is introduced according to (D.1.1) and (D.1.3), A and V have no decisive meaning, and that A and V do not depend on 8,' is interesting to note but no reason for a headache. This reminds us of a complementary property of the axion a or of (3)X. It features in V (see (D.1.44)), but it drops out of k C , (see (D.1.40) and Table D . l . l ) . Should we be alarmed that the axion doesn't contribute to the electromagnetic energy-momentum current? No, not really. Full-fledged theories of the Abclian axion field can be worked out. By the same token, in linear pre-metric electrodynamics, it is not alarming that $,J drops out from the Lagrangian V, and in the futurc we will take the possible existence of $,J for granted.
Let us finally collect the decomposition of the con~titut~ive tensor density of spacetime xzj" in the master formula
( l ) X i j k l - (1) k l i j
$,j
(D.1.71)
We can now forget the details of the derivations. The formulas (D.1.70) and (D.1.71) contain decisive inf~rlnat~ion about tlic decomposition of the constitutive tensor density of spacetime in terms of the principal part ( 1 ) X 2 ~ k 1(20 independent components), the skcwon field $,I (15 components), and the axion field cr (1 conlponent). In terms of K , ~ " , (D.1.70) rcads
What is then the possible physical meaning of of (B.5.13) that is defined for every vector field
Let us recall the "charge" = <"en by
This yields for the spacetime relation For its exterior derivative, we found
In contrast to the axion field, the skcwon field does contribute to the electromagnetic energy-momentum. In general, ( V k C , # 0 (see (D.1.40)). Therefore the liypotl~cticalskewon field is expected to have more impact on light propagation than the axion field. And this turns out to be true. What has been said about the axion field should also be strrssed here: The high degree of univsrsality of $,I. The metric g, to be derived from the principal part the skewon $ and the axion cr come up in a similar context (see (D.1.70)). They just emerge from three irreducible parts of the same quantity. For this reason, the hypothesis of the existence of the skewon field seems rcasonablc. However, ns we saw in (D.1.43), the skewon part of the Lagrangian vanishes identically: (2)V= 0. This is the usual ~rgulnentfor discarding ( 2 ) X . Since we conventionally assume that all information of a physical system is collected in its Lagrangian, we reject 8" , 0 as being unphysical. Nevertheless, this argument docs not forbid the existcncc of $,+ 0. It only implies that V is "insensitive" to $,I. In other words, if $,? # 0, then not all information about the system is contained in the Lagrangian. Rclnembcr that prc-metric eiectrodynalnics is based on the conservation laws of electric chnrge and magnetic flux and on an axiom about the (kinematic) electromagnetic energy-momentum current kC,. No Lagrangian is needed nor assumed. But, of coursc, the protro-Lagrangian A = -F A H/2 exists anyway
(see (B.5.18)),or in holonomic components (see (B.5.19)),
I
Here LEclcnote~the Lie derivative along E . Now wc substitute the linear relation (D.1.12), or fikl- x klmn Fmn/2, with fikl= cklij H i j / 2 , and find
We apply the Leibniz rule of the Lie derivative and rearrange a bit:
We substitute the irreducible pieces of
1
aiGi = pFkl + -8 L,
+
((l)xijkl
xijkl.Then we have 1
(3)Xijkl)
F ~ + ~4 (Z)Xiikl F ~F,, ~ E Fk l . (D.1.79)
If the vector ( is chosen according to E = e6 = n, then the charge & represents the energy of the electromagnetic field (cf. (B.5.59)). The vanishing of the Lie
11.1.7 Six-tlirr~crisioniilrcprcscri1,ntion of' tllc spilcclirnc rclnlion
derivatives - E , , ( ' ) ~ = = 0 has a clear physical meaning in that tllesc parts of the constitutive tensor density liavc the samr values on cvery lcnf of the spacetinlc foliation. 111 other words, they are constant in "time." In this casp in vacuum, i.c., for J" 0, we conclude that tlie energy Q is not conscivrtI because of the offending term ('IX ~ $ ' / 4= $z3pkfik'kJ. Hcre .ft3 = f z l k ' Fk1/2, and the dot sy~nbolizrsthe "time" derivative. Thus ( 2 ) X or, equivalently, the skewon field $,J iilduces a dissipative tern1 ill the energy balance equation carrying a first "timc" clerivativc. This obscrvatioii is consistent with the dropping out of the skewon field from the Lagrangiall since it is well known that dissipative phenomena in general cannot be dcscrihc~cl within a Lagrangian framework. It is the11 our hypothesis that $,J represrnt,~a. field that is odd undpr T transformotzons. Of course, we nlust investigate how these sketuon.s, as we m:ty call them in a prelin~inaryway, tlisturl-, the light cone and whether tlierc exists ~ ~ c r l i a pmerely s a viable subclass of the skewons.
We now concentrate on the GD-versions
We deco~nposeexcitation and field strength with respect to the GI cobasis and also into 1 3 (see (B.4.6) and (B.4.7)):
+
We recall the (3
+ 3)-dccompositio~~(A.1.79) of 6':
Since cvery longitudinal (spatial) 1-form call be decomposed with respect t o the coframc 1 9 ~whereas , cvery 2-form can be convcllicntly expanded with respect t o the 2-form basis i,, we have ( a ,b, . . . = 1 , 2 , 3 )
D.1.6 Principal part of the constitutive tensor In the decompositio~lformula (D.1.70), it is the principal part ( l ) X Z l " that wc haven't discussed so far. It has 20 intiependent co~nponentsand we might wonclcr whether there is, similar t o the axion and the skew011 parts, the possibility of introducing a simplified vcrsio~lof ( l ) X 1 ~ k l . We arc not aware of any mctliotl of representing (1)X13" 111:t ~ ~ s c f lnltcrnativc il way. Ncvcrthelcss, since a. symmetric second rank tensor has 10 indel~rndcrit component,^, one co~ildtry the ansntz
259
1
'
We silbstitute this into (D.1.82), (D.1.83) tind find
Now we can write H and F as column vectors. Using (D.1.87) and (D.1.88), wc call recast (D.1.81) into the for111 with two sym~lletricfielcls !lij = gji and llij = hji. 011r ansatz c e r t a i ~ ~ oboys ly the algebraic symmetries of ( l ) x i ~ k l ant1 d c ~ ~ e n on d s 10 10 independent ficltls. Howcver, (D.1.80) represents only a. special case of (l)Xij"',as we shall sc?e in Sec. D.4.4, where we construct exa.mples tliat turn out t o be more gener;tl. In spite of this ncgativc outcome of our co~lsitlcration,wc gct tlie itlea thi\t;, l~rovitlcdgij = hi', a symmetric scco~ldrank tensor may he hidden in (l)xijk'. But clearly, one has to fintl s~iitableconstraints that (')xij" 11as to fillfill in ortlcr t,o be able to get down to this symmetxic second rank tensor field.
+
The new entries here arc the co~lstitutivc3 x 3 matrices A, B, C, 'D introdnccd by
D.1.7 Six-dimensional representation of the spacetimc relation We :~lre:l(l~ have cliffcrcnt versions of the spacetimc re1:ttion: tlie operator vcrsio~i (D.l.1) wit11 (D.1.3), the ti-versions in (D.1.6), and the X-versions in (D.1.12).
a," .-.-
I 2
-6ncdX
Obcd
260
D.1. A linear relation between H and F
XOnOb
=
dba
,
(D.1.95)
xoabc
-
ebcdVda ,
(D.1.96)
-
eabd c c d1 €abe €cdf
(D.1.97)
xnbcd
261
(D.1.70), we find
If we resolve with respect to X, we find
xabOc
D.1.8 'Special case: Spatially isotropic skcwon field
B
fee
(D.1.98)
The decomposition of can be found in (D.1.15). In terms of the new matrices, we find straightforwardly (1)
=
( $ pa + I,
This is what we substituted into (D.1.102) ant1 (D.1.104). For explicit reference, wc can d s o altcrnat,ively display the spacetime rrlations of the sltcwon ant1 the axion as
(Pa +Pb a )
a
$I,
a)
and
By reordering (see (D.1.90)), we can easily fintl the corresponding irreducible ~ / ~ - ~ a rFor t s a. 3D traceless quantity, wc used the notation &Pa := Mab c Mc 6!/3. Note that for the corresponding 4D case wc use the slash / . Now we woultl like to express (D.l.lOO) and (D.l.lO1) in terms of $ , J and a , respcctivcly. We find (D.1.102) and
D.1.8 @Specialcase: Spatially isotropic skewon field Thc axion firlcl, as a pseudoscalar fiinctioii, is isotropic by definition. Thc skrwon field $ , 3 , as a second rank tensor fiold, is aiiisotropic. However, as a specific srll,casc, it can bc. 3D isotropic $," ha. We invcstigatc that case in this section. We hasten to atld that the principal part ( l ) X Z ~ k lcannot be isotropic, silice it has only upper indiccs ant1 no proportionality with a I
The corresporiding &-parts read with the 3D pseucloscalar fr~nctions = s ( x ) . Tlius,
(D.1.104) and
and Comparing (D.l.lO1) with (D.1.103) and (D.1.105), we easily recover the axion a (cf. (D.l.G8)2). As for the skewon $ contribution, using our master formula
262
D.1. A linear relation between H a n d F
Consequently, (Ij.1.110) ant1 (D.1.111) bccomc"
If we substitute (D.1.117) anti (D.1.112), (D.1.113) into (D.1.89), we can display the complete spacetime rclation as principal part $ isotropic skewon field $ axion ficld:
Propagation of electromagnetic waves: Quartic wave surface
Note how the isotropic skcwon and the axion act with different signs on E, and 3".Thereby they can be distinguished pl~enomenologically.
W e substitute the linear spacetime relation into the vacuum Maxwell equations and determine how electromagnetic waves propagate. W e find that the wave covectors lie o n quartic surfaces i f no constraints are imposed o n the constitutive ten,sor.
D.2.1 Fresnel equation
'Nieves and Pa1 (34, 351 had postulated and discussed such relations within a material medium. Accordingl~,,the spacetime relations ( D . l . l lo), (D. 1.1 1 1) are nnzsotropzc generalizations of the Nicves slid Pal ansatz. The off-diagonal terms with Poa and Sb0lead, respectively, t o magnetic and elcclric Faratlay-type effects of tlie spacetime under consideration; i.e., these tcrrns rotate t.lle ~ ~ I ' t r i z a t i oof n a wave propagating in such a spncetime.
As soon as t,he spacetime relation is specified, electrodynamics becomes a predictive theory and one can study various of its physical effects, such as the propagation of elcctro~nagneticdisturbances and, in particular, the phenomenon of wavc propagation in vacuum. Oftcn, for such a purpose, onc applies the gcomctrical optics approximation. Thc field strcngt,h is split into a background ficld plus a wave term of the for111 fi, e"' where thc amplitude f varies slowly compared to the phase @. It is assurned that the scale over which the electromagnetic background field varies is much larger than that of the variations of a. Then one derives a system of algebraic equations for the amplitude f that ultimately determines the charactrristics of the wave solutions in terms of the background field. An alternative method, which we use because of its mathematical transparcncy, is Hadamard's theory of weak discontinuities.' It yields the same 'The corresponding theory was developed in detail by Hadamard [13] and Lichnerowicz [28], among others; see also the remarkable paper of Kiehn, Kiehn, and Roberds [23].
I
264
D.2. Propagation of clcctromi~gncticwaves: Q u a r t i c wavc surface
D.2.1 Fresncl e q u a t i o n
physical rcsults. In the theory of partial differential equations, the propaga tion of tlistu~banccsis described by thc I-Iadamarcl discontinuities of solutions across a cliaractcristic hypersurface S , the wave front. One can locally dcfinc. S by the cquation cl)(xz) = const. Tlic Haclamarcl discontinuity of any funrtion 3 ( x ) across the liypersnrfacc S is determined as thc difference2 [3] (x) := F ( a + ) - 3 ( a - ) , wlierc a* := lim (x E ) are points on the oppositc sides of E-0
Accordingly, the jump equations (D.2.4) can bc put into tlie forni q~h=q~iE(f)=O, q ~ f = 0 .
Note tliat the axion drops out cornplctely even though it occurs in Maxwell's equations. Erom thc constitutive tensor rc only E = ( 2 ) nis left over witli 20 15 independent components. If we multiply the two equations of (D.2.7) by q, both vanisli identically. Hencc only 3 3 equations turn out t o be indepcndcnt. If E is specified, (D.2.7) represents six independent homogeneous algebraic equations for thc six independent components of the jump f . Wc can solvc (D.2.4)2 by
*
I
T h e wave-covector q shoultl not be mixccl up with tlie climcnsion of cliargc. Equations (D.2.1) ttnd (D.2.2) represent the Hadainard gconletrical coinpatibility contlit,ions. If we 11sc Maxwell's vacuum equations d H = 0 and d F = 0, t,licn (D.2.1) and (D.2.2) yield
I
I
I
1
-
'In Sec. D.2.1 only, brackets
-
T h e 1-form a is only determined up to a gaugc transforlnation (D.2.9)
with cp as an arbitrary scalar function (0-form). We substitute (D.2.8) into (D.2.4)i:
This is a 3-form. Hence we have four liomogcneous algebraic equations for a . However, because of (D.2.9), only threc equations are independent. In order t o isolate thcsc three equations, wc decompose a with respcct t o an arbitrary cofratnc 19" = e i a dx' according to a = apdp. Here a , P, . . . = 0 , 1 , 2 , 3 . Thus, because of tlic linearity of thc operator iE, wc liave
"Spatial" anholonomic indiccs run ovcr a , b, . . . = 1 , 2 , 3 . The trick is now to pick up a convenient specific coframe. At each point of spacetime, we can always clioosc t,o idcntify tlic zcrotli lcg of thc cofranic wit11 the wave covcctor:
Thc convenielicc of this choicc is immcdititcly evident from tlle fact that, witli respect t o this coframe, we have a = a6 q ab d h . Correspondingly, the gaugc transformation (D.2.9) affects only the zcrotth component, ag -+ ag+cp, whereas the spatial colnponents ab arc gauge invariant. Now, with respcct t o the cofranic chosen, the first term in (D.2.11) vanislics, since do A d o = 0, and we find
+
Tlierc are still four equations. But by mult,iplication (from tlic right) witli i y a , we find a vcctor-valucd 4-form. Again because 19' Ad0 = 0, only three equations are left over:
[I are used to cl~nractcrizcthe jump of a function. E l s c w l ~ r ~ ( ~
they denote the dinlension of the quantity cnclosccl by the brackets. "~icitlentally, one can also, soniewhat formalistically, cleterlninr the c n c r g y - ~ n o m e ~ ~ tof nm kC, + kcr,, I I 11, and F -+ f . One finds, ka, = (I, w , with the wave by the rel~lacelncl~ts the flow 3-form w := q A n A c.
+
a4a+'f'9,
Sillcc wc can solvc thcsc. cquations by h = q A c and f = q A a , wc recognize tliat tlicsc? 2-forms fulfill
that is, thc first tlirec invariants of the jnmps"~ and f vanisli (scc (B.2.28) (B.2.33)). We can say sonietliiiig moro d ~ o u ttlic jump 11 if wc apply the spacetirncb relation (D.1.47). Provided tlic components of tlic lincar operator rc (that is, of K and ru) arc continuons across S , we find I)y differentiating (D.1.47) and using (D.2.1) and (D.2.2),
+
+
S 3 x . We call [3] (1) the jump of tlic function 3 a t x. An ordinary elcct~omagncticwave is a solution of Maxwell's vacuunl equations d H = 0 and d F = 0 fol which the deni~atzvesof H ant1 F arc cliscontinlr011s acloss tlie wavc front hypcrsurfacc S . In ternis of H ancl F , wcl liave on tlic cliaractcristic liypcrsurface S
Hc1.c tlic 2-fornis 11,f clc~cril~c tlie ~ U I I I ~ (discontin~iitics) S of the differentials of the clcctromagnctic field across S, ant1 the wave-covcctor norlrial to tlic front is given by
(D.2.7)
n
I
A
d B A d'% iE(p9' A dh) ah = 0 .
(D.2.14)
266
11.2. IJropagation of electromagnetic waves: Quartic wave surface
267
D.2.2 @Properticsof the Tamm-Rubilar tensor dcnsity
Our problem is solved! These are three equations for the three spatial components of a , whereas the unphysical gauge-dependent ag is successfully expelled. In order to bring (D.2.14) into a more manageable form, we decompose the operator iE according to (D.1.4) and use the 4-volume element E^ as in (A.1.61):
with 0 := dct(e,"). This equation has a tensorial transformation behavior. Thlls, it is valid with respect to an arbitrary cofriime. We define the fourth-order Tamm-Rubilar (TR) tensor density of weight +1,
+ (2)X; furthermore,
Here the total symmetrization is extended only over the four indices z , j , k,1, witli all the dummy (or dead) sunnnation indices excluded. G is totally symmetric, G"'"'(~) = G ( ' J ~ ' ) ( ~A) .com~letelysymmetric tensor of rank p in n dimensions has, expressed as binomial coefficient, (la+;-') = ("i!:") independent components. Thus, E carries = 35 independent components. Note that we dcfined G with respect to the total constitutive tensor density x and ~ ' total , antinot only in terms of 2. However, bccausc X23k' = 2jk' ~ E ~ J the symmetry of c yicltls G(x) = G(2). An explicit proof is given below in (D.2.35). This is consistent witli the dropping out of the axion field in (D.2.7). With (D.2.21) and (D.2.22), we find tlic (extcndccl) Fresnel cquation%l~icli is generally covariant in four dimensions:
We recall the definition (D.1.9) of i # 0. Thus,
wa"nb = o ,
x and with
put
2 :=
('IX
wnb:= pa'',
(D.2.16)
("41T4)
The necessary and sufficient condition for the existence of non-trivial solutions for a.c, is the vanishing of the determinant W of the 3 x 3 matrix W:
+
1 4a8d -8h6e 4 c b f W := det W = - i , i , , & , f ~ " " ~ h e ~ ~ =f1. 3! 3! fabcidef x X X . (D.2.17)
This yields the Fresncl equation W = 0. However, to begin with, we rewrite W in a fully 4-dimensional covariant manner. First, we observe that the 3-dimensional Levi-Civita symbol &bc is related to the 4-dimensional one by means of ;abc G E^(jabc. Then we can extend one 6-component of the constitutive tensors to a fourth summation index. As a result, we find tlic itlentity
I
which holds true due to the (anti)symmetry properties of the Levi-Civita symbol and of the constitutivc tensor. This allows us to rewrite (D.2.17) as
It is always a quartzc equation in q, despite the fact that it was derived from a dctcrlninant of a 3 x 3 nlatrix quadratic in the wavc covcctors. Thus, the wavc covcctors q lie on a quartic Rcsnel wave surface, not exactly what we arc observing in vacuum at the present epoch of our universe. The rest of Part D is clcvotcd to finding conditions that make the quartic fiesnel equation factorize into two cluadratic ones and to tlctcilnining ~intlerwhich circumstancc~sthe F'rcsnel equation turlis out to bc n perfc>ctsquare.
D.2.2 @Properties of thc Tamm-Rubilar tensor density The various irreducible pieces of tlie constitutivc tensor density affect wavc propagation in different ways. Technically, this can be determined by stlidying how a certain irrctluciblc picce (")Xcont,ributes to the TR-tensor tlcnsit,y (D.2.22) and, thcrcby, to thc Frcsnel cquation (D.2.23), which governs the wave covectors.
NOWwe apply the same procedure to tlie second Levi-Civita symbol and finally obtain
Due to (D.2.12), the coordinate components of the wave vector are e,' = q,. Tlletcfore, (D.2.20) can, in coordinate componcnts, be rewritten as
!
'The original FrcsneI eqr~at.iondrscribcd rcfringence in anisotropic crystals (see 127, 44, 521). Our rxtcndcd vcrsion, first clerived by Obukl~ov,Rtkui, and Rubilar (scc 1361) also includes clectric a t ~ t lmagnctic Famday rotatiort, optical activity, ant1 Fizean-l+esnel phenomcna. Tamm [53] stt~diedthe relation between fourth- and second-order wavc geometry for a sl~ecialcase of ,z linear constitutivc law. Me also introducctl a "fourtll-order metric" of the type of Q. A derivation of the 4 D forrnl~la(D.2.22) was first found by Rut)ilar (451.
268
D.2. Propagation of electromagnetic waves: Quartic wave surface
269
D.2.2 @Propertiesof the Tamm-Ruhilar tensor density
We dcmonstratc here some algebraic properties of the TR-tensor density. The simplest is to check that
As dready mentioned above, we have the important relation
+
By direct substitution of (3)Xi.7" cuting the symmetrization, Emnp9 E,3tzl
For its proof, let us put in formula (D.2.28) q5 = ('IX ('IX and 1CI = (3)X. Then we can verify that all the mixed terms (D.2.29)-(D.2.34) vanish. Indeed, using ( 3 I X i j k l = a E i j k l , we find
= 0 czjkl into (D.2.22), we have, before exe-
x
(3)Xmnri (3) j p s k (3) lqttr
X
- 4 3 ri lq j p s k dl,, 6,. E -
-16a%jilk
.(~.2.25)
This vanishes up011syn~~ilctrization over the indices i , j,k, 1. A bit more involvetl is the proof of
Using the symmetry properties fintl imnpq irstlr (')x
m n r i (2)
--
X
(2)Xijk1 =
j p s k (2)
X
x
- (2) j i k l - -(2)Xijlk
- - f m ~ 7 ~ pErstlr q
- ( 2 ) X k l i j , wc
1qt.1~
(D.2.27)
f r 3 t l r ( 2 ) X r i m r ~(2) A
This is zero when we impose the symmetrization over i, j , k, 1. Similarly, we find
X s k j p (2) X tlilq
Both expressions vanish when we impose symmetrization over i, j , k, 1. The proof that all T's are equal to zero reduces to the above formulas in which we simply need to replace one of the 4 factors by the Levi-Civita E. Consequently, (D.2.28) yields G(q5 $) = G(4) G($) = G(4), since G(+) = G ( ( ~ ) x= ) 0, qed. The relation
(2) 11~91(2) k s p j (2) irmn
X'
X
X
x
- - imnpq irstzc (')x mnrl (2) k p s j (2) X iqtzr .
+
Upon symmetrization over the indices i, j,k, 1, we find G((')x) = - 6((')x) or ~ ( ( ' ) x )= 0. Now we want to turn to "composite" G's like ~ ( ( ' 1 ('IX ~ (3)X).As a preparatory step, let us take x = 4 q". Then, upon substitution into (D.2.22), we have quite generally
+
+
+
+
is a special case of (D.2.35). Just put ('IX = 0 and use (D.2.26). Let us pause for a moment with the algebrx and let us look back a t what we have achieved so far. Because of (D.2.24), (D.2.26), and (D.2.38), neither the axion part (2)X nor the skewon part (3)X (alone or together) can provide electromagnetic wave propagation, since the Rcsnel equation collapses under those conditions. Therefore the presence of the principal part of the constitutive tensor density is indispensable for the existence of nontrivial electromagnetic waves and ultimately for the existence of the light cone structure on spacetime. In any case, the TR-tensor density reads
Here the mixed terms 0, contain one $-factor and the T,'s two $-factors. Thesc terms read explicitly as follows:
I
We defined the 0,'s and the T,'s without symmetrization and apply symmetrization only afterwards in (D.2.28). Note that such index-stricken expressions can conveniently be manipulated by means of computer algebra in general and the Reduce system in particular.
We recover the result that the axion field does not influence the Fresnel wave surfaces a t all. Hence a possible axion field can easily remain unnoticed. But what about the skewon field? In TableD.l.l we marked already that it does influence the light propagation. Let us show this explicitly. Actually, wc can use (D.2.28) to find more exactly the contribution of the skewon t o the Frcsncl tensor. One can straightforwardly see that
270
11.2. I'rol)ngatiol~of elcct.rornagnct,icwnvcs: Quartic wavc s~lrf;~c,c.
As a comparison with (D.2.22) shows, the coillponcnts of 6 are cubic in X 2 ~ kBut 1 . the components X 2 ~ k 1can be, in turn, by means of (D.1.95)-(D.1.98), expressed in tcrms of the lnatriccs A, B, C, 'D. Upon substitution, we find, after some algebra:
The proof is analogous to the clcmonstration of tlie property G((')x) = 0 and is ljascd directly on the sl
We substitute hcrc the explicit form of the skewon part (D.l.G3) and find
In fact, t,llis formula subsumes the relations (D.2.24) (put = ('1 X = 01, (D.2.26) (put = (3)X = 0), and (D.2.38) (put ('IX = 0). Also (D.2.35) is contained in it since (qXdoesn't enter the right-hand side of (D.2.42). In otlirr words, we can forget those rclations that we derived earlier and just subscrihc. to (D.2.42). Our basic results on light propagation arc then that # O (otherwise there is no orderly wavc propagation), that the axion field a is left arbitrary (it doesn't inflncncc. light propagation locally), and that the skcwon field 8,' makes itself felt via equation (D.2.42).
I D.2.3 @Fresnelequation dccomposed into timc and spacc For prnctical calrtilntio~ls,it is colivcnicnt t80usc. a 1 Frc~siiclrqriation (D.2.23). If wc put s~lccessivrly
I
----
Actually, tlie Resnel equation (D.2.23) is expressed in terms of 2. Accordingly, the M1s,strictly speaking, should be displayed in terms of A, 13,C, 2).However, because of (D.2.35), the axion terms drop out, and we can just drop the tildes as well. Results (D.2.45)-(D.2.49) have been checked thoroughly by means of computer algebra."
+ 3 decomposition of the.
then (D.2.23) reads
Thus, we have expressed the 35 intlcpcndent components of 6 in tcrms of the 35 independent colnponents of the h4's. This comes about as follows: The M's arc totally symmetric expressions in their (spatial) indices. Since totally symlllctric n-lfp cxprcssioos of rank p in n dimensions (here n = 3) have = ( ) independent component,^, the M's altogether carry 1 CD 3 CD G @ 10 $ 15 = 35 independent components.
"-,
%ce Rubilar [45] who rlsccl the computer algebra systeln Maple with its tensor package GrTensor ( c f ,http: //@ensor. org).
First constraint: Electric/magnetic reciprocity
W e impose a jirst constraint on the constitutive tensor of spacetime. Th.e axionfree part of the linear spacetime relation is required to obey electric/magn.etic reciprocity. Mathematically, this implies an almost complex structure on spacetime thereby reducing the number of independent components of the axion-fme constitutive tensor from 35 to 18.
D.3.1 Reciprocity implies closure
+
The linear spacetime relation leaves us still with 35 1 independent components of the constitutive tensor rc, even though we recognized in tlie meantime t,liat tlie axion field is irrelevant for the propagation of electromagnetic waves. Clearly wc need a new method to constrain rc in some way. An obvious choice is to require electric/magnetic reciprocity for the spacetime relation (D.1.1) with (D.1 3 ) . We have discovered electriclmagnetic reciprocity
in (B.5.21) as a property of the energy-momentum current k ~ , of thc elcct,romagnetic field. Why shouldn't we apply it to (D.1.1). too? However, we should not forget that, according to (D.1.40), the axion ficld drops out of kC, (as it does from the TR-tensor density). Therefore the reciprocity of the energy-momentum current does not affect the axion field a t all. In other words, the concept of reciprocity is alien to the axion field. Accordingly,
D.3. F i r s t constraint: Electric/magnctic reciprocity
274
we apply reciprocity only to the axion-free part of the sp:tcetime relation, t,llat, is, t80
Let us perform an electric/magnetic reciprocity transformation (D.3.1) in (D.3.2) and recall the linear it,^ of E:
275
D.3.3 Draw t h e s q u a r e r o o t of t h e 6D negative unit m a t r i x
As we will see, the minus sign, which originates from (D.3.1)2, is very decisive: It will eventually yield the Lorentzian signature of the metric of spacetime. With the closure relation (D.3.8), the operator J defines an almost complex stm~ctureon the space M G of 2-forms, as discussed in Sec. A . l . l l . If we apply J to the 2-form basis (see (D.1.4)), we find
With F = FIG1, the spacetime relation can now be written as
By definition, the reciprocity transformatioll commutes with the linear operator E. Symbolically in the GD formalism, we can write (D.3.3)2 as
a.
a remarlmbly simple result were it not for tlie symbolic square root J = However, discussions of the square roots of matrices can be found in the literatur~.~ The scalar field A, as a look at (D.3.5) confirms, can bc considered as the absolute valuc of the axion-free constitutive tensor K,,"". We call X the dilat o n field3 Besides the axion a, it is the second scalar field possibly related to electrodynamics.
-
Obviously, thc spacetime relation is not electric/magnetic reciprocal for an arbitrary transforlnation function (as the energy-momcntum current is). Thr linear operator E is based on the measurable componellts E l K (or Z Z s M ) .If applied twice as in (D.3.4), there must not emerge an arbitrary function. In other words, we can solve (D.3.4) for C2 by taking its trace,
c
-
D.3.3 Draw the square root of the 6D negative unit matrix with the quadratic invariant
i2 (see (D.1.14)). Thus, we call A colnparisoll with (D.1.90)1 allows us to express J in terms of the constitutive fulictions according to
the c l o s ~ ~ rrelation1 e since applying the operator E twice yields, up to a nrgativc function, the identity l6 (= 6:< or = ;6: in components). In this sense, thc operation closes.
D.3.2
(J,
I
)=
1 =
X
"
) ( C b . ,
2" ,Dl,"
A"
Bha) D,,"
Because of (D.3.8), the 3 x 3 blocl<s A, B, C, D are constrained by
Almost complex structure and dilaton field
If we define a new dimensionless operator by
-
E =: X J ,
then
-
[A] = l/resistance
(D.3.7)
aritl, because of (D.3.G),
-
'Tortpin [tid], for X2 = const, just called the closllre relation (for v a ~ ~ i s h i naxion g ant1 skrwon fields) "electric and magnetic reciprocity."
2For tlie square root of such a negative unit matrix (see Ganttnncher [ l l ] , p. 214 e t seq.). %11iot1~the first to consider a scalar field coupled to electromagnetism (and t o gravity) have been Jordan [22] and Brans & Dicke [4]. 'I'he related notion of a "dilaton field" emerged seemingly during the late lSGOs as a field coupled to the dila[ta]tion current, tlie Noether currcnt of scale (or dila[trtjtion) transformations. Isham, Salam, and Stmthdee 1201 used this tinme. They consideretl the dilator^ as a Goldstone boson related to the breaking of scale invariance. T h e dilaton also features in the low energy limit of string theory. There, it is usually written as ((x) = e - b + ( x ) , with a constant b ant1 the dilation field d(x). For more recent liternttrre, see, e.g., Bekenstein (21 and Sandvik et al. [47].
276
D.3. First constraint: Electric/magnetic reciprocity
+
+
We have, for instance, CCaCbc BCaAbC = CC,Cbc Ab,BCn. Then we can put our scheme in matrix form
D.3.3 Draw the square root of the 6D negative unit matrix
277
By squaring J, one can directly verify the closure relation (D.3.8). The two arbitrary 3 x 3 matrices B and K parametrize the solution, which thus has 2 x 9 = 18 independent components. Clearly, it would be possible to substitute K via K = B C , but the matrix K turns out to be particularly useful in Chap. D.4. If we measure the elements of the 3 x 3 matrices A, B , C , V of Z and thereby, according to (D.3.11), also those of A, B , C, D, then closure is only guaranteed provided the relations (D.3.27)(D.3.29) are fulfilled. Therefore, closure has a well-defined operational meaning. If we assume that det A # 0, then an analogous derivation leads to
----
We are able to solve this closure relation. Assume that det B # 0. Consider (D.3.17). Then we can make an ansatz for the matrix C , namely
We substitute this into (D.3.17) and find
Before turning to the different subcases, a word to the physics of all of this appears to be in order. We know from experiments in vacuum that V w EO E and 3-t (11110) B. If we compare this with the spacetime relation (D.1.89), then we recognize that EO is related the 3 x 3 matrix 3 and po to the 3 x 3 matrix E: N
Next, we straightforwardly solve (D.3.16) with respect to A:
Now we turn to (D.3.18). Equations (D.3.20) and (D.3.22) yield,
Thus, we conclude that (D.3.18) is satisfied. Finally, (D.3.19) is left for consideration. From (D.3.22) and (D.3.21), we obtain:
Thus, in view of (D.3.25) and (D.3.26), equation (D.3.19) is also fulfilled. Summing up, we have derived the general solution of the closure equations (D.3.16)-(D.3.19). They read,
or if we put this in 6 x 6 matrix form,
Therefore it is safe to assume that det A # 0 and det B # 0. From this practical point of view the other subcases don't seem to be of much interest. We could take the matrices A, C, D of (D.3.27)-(D.3.29), multiply them by i, e.g., A = A , substitute them into the M's of (D.2.45)-(D.2.49) of the F'resnel equation, and then discuss the corresponding consequences. However, we get a much more decisive restructuring of the Fresnel equation if we require, in addition to the reciprocity relation, the vanishing of the skewon field to which we now turn.
1
I
Second constraint: Vanishing skewon field. Emergence of the lighCcone
W e found an almost complex structure on spacetime by linearity and electric/magnetic reciprocity. Now we impose a second constraint on th.e con.stitutitre tensor of spacetime by removing its skewon part. Thereby we reduce the n.umber of independent components of the constitutive tensor to 9 plus 1 corn,pon,ents. One goes for the axion field, whereas the 9 components mentioned reduce the quartic wave surface to a quadratic one that spans the light cone at each point of spacetirn.e.
D.4.1 Lagrangian and symmetry The removal of thc skewon field $ is equivalent to ( 2 ) ~= 0. Then rc, ant1 k are symmetric. Accordingly, we can alternatively require the operator E to be symmetric,
for arbitrary twisted or untwisted 2-forms q5 and $. As a consequence, J and K are symmetric, too. The axion part ( 3 ) ~= K - E is not touchcd by this constraint since it is symmetric by definition. This can be motivated in that we usually assume that a Lagrange 4-form exists for a fundamental theory, as we discussed in Sec. B.5.4. If wc do this in the context of our linear spacetime relation then, because H = - dV/dF, the Lagrangian must be quadratic in F, that is, V = -F A H/2. As we saw in (D.1.44), the skewon part drops out in such a quadratic Lagrangian (cf. Table
280
D.4. Second constraint: Vanishing skcwon field. Emcrgcncc of the light conc
D . l . l ) . Hence, if we want a11 information on the field equation t o be contained in tlic Lagrangian, then wc liavc to al~andona possible skewon contribution. A strict Lagrangian point of vicw - including a lack of dissipation - strangles tlie skcwon. In tlie lar~griagcof M % p c e , tlic G x G matrix 2'" is now symmetric and, as a cornparison witli (D.1.90) sliows, it can be represented with the help of 3 x 3 n1atricc.s A, B,C as
We can consider the duality operator # also from another point of vicw. I t is our clcsire to describe eventually empty spacetime with such a linear ansatz. Therefore we liavc t o reduce tlic number of independent components of zzjkl somehow. T h e only constants with even parity are the generalized Kronecker deltas. Recognizing that in tlie framework of electrodyiiamics in matter a linear ansatz with El, can describe anisotropic media, we need a condition in order to guarantee isotropy. A "square" of 2 does the job,
---
Equivalently, we have (see (D.1.90)),
with :6 as a generalized Kronccker delta. Alternatively, (D.4.10) can also be writatenas
Notc tliat we have V = C T in this case. Accordingly, thc spacetime relation (D.1.47) witli the symmetry (D.4.1) now reads H=E(F)+aF,
with
281
D.4.3 Algebraic solution of the closure and symmetry relations
E(F)r\F=Fr\E(F),
which is, for syinlnetric E, equation (D.3.8) in another disguise. However, in contrast to requiring reciprocity of tlie spacetime relation, this consideration does not yield the minus sign in (D.4.10) and (D.4.11) which, in turn, induces t,hc Lorcntz signature of the spacctime metric.
(D.4.4)
D.4-.3 Algebraic solution of the closure and symmctry relations D.4.2 Duality operator
In addition t o the allilost complex structure J , we arrived a t the symmctry of the constitntivc matrix (D.4.2). As a consequence, the constraints (D.3.16)(D.3.19), followiiig from tlic clos~irerelation, pick up the additional properties
\Vc know from Stc. C.2.4 that, in tlic framework of the almost complex structurcl of (D.3.7), the symmetry (D.4.1) introduces a duality operator. Therefore, wc. define a clunlity operator according t o 1 ":=J==E, X
Then they reduce t o with
J(4)r\$=cf,r\J($).
Then, wc also Ilavc self-ndjointness with respect to t,he 6-metric,
4%#w) = E ( # ( P ,
w).
(D.4.7) Naturally, we would likc to resolve these algebraic constraints.
Altcrnativtly, wc. can also write (D.4.6) as
Preliminary analysis Being a solutio~iof thc syst,em (D.4.13)-(D.4.15), tlie matrix C has very specific properties. First of all, becausc Tr k = 0, we find from (D.4.3), Tr C Tr CT = 0. Tlius, Tr C = 0. More generally, the traces of all odd powers of the matrix C
+
Thus, clcctric/magnctic reciprocity and symmetry of the linear ansatz eventually lcatl to tlie spacetime relation
I
282
D.4. Second constraint: Vanishing skewon field. Emergence of the light cone
283
D.4.3 Algebraic solution of the closure and symmetry relations
Indeed, multiplying (D.4.13) by C from the right and taking the trace, we find Tr(ABC) Tr(C3) = 0. On the other hand, if we transpose (D.4.13) and multiply the result by CT, then the trace yields Tr(BACT) Tr(C") = 0. Thc sum of the two last equations reads Tr (A(BC c T B ) ) +2Tr(C3) = 0. In view of (D.4.14), we then conclude that Tr(C3) = 0. The same line of argument yields generalizations to the higher odd powers. It follows from (D.4.16) that the matrix C is always degenerate,
where I( = -K T. In this case, the 6 3 = 9 degrees of freedom of the general solution are encoded in the matrices A (symmetric) and I? (antisymmetric). The transition between the two representations is established by means of the relation
det C = 0.
I? = B-'KA.
+
+
+
+
A
(D.4.17)
Indeed, recall that the determinant of an arbitary 3 x 3 matrix N (= Nba) reatls 1 det N = 6
A
zabc E ~ ' ~ ' ~ ' N ~ , ~ N ~ , ~ N ~ , ~
(D.4.24)
One can readily check that (D.4.20), (D.4.21) and (D.4.22), (D.4.23) are really the solutions of the closure and symmetry relations. Indeed, since the matrix B is nondegenerate, we find that the ansatz C = B-'K solves (D.4.14) provided K + K T = 0. Then, from (D.4.13), we obtain the matrix A in the form (D.4.20). Finally, from (D.4.20), (D.4.21) we get
CA = - B - ~ K B - ~- B - ~ K B - ~ K B - ~ K B - ' . Because of (D.4.16), we can immediately read off (D.4.17). Let us now analyse the determinants of A and B. When the term C 2 is moved from the left-hand side of (D.4.13) to the right-hand side, a direct computation of the determinant yields:
We used the formula (D.4.18) and the properties (D.4.16) and (D.4.17) to evaluate the right-lland side. Accordingly, the matrices A and B cannot be both positive definite. Moreover, when a t least one of them is degenerate, we find that necessarily Tr(C2)= - 2.
General regular solution Let us consider the case when both or a t least one of A and B are regular matrices, i.e., det A # 0, det B # 0. The general solution has been given in (D.3.30) and (D.3.31). Together with the symmetries (D.4.12), the general solution of (D.4.12)-(D.4.14) can be presented in one of the following two equivalent forms.
The right-hand side is obviously antisymmetric (i.e., the sign is changed under transposition). Hence equation (D.4.15) is satisfied identically. If, instead, we start from a nondegenerate A, then the analogous ansatz C = j ? ~ - 'solves (D.4.15), whereas B, because of (D.4.13), is found to be in the form of (D.4.22). This time, equation (D.4.14) is fullfilled because of (D.4.22) and (D.4.23). In the case in which both A and B are nondegenerate, the formulas (D.4.20), (D.4.21) and (D.4.22), (D.4.23) are merely two alternative representations of the same solution. By using (D.4.24), one can recast (D.4.20) and (D.4.21) into (D.4.22) and (D.4.23), and vice versa. However, if det A = 0 and det B # 0, then the B-representation (D.4.20), (D.4.21) can be used for the solution of the problem. In the opposite case, i.e., for det A # 0 and det B = 0, we turn to the A-representation (D.4.22) and (D.4.23). In these cases the equivalence of both sets, mediated by (D.4.24), is removed. The totally degenerate case is treated in the next subsection. It is useful to write the regular solution explicitly in components. As usual, we denote the components of the matrices as B = Bab, A = A ab, C = Cab,and .. = Aab. We the components of the inverse matrices as (B-l) = Bab and (A-') introduce the antisymmetric matrices by I( = Kab and I( = Kab.Then the component version of the B-representation (D.4.20), (D.4.21) reads: A
pb= -
+
~ a b- B
~ ~ K ~ ~ B c ~ K ~ ~ B ~ ~
1 (k2 ~ -
k b ),
(D.4.26)
= BacZCbd kd = det B cacdBcbkd.
(D.4.27)
-where K = - K T . An arbitrary symmetric matrix B and an arbitrary an.tisymmetric matrix I( describe the 6 3 = 9 degrees of freedom of the general solution.
(D.4.25)
~ a b+
det B
a a -bk
Cab= BaCKcb 1
284
D.4.3 Algebraic solution of the closurc and symmetry relations
D.4. Second constraint: Vanishing skewon field. Emergence of the light conc
Here wc introduced ka := ! j ~ ~ and~k,~:=l Bab ( ~ k b; ~ moreover, k 2 := kaka. Analogously, the A-representation (D.4.22), (D.4.23) reads:
Figure D.4.1: In this tree graph, we represent the different subcases of the lincar spacetime relation and the implications for the light cone structure. Note that the axion part (3)X drops out and doesn't play a role here whereas the principal # 0. The skewon part (2)X is of decisive part is always nonvanishing, importance. The 3 x 3 matrix K of (D.3.30) describes electric-magnetic mixing terms, as can be seen from (D.3.32). If K = 0, then we have simply 2) E and 3-t B. The 3 x 3 matrix b, in (D.3.30) denoted by B, since no mixing with the magnetic field was possible, represents a part of ( ' I X # 0. It can be of rank 1, 2, or 3.
1 + det (k2Aab - ka b ) , A A
= -Anb
A
N
acd
A
.
-
=Abc€
wherc k, := !ji,bcJ
1 kd=det A
^
A
Zcbd
ac
A zd,
A
and k2 := k, k a .
*
f rec'prOc.
@Degeneratesolution
f
-
no reclproc.
Besides the regular case outlined above, the closure and symmetry relations also a.dmit a degenerate case when all the matrices are singular, i.e., det A = det B = 0.
unknown condltlon
llght cone
unknown condltlon
blrefrlgence
-
4th-order Fresnel equatlon
(D.4.30)
Recall that we always have d e t C = 0 (see (D.4.17)). As mentioned in the context of (D.3.32), from a physical point of view this case is not too interesting. However, for the sake of completeness, we study it in this subsection. We do not give a detailed analysis of the degenerate case because, in a certain sense t o be explained below, it reduces to the regular solution. Nevertheless, let us outline the main steps that yield the explicit construction of the degenerate solution. The basic tool for this is the use of the "gauge" freedom of the systcrn (D.4.13)-(D.4.15), which is obviously invariant under the action of the general lincar group GL(3, R) 3 Ll,n defined by
mak(b)=3-
4lh-order Fresnel equatlon
rank(b)=Z-
blrefrlngence
r;utk(b)=l
degenerated llght cone
-
no apparent reduction
K*O
* general 4th-order Fresnel eq.
no reclproc.
+
This transformation does not change the determinants of the matrices, and hcnce by means of (D.4.31), the degenerate solutions are mapped again into degenerate ones. We can use thc freedom (D.4.31) in order t o simplify thc construction of the singular solutions. The va~iishingof a determinant det B = 0 means that the algebraic rank of the matrix B is less than three (and the same for A). A rather lengthy analysis then shows that the system (D.4.13)-(D.4.15) does not have real solutions when the matrices A or B have rank 2. As a result, we have t o admit that both A and B carry rank 1. Then direct inspection shows that the degenerate matrices A and B can be represented in the general form
Here va and u, are arbitrary 3D vectors and covectors. We substitute (D.4.32)
into (D.4.13). This yields, for the square of the C matrix, Cac CCr,= -6; (v Cu c) vaub. Taking into account the constraint Tr(C2) = -2, which arises from (D.4.19), we find the general structure of C 2 as
cat C C b= -6; + vaub,
I
with
v Cu c = 1.
(D.4.33)
If we multiply (D.4.33), with u, and v b, respectively, we find that u, and ub are eigenvectors of C 2 with eigenvalues zero; this is necessary for the validity of the equations (D.4.15) and (D.4.14). It remains to find the matrix C as the square root of (D.4.33)1.Although this is a rather tedious task, one can solve it with the help of the linear transformations (D.4.31). I t is always possible to use (D.4.31) and to bring the column va and the row u, into the specific form
286
11.4. Second constrsinl: Vanishing skewon field. Emergence of the light cone
D.4.4 From a quartic wave surface to the light cone
2A(abixd)
clet A
det A
+
@$icZd) detA
(D.4.45)
Then (D.4.33) can be solvctl explicitly ant1 yields Here every M is described by two lines, the first displaying the expression in the B-representation and the second in the A-representation. Substituting all this into the general Fresnel equation (D.2.44), we find Surnmari~ing,the general degenerate sollition is given by the matrices A, B of (D.4.32), with vR = Lb" gb, ua = ( L - ' ) ~ ~ ; ~and , the matrix Cab= LCa(L-'),,'l
I
tlet B
0
CCd.An arbitrary n ~ a t ~ r Lbn i x E GL(3, EX) embodies the nine degrees of freedom det A
of this solution.
Here g2J is a ~yrnmet~ric tensor fieltl. Let, for definiteness, tlet I3 > 0. Hence, det A < 0. Then from (D.4.46) we read off it,s components as
D.4.4 From a quartic wave surface t o the light cone Having takcri care of closure ant1 syln~netryin the last section and having dctcrrnincd the explicit form of the matrix J, we can come back t o the Fresncl equation (D.2.44) and its M coefficient,^ (D.2.45)-(D.2.49). The latter can now bc calculat,cd. The regular nncl degenerate solritions should be considered scparately. Let us begin with the rcgular case.
..
g'J =
Starting fro.11 (D.4.26)-(D.4.27) or from (D.4.28)-(D.4.29), direct calculation yicltls for the coefficients of the FYcsncl equation (D.2.45)-(D.2.49):
1
1 det B
=-
~ c b = - -
det B = -2a.b
4k"k0+2Bnh ((1 -
+
x)
(
1 1 - (det B)-' Bedkc kd -
&zE
- k"
- (clet B ) 13""
)
(D.4.47)
One can prove that this tensor 1 1 ~ 9Lorentz signature. Hence it can be understood as the mrtrzc of spacrtzm~.For a flow diagram indicating the different constraints that led to the light cone, see Fig. D.4.1. Thus, from our general analysis, we indeed recover the null or light cone stnicturc q,qz = q2qJg" = 0 for the propagation of electromagnetic waves: Provided the linear spacctirne relation satisfies reciprocity and symmetry, the quartic wave surface in (D.2.44) reduces t o two coinciding quadrntzc wave surfaces, that is, to the light cone for the metric g". A visualization of the corresponding conforinal manifold is given in Fig. D.4.2.
Regular case
M"
+ 2qoqnka - q,qb A
det A
@Degeneratecase The salne conclusion is also true for the clegcnerat,e caw. Slibstitliting (D.4.32) into (D.2.46)-(D.2.49) and tising (D.4.33)-(D.4.35), we find
det B
L*$),
dct A
Inserting this into (D.2.44), we obtain
288
D.4. Second constraint: Vanishing skewon field. Emergence of the light cone
D.4.4 From a quartic wave surface to the light cone
289
Consequently, (D.2.44) reduces to
where cu := Aabqaqb,P := Babqaqb,and y := Mabqaqb.Assuming that (D.4.57) describes a light cone, one concludes that the roots for q: should coincide. Thus necessarily
Let us write (det A det B ) = s I det Adet BI, with s = sign(det A det B). Then (D.4.58) yields
where X is an arbitrary scalar factor. We recall the definitions of cu,P,y and find
Figure D.4.2: Null cones fitted together to for~na conformal manifold (see Pirani and Schild [41]).
Consequently, M = detA = sX6/det B and M a b = 2X4Bab. Thus, (D.4.57) simplifies to 2
sX 2 (X2qi + s qaqb Babdet B) = 0.
det B This time the tensor field
gij is
described by
For s = -1, we immediately recognize that the quadratic form in (D.4.61) can have a signature of either (+ - - - ) or (+ +-). Similarly, for s = 1, the signature is either (+ ++) or (+ + --). Therefore, the Fresnel equation describes the correct (hyperbolic) light cone structure only in the case s = -1. Finally, one can verify that the above solutions satisfy
+
+
This tensor is nondegenerate and it has Lorentzian signature.
@Isreciprocity also necessary for the light cone? Reciprocity is sufficient for the light cone to exist. Is it also a necessary condition for the reduction of the quartic geometry (D.2.44) to the light cone? Such a conjecture is substantiated by some particular examples, but its general proof is still missing. For the special case of vanishing C , that is, for C = 0, we are also able to prove necessity. Here we do not assume closure of iZ;, but we keep its symmetry. Putting C a b = 0, we find from (D.2.46)-(D.2.49) that M a = 0 and Mabc= 0, whereas
(D.4.61)
which reproduces the closure relation (D.3.4) for s = -1.
Extracting the metric by an alternative method
W e discussed wave propagation i n linear pre-metric electrodynamics and f0un.d that reciprocity and s y m m e t y enabled us to obtain the metric of spacetime from. the spacetime relation. Here we present an alternative construction of the Lorentzian metric from the constitutive ten.sor of spacetime. As above, the crucial point is a well-known, mathematical fact. Here it is the one-to-one correspondence between the duality operators # and the conformal classes of the metrics of spacetime. Henm, with the linearity condition (D. 1.6), together with reciprocity or closure (0.3.4) and symmetry (D.4..?), we can construct, up to a conformal factor, the metric of spacetime from the components of the duality operator # . I n this sen,se, tlre metric is a concept derived from the electrodynamic spacethe relation. At the center of th,is derivation is the triplet of (anti).relf-dual 2-form.s, wh,iclt. provides the basis of the (anti)self-dual subspace of the complexified space of all e-forms MY(@).
D.5.1 @Tripletof self-dual 2-forms and metric In Sec. C.2.5 we saw that the basis of the self-dual 2-forms can be described either by ((2.2.45)or by ((2.2.46). These triplets are linearly dependent, and one can use any of them for the actual computation. For example, one can take as (8)
the fundamental triplet work with
s(") =
ab
s(") =
Pa, with
B (:Ib, with G
ab
G~" ab
~ ~ ~Alternatively, 1 4 . we can
= B /4. Both choices yield equivalent
292
D.5. Extracting the metric by an alternative method
results if A and B arc non dcgcneratc. For definiteness, let us choose the second option. Then from (C.2.46), in thc B-representtation, we have explicit,ly
=
2
( - i dzo A dx" + i(det B)-' khd s b A dra
(D.5.1)
One can find the determinant of this expression and verify that ( i m ) = Jq = 1 in accordance with (C.2.74). If instead of the R-repre~cnt~ation (D.4.26), (D.4.27) of the solution of the closure and sylnlnetry relations, we start from the A-representation (D.4.28), (D.4.29), then thc Schonberg-Urbantke formulas yield an alternative form of the spacctime metric,
-
1 gij
Hcre r0= a and n." are the spatial coortlinates, with a , b, c, . . . = 1,2,3. A rlsrful c:tlc~llationaltool is provicled by a set of 1-forms := a, _r s(") = s!;i)dr~.Witli tlieir Iielp, wc can r e d off tlic contractions oeeclecl in tire Sclionhcrg-Urbnntkc formulas (C.2.73), (C.2.74) from tlie exterior products
s:"'
293
D.5.1 @Tripletof self-dual 2-forms and metric
=
- kb
Jq&$
- (det A) Ash
(D.5.10)
,
Thc triplet of the self-dual 2-forms s(,)is defined up to an arbitrary scalar factor: By lnultiplying them with an arbitrary (in general complex) function h ( a ) ,one preserves tlic completeness condition (C.2.52), (C.2.53). Correspondingly, the determinant of the mctric is rescaled by a factor h" wl~ereasthe metric itself is rcscalcd by a factor h. I11 other words, the whole proceclurc defines a conformal class of metrics rather than a mctric itself. Clearly, one can always clioosc~the conformal fact,or h so as to eliminate the first factor in (D.5.9). For tlic metric (D.5.9), the line element of spacctime reads explicitly:
Hcrc, as usual, Vol = d r o A dal A dx2 A dx3. From (D.5.1) we have Howcvcr, as just eliscussed, a confornial factor is irrelevant. Therefore we may limit oursclvcs t,o tlic line element
- dzO6;
+ i BnCCcbd
dsd
+ (dct B)-' kc dzc 61: - (det B)-' kl)d s n
(D.5.4)
If wc consider the 3 x 3 matrix R, wc call clistinguisll four clifferctlt c ~ e s with t11c signatures (+ + +), (- +), (- - +), and (- - -), respectively. Let us demonstrate that a11 these cases lead to a Lorentzian signatflure. For (+ +), the determinant is positive and the exprcssion Babdn."drb is posit~vctlefinitc. Thus, we can lend off the Lorentzian sigrlaturc immediately: a is the time cooltlinatc, the r n ' s the tlirce spatial coordinates. FOI(- +), tlic determinant is negative and the expression Bat)dxndx" is 2 lndcfinitc.. Therefore tlie square of onc d r coordinate differential, say (dr') , carries a positive sign anel can be identified ns the time coordinatc, whcrcas a can he identified as the space coordinatc. For (- -+), the determinant is positive ngdn and the expressio~i-Bob dsadxl) is indefinite with signature (+ -). Therefore s3is the timc coordinate in this Case. For (- - -), the determinant is negative and the expression -Bar, drarls"s positivc definite. In other worcls, this corrcsponcls to the case (+ +) with an overall sign change. Accorclingly, tlie structure in (D.5.12) is rather robust and thc quantity k, doesn't influence tlie Lorclltziall signature of (D.5.12).
+
By a. ratlier lcngtlly calcl~lationwe find
+
A s(Q A
i,l,,
SF)=
s?) A s ( ~ A s) :)
=-
3% 4
- ((let B)-
' k,,
4
+
(D.5.7)
VOI,
( - (
~ (D.5.8)
Wc 11s~.(D.5.5)-(D.5.8)t o g c t h ~ with r (D.5.2). Then, the Scllonbcrg-Urbantke formrilas (C.2.73), (C.2.74), for the con~ponentsof the spacetime metric yield Q13
=
1
det B
- kt,
+
+
294
D.5. Extracting the metric by an alternative method
It is quite satisfactory to note that the Schonberg-Urbantke form a1'ism produces the same Lorentzian metric that we recovered earlier in Sec. D.4.4 whcn discussing the reduction of the fourth-order wave surface to thc (second-ordcr) light cone. In that approach, since the wave vector is a 1-form, we obtained tllc contravariant components (D.4.47) and (D.4.48) of the metric, which are just, the inverses of (D.5.9) and (D.5.10), respectively. However, our new "reduction" method of Sec. D.4.4 also produces a spacctime metric for the degcneratc solution of thc closure relation (see (D.4.54)). The Schonberg-Urbantke formulas are inapplicable to the degenerate solutions. In this sense, the reduction mcthotl is farther rcaching.
D.5.2 Maxwell-Lorentz spacetime relation and Minkowski spacetime Let us take a particular cxample for the spacetime relation. Wc assume that we are in a suitable spccific frame such that we measure
295
D.5.3 Hodge star operator and isotropy
the metric (D.5.9), (D.5.10) in the form:
This is, up to the factor I/&, the standard Minkowski metric in orthonormal coordinates. If we include 1/fi, then we find the Minkowski metric density of weight -112.
D.5.3 Hodge star operator and isotropy The inverse of (D.5.9) is given by .. gEJ
=
(
-.
-
.~,. ,
1 - (det B)-' k2
I (
-.
-(det B ) B a
'
.
(D.5.18)
With the help of (D.5.9) and (D.5.18), we can define the corresponding Hodge star operator attached to the metric extracted by means of the SchonbergUrbantke formalism. Its action on a 2-form, say F, is described by (C.2.92) or, explicitly, by This set is known as the Maxwell-Lorentz spacetime relation in the framc chosen. The components of the constitutive tensor of spacetime in (D.1.6), for thc example discussed, are independent of the spacetimc coordinates and arc given by (D.5.13) in terms of a pair of basic constants EO,PO,with (a, b, c = 1,2,3)
*Fij:= 6iijklg k m g l n ~ m n . 2
Such a Hodge duality operator (see (C.2.33)) has the constitutive tensor of spacetime
ailk1= Cijmn R
We can read off K , , ~ ' from (D.5.13). Then a direct computation shows that thc quadratic invariant (D.3.5) is cqual to X2 = ~ ~ / with p ~[A], = q 2 / / SI = ~ 1/Q. Consequently, the duality operator # of (D.4.6), as defined by the spacetimc. relation (D.5.13), is given by the G x G matrix
-
-
(cf. (D.l.G)). Here we denote c := I/-, with [c]= velocity. Thus we havc the matrices A = -B-', B = c13, whereas C = 0. One then imrnediatcly finds
(D.5.19)
kmnk1/2 with
..
Note that x v k l is invariant under conformal transformations gij + ex(x)gij; therefore, only nine of the possible ten components of the metric can ever enter gijkl. We can compare %jklwith the original gjkl of the linear spacetime relation. For this purpose, we have to substitute the metric components (D.5.9) into the A, B , C blocks (C.2.95)-(C.2.97) of the Hodge duality matrix (C.2.94). Then inspection of (D.4.26)-(D.4.29) demonstrates the exact coincidence
R
Summing up, it turns out that x I J = %IJ; i.e., the metric extracted allows u s to write the original duality operator # as the Hodge star operator associated
D.5. Extracting the metric by an alternative method
296
with that metric. Therefore, the original linear constitutive tensor (D.1.70) for a vanishing skewon field can then finally be written as
(see also (D.4.9)), with the dilaton field and the axion field a. Given a metric, we can define the notion of local isotropy. Let Til...ip be thc contravariant coordinate components of a tensor field and TaI..."p := eil " 1 . . . eipmpT i l , , , i p its frame components with respect to an orthonormal frame e, = \e A tensor is said to be locally isotropic a t a given point if its frame components are invariant under a Lorentz rotation of the orthonormal frame. Similar considerations extend to tensor densities. There are only two geometrical objects that are numerically invariant under (local) Lorentz transformations: the Minkowski metric oaP = diag(+l, -1, -1, -1) and the Levi-Civita tensor density c"Py< Thus
ai.
D.5.4@Covarianceproperties
297
the duality operator is tiescribed by the coniponents of tlie spacctimc matrix # , o ~ ~=~ /8, 2 61 6 1 L p# f J k l . Accordingly, the components of the cxtractetl spacctime metric rend gap = 8;s; g,,. Because of the tensorial nature of the definition of the duality oprrator (D.1.46), a linear transformation (A.1.92) of the basis, 8" = Let" 19"' , yields the correspontling transformation of the duality components,
Recall that L,,"' is the inverse of La,". We now demonstrate that thc Schonberg-Urbantkc construction is completely covariant ant1 that the extracted metric (C.2.73) transforllls as
untler the lincar transformation (A. 1.92), (D.5.24). is the most general locally isotropic contravariant fourth-rank tensor density of . 4 weight +1 with the symmetries ' T a p @ = -'TPny6 = - T a O 6 y = l y 6 c u o Here and cp are scalar and pseudoscalar fields, respectively. Accordingly, in view of (D.5.22) we have proved that the constitutive tensor density X i j k i with the constraints of reciprocity and symmetry is locally isotropic with respect to the metric (D.5.9).
The proof is technically simple and straiglitforwartl, but it is somewhat now lengthy. We have prcparcd the nccrssary tools in Scc, A.l.lO. Coml~inil~g the matrix equations (A. 1.95))(A.1.98),and (C.2.37), we find that, with rcsl~cct t o the new coframr, the duality operator
is described I>y tllc new matrix components
D.5.4 @Covarianceproperties The tliscussion above was confined to a fixed coordinate system. However, one may ask: What happens when the coordinates are changed or, more generally, when a local (co)frame is transformed? Up to now, we considered only a holonomic coframe 8 , = 6; dx2,but in physically important cases one often needs to go to nonliolonomic coframes. In this section we study the covariance properties of the Schonberg-Urbantke construction. The behavior of the metric (C.2.73), (C.2.74) under coordinate ancl frame transformations is by no means obvious. Although the fundamental completeness relation (C.2.52), (C.2.53) looks covariant, one should recall that the index (") of the self-dual S-forms comes, in a noncovariant manner, from the 3+3 splitting of the basis 6'. A transformation of the spacetime coframe (A.1.92) acts on both types of indices in the coefficients S$ that enter the Schonberg-Urbantkr formulas (C.2.73), (C.2.74). Thus, the determination of the new components of the metric with respect to transformed frame becomes a nontrivial problem. For tlie sake of generality, we consider an arbitrary linear transformation (A.1.92) of the coframe that includes the holonomic coordinate transformation as a particular case. With respect to the original coframe 6" = 6; dx',
This is the clircct matrix rcmakc of tlic tensorial version (D.5.24). Tlic rn;tt,riccs P, Q, PV, Z are drscribcd in (A.1.96) and (A.1.97). Explicitly, (D.5.27) yiclcls
+ Q'CW + w ~ c " Q ) , (P'BP + Z'AZ + Z'CP + P T C T Z ) , ( Q ~ C+ P W'C~Z + Q ~ A + Z W'BP).
A' = (det L)-' (QTAQ + I/V'BI/V
(D.5.28)
B'
(D.5.29)
= (tlct L)-'
C' = (dct L)-'
(D.5.30)
It is convenient to consitlcr tlic thrrc subcascs of thc lincw transforniation (A.1.101)-(A.1.103) bccausc their product (A.l.lOO) dcscri1)cs a general 1inc.w tranformation. For L = L 1 , we 11:tvc (A.1.104) with only the matrix Iv"" = ca'"'U,. being. nontriviixl. Thcn (D.5.28)-(D.5.30) yield
D.S. Extracting the metric by an alternative method
298
D.5.4@Covarianceproperties For the antisymmetric matrix K , this yields:
Comparing this with the B-representation (D.4.20) and (D.4.21), we see that the transformation (D.5.31) means a mere shift of the antisymmetric matrix: K 1 = I< + B w T B ,
or equivalently, k', = k, - (det B ) U,.
(D.5.32)
Substituting this into (D.5.9), we find the transformed metric: (det B ) Ub -kaUb - kbUa (det B ) UaUb (D.5.33)
0 (det B ) U,
ging = gap +
+
Comparing with (A.1.101), we get a particular case of (D.5.25): g'a,pl = Lala Lp)P gap ,
with
L = L1.
=
A,
BI
=
B+Z~AZ+Z~C+C~Z,
C'
=
C
(D.5.35)
+ AZ.
I?' = I? + AZA,
/.
A
k: = k,, - (det A ) A,,, v b .
or cquivalently,
(D.5.36)
c
2% V' - (det A) Acd V V +-( (clct A ) AacVC ,Ax3
d
)
(det B ) AbdVd 0 (D.5.37)
with
L = L2.
(D.5.38)
Finally, for L = Lg we have (A.1.106). Then (D.5.28)-(D.5.30) reduce to A'
=
B'
=
det A
A-I
A(A-~)~,
-ATBA, det A
k b.
(D.5.40)
C a = -
,
1 Aa det A
(8)
Eb.
Hence, for the S-forms (D.5.1), we find
and consequently, ab
S ~ ( a )r\ S'(b)
-
1 Bab~ det L3
( /\ ~s ( 1~ )
Using this in (C.2.74), one finally proves the invariance of the determinant:
Jdetsl=J;iets.
(D.5.44)
1-
AaCi\bd
- A ~ ~ A ~ Q ~ BCd+ (det B)-' kc kd]
Thus, the metric transforms as a tensor density,
Wc compare with (A.1.102) and recover a subcase of (D.5.25), glatp, = LaraLp,P gap,
kIa = AoO(A-')ba
Taking this result into account when substituting (D.5.39) and (D.5.40) into (D.5.9), one obtains the transformed metric
Substituting this into (D.5.10), we obtain the transformed metric: !?Iao= gap
(8)'
B'
Contrary t,o the first case, it is now more convenient to proceed in the Arcprescntntion. From (D.4.22) and (D.4.23),wc then see that the transformation (D.5.35) means a mere shift of the antisymmetric matrix
or equivalently,
As we saw above, the analysis of the case (A.1.104) was easier in the Brepresentation, whereas the A-representation was more suitable for the treatment of the case (A.1.105). However, the last case (A.1.106), (D.5.39) looks the same in both pictures. For definiteness, let us choose the B-representation. A new and nontrivial feature of the present case is that the transformation L3 is not unimodular, det Lg = AoOdet A # 1. Recall that det L1 = det L2 = 1. As a consequence, one should carefully study the behavior of the determinant of the metric defined by the second Schonberg-Urbantke formula (C.2.74). From (A.1.98) we have the transformation of the self-dual basis (C.2.46):
(D.5.34)
Wlicn L = Lz, the matrix Zar, = Zabc VC describes the case (A.1.105). Then, from (D.5.28)-(D.5.30), wc find: A'
AoO ATKA, K' = det A
1 g',lp, = & L a Lala~B'Pgap
with
L =L ~ .
This transformation law is completely consistent with the invariance of the determinant (D.5.44). Turning now to the casc of a general linear transformation, one can use the factorization (A.l.lOO) and perform the three transformations (A.1.101)(A.1.103) one after another. This yields subsequently (D.5.34), (D.5.38), and (D.5.46). The final result is then represented in (D.5.25) with an arbitrary transformation matrix L.
300
D.5, Extracting the metric by an alternative method
Reducing the degenerate case to the regular one Using the formalism above, we can show t,hat the degenerate solutions of the closure relation, A, B, C , can always be transformed into the regular configurations. Indeed, let us take the degenerate solution (D.4.32), (D.4.34), (D.4.35), which is explicitly given by the three matrices
Fifth axiom: Maxwell-Lorentz spacetime relation
Consider a simple transformation of the coframe 19" = La!" 19"' by means of the L = L1 matrix (A.l.lO1) with U, = (0,0,1). This induces the transformation of the 2-form basis (A.1.95) where the matrix W of (A.1.104) reads explicitly
The other matrices are P = Q = 13 and Z = 0. Then, using (D.5.31), wc find the transformed spacetime relation matrices as
D.6.1 The axiom
With respect to the transformed basis, as we imlnediatcly see, the consti0 tut,ive matrices become nondegenerate, dct A' # 0, and we can usc thc Arcprcsentation to describe this configuratiori and to construct the corresponding metric of spacetime. Evidently, the gcneral dcgencrate solution can also bc reduced to the regular case. Then the linear transformation above, with L = L1, should be supplementecl by the appropriate GL(3, R) transformation (D.4.31). We thlis conclude that the separate treatment of the dcgcncrate case is in fact not necessary: T h r dcgcncracy (D.4.30) merely reflects an unfortunate choice of the frame that can be easily reinovcd by a linear t,ransformation.
Summarizing the content of Part D, we can now give a clear formulation of the fifth axiom. In Maxwell-Lorentz electrodynamics, the 2-forms of the electromagnetic excitation H and the field strength F arc related by the universal, locad, lin.ear spacetime relation
The linear operator
(here a,, b are functions and 4, 2-forms) can be represented in its spacetime components such that (D.G.1) reads $J
Under the linear group, the operator K can be decoinposed irretlucibly into three parts,
Here a is the axion field. In terms of components, the corresponding spacetime relatioil reads
D.6. Fifth axiom: Maxwell- Lorentz s p a c e t i m e relation
302
with the principal part ('In with twenty and the (traceless) skewon field $ with fifteen independent components. We demand electric/magnetic reciprocity of the spacetime relation H = E ( F ) and find the closure property
x2
= -Tr(EE)/6. We define J = E l i . Then, at this stage, the spacetime with relation has the form1
H = ~ J ( F ) + C Y F , with
JJ=-1.
(D.6.7)
Furthermore, we demand the vanishing of the skewon field $, i.e., the symmetry of the operator tc or, equivalently, of E,
D.6.2 O n thc Poincilri! ancl t h e Lorcntx groups
303
with tlic electric c0nsfan.t E" z 8.854 x 10-l2 s/(R m) (permittivity of vacuum) R s / m (permeability of vxcunm). anti the magnetic constant = 47r x So much about the vacuum impedance of the spacetimc relatioil (D.G.9). What are wc going to tlo about the axion ct in the same equation? Well, we remove it by putting 1 a = - Tr tc = 0 G
(no axion) .
Then, malting use of tlie extracted metric and of the corresponding Hodgc star operator *, thc Maxwell-Loren& spncetim.e relation. can bc written as
1-
(fift11 axiom) ,
or in spacetime components, as Then J becomes a duality operator, the Hodge star operator * that defines a metric tensor field up t,o a11 uncletermined (conformal) f a ~ t o r : ~ An alter~iativcforrnulat,ion is (raising the indices of F,,) Accordingly, linearity, reciprocity, and symmetry provide a unique light cone structure for tlie propagation of electromagnetic waves (see Fig. D.4.2). As a result, the spacetimc metric with tlie correct Lorenttian signature is, up to a conformal factor, reconstructed from the constitutive tensor nZjk1of spacetime. Maxwell-Lorentzian clcctrodynamics is specified, among the models defined in (D.G.9), by a constant universal dilation factor =
Xo = const
(no dilaton) .
(D.6.10)
The actual value of this universal constant 3 characterizing the vacuum has to be taken from experiment, basically from a Weber-Kohlrausch type of experiment relating electric to magnetic effects. In the SI system, its value is
we could write (D.6.7)1 syrnbol'With the t ~ d r r s t a n d i n gthat H and F also live in icnlly it3 H = a F a F . Isn't that a re~nnrkablysimple expression after having only applied linenrity and reciprocity? 'Sows [50] hns proposed a "nonlinmr Maxwell tl~cory"with H = A *F, i.e., with a = 0. His dilaton - he calls it the filling factor, following the terminology of the $HE (see our equation (B.4.54)) - obeys a nonlinear wave equation f + a * ( F A ' F ) f = v f with the constants n nnd v. For vacuuln, J = 0, Sowa found interesting vortex-like solutions of his theory. "p-to-clatc information about the fundamental constants in nature (including the electric ones), their measurement, nntl their SI values has been provided by Flowers & Petley [lo] and by Mohr & Tnylor [29].
+
-
Accordingly, tllc experimentally well-cstablislicd Maxwell-Lorcntz elcct,rodynamics is dist,inguislicd from other models hy linearity, r~ciprocit~y, symnicxtry (no skcwon), no dilaton, and no axion.
D.6.2 On thc Poincark and the Lorcntz groups The light cone is invariant undcr tllc 15-pttrt~metrrgroup of local col~folnii~l transformitt ions." Tlic 10-parameter Poincar6 group (inhornogcnc~ousLo1 c.nt z group) is a subgroup of tlic conformal group 111 our consitlcri~tiolis,we ~icvc>r recover tlie rigid (global) Poincard group unlrss the mctric is h4inkowski;ui. In our whole axiomatics in Part B alid Part D we did not use tlic Poinc;ti6 group nor its G-paramrtcr Lorrntz subgroup at rill. In tlir conventional app~oacli to clcctrotlynamics, tlic Poincnr6 group is an cssclitial ingrctliclit for cooltil~g 1111 the formalism of clcctrotlynamics. I11 thr general covariant approacli, witli electric charge and magnetic flux conservation as its basis, which wc followc~d, the mctric is tlist,illctl from a linear spacctime relation witli reciprocity ant1 symmetry as adtlitives. The mctric, and thus the gravitational potcwtial, is a derivccl conccpt. The mctric gets its meaning from clcctrodynamics; it is not i) fundamental field.
304
D.6. Fifth axiom: Maxwell-Lorentz spacetime relation
D.6.3 @Extensions.Dissolving Lorentz invariance? We formulated our deductions of the Maxwell-Lorentz spacetime relation already in such a way that the possible generalizations of the Maxwell-Lorentz framework are apparent: Have a dilaton, have an axion, have a skewon, either one of them, some of them, or all of them. But all of these possible generalizations to dilaton electrodynamics, axion(-dilaton) electrodynamics, and so on take place in the framework of the linear spacetime relation (D.G.1) cum (D.6.2). This linearity gave birth to the axion, the skcwon, and an unstructured prin/- (1) xk / %. ~The subsequent assumption of reciprocity and cipal part ( l ) X ~ 3 , symmetry created the metric g of spacetime. Had we given up linearity, that is, had we allowed nonlinearity at a fundamental level in the spacetime relation (D.G.l), thcn thc metric and their siblings axion and skewon would bc irretrievably lost. In fact, we arc not aware of any attempt to construct a nonlinear electrodynamic model a t this fundamental level, and we see no way of doing so. However, if we take the mctric for granted, i.e., fundamental linearity, reciprocity, and symmetry, then, with the help of the metric, one can construct nonlocal and nonlinear nlodels that, nevertheless, are based on fundamental linparity b la (D.6.2). Presupposing a metric of spacetime implies a linear spacetime relation willy-nilly. To fix this linear relation later by some decorative nonlinear and nonlocal terms doesn't change the underlying rationale. For this reason, we discuss these nonlocal ancl nonlinear models, which respect the linear spacetime relation in the sense described, in Chap. E.2. Let us come back to the possible extensions of the Maxwell-Lorentz electrodynamics by means of introducing dilaton, axion, and/or skewon fields. Therc are two distinct cla.~ses:Those that respect the light cone (one could call them the harmless extensions) and those that don't. To the former belong axion, dilaton, ant1 axion-dilaton electrodynamics, to the latter skewon electrodynamics (that is, with $ # 0) with possible admixtures of axion and/or dilaton fields. As soon as one mixes a skewon field into the spacetime relation (see (D.6.5))) one cannot recover the light cone any longer and the Lorentz group is dissolved. Several such attempts arc discussed in the l i t e r a t ~ r e If. ~one wants t o look for possible new physics violating Lorentz invariance, then the assumption of a skcwon appears to be tlle most natural possibility. As a first attempt one could try to simplify (D.6.5) by brute force to
D.6.3 @Extensions.Dissolving Lorentz invariance?
305
Moreover, recently, there was some observational evidence from astronomy that Sommerfeld's fine structure constant may depend on (cosmic) time:
(e = elementary charge, h = reduced Planck constant). Even though these observations did not find independent support, the posed question may be interesting for future. Provided that we want to uphold the pre-metric Maxwellian structure (B.4.1), then, in the light of (B.4.2), only EOC could be time dependent: The speed of light would not be necessarily constant 6 We saw that EOC codifies a response of spacetime to the propagation of electromagnetic disturbances. Dirnensionwise, we have [ E ~ c=] l/resistance, that is, it represents an impedance of spacetime ("vacuum"). It is conceivable that such a "vacuum" impedance picks up some time dependence or, more generally, becomes a field dependig on time and space. Such an approach would dismantle local Lorentz invariance and thereby general relativity too. s
and then investigate the influence of $ on light propag a t'lon. "ee,
for instance, KostelcckJ; [25]
"his
is the main result of
a recent article of Peres [40]. Here we partly follow his arguments.
References
[ I ] A.O. Barut and R. Rgczka, Tlzeory of Group Representations and Applications ( P W N - Polish Scientific Publishers: Warsaw, 1977). [2] J.D. Bekenstein, Fine-stm~ctureconstant ~~ariability, equivalence principle and cosmology, Eprint Archive gr-qc/0208081, 18 pagcs (August 2002). [3] C.H. Brans, Complex 2-forms representation of the Einstein, equations: The Petrov Type 111 solutions, J. Math. P11,y.s. 12 (1971) 1616-1619. (41 C.H. Brans and R.H. Dicke, Mach's prin.ciple and a relativistic t h e o ~ ~ofj gravitation, Pti-ys. Rev. 124 (1961) 925-935. [5] R. Capovilla, T . Jacobson, and J . Dell, General relativity witl~outthe metric, Phys. Rev. Lett. 63 (1989) 2325-2328. [6] L. Cooper and G.E. Stedman, Axion detection by ring lasers, Phys. Lett. B357 (1995) 464-468. [7] R.H. Dicke, The theoretical significance of experimental relativity, (Gordon and Breach: New Y o r k , 1964). [8] B. Fauser, Projective relativit?~:Present status and outlook, Gcn. Relat. Grnv. 33 (2001) 875-887. (91 G.B. Field and S.M. Carroll, Cosmological magnetic fields from primordial helicity, P11.y.s. Rev. D62 (2000) 103008, 5 pages.
308
Part D. The Maxwell-Lorentz Spacetime Relation
[lo] J.L. Flowers and B.W. Petley, Progress in our knowledge of the fundamental constanh in physics, R,ep. Progr. Phys. 64 (2001) 1191-1246. [ll] F.R. Gantmachcr, Matritenrechnung. Tcil I: Allgemeine Theorie (VEB Deutschcr Vcrlag der Wisscnschaften: Berlin, 1958). [12] A. Gross and G.F. Rubilar, On the derivation of the spacetime metric from linear electrodynamics, Pl~ys.Lett. A285 (2001) 267-272. [13] J. Hadanlard, L e ~ o n ssur la propagation des ondes et les e'quations de 1'lz?jdrod~jnaml:q~~e (Hermann: Paris, 1903). [14] G. Harnett, Metrics and dual operators, J. Math. Pliys. 32 (1991) 84-91. [15] G. Harnett, TIze hivector Clifford algebra and the geometry of Hodge dual operators, J. Pl;ys. A25 (1992) 5649-5662. [16] M. Haugan ancl C. Liimmerzahl, On the experimental foundations of the Maxwell equations, Ann. Physik (Leipzig) 9 (2000) Special Issue, SI-119SI-124. [I71 F.W. Hchl, Yu.N. Obukliov, G.F. Rubilar, On a possible new type of a T odd skeuron field linked to electromagnetism. In: Developments in Mathematical and Experimental Ph.ysics, A. Macias, F. Uribe, and E. Diaz, eds. Voluinc A: Cosmology ancl Gravitation (Kluwer Academic/Plenum Publishers: New York, 2002) pp.241-256. [18] G. 't I-Iooft, A clriral alternative to the vierbein field in general relativity, Nl~cl.Pliys. B357 (1991) 211-221. [I91 K. Huang, Quarks, Leptons Ed Gauge Fields, 2nd ed. (World Scientific: Singapore, 1992). [20] C.J. Isham, Abdl~sSalam, and J . Strathdee, Broken chiral and conformal sym.metry in an effective-Lagrangian formalism, Phys.Rev. D2 (1970) 685690. [21] A.Z. Jadczyk, Electromagnetic permeability of the vacuum and light-cone structure, B1111. Acad. Pol. Sci., Sdr. sci. pli,ys, et astr. 27 (1979) 91-94. [22] P. Jordan, Schwerkraft und Weltall, 2nd ed. (Vieweg: Braunschweig, 1955). [23] R.M. Kielin, G.P. Kichn, and J.B. Roberds, Parity and time-reversal symmetry breaking, singular solutions, and Fresnel surfaces, Phys. Rev. A43 (1991) 5665-5671. [24] E.W. I
References
309
[25] V.A. Kostclcck$, Topics In Lorrntt and C P T violation, Invitctl tiilk at Intern. Conf. on Orbis Scicntiae 2000: Coral Gables, Ft. Lauderclalr, F l ~ r i d i ~ , 14-17 Dec 2000, Eprint Archive hep-phl0104227 (2001). [26] A. Kovetz, Electroinagn,etic Theory (Oxford University Press: Oxfortl, 2000). [27] L.D. Landau, E.M. Lifshitz, and L.P. Pitaevskii, Electrodynamics of continrlous media, 2nd edition (Pergamon Press: Oxford, 1984) 460 pagcs. [28] A. Lichnerowicz, Relativit?l theory and mathematical physic.^, in: "Astrofisica e coslnologia gravitazionc qnanti c relativith", Ccntcnario di Einstchi (Giunti Barbera: Firenze, 1979.) [29] P.J. Mohr and B.N. Taylor, CODATA recom.mended values of the firndam.en.tal physical con8tan.t~:1998, Rev. Mod. P11.ys. 72 (2000) 351-495. [30] J.E. Mootly and F. Wilczck, New m.acroscopic forces? P1,y.~.Rev. D30 (1984) 130--138. [31] W.-T. Ni, A non-m.etric theory of gravity. Dept. Physics, Montan;~St,atc University, Bozeman. Prcprint Decemher 1973. [This paper is rcfcrrcd to by W.-T. Ni in B~lll.Amcr. P11.y.s. Soc. 19 (1974) 655.1 The paper is available via h t t p : / / g r a v i t y 5 .phys . n t h u . e d u . tw/webpage/article4 /index.html . [32] W.-T. Ni, Equivalence principles and electromogn,etism, Phys. R.cv. Lct,i,. 38 (1977) 301-304. [33] W.-T. Ni, Eq~iivnlrncrp1711riplrs nnd prerlston expr77mrnfs. In Pi.rcision Mcasrlreinent alld Fuildarnci~talCor~starits11,B.N. Taylor, W.D. Phillips, ctls. Nat. Bur. St,antl. (US) S p c ~ Publ. . 617, US Govcrnmcnt Printing Office, Washington, DC (1984). [34] J.F. Nicves and P.B. Pal, P and CP-odd terms in tile photon self-enerqy within. a medium., P11.y.s. Rkv. D39 (1989) 652-659. [35] J.F. Nicvcs ant1 P.B. Pal, The third electromagn.etic constant of an isotropic medium., Ani. J. Phys. 62 (1994) 207-216. [36] Y1i.N. Obukhov and F.W. Hehl, Space-time rn,etric from lin.ear electradyn.amics, Phys. Lett. 13458 (1999) 466-470; F.W. Hehl, Yu.N. Obultliov, and G.F. Ruhilar, Spacetime metric from linear electrod?ln,amic.~ II, Ann. d. Pl~jfs.(Lcipzig;) 9 (2000) Special issue, SI-71-SI-78; Yu.N. O l ) ~ ~ l < l ~ o v , T . F~lltni,and G.F. Rubilar, Wave propagation in li~zearelectrod?jratr7rl.ics, P11.ys. Rev. D62 (2000) 044050 ( 5 pages); G.F. Rubilar, Yl1.N. Ol~ul
Part D. The Maxwell-Lorcnta Spncetime Relation
Light propagation in gen.erally covariant electrodynamics and the Fresnel equation. Invited talk at Journkes Relativistes, Dublin, Ireland, 6-8 Sep 2001. Int. J. Mod. Phys. A17 (2002) 2695-2700. Yu.N. Obukhov and G.F. Rubilar, Fresnel analysis of wave propagation in nonlinear electrodynamics, Phys. Rev. D66 (2002) 024042 (11 pages). [37] Yu.N. Obukhov and S.I. Tertychniy, Vacuum Einstein equations in terms of curvature forms, Class. Quarltum Grav. 13 (1996) 1623-1640. [38] R.D. Peccei and H.R. Quinn C P con.servation in the presence of pseudoparticles, P1l.y~.Rev. Lett. 38 (1977) 1440-1443. [39] A. Peres, Electromagnetism, geomety, and the equivalence principle, Ann. Phys. (NY) 19 (1962) 279-286. [40] A. Pcres, The speed of light need not be constant, Eprint Archive: grqc/0210066, 3 pages (Oct 2002). [41] F.A.E. Pirani and A. Schild, Conformal geomety and the interpretation of the Weyl tensor, in: Perspectives in Geometry and Relativity. Essays in honor of V. Hlavat3;. B. Hoffmann, editor. (Indiana University Press: Bloomington, 1966) pp. 291-309. 1421 C. Piron and D.J. Moore, New aspects of field theory, Turk. J. Ph-ys. 19 (1995) 202-216.
References
311
[50] A. Sowa, The nonlinear Maxwell theory: An outline, Eprint Archive physics/0103061, 22 pages (March 2001). [51] G.E. Stcdman, Ring-laser tests of fundamental physics and geophysics, Reports on Progress in Physics 60 (1997) 615-688. [52] G. Szivessy, Kristalloptilc, in: "Handbuch der Physik", Eds. H. Geiger and K. Scheel, Vol. 20 (Springer: Berlin, 1928) 635-904. [53] I.E. Tamm, Relativistic crystal optics and its relation to the geometry of a bi-quadratic form, Zhurn. Ross. Fiz.-Khim. Ob. 57, n. 3-4 (1925) 209-224 (in Russian). Reprinted in: I.E. Tamm, Collected Papers (Nauka: Moscow, 1975) Vol. 1, pp. 33-61 (in Russian). See also: I.E. Tamm, Electrodynamics of an anisotropic medium in special relativity theory, ibid, pp. 19-31; A short version in German, together with L.I. Mandelstam, ibid. pp. 62-67. [54] R.A. Toupin, Elasticity and electro-magnetics, in: Non-Linear Continuum Theories, C.I.M.E. Conference, Bressanone, Italy 1965. C. Truesdell and G. Grioli coordinators, pp. 203-342. [55] C. Truesdell and R.A. Toupin, The classical field theories, in: Handbuch der Physik, Vol. 11111, S. Fliigge ed. (Springer: Berlin, 1960) pp. 226-793. (561 H. Urbantke, A quasi-metric associated with SU(2) Yang-Mills field, Acta Phys. Austriaca Suppl. XIX (1978) 875-876.
[43] E.J. Post,, The constitutive map and som,e of its ramifications, Ann. Phys. ( N Y ) 71 (1972) 497-518.
[57] C. Wang, Mathematical Principles of Mechanics and Electromagnetism, Part B: Electromagnetism and Gravitation (Plenum Press: New York, 1979).
[44] G.N. Ramacliandrari and S. Ramaseshan, Cystal optics, in: "Handbrich dcr Pllysik", Ed. S. Fliigge, Vol. XXV/1 (Springer: Berlin, 1961) 1-217.
[58] S. Wcinberg, A new light boson? Phys. Rev. Lett. 40 (1978) 223-226.
[45] G.F. Rubilar, Linear pre-metric electrodynamics and deduction of the light-
cone, Thesis (University of Cologne, June 2002); see Ann. Phys. (Leipzig;) 11 (2002) 717-782.
[46] V, de Sabbata and C. Sivnram, Spin and Torsion in Gravitation (World Scientific: Singapore, 1994). [47] H.B. Sandvik, J.D. Barrow, and J . Magueijo, A .simple cosmology with a vaying fine structure constant, Ph,ys. Rev. Lett. 88 (2002) 031302 (4 pages). [48] M. Schonberg, E1ectrom.agnetism and gravitation, Rivistn Bra~ileiradc Fisica 1 (1971) 91-122. [49] P. Siltivie, ed., Azion,s '98, in: Proc. of the 5th IFT Workshop on Axions, Gainesville, Florida, USA. Nucl. Phys. B (Proc. Suppl.) 72 (1999) 1-240.
[59] F . Wilczek, Problem of strong P and T invariance in the presence of instantons, Phys. Rev. Lett. 40 (1978) 279-282.
[GO] F. Wilczck, Two applications of axion electrodynamics, Phys, Rev. Lett. 58 (1987) 1799-1802.
Part E
Electrodynamics in Vacuum and in Matter
314
Part E. Electrodynamics in Vacuum and in Matter
After the spacctime relat,ion is added m fifth axiom, the formulation of electrodynamics, as a predictive theory, is complete. In Part E,we present the basic aspects of standard Maxwell-Lorentz electrodynamics in vacuum and outline the possible generalizations of the theory when a, metric is prescribed and the spacetime relation becomes nonlocal and/or nonlinear. The electromagnetic spacctime relation is universal. It is determined by the electric and magnetic properties of spacetime itself and is intimately related to the metric structure of spacctime. Additionally, we have t o take into account the electric and magnetic properties of the material continuum as well as its state of motion. This leads to the necessity of introducing the last, namely the sixth, axiom, which expresses the splitting of the current into two parts, one referring to the bound charge and the other to the free charge. Thereby a constitutive relation emerges. Throughout this part, we consistently distinguish between the spacetime relation and the constitutive relation with particular attention paid to the fundamental theoretical and experimental aspects of thc electrodynamics of moving bodies.
Standard Maxwell-Lorentz theory in vacuum
W e descr.ibe standard Mamuell-Lorentz electrodynam,ics based on tlre jiftlr. axiom th,at we p ~ forward ~ t in. Ch.ap. D.G. The field equation.^ as well as tlre action formulated i n a gen.erally covariant manner on a n arbitranj curved spacetime. For a given spn.cetime metric, we i n t r o d ~ ~ c(1e special foliation adapted to tire m.etric s t r ~ ~ c t u mT/~,i.s . gives rise to tlre effective permittivity and perrn,enbility fl~nctior~s of the vac~~urn. which arc! expressed in, t e r n s of the componen,t.c of the m.etric. F?~rtl~~erm.ore, we use tlre metric in, order to prove the s?lrn,m,et~l of t 1 1 ~ elcctro~nagn,cticen,ergy-n~,on~ent~~m.
E.1.1 Maxwell-Lorentz equations, impedance of the vacullm When the Adaxwell-Loren.tt spacetime relation (D.6.13) is subst,it,utctl illto t,l~c Maxwell equations (B.4.9), (13.4.l o ) , wc fintl thc Maxwell-Lorcntz equntiolls
of standard clectrotlynamics. According to (D.G.lI ) , the cliarartclistic. ~rnpcrlr~nrr, (or wave resistance) of vacuum l / X o is fixed by cxpcrilnent.' ' I n q u a l i t ~ ~firltl n ~ tllcory, the electromagnet ir coupling constant is give11 by t l~c,tlirnensionlrss fine structllre constant cur := e 2 / ( 2 ~ o h c = ) 1/137.036, with e = elelncl~tarycharge and II = Planck's constant (sre Pohl [28, 1,. 2001). We can express Xo in tcrrns of rrr :~ntlfind Xo = e2/(2fffh).
316
E.1. S t a n d a r d Maxwell-Lorentz theory in vacuum
The impedance l / X o rcpresents a fundamental constant that describes the basic elcctromagnetic propcrty of spacetime; that is, one can interpret spacetime as a special type of medium (sometimes called vacuum, or aether, in the old terminology). In this scnse, one can understand (D.6.13) as the constitutive relation for spacctime itself. The Maxwell-Lorentz spacetime relation (D.6.13) is universal. It is cqually valid in Minkowski, Riemannian, and post-Riemannian spacetimes. The electric constant €0 and the magnetic constant po (also called permittivity of vacuum and permeability of vacuum, respectively) determine the universal constant of nature
E.1.2 Action
317
The left-hand side of this equation, in terms of components, can be determined by substituting (C.2.110) and (C.2.107):
Accordingly, curvature dependent terms surface in a natural way, both in (E.1.6) and in (E.1.8).
E.1.2
Action
According to (B.5.84), the excitation can be expressed in terms of the electromagnetic Lagrangian V by which yields the vacuum velocity of light. Thc homogeneous Maxwell equation can be solved by F = dA. Using this and (E.1.2), wc can recast the inhomogencous Maxwell equation (E.1.1)l in thc form 1 c 0 d * F = ~ o d * d A= - J . C
(E.1.3)
Recalling thc dcfinitions of the codifferential (C.2.107), dt := *d*, and of the wave operator (d'Alembertian) (C.2.1lo), we can rewrite (E.1.3) as
Because of the Maxwell-Lorentz spacetime relation (D.6.13), the excitation H is linear and homogeneous in F. Therefore the action V is homogeneous in F of degree 2. Then by Euler's theorem for homogeneous functions, we have
or with (E.1.9) and (D.6.13),
(E.1.4) Because of gaugc invariance, one can imposc the Lorenz condition2 dtA = 0. Thcn wc find a wave cqliation for the clcctrornagnctic potential 1-form: 1 UA = - *J, with
d t =~ 0.
(E.1.5)
EoC
I11 cornponcnts, the left-hand side reads:
k is ,thc ~Ricci~ tensor, see (C.1.51). The tilde denotes the Here Ric,, := ~ covariant differentiation and the geometric objects defined by the Levi-Civita connection (C.2.103). Also F obcys a wave equation. Wc take the Hodge star of (E.1.3) and diffcrcntiate it. We ~ubstitut~e d F = 0 and find
This is the twisted Lagrangian 4-form of the electromagnetic field b la MaxwellLorentz.
E.1.3 Foliation of a spacetime carrying a metric. Effective permeabilities In the prcvious sections, the standard Maxwell-Lorentz theory was presented in 4-dimensional form. In order to visualize the scparate electric and magnetic pieccs, we have to use the (1 3)-decomposition technique. In the presence of a metric, it becomes necessary to further specialize the vector field n , which is our basic tool in a (1 3)-decomposition. Before we introduced the metric, all possible vectors n described the same spacetime foliation without really distinguishing between "time" and "space" (see Fig. B.1.3), since the very notions of timelike and spacelike vectors and subspaces were absent. Now we choose the three functions n a in such a way that
+
+
g(n, aa) = g(a)a 'Ludvig Valcntin Lorenz (Danish physicist, 1829-1891): Lorcnz condition; Hendrik Antoon Lorentz ( D u t c l ~physicist, 1853-1928): Lorentz transformation, Maxwell-Lorentz electrodynan~ics.T h e Lorentz-Lorcnz formula in molecular polarizability.
+ gabnb = 0,
(E.1.12)
where gab := g(da, a h ) , g(,la := g(d,, a,), and g(,)(,) := g(&, 8,). F'rovided
E.1. Standard Maxwcll-Lorentz theory in vitcuum
319
E.1.4 Elect,romagnetic energy and momentum
Then the 3-dimensional version of the Maxwell-Lorentz spacetime relation reads V=E,EO*E
and
B = p,po5a,
(E.1.18)
where we introduced the effective electric and magnetic permeabilities
see (D.G.13) and (E.1.2). In general, these quantities are functions of the coordinates since N , according to (E.1.13), is determined by the geometry of spacetime. Thus the gravitational field acts, via its potential - the metric - o n spacetime and makes it look like a medium with nontrivial polarization properties. In particular, thc propagation of light, described by the Maxwell equations, is affcctcd by these refractive properties of curved spacetime. In flat Minkowski spacc in the frame chosen, N = c, and hence, E, = p, = 1.
+
Figure E.1.1: The vector field n adapted t,o the (1 3)-foliation on a manifold with metric: The vector n is orthogonal to the hypersurface lr, (compare wit,li Fig. B.1.3). the vector field n is timelike, and thus we can indeed consider a as a local tin)(. coordinate. The condition (E.1.12) guarantees that the folia h, of constant a arc orthogonal to n (see Fig. E . l . l ) . Thus they are really 3-dimensional spactllikc hypcrsurfaces. The metric
E.1.4 Electromagnetic energy and momentum If the spacetime metric g is given, then there exists a unique torsion-free and metric-compatible Levi-Civita connection r n P , see (C.2.103), (C.2.136). Consider the conservation law (B.5.52) of the energy-momentum. In a Riemannian = 5E J J fi commutes with the Hodge space, thc covariant Lie derivative operator, Ec* = *LC.Thus (B.5.53) straightforwardly yields
ec
-
+
Therefore in general relativity (GR), with tlic Maxwell-Lorcntz spacetime relation, (B.5.52) simply reduces to is evidently a positive definite Riemannian metric on h,. We denote by 5 tlir Hodgc star operator defined in terms of the 3-dimensional metric (E.1.14). Applying the general definitions (C.2.85) to our foliation compatible cofrnmc. (B.1.33), we find relations between the 4-dimensional and the 3-dimensionti1 star operators:
Thc cnergy-momentum current (B.5.7) now reads
X
k ~ , = o [ ~ ~ ( e , ~ * ~ ) - ( c , , ~ ~ ) (E.1.22) ~ * ~ ] . 2
In t,llc absence of sources (J = 0), we find the energy-momentum law EkC, = 0.
-
(P)
Here 2 is an arbitrary transversal (i.e., purely spatia,l) pform. Note that "* = 1 for all forms. We substitute the (1 3) decompositions (B.1.39) and (B.2.9) into (D.G.l3),
+
-
(E.1.23)
In the flat Minkowski spacetime of SR, we can globally choose coordinates in 0. such a way that I',@ f 0. Thus D 2 d and d k C, As we already know from (B.5.20), the currcnt (E.1.22) is trace1e.s.s (79" A kC, = 0). Moreover, we can now use the metric and also prove its symmetry. We multiply (E.1.22) by 1.9~ := goy t97 and antisymmetrize:
E.1. Standard Maxwell-Lorentz theory in vacuum
E.1.4 Electromagnetic energy and momentum
321
Because of ('2.2.133) and (C.2.131), the first term on the right-hand side can be rewritten,
It is a reflection of the symmetry of kT,p that the energy flux density 2-form s of (B.5.62) and the momentum density 3-form p, of (B.5.63) are closely interrelated. Indeed, using the spacetime relation (E.1.18) in the definition (B.5.63), we obtain:
i.e., it is compensated by the third term. We apply the analogous technique to the second term. Because **F= -F, we have
Here we used the identities (C.2.133), (C.2.131) and (E.1.2). Taking into account (E.1.19) and comparing with (B.5.G2), we finally find
In other words, the second tcrm is compensated by the fourth one and we have
Alternatively, we can work with the energy-momentum tensor. We decompose , respect to the 77-basis. This is now possible since a metric the 3-form k ~ with is available. Because 29" r\ % = 6: 77, we find
see (B.5.37),(B.5.38).Thus,
Its tracelessness T YY = 0 has already been established in equation (B.5.20). Its symmetry
-
can be either read off from (E.1.28) and (E.1.27) or directly from (B.5.49) with F2J,A manifestly symmetric version of the energy-momentum tensor can be derived from (E.1.28)2 and (E.1.22):
Thus kT,p is a traceless symmetric tensor(-valued 0-form) with nine independent components. Its symmetry is sometimes called a bastard s y m m e t y since it interrelates two indices of totally different origin, as can be seen from (E.1.27). Without using a metric, it cannot even be formulated, see (B.5.38).
In a Minkowski space, we have N = c. This is the electromagnetic version of the relativistic formula m = $ E in a field-theoretical disguise. The energy density (B.5.61) becomes explicitly positive when we substitute the spacetime relation (E.1.18) in it:
Electromagnetic spacetime relations beyond locality and linearity
Here we outline th,e structure of electrod?jn.amics that arises when we keep the first four o.n:ioms but generalize the fifth, one. Particl~larezamples of spacetime re1n.tion.s are described that are either n.onlocal or nonlinear. I n these m,odels tlie eristence of a metric of spacetim.e is presupposed.
E.2.1 I
324
E.2. Electromagnetic spacctime relations beyond locality a n d linearity
dynamics is untouched; that is, the first four axioms stand firm. If we turn the argument around: The limits of the Maxwell-Lorentz spacetime relation become visible. The fifth axiom is built on shakier grounds than the first four axioms. Obviously, besides nonlinearity in the spacetime relation, the nonlocality can be and has been explored. This is further away from present-day experiments but it may be unavoidable in the end. We turn first to a discr~ssionof nonlocality.
Volterra technique. It turned out, however, that the integral in (E.2.3) diverges in the convolution case under certain physically reasonable conditions. Preserving the main ideas of Mashhoon's approach, one can abandon the convolution condition and assume that, apart from the local term proportional ') only on the to the delta function 6 ( r - TI), the kernel K , ~ ~ ~ ( T , Tdepends value of TI. This is the case of "kinematic memory". Let u be the observer's 4-velocity and I',P the linear connection 1-form of the underlying spacetime. Then the general form of the kernel can be worked out e ~ ~ l i c i t l y : ~ I<,~'*(T,
A spacetime has dispersion properties when the electromagnetic fields producr noninstantaneous polarization and magnetization effects. The most general linear spacetime relation is then given, in the comoving system, by means of tlir Volterra integral
The coefficients of the kernel I
The spacetime relation is then replaced by1
The response kernel in (E.2.3) is now defined by the acceleration and rotation of the observer's reference system. It is a constitutive law for the vacuum as viewed from a non-inertia1 frame of reference. Mashhoon originally imposed the additional assumption that the kernel is ( T7'). This is called "clyof a convolution type, i.e., I < , ~ Y ~ ( T , T=' ) I < , ~ Y ~ namic memory" .2 Then the kernel can be uniquely determined by means of tlic 'see Mashhoon [22, 231. 'The notions "dynamic" and "kinematic" menlory were introduced and extensively discussed by Chicone and Mashhoon 18, 91.
325
E.2.3 Heisenberg-Euler
1
TI)
=2 e,ppI6 (621 6 ( r - r') -
uJ I',,'~(T'))
.
(E.2.4)
The influence of noninertiality is manifest in the presence of the connection 1form. Clearly a metric is here assumed beforehand. The kernel (E.2.4) coincides with the original Mashhoon kernel in the case of constant acceleration and rotation, but in general the two kernels are different. The kernel (E.2.4) yields convergent integrals and is thus a reasonable choice. Only direct observations could establish the true form of the nonlocal spacetime relation. However, such nonlocal effects have not been confirmed experimentally as yet.
E.2.3 Heisenberg-Euler Quantum electrodynamical vacuum corrections to the Maxwell-Lorentz theory can be accounted for by an effective nonlinear spacetime relation derived by Heisenberg and Euler [15]. To first order in the fine structure constant af= e2/4n~otic x 11137, it is given by4
where the magnetic field stren g th 5 Bk = m 2 c 2 / e hx 4.4 x lo9 T , with the mass of the electron m. Again, post-Riemannian structures don't interfere here. This theory is a valid physical theory." According to (E.2.5), the vacuum is treated as a specific type of medium the polarizability and magnetizability properties of which are determined by 3See Hehl and Obukhov [14]and Muench et al. [25]. 4See, for instance, Itzykson and Zuber (181 or Heyl and Hernquist 116). S ~ e us t check t h e dimensional consistency of (E.2.5). We abbreviate the dimensions of length and magnetic flux by e and +, respectively. If we assume that the line element ds2 has dimension [ds 2 ] = e2, then the metric volume element 1) has [v] = e4. Accordingly, since '1) = 1, [*I4 = 1/e4. As we indicated with the subscript 4, the latter formula is oly valid if t h a t is, [*Iz = 1. Hence in (E.2.5), we applied to a 4-form. For a 2-form, we have [F] = [*F], find [* (F A *F)]= [F][ * ~ ] / = e ~(+/ez)' 2 (wb/m2)' = T'. 6Recently, Rrodin et al. [7] proposed a possible direct measurement of the Heisenberg-Euler effect.
326
E.2. Electromagnetic spacetime relations beyond locality a n d linearity
Both (E.2.6) and (E.2.5) are special cases of Plebafiski's more general nonlinear electrodynamics [27]. Let the quadratic invariants of the electromagnetic field strength be denoted by Z ~ : =2~ * ( F A * F ) and
Z 2 : =2~ * ( ~ h F ' ) ,
(E.2.7)
where Z1 is an untwisted and Z2 a twisted scalar (the Hodge operator is twisted). The relations to our 4-forms of Part B and their (1 + 3)-decompositions, see (B.2.28) and (B.2.29), are given by
Figure E.2.1: Spherically symmetric electric field of a point charge in the BornInfeld (solid line) and in Maxwell-Lorentz theory (dashed line). On the axes we have dimensionless variables n: := r/ro and y := E/E,. the "clouds" of virtual charges surrounding the real currents and charges. As already explained, the first four axioms remain untouched by (E.2.5).
The nonlinear Born-Infclcl theory represents a classical generalization of the Maxwcll-Lorentz theory for accommodating stable solutions for the description of "electrons". Its spacetirne relation reads7
Because of the nonlinearity, the field of a point charge, for example, turns out to be finite a t r = 0, in contrast to the well-known l / r 2 singularity of the Co~ilombfield in Maxwell-Lorentz electrodynamics (see Fig.E.2.1). The dimensional parameter f, := E,/c is defined by the so-called maximal field strength achieved in the Coulomb configuration of an electron:' E, := e/4n~or;, where r 0 := Jr,, with the classical electron radius re := e2/4n.comc2 = crrh/mc and a nu~nericalconstant J = 1. Explicitly, we have E, z 1.8 x loz0 V/m anti f,= E,/c = 6.4 x 10" T . The spacetime relation (E.2.6)-like (E.2.5)-leads to a nonlinear equation for the dynamical evolution of the field strength F. As a consequence, thc characteristic surface, the light cone, depends on the field strength, and the superposition principle for the electromagnetic field doesn't hold any longer. 'See Born and Infeld [4], Sommerfeld [33], and Gibbons and Rasheed [ll]. RIn quantum string theory, thc Born-Infeld spacetime relation arises as a n effective moclcl with f, = l/2ncr', where a' is the inverse string tension constant.
Then Plebafiski postulated a nonlinear electrodynamics with the spacetirne relation"
where U and V are functions of the two invariants. Note that in the Born-Infelcl case U and V depend on both invariants whereas in the Heisenberg-Euler case we have UHE(Z1)and VHF,(Z2).Nevertheless, in both cases U is rcquircd as well as V. And in both cases, see (E.2.6) and (E.2.5), U is an even function and V and an odd one so as to preserve parity invariance. If one chooses V(ZI,1 2 ) to be an even function, then parity violating terms would emerge, a case that is not visible in experiment.
%trictly, Plebariski assumed a Lagrangian that yields the Maxwell equations together with * H v(Z1,T2) H.T h e latter law, apnrt from singular the stmctuml relations F = u(Z1,Z2) cases, is equivalent t o (E.2.10).
+
Electrodynamics in makter, constitutive law
When considerin.g electrom.agnetic ph,en.om.ena inside a material m,edium with nontrivial electric and magnetic properties, we have to distin g uish between bound an,d free charqes and the corresponding currents. Tlzis leads to the sixth axiom, which describes the splitting of the electromagnetic current and introduces the polarization and magnetization forms. The Maxwell equations are th,en form,ulaterl i n tcrm,s of the external excitations (~uhichcorrespond to the fme clrarqes and currents). The electric and magnetic susceptibi1itie.s of matter are defined by linear constitutive relations. The fundamental physical consequence of tlre c7rri.en,t-splitti7lg axiom is the correspon,ding split of the energy-momentum current, which tumz.s out to be the sum of free and bound (material) enerqymomenta,. In order to illustrate the physical contents of the latter, 7ue study the expmim,ents of Walker t 4 Walker a,nd .lames ~ u h omeasured a pon.deromotive force that ciffects a matter with not~iuialelectric and magnetic properties placed i n a,r1 en:ternnl electromagnetic field.
E.3.1 Splitting of the current: Sixth axiom this chaptcr wc prcsent a consistent microscopic approach to the electrodyliari~icsof continuous mcdia.' Besidcs the field strength F, the excitation H is
111
'In a grcnt nu~nberof texts on clcctrodynamics, the electric ancl magnetic properties of ~ n r d i nnrc described following the macroscapzc averagzng scheme of Lorentz [21]. I-Iowever, this formalism 11n.s a number of serious limitations; see the relevant criticism of Hirst 1171, for
E.3. Electrodynamics in matter, constitutive law
330
a microscopic field in its own right, as we havc shown in our axiomatic disciission in Part B. The total current density is the sum of the two contributions originating "from the inside" of the medium (bound or material charge) and "from the outside" (frce or external charge): J
=
Jmat + ~ e x t
(sixth axiom a).
(E.3.1)
Accordingly, the bound electric current inside matter is denoted by mat and the external current by ext. The same notational schemc is also applied to thc t excitation If,so we have HInRtand H P X . Bound charges and bound currents are inherent characteristics of mattcr determined by the medium itself. They only emerge inside the medium. In contrast, free charges and free currents in general appear outside and inside mattcr. They can bc prepared for a specific purpose by a suitable experimental arrangcment. We can, for instance, prepare a beam of charged particles, described by J PXt , and scatter thcm at the medium, or we could study the reaction of n medium in response to a prescribed configuration of charges ant1 currents, JPX' . We assume that the cllargc bound by matter fulfills thc usual charge conscrvation law separately: d
mat = 0
(sixth axiom b).
331
E.3.2 Maxwell's equations in matter
The minus sign is conventional. Thus, in analogy to the inhomogeneous Maxwell equations (B.1.44), (B.1.45), we find - d-p =
dM+p=jmat.
(E.3.6)
The identifications (E.3.5) are only true up t o an exact form. However, if we require Vmat = 0 for E = 0 and Rmat= 0 for B = 0, as we do in (E.3.56), uniqueness is guaranteed.
E.3.2 Maxwell's equations in matter The Maxwell equations (B.4.9),
are linear partial differential equations of first order. Therefore it is useful to define the external excitation
(E.3.2)
Wc call (E.3.1) and (E.3.2) the siztlt asxiom. It specifies thc propcrtics of t,hc classical material medium. In vicw of the relation d J = 0, resulting from tlic first axiom, the assumption (E.3.2) means that thcre is no physical exchange (or conversion) between the bound and free charges. The sixth axiom certainly docs not exhaust all possible typcs of material media, but it is valid for a sufficient,ly wide class of mcdia. Mathematically, (E.3.2) has the same form as d J = 0. As a consequence, wc can repcat the arguments of Sec. B.1.3 and find the corresponding cxcit a t 'ion HlnRt as a "potential" for thc bound current:
The external excitation lH = (fj,9)can be understood as an auxiliary quantity. We differentiate (E.3.8) and eliminate d H and dHmRtby (E.3.7) and (E.3.3), respectively. Then, using (E.3.1), the inhomogeneous Maxwell equation for matter finally reads
or in (1
+ 3)-decomposed form,
Ram (E.3.8) and the universal spacetime relation (E.1.18) we obtain The (1
+ 3)-decomposition, following t11c pattern of (B.1.39), yields
9
=
f i = -
*E + P ( E , B ) ,
(E.3.12)
'B - M ( B , E ) .
(E.3.13)
.E~.EO
1 PUP0
The conventional names for these ncwly introduced excitations are polarization 2-form P and magnetization 1-form M , i.e.,
example. An appropriate modern presentation of the microscopic approach to this subject hns been given in the textbook of Kovetz [20].
The polarization P ( E ,B) is a functional of the electromagnetic field strengths E and B. In general, it can depend also on the temperature T and possibly on other thermodynamic variables specifying the material continuum under consideration; similar remarks apply to the magnetization M ( B , E ) . The system (E.3.10), (E.3.11) looks similar to the Maxwell equations (E.3.7). However, the former equations refer only to the external fields and sources. We stress that the homogeneous Maxwell equation remains valid in its original form.
E.3. Electrodynamics in matter, constitutive law
332
E.3.3 Energy-momentum currents in matter
Here the "free-cltarge" energy-momentum 3-form of the electromagnetic field and the supplementary term are, respectively, given by
In a medium, the total electric current J is the sum (E.3.1) of the external or free charge J eXt and the material or bound charge Jmnt. When both of these currents are nontrivial, the (ponderomotive) force acting on the medium is the sum of the two Lorentz forces,
frt
Here the Lorentz force density = (e, J F ) A JeXt describes the influence of the electric and magnetic fields on the external (free) current J eXt and the Lorentz force density fCRt= (e, J F ) AJmnt specifies the action on the material (bound) charges. Recalling the (1 3)-analysis of Scc. B.2.2, we can write now the total longitudinal force as the sum f, = d a A (8, bk,) with
+
= -j m a t A
'E,
:=
1 [FA (e, 2
-
J
N ) - N A (e,
J
F ) ],
(E.3.20)
This energy-momentum describes the action of the electromagnetic field on the free charges (hence the notation where the superscript "f' stands for "free"). First, let us analyze the supplementary term. In (1 3)-decomposed form, we have F = E A da + B and N = -3 A do + 9. The Lie derivatives of these 2-forms can be easily computed,
+
+
' l oc ---jextAE, I-,/i
333
E.3.3 Energy-momentum currents in matter
' k , = p e x t A ( e , ~ E ) + j e x t A ( e , ~ B )(E.3.15) ,
El
"lc, = pmntA (e,
J
E)
+ jmatA (e, J B ) . (E.3.16)
Starting from the left equality in (E.3.14), we find that the energy-momentum current of the electromagnetic field in matter has the form (B.5.7). We may call (B.5.7) the total energy-momentum current in a material medium. Upon substitution of the Maxwell-Lorentz spacetime relation (D.6.13), it reads2 k ~ , = $ [ ~ A (e,
J
and similarly,
*F)- *FA (e,
J
F)]
+
Here fl,,, := de, J e, J d is the purely spatial Lie derivative. Substituting (E.3.22), (E.3.23) into (E.3.21), we find
(E.3.17)
(see (E.1.22)). Looking at the right equality in (E.3.14), we observe that this total energy-momentum current is the sum of the two separate contributions on the right-hand side:
The structure of the free-charge energy-momentum is revealed via the standard (1 3)-decomposition:
+
'C6 We can study these two contributions separately because there is, as we ,zssumctl in Sec. E.3.1, no physical mixing between the free and the bound currents; ill particular, they are conserved separately. The corresponding analysis gives rise to two energy-momentum currents that are specified below. In actual observatio~lsin media, we are inspecting how the electric and magnetic fields act on the external or free chayqes and currents. By using the inhomogeneous Maxwell equations in matter, namely (E.3.9) or (E.3.10), (E.3.11), and repeating the derivations of Sec. B.5.3, we obtain
'c,
'u-~UA'S, = -$I,-daAfs,.
=
(E.3.27)
Here, in complete analogy with (B.5.59), (B.5.60) and (B.5.61)-(B.5.64), we introduced the energy density 3-form
the energy flux density (or Poynting) 2-form
the momentum density 3-form 2'I'his energy-morncntum current was proposecl in [29] and [26].
(E.3.26)
ba := -
B A ( ~ ~ J ~ ) ,
E.3. Electrodynamics in matter, conslitt~tivclaw
334
and tlie stress (or momentum flux density) 2-form of the clectromagnetic ficld f
1 2
:= - [(e, J
~
,
E) A D - (e,_rD) A E + ( ~ , J A ) A B - ( e , ~B) A A ] . (E.3.31)
In tlic abscnce of free cliarges and currents, we have thc balance equations 'X, = 0. In tht. for the electromagnetic ficld energy and momentum d fC, (1 3)-decomposed form this reads, analogously to (B.5.71), (B.5.72):
+
+
Hcrc, in cornplctc analogy witli (B.5.59), (B.5.60) and (B.5.61)-(B.5.64), wc introducctl the bound-charge energy density 3-form
the bound-clrarqe enwrgy flux density 2-form
the bound-charge momen.t~imdensity 3-form Tlie Minkowsl
and thc bound-charge stress (or momentum flux density) 2-form of tlie electromagnetic field
Here wc i~itroducea new material energy-momentum 3-form of tlie electromagnetic field and the corresponding supplementary term:
The (1 3)-dcconil~oscdbalance equations for tlie bound-charge cnergy-momenturn are analogous to (B.5.71), (B.5.72) ancl (E.3.32), (E.3.33). Writing the purely longitudinal force 4-form as f:"' = du A "k,, we find:
+
I'c,
:=
1
-
2
[F A (e,?J H1lla" ) - HmatA (e,
J
F )] ,
(E.3.35)
Tlic material energy-moii~ent~iin describes thc action of the clcctrornagnctic ficltl on tlic bouncl charges (liencc tlie notation where superscript "b" stands for "bouncl"). I11 (1 3)-tlecomposcd form, we have Hnl'"'= - (M~ d uP) antl, as usual, F = E A do B. Thus tlic energy-momentum (E.3.35) is ultimately cxpressctl in terms of tlie polarization P ancl magnetization M forms (E.3.5) \Vlien there are 110 frcc charges and currents, thc energy-mornentumlneiti (B.5.7) reduces to (E.3.35), and tlic latter should bc used for tllc computation of thr forces acting on dielectric and magnetic matter. It is strniglitforward to perform, as in (E.3.22)-(E.3.25), a derivation of tlic supplcmcntary tcr1n. The result reads
+
+
+
"x, - - 'X, + X,.
(E.3.37)
Here X, is given by (B.5.8). This relation is valid for all constitutive laws and spacetime relations. Tlie (1 3)-decornl~ositionyields a stxucture sinlilar to (E.3.26), (E.3.27):
+
In absc~iccof the frcc cliarges, tlic integral of tlie 3-form of the force density (E.3.45) over thc 3-tlimensional tlomain R"'"', occupied by a material botly or a nirtlium, yicltls the t,otill 3-force ~ c t i n g011 the I~odyor the medium,
I<,
=
/
Ilk,.
(E.3.46)
n m n t
@Abraham versus Minkowski The tliscussion 011 the correct e~iergy-momentumcurrent in macroscopic electrotlynamics is quite o l d . ~ c v e r t h e l c s sup , to now this question wm not settled tlicor~tically,~ nntl not cveli hy cxpcriinent,~was it possible to make a definite ;inti dccisivc choicc for rlcctromitgnetic energy and momcntunl in material "I'hc beginning of this dispute gocs back to Abraham [ l ] who raised sor~icobjections agailist Minkowski's work [2/1];for a good rcvicw scc Rrevik [GI. 4 S (21, ~ for ~ exalnplc.
336
E.3. Electrodynamics in matter, constitutive law
media. Here we recall the definitions of the Minkowski and Abraham cnergymomenta and compare them with our energy-momentum current. The Minkowski energy-rnomcntum is defined by (E.3.20), (E.3.26)-(E.3.27). Its properties are describcd in this section above. The corresponcling force density reads:
When the free charges arc abscnt, the Minkowski forcc tlensity, in view of (E.3.33), reduces mcrely to Mk, = - (fXa)I. In Maxwell-Lorentz electrodynamics, the presence of the spacetime metric allows to discuss the symmetry of the energy-momentum tensor. The latter is defined by fC, =: fTapqp (see (E.1.28)). Let us study this question in (flat,) Minkowski spacetime for the case of a medium that is a t rest in the laboratory frame. Using the definition (E.3.20), we find for the off-diagonal components:
E.3.4 Linear constitutive law
337
Here the Abraham stress 2-form is introduced by
= h a A A s b . The corresponding One can straightforwardly verify that dxb~"Sa energy-momentum tensor, introduced by means of "C, =: "Tap qp, is explicitly symmetric:
The electromagnetic Abraham force density is then given by If the upper index is lowered by means of the spacetirnc metric, fTikgkjl we find indeed that
since 'Tohgba = qabcfjhEc,whereas fTaO goo = EP qabcfjhEc. The extra factor is the square of the refractive index n 2 = EP of matter. Moreover, the Minkowski stress (E.3.31) is also nonsymmetric: One can easily prove that &!,A 'Sa # ds, A 'Sb in general. However, for an isotropic medium with the constitutivc law (E.3.58) thc stress becomes symmetric. It is, nevcrthcless, intcrcsting to obscrvc that thc Minkowski cncrgy-momcntum tensor is symmetric provided we usc thc optical metric (to be specified ill (E.4.33) below) for the lowering of the upper index:
This can easily be verified by means of (E.4.35). The generalized symmetry (E.3.50) also holds true in an arbitrarily moving medium. This fact highlights thc fundamental position that the optical ~netricoccl~piesin tlie MaxwcllLorentz theory. Thus, it is evident why thc Minkowski energy-momcntu~~~ turns out to be thc most uscful one for tlie cliscussion of optical phcnoincna in material media. The symmetric Abraham energy-momentum was proposed a replaccnlent of the nonsymmetric Minkowski cncrgy-momentum. Contrary to a widely hcld view, thcrc does not exist a derivation of the Abraham current "C, from first principles. In contrast, the Minkowski current emerges naturally from thc Lorentz force axiom. The (1 +3)-decomposed dcfinition of thc Abraham energymomentum form reads:
Within the framework of our axiomatic approach, the Abraham energymomentum appears to be an unnatural object. As we established, the energymomentum current is well defined (both in vacuum and in matter) already in the pre-metric formulation of electrodynamics. The fact that for the introduction of the Abraham energy-momentum one necessarily needs the metric makes far less fundamental than the Minkowski energy-momentum current. Below we demonstrate that such an object also lacks substance from an experimental point of view.
E.3.4 Linear constitutive law "It should be needless to remark that while from the mathematical standpoint a constitutive equation is a postulate or a definition, the first guide is physical experience, perhaps fortified by experimental data. " C. Truesdell and R.A. Toupin (1960) In the simplest case of a homogeneous isotropic medium at rest with nontrivial polarizational/magnetizational properties, we have the linear constitutive laws
with the electric and magnetic susceptibilities ( x E , x R ) . If we introduce the material constants
338
E.3. Elcctrodynnmics in mnttcr, constitutive law
wc can rewrite the constitutive laws (E.3.56) as
In curved spacetime, the quantities ( E E ~and ) (pp,), in general, are functions of coordinates, but in flat Minkowski spacetime they are usually constant. However, E # p, contrary to tlie effective gravitational permeabilities (E.1.19). In the general case, their values are determined by the electric and magnetic polarizability of a material medium. A medium characterized by (E.3.58) is called simple. For a conductive medium, one usually adds one more constitutive relation that specifies tlie dependence of the free-charge current on the electric field, namely, Ohm's law:
E.3.5 @Experimentof Walker & Walker
339
where Vol is the 4-form of the spacetime volume. Thus fX, vanishes for homogeneous media with constant electric and magnetic permeabilities. Furthermore, within Maxwell-Lorcntz electrodynamics with a general linear constitutive relation, we recover (E.3.62) with the opposite sign for the supplementary bound charge force (E.3.37). The Abraham energy-momentum is also simplified, and its comparison with the Minkowski energy-momentum current becomes straightforward. For isotropic media with the constitutive law (E.3.58), the Abraham stress (E.3.53) coincides with that of Minkowski (E.3.31), *S, = fSa.As a result, the comparison of (E.3.47) and (E.3.55) yields the relation:
The expression Here a is the conductivity of the simple medium. Thc constitutive law (E.3.58) can be written in an alternative way by using the foliation projectors explicitly:
This form is particularly convenient for the discussion of the transition to vacuum. Then E = p = 1 and (E.3.60) reduces immediately to the universal spacetime relation (D.6.13). For anisotropic media the constitutive laws (E.3.58), (E.3.59) are further generalized by replacing E , p, a by the linear operators E , p , u acting on the spaces of transversal 2- and 1-forms. The easiest way to formulate these constitutive laws explicit,ly is to use the 3-vcct,or components of the electromagnetic ficltl strengths and cxcitations that were introduced earlier in (D.1.87), (D.1.88). In this description, thc linear operators are just 3 x 3 matrices E = E," and p = l~,!,. Onc can writc cxplicitly
In gcncral, the matrices E " ~and pnb clepend on the spacetime coordinates. Thc contribution of thc gravitational field is included in the metric-dependent factors N and fi= fi. In Minkowski spacetime, we have N = c, f i = 1. Aftcr spccifying the material constitutive relation, we can make the structurc of thc cnergy-momentum simpler and more transparent. In particular, in Minkowski spacetime for matter with the general linear constitutive law (E.3.61), we find for the supplementary force term (E.3.21):
is usually called the Abraham term. Substituting (E.3.58) into the definition of thc Minkowski momentum density (E.3.30) ancl using thc identities (C.2.133) and (C.2.131), we find:
Accordingly, the Abraham term (E.3.64) reads: Ep - 1 f EP-1 &, A (E A 9 i=Ak, := -& , ~ c2 c2
+ E A A) .
(~.3.66)
E.3.5 @Experimentof Walker & Walker Let us consider an explicit example that shows how the energy-momcntum current (E.3.18), (E.3.20) and (E.3.35) works. For concreteness, we analyze the experiment of Walker & Walker who measured the force acting on a dielectric disk placed in a vertical magnetic field B, as shown in Fig. E.3.1. Tlic time-dependent magnetic field was synchronized with the alternating voltage applied to the inner and outer cylindrical surfaces of the disk at radius pl and p2, respectively, thus creating the electric field along the radial direction. The experiment revealed the torque around the vertical z-axis. We derive this torque by using the bound-charge energy-momentum current. We have the Minkowski spacetime geometry. In cylindrical coordinates (p, cp, z), the torque density around the z-axis is given by the product pbk,. Hence the total torque is the integral
Nz =
/ disk
pbk9.
(E.3.67)
-
E.3. Electrodynamics in matter, constitutive law
340
torque
t"
\
E.3.5 @Experimentof Walker & Walker
One can vcrify hy substit~ltionthat the clectromagnctic field (E.3.68)-(E.3.72) rcprescnts a11 exact solut,ion of the Maxwell cquatiolls plus the constitutive relations (E.3.58). In the actual Walker & Walker experiment" the disk, made of barium titanate with E = 3340, has 1 x 2 cin height and tlic internal and external radius pl = 0.4 cin and pa x 2.6 cm, respectively. The oscillation frequency is rather low, namely w = 60 Hz. Correspondingly, one can verify that everywhcre in the disk wc have. pwn zwn (E.3.73) lo-7 << 1.
--- C
C
Thcn tlic field strciigtlis read approximately
Using tlie constitutive relatioils (E.3.56), we find thc polarization 2-form Figure E.3.1: Experiment of Walker & Walker. A dielectric disk, with inner ratlilis pl and outer radius p2, is placed in a vertical magnetic field B between the poles of an electromagnet. Assuming the harmonically oscillating electric and magnetic fields, we find for the cxcitations inside the disk
P=
11
=
cos(wt) ---Po
&&
[al
~ ~ (- 7(-rn) ) dz-
P
sin
pcip
I
Hcre Xo = and TL := fi is the refractive index of the medium. The disk consists of nonmagiietic dielectric material with p = 1. Here w is the oscillation frcquency and J o , J1arc Bessel functions. Tlic two integration constants
determine, respectively, the magnitude Bo of the oscillating magnetic field and tlie amplitude Uo of the voltage applied between the inner (p = pl) and thr outer (p = p2) cylindrical surfaces of the disk, AU = Uo sin(wt). The field strength forms look similar:
I
ca2 a1 w pdz A dp + -d q A dz .
n
The 1-form of magnetization is vanishing, M = 0, since p = 1. For the com~>ut,:~tion of the torque around the vertical axis, we need only the azimuthal componeiit,~of tlie molne~ltuln(E.3.42) and of the strcss form (E.3.43). Note that e, = 4. A simple calculation yiclds:
, R
sill(wt)
=
~ 0 x 1 sin(wt) 2 COS(W~)-"la2
dp
pdp A clz,
(E.3.77)
nP
As a rcsult of t,hc cylindrical syinmctry, tlir permittivity and pcrrncability do not tlrl>c~~d on the ang~rlarcoordinat,e and, accordingly, (E.3.62) yields " X , = - fX, = O. St~l,stitutingall this into (E.3.45) and subsequently computing thc intcgral (E.3.67), wc find the torqur
We :~lrcntlysubstituted the values of the integration constants (E.3.70);1 is the hright of t,hc disk, whereas pl and p2 are its inner and outcr radius, as shown on Fig. E.3.1. Formula (E.3.79) has been verified cxpcrimcntally by Walker & Wallccr. It is rather curious that this fact was consiclcred an argument in favor of tlie Abraham energy-momentum tensor"11at was introduced in tlie previous section. A ccrtaili formal coincidcncc is taking place, indeed, in the following sense: 5Sec Walker and Walker [XI. %ee, c.g., [3G] and (21.
342
E.3. Electrodynamics in matter, ~onst~itutive law
E.3.6 @Experimentof James
Recall that our starting point for deriving the bound-charge energy-momentum was the Lorentz force (E.3.34). Quite generally, for the %force density (E.3.34), we have from (E.3.3)-(E.3.6):
In the Walker & Walker experiment, we have M = 0. By differentiating (E.3.7G), one can prove that dP = 0. Thus the last line in (E.3.80) vanishes. Accordingly, the force density reduces to
which resembles the Abraham term (E.3.66). Howcver, our derivation was not based 011 the Abraham energy-n~omentum,and moreover, thc argument of thc symmetry of the energy-momentum tensor is absolutely irrelevant. As one can see, our bound-charge energy-momentum is manifestly asymmetric since t l i ~ energy flux (E.3.41) is plainly zero whereas the momentum (E.3.42) is nonvanishing.
Figure E.3.2: Experiment of James. A ferrite disk is placed in an azimuthal magnetic field B created by an electric current flowing along the z-axis. A voltage, applied between the inner and outer cylindrical surfaces, creates the radial electric field E. Here n 2 = EP and
p o ~ 2= ~ / E o .Correspondingly,
the field strength forms are:
E.3.6 @Experimentof James As another cxamplc we considcr thc experiment of ,James7 which is in many respects very similar to the one of Walker & Walker. James also placcd a disk into crossetl electric and magnetic fields (see Fig. E.3.2). The small cylinders were made of a composition of nickel-zinc ferrite with /I = 16 or 43 and E % 7. Like in the experiment of the Walkers, the radial electric ficld was created by means of an oscillating voltagc AU = Uo sin(w,t) applied between the inner and the outer cylindrical surface of the disk. However, instead of an axial magnetic magnetic field was produced inside matter by an alternating field, an azim~~thal I = I. sin(w,t) electric current in a conducting wire placed along the axis of thc disk. The frequencies w, and w,, are different (in the Walker & Walker experiment, electric and magnetic fields were oscillating with the same frequency). The resulting configuration for the excitations inside the disk is
he results of his observations are only described in his thesis One can find, though, a somewhat detailccl discussion in [ 5 ] .
and his short
paper [19].
Like in (E.3.74)-(E.3.75), we limit ourselves again to an approximation: The forinulas (E.3.82)-(E.3.85) hold true for zw,n zwin <( 1 (E.3.86)
--C
C
that is fulfilled everywhere inside the disk. The length (height) 1 of the disk, as well as the inner and outer radii pl and p2, are of of order 1-3 cm, whereas the frcquencies wi and w,, were varied in the course of the experiment between 10 and 30 kHz. We can verify that (E.3.82)-(E.3.85) represent a solution of the Maxwell equations. The integration constants are determined by the magnitudes of the electric current loand the voltage Uo:
James measured the force acting along the z-axis of the disk in the crossed electric and magnetic fields (E.3.82)-(E.3.85). Let us derive the theoretical value of this force by using the general expression for the force density (E.3.14). There are no free charges and currents inside matter, pext = 0 and jext= 0. As a result,
the corresponding part (E.3.15) vanishes and we are left with a contribution of the bound chalge and current (E.3.16). Using tlie constitutive relations (E.3.56), we find the polarization 2-form and tlie magnetizat.on 1-form:
+(
- 1) w,, sin(wit) cos(w,.t)]
.
(E.3.94)
Lrt us now evaluate the Al,rnllam terln. Substituting (E.3.84) and (E.3.83) int,o (E.3.29), we fintl the energy flux density 2-form:
M
=
" [ -d p
Po
-
- cos(w,t) zw, an :
+ as sin(wit)] .
Bound charge knd current densities are obtained by exterior differentiation accorcling t o (E.3 6). Since dP 0, we find vanishing charge = 0. The bound current is nontrivial:
--
1
I
Differentiating this ant1 using (E.3.66), we obtain the additional contribution to the total forcc: Inserting this t ~ g e t h e rwith (E.3.85) into (E.3.16), we find the force density:
'kL
1
=
jl'lLt
A (e, J
1
B) = (EP - 1 )- dp ~ A~d p A dz P
[
This theoretical prediction was actually verified in the experiment of James. For comparixon, let us derive the alternative theoretical predictions based on the Minkowski and Abraham energy-momentum currents. In the absence of free charges in matter, the Minkowski force density (E.3.47) reduces t o Mk, = - ( f X , ) ~ Sincc . the permittivity has the constant valuc of E # 1 inside the body, i.e., for -112 < z < 112, and drops to E = 1 outside of that interval, the derivative of such a s;epwisc function reads a , ~ ( z )= (E- 1) [6(z 112) - 6(z - 1/2)]. A similar relation holds for tlic derivative of the permeability function: 8,p(r) = ( j -~ 1) [6(z 1/2) - 6 ( r - 1/2)]. Correspondingly, froni (E.3.62) we find for the Minkowski forcc
+
+
-
I ( ' X , ) ~= disk
-:I disk
~ k =,
1 -&AG
/LE - 1
c2
disk
cos2(w,,t) z
We substituted (E.3.87) here. According to James, we choose wi = w,, f wo witli wo the mechanical resonance frequency of the disk. Then, specializing t o the component that varies witli the frequency wo, we find tlie final expression for tlie force
=
/ disk
- w:a;
The total force is computed as the integral (E.3.46) over the disk:
M~<,
Ax, =
[ ~ ~ E ~ m E 8 , ~ + p ~ f i ~ ~ f i 8 , p ]
In Jamcs' experiment, we put w, = w,, f wo ant1 sclcct only the component varying with the mechanical resonance frequency wo of the body. Then (E.3.G3), (E.3.94), and (E.3.96) yicltl the Minkowski and t,hc Abraham forces, rcspcct,ivrly:
All t,llc>tl~eo~eticnl c.xprcsslons fol the elcc.tromiiglict,ic force look simili~r:colilpare (E.3.93) wit11 (E.3.97) and (E.3.98). Ilowcvcr, the c~ucialdiffcrcl~ccis rc.vealctl w l ~ r nwe take into account that Jamc.s ~ n c a s ~ ~ not r e d t,lir force itself but a "rcducc~tlforcc" drfinetl as t,he liican value
With liifih accuracy, Ja~ilesobscrvc(1 the vnnzsll7nq of the rcducctl force in his cxpcrilncllt, I<;"" = 0. This ohservation is in colnplrte agreement with thc tlieorctical dcrivat,ioli (E.3.93) 1,nscd on our encygy-moment~i~n cllrrcnt (E.3.17), whereas l~ot,li,tlic expressions of Minkowski (E.3.97) and of Abraham (E.3.98)) clearly contradict t,liis experiment.
Electrodynamics of moving continua
The adapted spacetime foliation, which is associated with the metric structure, determines what can be considered as a laboratory frame of reference i n spacetime. When a material medium moves with respect to such a laboratory foliation, 3)its 4-vector field of the average velocity of matter induces another (1 splitting of the spacetime that we call the material foliation. W e establish the relation between th,e components of the electromagnetic field with respect to the two foliation,s and derive the Minkowski constitutive law for arbitrarily moving matter. The m.otion. of the material medium results, i n particular, i n the "dragging of light" pli,en,om.enon that tells us the electromagnetic waves i n a moving medium propagate along the light cone of the so-called optical metric rather than along the cone of the original spacetime metric. Furthermore, we determine the generation of an electromagnetic field inside a moving medium. W e discuss the two corresponding experiments of Rontgen and Wilson & Wilson who discovered this important physical effect. Finally, we discuss Maxwell-Lorentz electrodynamics as seen by a noninertial observer.
+
E.4.1 Laboratory and material foliation The elcctric and magnetic parts of current, excitation, and field strength are only determined with respect to a certain foliation of spacetime. In Sec. B.1.4 wc assumed the existence of a foliation specified by a formal "time" parameter a and a vector field n. We know how to project all the physical and geometrical objects into transversal and longitudinal parts by using the coordinate-free (1
+
348
E.4. I
3)-decomposition technique of Sec. B.1.4. This original spacctimc foliation is calletl tlic laboratory foliation. Moving macroscopic matter, by means of its own velocity, defines another (1+3)-splitting of spacetime that is different from the original foliation tiiscussetl above. Here we describe this material folzation and its interrelationship with t,hc laboratory foliation. Let us denote a 3-dimensional matter-filled domain by V. Mathematically, we start with a 3-dimensional arithmetic space R 3 equipped with the coordinat,cs Ea, whcre a = 1,2,3. We consider a smooth mapping z(0): R3 + V E X4 into spacetime that defines a 3-dimensional domain (hypersurface) V representing the initial distribution of matter. The coordinates [a (known as the Lagrangc coordinates in continuum mechanics) serve as labels that denote the elcmcnts of the material medium. Given the initial configuration V of matter, wc parametrize the dylialnics of the medium by tlie coordinate 7, which is defined as the proper time mesurcd along an clement's world line from the original hyl>ersurfacc V. The resulting local coordinates (7,<") are 11sua11y called the normalized comoving coortlinates. The motion of mattcr is thus described by the functions r 2 ( r , < " ) ,ant1 we subsequently define the (mean) velocity 4-vector field by
The technique of the (1$3)-splitting is similar to that described in Sec. B.1.4 for the laboratory foliation. Namely, following the pattern of (13.1.22) and (B.1.23), one defines the decompositions with respect to the material foliation: For any p-form 9 we denote the part longitudinal to the velocity vector u by - ( @ : = d r ~ 9 ~ , QF : = u J @ , and the part transversal to the velocity
@ : -
J
(
A
TL
) = (1 -
(E.4.4)
by
)
UJ
9 -0.
(E.4.5)
Please note that the projectors arc denoted now differently (-' and - ) in order to distinguish them from the corresponding projectors (Iand -) of the laboratory foliation. With the spacetime metric introduced on the X4 by means of the MaxwellLorentz spacetime relation, we assume that the laboratory foliation is consistent with the metric structure in the sense outlined in Sec. E.1.3. In particular, taking into account (E.1.12) and (E.1.13), one finds tlie line element with respect t o the laboratory foliation coframe: ds 2
=
N~ do 2 + gab clxn dzh = N~ da 2 - ("gab dza dzh.
(E.4.6)
Now it is straightforward to find the relation between the two foliations. Technically, by using (E.4.4) and (E.4.5), one just needs to (1 3)-decompose the basis 1-forms of the laboratory coframc (do, @) with respect to the material foliat,ion. Taking into account tliat, in local coortlinates, n = 8, rind, and similarly t i = ~ l ( ~ + ) aP ,I L ~ ~ tlic ~ , , result, in a convenient matrix form, reads:
+
By construct,ion, this vector field is timelike and it is normalized according to
Evidently, a family of observers conloving with lnattcr is characterized by tlic SR~IIC timelikc congruence r'(r,En). They arc making physical (in particular, clcctrodynamical) mcasurcmcnts in tlicir local reference frames tlrifting with tllc 111atclialmotion. By thc hy11othesz.s of localzty it is assunictl tliat tlic instruments in the comoving frame arc not affcctcd in an appreciable way by the local accclcration they experience. They measure the sarnc as if they were in a suitnbl(1 colnoving instantaneous znerflal frame. After these preliminaries, we are ready to find the relation betwren tlic two foliations. Technically, tlir crucial point is to express the labomtory cofranic. (du, &) in terms of the coframe adapted to thc matcnal foliation. Recall that according to our conventions formulated in Sec. B.1.4, c(rc" = d.ra - n" do is the. transversal projection of the spatial coframc. Tlic motion of a medium uniquely determines tlic (1 3)-decomposition of spacetime through a matenal folzatlon that is obtained by replacing n, da Ijy u, d r . Note that the proper time differential is d r = c - ~ u,dz'. Thus evidently
+
We consider an arbitrary lnotion of matter. The velocity field u is arbitrary, and one does not aSsulne tliat the laboratory and moving reference systenls arc related by a Lorentz transformation.
+
Here we introduced for the relative velocity 3-vector the not a t'lon
furthermore y :=
1
Jq'
(E.4.9)
Observe that (E.4.7) is not a Lorentz transformation since it relates two frames that arc both noninertial in general. As usual, the spatial indices arc raised and lowered by the 3-space metric of (E.1.14), ("gar, := -gab. Ill particular, we have explicitly v, = ( " ) g a b ~ h l l d u2 := u,vn. By means of the normalization (E.4.2), we can express the zeroth (time) cornponcnt of the velocity as u ( " ) = y(c/N). Hence the explicit form of the mattcr 4-velocity reads:
E.4. Electrodynamics of moving continua
350
E.4.2 Electromagnetic field in laboratory and material frames
Table E.4.1: Two foliations
vector field "time"
laboratory frame
material frame
n
u T
0 I
I
I
I
1q
-'9
time coframe
da
d7
3 0 coframc
a dx -
4 0 line clement
cis2 = N2du2
longitudinal
dx -
a
ds2 = c2dr2
Let us consider the case of a simple medium with l~oinogcneousand isotropic electric and magnetic properties. The constitutive law (E.3.60) for such a medium at rest with respect to the laboratoy frame has to be understood as a result of a 1abor;itory foliation. A moving medium is naturally a t rest with respect to its own material foliation. Conscquent,ly, tlic constitutive law for such a silnplc medium reads
How does tJie constitutivc law look as seen from the original labortitory frame? For this purpose we will use tlie results of Scc. D.5.4 and the relations between thc two foliations estal>lishcd in the previous scction. To begin with, recall of how tlie excitation ant1 the field strength 2-forms dccompose wit,ll respect to the the laborutoql frame,
ant1 a~~alogorisly, with rcspcct to the matem:al frame: Here (e6,e,) is tlie frame dual to the adapted laboratory coframe ( d a , b a ) , i.c., ec = n,ea = 8,. When the rclativc 3-vclocity is zero v" = 0, the material and the laboratory folintioiis coincide bccause the corresponding foliation I-forms turn out to be proportional to each otlicr v = (c/N) n. Substituting (E.4.7) into (E.4.G), we find for the line element in terms of the new variables
Clcarly, we preserve the same sylnbols If and d 011 tllc left-liantl sidcs of (E.4.14) ant1 (E.4.15) because tlicsc arc just the samc physical objects. In contrast, tlic right-hand sidcs arc of course tliffcrent, hence we usc prirncs. The constit,utivc law (E.4.13), ;iccortling to the rcs~iltsof the prcvior~ssection, can hc rewritten as
cab
Comparing this with (E.4.6), wc recognize that the transition from a laboratory frarne to a moving material frame cliangcs the form of the line element from (E.4.G) t,o (E.4.11). Consequently, this transitioli corresponds to a linear hoinogcrleous transformation tliat is aiiholonornic in general. It is not a Lorcntz transformation which, by definition, preserves the forin of the metric coefficients. The metric of the material foliation has the inverse
For its determinant one finds (dctFab)= (det gab) y-2.
Rcrc the Hotlgc star ;corrcsponds to tlic metric of the matenal foliation. (Plcasc do not mix it up with tlic Hotlgc star 5 clcfincd by the 3-space nictric ( ' ' ) g , , ~ lof tllc l i \ l ) ~ r ~ tfoliation.) ~~.y Now, (E.4.1G) can be prcsentctl in t l l ~ cquivalcnt matrix form
The coniponcnt,s of the constitutivc matrices read explicitly
1C.4.2 Elrctrornagnctic firltl in laboratory and material frames
In order to find the constitutive law in the laboratory frame, we have to perform some very straightforward mallipulations in matrix algebra along the lines described in Sec. D.5.4. Given is the linear transformation of the coframes (E.4.7). The corresponcling trnnsformation of t,hc 2-form basis (A.1.95) turns out to be
We use these results in (D.5.27)-(D.5.30). Then, after a lengthy matrix computation, we obtain from (E.4.18) the constitutive matrices in the laboratory foliation:
The resulting constitutive law
359
3-velocity that enter (E.4.27). Direct inspection shows that the constitutive law (E.4.25), (E.4.26) of above can alternatively be recast into the pair of equations:
These are the famous Minkowski relations for the electromagnetic field in a moving meclium.' Originally, the constitutive relations (E.4.28), (E.4.29) were derived by Minkowski with the help of the Lorentz transformations for the case of a flat spacetime and a uniformly moving medium. We stress, however, that the Lorentz group never entered the scene in our derivation above. This dcmonstratcs (contrary to the traditional view) that the role and the value of the Lorentz invariance in electrodynamics should not be overestimated. The constitutive law (E.4.25), (E.4.26) or, equivalently, (E.4.28), (E.4.29) describes a moving simple medium on an arbitranj curved background. The influence of the spacetime geometry is manifest in E ~pg, and in which enters the Hodge star operator. In flat Minkowski spacetime in Cartesian coordinates, these quantities reduce to Eg = pg = 1, ( 3 ) g , b = dab. The physical sources of the electric and magnetic excitations 9 and d are free charges and currents, Recalling the definitions (E.3.8) and (E.3.5), we can find the polarization P and the magnetization M that have bound charges and currents as their sources. A direct substitution of (E.4.25) and (E.4.26) into (E.3.8) yields:
can be presented in terms of exterior forms as:
Here we introdr~cedthe 3-velocity 1-form
The 3-velocity vector is deconlposed according to vn e,. If we lower the index v" by means of thc 3-metric (")CJ,~,, we find the covariant components v, of the
Here XE alld X B are the electric and magnetic susceptibilities (E.3.57). When the matter is a t rest, i.e., v = 0, the equations (E.4.30), (E.4.31) reduce t o the rest frame relations (E.3.56). 'See the discussions of various aspects of the electrodyna~nicsof moving media in [3, 6, 20, 30, 31, 351.
354
E.4. Elcctrodynarnics of moving continua.
E.4.4 Electromagnetic field generated in moving continua
E.4.3 Optical metric from the constitutivc law A direct check shows tliat the constitutive matrices (E.4.21)-(E.4.23) satisfy the closure relation (D.3.4), (D.4.13), (D.4.14). Consequently, a metric of Lorcntzian signature is induced by the constitutivc law (E.4.24). The geilcral reconstructioll of a metric from a linear constitutive law is given by (D.5.9). Starting from (E.4.18))we immecliatcly find the induced metric in the material foliation:
Making use of relation (E.4.7) between the foliations and the covariance properties proven in Sec. D.5.4, we find tlie explicit form of the iilduced metric in the laboratory foliation:
Here g,, are the components of the metric tensor of spacetime and uz are tllr covariant components of the 4-velocity of matter (E.4.10). Note that g" uzL, u, = c2, m ~islial.The contravariallt inducecl metric reads:
(2) vacuum
Figure E.4.1: Two regions divided by a surface S.
E.4.4 Electromagnetic field generated in moving continua Such an indr~ccdmctric' ; :JC is usually calletl the optical rnctmc in ordcr to distinguish it from the true spacetimc metric g T 3 .It describes the "drnggiilg of ) . adjective "optical" expresses tlic. the nether" ("Mitfiihrung tles ~ t l i e r s " ~The fact tliat a11 the optical effects in moving matter are determined by the Fresiirl equation (D.2.44), which reduces to the equation for the light cone detcrminctl by the mctric (E.4.33) in the present case. The nontrivial polarization/magnetimtion properties of matter arc manifcst,ly present cvcil when t,lle medium in the laboratory frame is at rest. Let 11s coilsider Minkowski spacctiinc with g,) = diag(c2,-1, -1, -I), for example, ant1 a lnediuin a t rest in it. Then u = dt or, in components, 11' = (1,0,0,O). WP substitute this into (E.4.33) ailcl find the optical metric
E~idently,the vclocit,y of light c is replacctl by c l n , with n as the refractive index of the dielectric and inagnetic media.
Let us collsiclcr an explicit example that demonstrates the power of the generally covariant const,itutivc law. For simplicity, we study electrodynamics in flat M~nl
2Sce Gordorl [I21 who first introduced the optical metric.
E(2)
=
B' dz 2 A dx" B2dz" d.rl + B3 dzl El d r l + E2dx2 + E3 dx3,
A dx2,
(E.4.36)
356
E.4. Electrodynixmics of moving continua
do not depend on t,.?.'.Then, the constitutivc law (E.4.24)-(E.4.26) yiclds that the components 9") anrl Di2) of the magnetic and electric excitations forms
are also constallt ill space and time. The spatial indices are raised and lowered by tlie spatial metric ( 3 ) g a b = Snh. These assumptions evidently guarantee that both the homogeneous and inhomogeneous Maxwell equations, d F = 0, dl?f = J, arc satisfied for tlic trivial sources J = 0 ( p = 0 and j = 0) in the second region. Let us now verify that the motion of matter generates nontrivial electric and magnetic fields in the first region. In order to find their configurations, it is necessary to use the constitutivc law (E.4.24)-(E.4.26) and the boundary conditions at the surface S . Recalling the jump conditions on thc separating surfacc (B.4.25), (B.4.26) and (B.4.27), (B.4.28), we find, in the absence of frce charges ant1 currents:
I':./L.5 'l'l~cc,xl)critr~c:nLsol' It611tgcniintl Wilson & Wilson
357
Here = - C B A , witli cii = 1 (and tlic same for cA'), Tlicsc equations should be taken on tlie boundary surface S . A simple but rather lengthy calculation yields the inverse relations:
UA'E
=
(h -pz)
UAD
&oER
The three cquations (E.4.43), (3.4.44) taken on S, together witli the three eqnations (3.4.39) specify all six components of tlie electric and magnetic field strengths E and B 011 the boundary S in terms of tlie constant values of the field strengths (E.4.3G) in the matter free region. Tlic standarc1 way of finding the static clcctromagnctic field in region 1 is as follows. Tlie [(I 3)-decomposed] homogeneous Maxwell cquations rlE(l) = 0, dB(1)= 0 arc solved by E ( l )= dp, B(l)= d d . Substituting this, by using the constitutivc law (E.4.25), (E.4.26), into tlie inhomogeneous Maxwell cquations dfi(l) = 0, dB(l) = 0, we obtain the four secontl-order differential cquations for tlie four independent components of the clcctromagnctic potent,ial p ( x ) , d ( r ) . Tlic r~niqucsolution of the resulting partial differential systcm is tletcrminctl by the boundary conditiolis (E.4.39) and (E.4.43), (E.4.44). I11 the gc.ncl.al cnsc, this is a highly nont,rivial problcm. Howevcr, tlicrc. arc two spc3cinl c:~sc.s of physical importnncc for whicli t,hr solutiori is straiglitforwartl. Tliry tlcscribc tlic expcrimcnts of Itontgen and the Wilsons with lnoving dielectric bodies.
+
Since tlic matter is confined to tlie first region, we conclude that tlie 3-velocity vector on thc bountlary surface S has only two tangential components:
Let us assume that the two tangential vectors are mutually orthogonal and have unit lcngth (which is always possible to acliicvc by the suitable choice of the variables EA parametrizing the boundary surface). Tlie solution of tlie Maxwell equations dl: = 0 and d H = 0 in the second region is uniquely defined by tlie continuity conditions (E.4.38), (E.4.39). Let us write tlicni down explicitly. Applying TA J t,o (E.4.25) and v A to (E.4.26), we find:
E.4.5 Thc experiments of Rontgen and Wilson & Wilson In both cases, we choose th[> I~ountlaryas a plane S = {x"
01, so t1i;lt thc
tangential vcctors ant1 thc normal 1-foim ore 71 =
al,
72 = 82;
u = d~3 .
We assume tliat the uppcr half-space (corresponding to "x matter moving witli t,hc horizontal velocity v = vl d r l
+ v2 d.r2.
(3.4.45) 0) is filled witli
(E.4.46)
E.4. Elcct,rodynamics of moving continua
E.4.5 Thc experiments of Rontgen and Wilson & Wilson
t x3
Figure E.4.2: Experiment of Rontgen.
Figure E.4.3: Experiment of Wilson & Wilson.
Let us consider the case when the magnetic field is absent in the matter-free region, whereas the electric field is clirectcd towards the boundary:
the nonrelativistic approximation (neglecting terms with v2/c2), the formulas (E.4.49), (E.4.50) describe a solution of the Maxwell equations provided dv = 0. This includes, in particular, the case of the slow uniform rotation of a small disk. The magnetic field generated along the radial direction can be detected by means of a magnetic needle, for example.
Then, from (E.4.37), we have
Wilson and Wilson experiment
Rontgen experiment
In the "dual" case, the electric field is absent in the matter-free region whereas a magnetic field is pointing along the boundary: It is straightforward to verify that, for uniform motion (with constant v), the for~ns
B(2) = B' dz2 A dz3
+ 'B
dz3 A dxl,
E(2)=
0.
(E.4.52)
Then, from (E.4.37), we find
A solution of the M
This situation is described ill the left part of Figure E.4.2: A magnetic field is gcneratctl along the x 1 axis by the motion of matter along the x 2 axis. In order to simplify the derivations, we have studied here the case of the uniform translational motion of mattcr. However, in the actual experiment of Rontgenqn 1888 lir observed this cffect for a rotating dielectric disk, as shown sche~naticallyin the right part of Figure E.4.2. One can immediately see that in
This situation is depicted in the left part of Figure E.4.3. There, without restricting generality, we have chosen the velocity along x 2 and the magnetic field B(2) along 2'. Then the electric field generated is directed along the x 3 axis. %ee Riintgen [32] and t h e Inter thorough
experimental study of Eichenwnld [lo].
E.4. Electrodynamics of moving contintla
360
E.4.G @Noninertial"rotating coordinates"
361
The elect,ric ancl magnetic excitations in matter arc obtained from the constitutive law (E.4.25), (E.4.26) which, for (E.4.54) and (E.4.55), yields
Like the experiment of Rontgen, the experiment of Wilson & Wilson4 was actually performed for rotating matter, not for the uniform translational motion described above. The true schcmc of the experiment is given on the right side of Figure E.4.3. In fact, thc rotating cylindcr is formally obtained from thc left figure by identifying x 1 = z, x 2 = 4, "x p with the standard cylindrical coordinates (polar angle 4, radius p). Usually, one should be careful if one uses curvilincar coordinates in which tlie components of the mctric arc nonconstant. However, the use of cxterior calculus makes all computations transparent and simple. Wc leave it as an exercise to the reader to verify that the Maxwc.11 equat,ions yicld the following cxact solution for the cylindrical configuration of the Wilsons cxpcrimcnt: $2)
-
From the definitions of these quantities in (E.3.5), we obtain, merely by taking the exterior differential, the charge and currcnt densities:
It is these cl~argeand current densities that generate the nontrivial electric and magnetic fields in the rotating cylindcr of thc experiment of the Wilsons. The bound current and charge density (E.4.65), (E.4.66) satisfy the relation
1
Po B dz,
E.4.6 @Noninertial"rotating coordinates"
Thc bountiary conditions (E.4.39), (E.4.43), (E.4.44) are satisfied for (E.4.57)(E.4.Gl). Note that now v = dp, 71 = a,, 7 2 = a&.Tlie velocity 1-form reads
with constant angular velocity w . Thc radial electric ficld (E.4.61) that is gencrated in thc rotating cylinder can be detected by measuring the voltage bctwccli the inner and the outer surfaces of the cylinclcr. One may wonder what physical source is behind the electric and magnetic ficlds that arr generated in moving matter. After all, we have ass~~nlctl tliijt there arc no frce charges and c ~ ~ r r e ninside ts rcgion 1. However, we havc b o ~ ~ n t l charges ant1 currents therein described by tlic polarization ant1 magnetization (E.4.30) and (E.4.31). Sr~bstitr~ting (E.4.60), (E.4.61) into (E.4.30), (E.4.31), we find:
How is thc Maxwell-Lorentz electrodynamics seen by a noninertial observer? We need ii procedure of two steps for the installation of such an observer. In this section the first step is done Ily introducing suitable noninertial coordinates. We assumc tlie absence of tlic gravitat,ional ficld. Then spacetime is Minkowskian and a global Cartesian coordinate system t, x a (with n = 1 , 2 , 3 ) can bc introducctl that spans tlic inertial (refcrencc) framc. Tlie line elcment of sl)acetimc rratls
As usual, tlie electromagnetic excitation and the field strength are given by
Assuming matter to be at, rest in the inertial framc, we have the constitutive
1aw
Equivalently, we hilve ho = ~ c t = b
-
"ee Wilson & W i l s o ~1371. ~
11
Jsand tlre constitrltivc matrices
- - S~'J c
Bob=
C
- Sab,
n
Cab = 0.
The corresponding optical mctric is given by (E.4.35).
(E.4.71)
E.4. Electrodynamics of moving continua
362
Now we want to introduce nonincrtial "rotating coordinates" (t', XI") by
t = t',
xn = Lbaxth1
E.4.7 Rotat.ing obscrvcr
3G3
We s ~ ~ b s t i t u(E.4.78) te into (D.5.28)-(D.5.30) and find the constitr~tivcmatrices in tlic rotating natural frame as
(E.4.72)
witli the 3 x 3 matrix Lbn= nn nb
+ (6;
- na nh)cos p
+ incb n sin p . C
(E.4.73)
The matrix defines a rotation of an angle p = p(t) around the direction specified 1. The Latin (spatial) by the constant unit vector f i = n a , with babnan" indiccs are raiscd and lowcrcd by means of the Euclidean metric dab and dab (iacb = dadiffcb,for example). We put "rotating coordinates" in quotes since it is strictly spcalting the natural frame (dt', dx'") attached to the coordinates (t',x'") that is rotating witli respcct to the Cartesian frame. The electroniagnctic 2-forms H and F are independent of coordinates. However, their components arc different in different coordinate systems. In exterior calculus it is easy to find thc components of forms: one only needs to substitute the original natural coframc (dt, dzn) by the transformed one. A straightforward calculation, using (E.4.73), yields
Substit,uting thcsc diff(.rcnt,ials into (E.4.68), wc fincl the line element in rotating coordinates: ds
=
vd
nbncirhd -, C
(E.4.82)
with n = &Z. Tliesc matrices satisfy the algebraic closure relation (D.4.13), (D.4.14). Thus tlicy define an inducccl spacetime metric. The Iattcr is obtained from (D.5.25) by using (E.4.35) and (E.4.74):
E.4.7 Rotating observer Howcvcr, the brhrivior of firltls with respcct to a rotating fra111e is usually of minor physical interest to us. Thc obscrver rathcr lilcasurcs all pliysical q ~ a n t ~ i t i c s witli respcct to a local framc 29" that is anholonornic in general, LC., d79" # 0. Thr observer is, in fact, comot~lngwith that frame and the components of excitation and ficld strength should be dctcrmined with respcct to 29". Consrquc.ntly, t,lie obscrvcr's 4-velocity vector reads
Here the angular velocity 3-vcctor is defined by
2
Clab =
The "Lorcnt,~"factor
+
~ ~ ( d f[I' ) ~( ~ ' . P / c-) ~ (w'. w'/c2)(?' . i')] -2dt'dn" . [3 x
n"] - d?' . d?'.
(E.4.76)
The electromagnetic excitation and the ficld strength arc, as usual, decomposed wit,h rcspcct to the rotating frame:
IH = -41A dt + 9', F
=
E'
A dt
+ B'.
(E.4.77)
Note that dt = dt'. The const,itutive matrices are derived from (E.4.71) by means of t,hc transformation (D.5.28)-(D.5.30). Give11 (E.4.74), we find the matriccs (A. I.%), (A.1.97) as
is dual t.o tlic corresponding cofriinic 19" with
Exprrssccl in terms of this coframc, the nlctric (E.4.76) rcacls
Hereafter we use the abbreviation
v' := [w'x ?'I.
is tlctcrminetl for the metric (E.4.76) by thc normalization condition g(eii,eG)= e ' i j ~ l ~ g=: c2. ~ Note that 8 = dabv" ub. The observer's rotating framc e,, witli (E.4.84) and
(E.4.79)
E.4. Electrodynamics of moving continua
364
Combining (E.4.74) with (E.4.87), we obtain the transformation from the inertial (dt1d.ra) coframc to the noninertial (d6, 29)' frame as follows:
E.4.8 Accclerating observer
365
Here a 2 := a' a' is the magnitude of acceleration, and the unit vector fi, with 13.5 = nbnb= 1, gives its direction in space. Recall that we are in the Minkowski spacetime (E.4.68). The accelerating coordinates (t',xra) can be obtained from the Cartesian ones (t, x a ) by means of the transformation
With respect to d " , the electromagnetic excitation and field strength read
Using (E.4.89) in the transformation formulas (D.5.28)-(D.5.30), the constitutive law in the frame of a rotating observer turns out to be defined by the constitutive matrices
t
=
xa
=
1
- sinh d, na x'" c
K ba xlb + cna
Here we introduce the 3 x 3 matrix
+
i
K b a = (6; - n a n b )
1
d r cosh d , ( ~ ) ,
d r sinh d,(r).
+ n a n b cosh d,.
(E.4.99) (E.4.100)
(E.4.101)
The scalar function q5(t1) determines the magnitude of the acceleration by dd, a ( t I ) = c-. dt' The matrices (E.4.91)-(E.4.93) satisfy the algebraic closure relation (D.4.13), (D.4.14). The corresponding optical metric is obtained from (D.5.25), (E.4.35), and (E.4.89) as
In exterior calculus, the constitutive law (E.4.91)-(E.4.93) in the rotating frame reads
Here *' denotes the Hodge operator with respect to the corresponding 3-space vn u b (see (E.4.88)), and we introduced the velocity 1-form metric Sob
Differentiating (E.4.99), (E.4.100), we obtain the transformation from the inertial coframe to the accelerating one:
Substituting (E.4.103) into (E.4.G8), we find the metric in accelerating coordinates:
This is one of the possible forms of the well-known Rindler spacetime. It is straightforward to construct the local frame of a noninertial observer that is comoving with the accelerating coordinate system. With respect t o the original Cartesian coordinates, it reads:
+$
1 eg = - u , C
na ec = - sinh d, 0, Kabdxb. C
+
(E.4.105)
Here
u = cosh d, dt+ cn a sinh d, dxo
E.4.8 Accelerating observer Let us now analyze the case of pure acceleration. It is quite similar to the pure rotation that wm considered in Scc. E.4.6. More concretely, we study the motion in a fixed spatial direction with an acceleration 3-vector parametrized as
(E.4.106)
is the observer's velocity 4-vector field, which satisfies g(u, u) = c 2 . Clearly, the vectors of the basis (E.4.105), in the sense of the Minkowski 4-metric (E.4.68), are mutually orthogonal and normalized: g(e6, eB)= 1,
g(eijlea) = 0,
g(ec, e&)= - dab.
(E.4.107)
E.4. Electrodynamics of moving continrla
366
367
E.4.9 T h e proper reference frame of t h e noninertial observer
Because of (E.4.103), the coordinate bases are related by
( i + a . y c 2~) d t l , dz'" + [(3 x ?'la dt'.
1 9 ~= 19'"
=
(~.4.115) (E.4.116)
The corresponding basis vectors of the (dual) frame are Thus, the accelerated frame (E.4.105), with respect to the accelerating coordinate system, is described by the simple expressions
e;; According to the definition of the covariant differentiation, see (C.1.15), on(. has V,e, = r,fl(u)ep. This enables us to compute the proper time derivatives for the frame (E.4.105), (E.4.109):
This means that the frame (E.4.105) is Fermi-Walker transported along thc observer's world line. The coframe dual to the frame (E.4.105), (E.4.109) reads, with respect to thc accelerating coordinate system,
29'
=
(1
+ a'. ?/c2) cdt',
79'
= dda.
(E.4.112)
E.4.9 The proper reference frame of the noninertial observer ( "noninertial frame") The line elelncnts (E.4.76) and (E.4.104) of spacc.time reprcscnt the Minkowsk~ space in rotating and accelerating coordinate systems, respectively. Both arc particular cases of the line element,
ns2
=
c2(dt1)2[(I
+ a . i 1 / c 2 ) 2 + ;( . P / c ) ~- (2. ; / C ~ ) ( T.~TI)] .
- ant1d k [(3 x
2'1- d.rl . d z t .
(E.~.II~)
1
e6 = =
c(l
+ a'
?'/c2)
(a,, - [d x ."I" a,l(.) ,
a,,,..
(E.4.118)
With respect to the local frame chosen, the components connection read:
rGb
=
ri;
(l3.4.117)
r,fl of the Levi-Civita
-
(a"/c 2 ) = 1 + a . .?,"/c2 8' = (ab/c) dt',
(E.4.119)
We can readily check that dl?,@ = 0 and that the exterior prodr~ctsof the connection 1-forms are zero. As_a result, the Riemannian curvature 2-form of the metric (E.4.113) vanishes, R,@ = 0. Thus, indeed, we are in a flat spacetime as seen by a noninertial observer moving with acceleration a' and angular velocity 3. After these geometrical preliminaries, we can address the problem of how a noninertial observer (accelerating and/or rotating) sees the electrodynamical effects in his proper reference frame (E.4.117), (E.4.118). In order to apply t,hc results of the previous sections, let us specialize either t o the case of pure rotation or of pure acceleration. To begin witli, we note that the constitutive relation has its usual form (E.4.70), (E.4.71) in the inertial Cartesian coordinate system (E.4.68). Putting a' = 0, we find from (E.4.115), (E.4.116) the proper coframe of a rotating obscrvcr: A
29' = dt',
1 9 ~= dz'"
+ [(3 X .?'la
dtl.
(E.4.121)
Combining this witli (E.4.74), we find the triinsforlnatioll of the inertial coframe to the noninertial (rotating) one,
Here the 3-vectors of acceleration 2 (= a h ) and of angular velocity (3 (= w ' ) can be arbitrary functions of time t'. The line clement (E.4.113) reduces to thc diagonal form ds2 = 88 8 196
,gy
@J
- 84 8 292
-
,93
@J
83
(E.4.114)
in terms of the noninertial orthonorma1 coframe:' 'See Ilchl & Ni [13] who tliscussed the Dirac equation in this accclrrating and rotatirlg cofrntne.
Correspondingly, substituting this into the transformation (D.5.28)-(D.5.30), wc immediat,ely find from (E.4.70) the constitutive law in the rotating observer's frame:
E.4. Electrodynamics of moving continua
368
The same result holtls true for an accelerating observer. If we put L3 = 0 in (E.4.115), (E.4.116), we arrive a t thc coframe (E.4.112). Combined with (E.4.103), this yiclds thc transformation of thc incrtial coframe to the noninertial (accelerating) onr:
(Za)=(
cos11dJ cn" sin11q5
W;;dJ)(;)
When we use this in (D.5.28)-(D.5.30), the final constitutive law again turns out to bc (E.4.123). Summing up, despite the fact that the proper coframe (E.4.115), (E.4.116) is noninertial, the constitl~tivcrclation rcmains in this coframe formally the same as in the inertial coordinate system.'
E.4.10 Universality of the Maxwell-Lorentz spacetime relation The usc of foliations and of exterior calculus for thc dcscription of the reference frames enablcs us to cstablish thc univcrsality of the Maxwell-Lorentz spncetime rclation. Let us put p = E = 1 (hencc 71 = 1) illto thc formulas above. Physically, this corresponds to a transformation from one framc (a-foliation) to another framc (7-foliation) that moves with an arbitrary velocity u relative to the first one. Then the relation (E.4.13) reduces to
On the other hancl, from (E.4.21)-(E.4.23), we find for the constitutive matrices:
Equivalently, from (E.4.24), wc read off that
or returning to exterior forms, 9 = EOER 'El
1
Fj = 'B. Po P g
'This shows that it is 111islendingto associate the "Cartesian form" of a constitutive relation with inertial frames of reference. I
E.4.10 Universality of the Maxwell-Lorentz spacetime relation
This is nothing but a (1 foliation:
+ 3)-decomposition
369
with respect to the labora.t,ory
Comparing (E.4.125) with (E.4.129), we arrive a t the conclusion that (E.4.125) and (E.4.129) are just different "projections" of the generally valid MaxwellLorentz spacetime relation IH = X o *F. Since here mntH= 0, we have
In this form, the Maxwell-Lorentz spacetime relation is valid always and evemjwhere. Neither the choice of coordinates nor the choice of a specific reference play any role. Consequently, our fifth axiom has a univcrsal frame (f~liat~ion) physical meaning.
[I] M. Abralinm, Z ~ r rElelctrodyn,amib bewegter I
171 G . Brotlin, M. Mnrkl~lntl,L. Stcnflo, Proposal for detection of QED I I ~ ~ I L ?io?1.li71.earities in Maxwell's equations b y tlle use of waveguirles, P11.ys. Rev. Lctt. 87 (2001) 171801 (3 pngcs). [8] C. Chicolic! ant1 B. Mnshhooli, Acceleration,-induced non.locality: Kinetic mernor~jversus rl?jnam,ic mernor?j, AIIII.Pltys. (Leipzig) 11 (2002) 309332.
I L ~
Rcfcrences
(91 C. Chicone and B. Mashhoon, Acceleration-in.du,cednonlocality: Uniqueness of the kern,el, Ph,ys. Lett. A298 (2002) 229-235. [lo] A. Eichenwaltl, ~ b e die r ma.gneti.schen Wirkungen bewegter Kiirper im elektrostatischen Felde, Ann. Pltys. (Leipig;) 11 (1903) 1-30; 421-441. [l11 G.W. Gibbons and D. A. Rashcecl, Magnetic duality rotation,^ in nonlinear electrodynamics, Nncl. Pllys. B454 (1995) 185-206.
[12] W. Gordon, Zur Lichtfortpflanzung n.ach der Relativitiitstheori.e, Ann. Phys. (Leipzig) 72 (1923) 421-456. [I31 F.W. Hehl and W.-T. Ni, In,ertial effects of a Dirac particle, Plys, Rev. D42 (1990) 2045-2048. [14] F.W. Hehl and Yt1.N. Obukhov, How does the ~lectromagnet~ic ficltl couple to gravity, in particular t,o metric, n~nmetricit~y, torsion, and curvature? In: Gyros, Clocks, Interferometers . . . : Testin,g Relativistic Gravity in Space. C. Lammerzahl ct al., eds. Lccturc Notes in Physics Vo1.562 (Springer: Berlin, 2001) pp. 479-504; see also Los Alamos Eprint Archive gr-qc/0001010.
[23] B. Mashhoon, Nonlocal electrodynamics, in: Cosmology and Gravitation, Proc. VII Brazilian School of Cosmology and Gravitation, Rio de Janeiro, August 1993, M.Novello, editor (Editions Frontihres: Gif-sur-Yvette, 1994) pp. 245-295. [24] H. Minkowski, Die Grundgleichungen fiir die elektromagnetischen Vorgange in bewegten Korpern, Nachr. Ges. Wiss. Got tingen (1908) 53-111. [25] U. Muench, F.W. Hehl, and B. Mashhoon, Acceleration induced nonlocal electrodynam,ics in Minkowski spacetime, Phys. Lett. A271 (2000) 8-15. [26] Yt1.N. Obukhov and F.W. Hehl, Electromagnetic energy-momentum and forces in matter, P11,ys. Lett. A311 (2003) 277-284. [27] J. Plebahski, Non-Linear Electrodynamics: A Study (Nordita: 1968). Our copy is undated and stems from the CINVESTAV Library, Mexico City (courtcsy A. Macias). [28] R..W. Pohl, Elektrizitatslehre, 21st ed, (Springer: Berlin, 1975). [29] P. Poincelot, Sur le tenseur klectrodynamique, C. R. Acad. Sci. Paris, Sdrie B 264 (1967) 1179-1181; P. Poincelot, Sur le tenseur d 'impulsion-knergie e'lectromagnetique, C. R. Acad. Sci. Paris, S6rie B 264 (1967) 1560-1562.
[I51 W. Hciscnberg and H. Euler, Folgem~n.gennus der Diracschen Tli,eom:e ties Positrons, Z. Phys. 98 (1936) 714-732.
[30] C.T. Ridgely, Applying relativistic electrodynamics to a rotating material medium,, Am. J. Pl1.y~.66 (1998) 114-121.
(161 J.S. Heyl and I,. Hernquist, Birefringence and diclrroism of the QED vacuum, J. P11.ys. A30 (1997) 6485-6492.
[31] C.T. Ridgely, Applying covariant versus contravariant electromagnetic tensors to rotating media, Am. J . Ph.ys. 67 (1999) 414-421.
(171 L.L. Hirst, The m.icroscopic m.agnetization: Concept a.nd application,, Rev. Mod. Phys. 69 (1997) 607-627.
(321 W.C. Rontgen, Ueber die durch Bewegung cines im homogenen electrischen Felde l~efindlich,enDielectricums hervorgemfene electrodynamiscl~eKraft, Ann. Phys. (Leipzig) 35 (1888) 264-270.
118) C. Itzykson and J.-B. Zubcr, Quantum Field Theory (McGraw Hill: New York, 1985). (191 R.P. James, Force on permeable matter in time-uar~jin,gfields, Ph.D. Tl~csis (Dcpt. of Electrical Engineering, Stanford Univ.: 1968); R.P. James, A "simple.qt case" experiment resolving the Abral~am-Minkow.k controvers?~ on electrom.agnetic momentum in matter, Proc. Nat. Acad. Sci. (USA) 61 (1968) 1149-1150. [20] A. Kovctz, Electromagnetic Theor?/ (Oxford Univrr~it~y Press: Oxfortl, 2000).
1
373
[21] H.A. Lorentz, The Theory of Electrons and its Appli~at~ions to the Phcnonlena of Light and Radiant Hcat. 2nd ed. (Teubncr: Lcipzig, 1916). (221 B. Mashhoon, Nonlocal theory of accelerated observers, Phys. Rev. A47 (1993) 4498-4501.
1331 A. Sommcrfcld, Elektrodynamik. Vorlesungen iibcr Theoretische Physik, Band 3 (Dieterich'sche Verlagsbuchhandlung: Wiesbadcn, 1948) P.285 et scq. English translation: A. Sommerfeld, Electrodynamics, Vol. 3 of Lectures in Theoretical Physics (Academic Press: New York, 1952). (341 C. Truesdell and R.A. Toupin, The classical field theories, In: Handbuch dcr Physik, Vol. 11111, S. Fliigge ed. (Springer: Berlin, 1960) pp. 226-793. [35] J . Van Bladel, Relativity and Engineering. Springer Series in Electrophysics Vo1.15 (Springer: Berlin, 1984). [36] G.B. Walker and G. Walker, Mechanical forces in a dielectric due to electromagnetic fields, Can. J. Phys. 55 (1977) 2121-2127. (371 M. Wilson and H.A. Wilson, Electric effect of rotating a magnetic insulator in o, magnetic field, Proc. Rosy. Soc. (London) A89 (1913) 99-106.
Elrctrodynamics dcscribcs only one out of four interactions in naturc. And classical electrodynamics covers only the nonquantum aspects of thc clcctromagnetic field. Therefore electrodynamics is related to the other fields of knowledge in physics in a multitude of different ways. In this tour d'horizon, wc look out in different directions for neighboring clisciplincs of classical electrodynamics and indicate how they are related to classical electrodynamics. We first explore thc classzcal domain, namely the rrl a t ion ' of electrodynamics to gravity and topology. Subsequently, wr turn to quasiclassical sources of the electromagnetic field in the context of sul~erconcl~lct ivity (Ginzburg-Landau theory) and first quantized Dirac theory. This allows us to construct the electric current 3-form in terms of a complcx-valuctl classical scalar and spinor field, respectively. The Maxwell equations remain untouched otherwise. The quantum domain represents new territory, which wc only namc: thc Quantum Hall Effect (Jain's composite fermioil picture fits niccly into our approach to electrodynamics), the quantum version of clcctrotlyliamics, that is, quantum electrodynamics (QED), and the unified electroweak Glashow-SalamWeinberg model. The latter unifies the electromagnetic with tllc weak intcraction.
How does gravity affcct clcctrodynamics?
111otlier words, tlie elcctrolnagllctic fielcl cntcrs tlic gravity scene explicitly via MAX
T h e only otIier Itnown interaction, bcsidcs the clcctromagnctic one, that can bc describcd by mc:ins of the classical field concept, is t,hc gravitational interaction. Einstcilils theory of gravity, general relativity ( G R . ) , ~clcscribes the ~ I ~the ~ ~ macrophysical ~ domain. In G R , spacetirno gravitational field S I ~ C C C S S in is a 4-dilnc:nsional Riclniillllian manifole1 with a, metric g of Lorcl$xian signatllre. T h e ~nctricis tlie gr;lvit,;l.t.ionalpotentikt1 T h e curvature 2-forni Rap, subsumin!? t o secontl dcriv;itivcs of the metric, represents tho tidill forces of t l i ~grnvitiltiorial ficld. Einstein's ficlcl equation, witli respect, t o an arbitrary cofi.:ime 19", reads"
T h e tilde labels R.icmallliian objccts, G is Newt,onls gravitational constant;, alltl A cosmological const,ant, which we ncglcct in our future consitlerations. T h e source 011 tllc riglit,-halit1 side is a s?,m.metri.cenergy-momcnt~lmcurrent of LLlllatt,el.,"
cl1ll)odying all nongravitational contributions t o energy. Elcctro(Iynamics fits s~~ioot,lily into this picturr. T h e Maxwell lllaili ah~olutclytllr sanic,
C ~ a,U t '1011s
r('-
After 41, thc first, four ;t.xioms of (~1ectrotIynnmicsclori't tlcpend on tlio liiot,ric. Co~ls(?qucntly, gravity 1c;tvos them r~nn.fict,ccl.I-Iowevcr, t,hc Hotlgc stn,r * in t,ll(' spacetime rcl:ttion, t,hc fifth axiom,
"feels" thc dynamicnl nictric g fulfilling the Eilistein equation. Hence it is via h~axwcll-~orentx spacctillic relation that gravity maltcs itself felt. Tllc elcctrom;ignrtic firltl, in the frnnlcwork of GR, belongs t o the mattc'l sitlc,
7 C o n l l ~ a rwit,lr ~ 1 . i ~ n ~ l&i ~1,ifsllitx ~1 1211, for exa111~1e. EinsLCin's P r i n c ~ t o l lIcct~lres[7] sf.ill give a good itlca, of the untlcrlying principles ant\ s o n ~ oof t h e main rcsnlt,s of GR. Frarlkrl (81 has writt.cn a little book on G R untlcrlining its gcomctricnl character and, in pnrticlllnr, dcvcloping it in t,crms of exterior calclrlns. = *On. 'IIecall vnpr = *(onA oA A 0 7 ) ,
Max
its cncrgy-moli~cntumcrlrrcnt C ., Since dI,A C 01= 0, this is also possible in a smootlr way. Thrsc arc tlic basics of tlie gravito-electromagnetic complex. Let us illust,rate it by an example.
W consiclcl
n point source of Inass m and charge Q and study its gravitational ant1 c~lr~ctromngnetic fieltls in the clcctro-vacuum, that is, outside tlie mass m and the charge Q. T h e Einstrin equation in this casc reads
For t,lic? clcctromagnctic equations one shoulcl compare ( 0 . 3 ) and ( 0 . 4 ) . s?jmm.etr?l:ccofmm.e and express To solvc such a problem, we take a .spl~,er?l:call?j it in polar coordiliatcs r , 8 , d :
It contains the 0-form f metric rc.nds
=
f (7.) : ~ n dis i~~s11111ccI t o be orthonormal; i.e., tlic
Lct us first, turn t o tlie clcctroma.gnetic c;~sc?~uitlro7rtgravity (pure electric case). Then wa have a. Minlcowslci spacct,i~newitli f = 1. T h e spherically s y ~ l i ~ n e t r i c cl(:ct:rolriagl~ct~ic ficld is then described by tlie Coulomb ansata:
Herc (1 is a. const;tnt. T h e liomogenc~o~ls Maxwell equation d F = 0 as well as t h r inliomogcncous equation for the vacuu~ncasc d * F = 0 arc both fulfilled. T h e c~iergy-momelitulncurrent call be tlctcnnincd by s l ~ b s t i t ~ ~ t i(0.10) lig into (E.1.22):
Clearly, since f = 1, Einst,ein's cqr~ation(0.1) is not fulfilletl: T h e geolnctric left-lii~litlsiclc vnliishcs ant1 docs not counterbnlancc the no~itrivialright-hand s i d ~(0.11).
The intcgration constant q is relatcd to the total charge Q of tlic source. which, by means of (B.1.1), (B.1.45), and thc Stokes t,licorcm, turns out to be
where the 3-dimensional domain !& encloses the source. From (0.10) and (0.4) we find t,hc excitation 4 192 IJ = 2, = A, -
r\
19.9 = A, q sin 0 (10 A d(b.
This, together with the ortl~onormalcoframe (0.7) and the elcctric potential (0.15), represents the Reissner-Nordstrom solution of GR for a massive charged LLparti~Ic." The elcctromagnetic ficld of the Reissner-Nordstrom solution has the same innoccnt appearance as that of a point charge in flat Minkowski spacetime. It is clear, however, that all relevant geometric objects, coframe, metric, connection, curvature, "feel" - via the 0-form f - the presence of the electric charge. If the cliargc satisfies the incquality
(0.13)
7.2
If intcgratetl according to (0.12), we havc
&&
and c = I/-. We recall (D.G.11) and (E.1.2), that is, Xo = we find the standard SI form of the Coulomb potential (0.9):
Tlwn
Let us now turn to the gravitational casc for an electrically uncl~n~gerl sphere (pure gravity casc). Then, as is known frorn GR, wc havc thc Scliwarzscliiltl solution with
Here m is the mass of the source. It is an easy cxcrcisc in C O I I I ~ L Ialgebra ~ ~ ~ ~ to derive this solution by substituting the splicrically synlnictric fra~nc(0.7) illto the vacuum Einstein equation vnpyA Roy = 0 and l>ysubsequently integration tlic eincrging second-order ordinary cliffercntial cquat,ion (ODE). GR is a nonlinear field theory. Nevcrtl~cless,if wc now trvat thc con~hir~c~l case with c1e~tro1nag1~eti~ nnd gravitational fielcls, wc can sort of superimposc~ tlie single solutions bccitrlsc of our coordinntr and frame invariant prcscntatio~~ of electrotlynamics. Wc now havc f # 1, but we still l
-
+
then the spacetime metric (0.8) has a horizon that corresponds to the zeros of the function (0.17). However, as it is clearly seen from (0.9), (0.10), and cspccially from ( 0 . 1 I ) , the electromagnetic field is regular everywhere except at the origin. The emerging geometry describes a charged black hole. When the cliargc is so large that (0.18) becomes invalid, then the solution is no black hole but rat,hcr describes a bare (or naked) singularity. These rcsults on spherical symmetry can be straightforwardly generalized to gauge tlicories of gravity with post-Riemannian picces in thc linear connection (scc [26, 131).
Rotasting sou,rce: Kerr-Newm,an solution When a source is rotating, its elcctromagnetic and gravitational fields are no longer s~~hcrically symmetric. Instead, the Reissner-Nordstrom geometry discussed above is replaced hy thc a,xially symmetric configuration described by the cofran~e'
-
sin 0
-[ - a d
a
+ (r2 + a 2) d 4 ] ,
where A = A ( r ) , C = C(r, 0), and n is a constant. The lattcr is directly related to the angular momentum of the source. % Reduce-Excalc program for such a c o f r a ~ n ecirn be found in Sec. (3.2.2(including the cosmological constant for ffsqrt# 1). Puntigam et at. (271 derived the electrically charged Kerr nictric, the I<err-Newnian metric, by means of Excalc from the (uncharged) Kerr metric in a manner analogor~st o the derivation of the Reissner-Nordstrom metric from the Schwarzschild nictric shown above.
The electromagnetic potential 1-form reads
with A6 = A6(r,0). Substituting the ansatz (0.19), (0.20) into the EinsteinMaxwell field equations (O.G), (0.3), and (0.4), one finds:
In GR, knowledge of the Killing vectors provides important inforniation about the gravitating system. In particular, for a compact source, the total mass A4 and the total angular momentum L can be given in terms of the Killing vectors by means of the so-called Komar formula^:'^
Accordingly, the electromagnetic field strength reads: F=dA=-
((a 2 cos2 0 - r 2 ) d6 A d i ~ T & ~ C ~
+
All colnponents of the spacetime metric g,j depend only on the two coordinates r, 0. Accordingly, the Lie derivatives of the metric with respect t,o tlic tangent vector fields dt and vanish, LCgij = 0. These are Killing vectors of the metric. Thus the Kerr-Newman metric possesses the two Killing vectors
2a 2 r sin 0 cos 0
a
+
Wc denote p2 := (r 2 a2)2- a 2 A sin2 0 and introduce the 4-vector field n of t,he adapted spacetime foliation by
The integrals are taken over the spatial boundary of a sphere with infinite radius. Furthermore, we used tlie canonical map ((3.2.3) to define the I-forms (t)
(t)
k=g(E), Then we can write tlie metric of spacct,ime in the standard form: ds2 = o,p T9@ 8 1 9 ~= N 2 dt 2 -
(3)9ab drca dxb.
(0.26)
(4)
(6)
Ic=i?([)
in terms of the Killing vector fields. It is sufficient to use in tlie Komar expressions of (0.31) the asymptotic formula (0.29). Then, for the Kerr-Newman metric, we can prove
Hcrc, making use of (0.19) and (0.21), (0.22), we have This explains the physical meaning of the parameters m and n in the KrrrNewman solution. If we put a = 0, wc fall back to the Reisslicr-Nortlstro~n solution.
Electrodynamics outside black holes and neutron stars
wlicreas the transversal spatial 3-coframe reads, as usual, dr = -
dr,
$B = do,
d$ = d$ - n%t.
-
(0.28)
For large distances, we find from (0.19), (0.21), and (0.22) the asymptotic line element of spacetime as
Neutron stars and black holes arise from the gravitational collapsc of ordinary matter. The gravitational effects become very strong near such objects, ant1 GR is necessary for the description of the corresponding spacctirnc geometry. Normally, the total electric charge of the collapsing matter vanishes. Then we arc left, in general, with the Kerr metric, which can be ol~tainrtlfrom (0.19)(0.29) by putting Q = 0. Near tlic surface of a neutron star and outside a black hole, onr can expect many interesting elcctrodynamical effects. To describe theni, we ncetl to solve Maxwell's equations in a prescribed Kerr metric.
It is an~azil~gly simple to find tllc exact solution of the Maxwcll cql~ations in the Kclr gcornctly. The crucial points arc: (i) the Kcrr gcomctry desc~ibes a. (mattel-fire) vacuum spncetimr and (ii) there exist the two Killing vcctol fielcls (0.30). It is straightforwr~rrlto prove that every Killing vcctor E tlcfincs, in a ViLcullm spacctimc, n 11a111ioiiic 1-form k = g([) t h a t satisfies k = 0 ant1 d t k = 0. Recalling the Maxwell equations in the for111 of the wave equation (E.1.4), we immccliatcly find that the ansatz
In the mcmbranc the physics outsitle a rotating black holc is dcscribecl in terms of the horizon understood as a conducting membrane with slirfacc charge and current density, as well as witli surface resistivity. I11 particlilar, it is possible to develop a mechanism of extracting (rotational) energy from :t black hole by means of external magnetic fields. In realistic situations, one has a plasnia around neutron stars and black holes. Tlirrcforc pl:~~lnii or, Inore s~ecifically,magiietohydrodynmics has t o 111. t~pplicd.
Force- free electrodyn,am,ics yields an exact solut,ion of the Maxwcll eql~atiolison the background of the Kcrr metric. Here Bo is constant and the cocfficicnt in t,he first term is chosen in nccordancc with (0.31) and (0.33) in order to guarantee a totally vanishing charge. Su11st)ituting (0.32) into (0.34), we fincl explicitly
Near a black hole or a ncl~tronstar, the energy-inoinent,uni of the clectromagciitic ficld llcavily tlominatcs the energy-momentum of matter and, thus, approximately force-free fields can naturally emerge in the plasma of electrons ant1 p o ~ i t r o n s . 'In ~ Scc. B.2.2, we have defined such electromagnetic fields by thc coiitlit,iol~of the vanishing of the Lorcntz force (B.2.21). Using the spacetime rclntion ( 0 . 4 ) , we can now develop a morc substantial analysis of such a sit,uation. T h e force-free condition now reads:
Tlic pl1ysic:r.l interprct,atiol~is straigl~tforward:The 1-form potcl~tial(0.35) is a kind of superposition of the Co~rloiiih-typeelectric piece (the first term on
the right-haritl side, cf. (0.20), ( 0 . 2 3 ) ) and thc asyn~ptot,icallyhornogeneo~is niagnctic piccc (t,hc sccontl torm). With respect to the coorrlinatc foliation (for which 11 = a,), the electromagnetic ficld strength reatls F = dA = E A d t B, witli
+
To begin witli, let us recall the sourceless solution considered above. Tlie ~~iiignctic ficlcl (0.37) lins the evident structure
with
-12 (r2- 2)sin 8 cos 8 ole] ,
(0.36) Thus, t l ~ emagnctic ficld is nianifcstly axially symmet,ric, that is, its Lie tlerivativc. with rcspect to the vcctor ficld arbvanishes:
+ a 2 + a 2 cos48) dB] A dgi
- 21' sin 0 cos 8 (2r2cos20
+Do [r sin28 01--t (r2 + n 2 ) sin cos 0 do]
A dql.
(0.37)
For large tlistanccs, the last line ill (0.37) dominates, yicl(1ing asymptotically const,ant magnctic ficltl dircctctl along tlic z-axis, tlic honiogcnc~o~~s
Hcrc we performed the 11s11altr:tnsfor~-nationfrom spl~cricnlpolar coortlinntt.~ ( r , 0,ql) to Cnitcsian rectangular coordinates ( r ,?I,z). Tlic clcctric field vanishes for n = 0, i.c., fol a non-lotatiiig black holc. can draw a clircct parallel t o the Wilson & Wilson experiment where tlic magiic3tic fieltl ~ntluccsan clcctric field inside tlic rotating body. Tlie sp:~cctimcof a rotating Kerr gcomctry acts similarly and intluccs a11 clcctric ficld around tlw blnclc holr.
Hrrc we rrscd the Maxwell rquation tfB = 0, which is identically fillfilled for any function Q that does not tlcpcntl on 4, i.c., 3 = 3 ( r , 0 ) . Rt.turning t o the problem ~lnclcrconsideration ( ~ u z t hnontrivial p l a s ~ n source), a w r also demand t,liat the rnagnctic field be axially synimetric. As a morc general structure of tlic ficld we then expect
" ~ e r ,c.g., Stral~lnnlln1281 nntl t l ~ cliteratl~recited thcrein. "Sec, c.p., Cap [3], [lC,], and also I<noepfel [In]. '"or rccrr~tdcvelopments see, e . g . , I
Clearly, this 2-form satisfies the M
where tlie vector field
can be interpret,ecl as the vorticit,~of the magnetic field lines. The function R docs not tlepcnd on the angular coorclinate 4. Furthermore, substituting (0.43) into (0.44), we find
The Maxwell equation d E = 0 is fulfillctl if R = R ( 9 ) . Combining (0.43) and (0.46), we find as general ansntz for the electromagnetic field strength 2-form
This must be inserted into the "force-free condition" (0.39), which will hc. cvaluatecl with respect to the natural (or coortlinatc) frame e, = 6k8,. Direct inspectlion shows that ccluation (0.39) is identically fulfilled for at and d,,, provided I = I(11~).Substituting (0.47) into (0.39) for (3, and aO,we fincl i i nontrivial tliffcrcntial equation
Hcrc the f~nct~ions a = a(r,B), ,O = ,8(1., B), = y(r, B) arc constructed from R and fro111 the ComIloneilts of tLhcspacetime nlctric according to
~ ~ / (0.48) ~ . is cnllcd the Grad-Shafmnou Recall that (3)933 = ~ i n ~ 0Equation equatzon and, with the given f~~nctions R = R(q1) and I = I ( ~ J )the , solrition ql of (0.48) dcscribcs cornplrtcly the force-free elc~tromngnct~ic firld colifigura t '~ n n The corresponding distribution of the charge and curre~itdensity is dcrivctl fio111 the Maxwell equation J = d H : Explicitly, thr cliargc density 3-form rcwtls
whereas the electric current 2-form turns out t o be
With these few examples, we wanted to demonstrate that the electrodynamic formalism, which we developed in this book, is ideally suited for "marrying" clcctrodynan~icsto general relativity; this includes interesting astrophysical scenarios. There seems to be no better way than tlie one outlined above for the investigation of the mutual influence of the electromagnetic and the gravitational field on each other.
Remarks on topology and electrodynamics In our book, we did not discuss genuine topological aspects of electrodynamics. Howcvcr, topology can play a very important and nontrivial role in electrodynamics and in magneto hydrodynamic^.^^ A word of caution is in order: One should carefully distinguish the physical situations in which an underlying spacetime (or space1" has a complicated topology from the case in which the clcctromagnetic field configuration is topologically nontrivial.'' Usually the decisive role is played by the pure gauge contribution to the electromagnetic potential 1-form. A manifest example is given by the force-free magnetic fields that approximately describe the twisted flux tubes in the models of solar prominence (sheets of lumino~isgas emanating from the Sun's surface)." Recall equation (B.2.23); whicli determines a force-free magnetic field. It is identically fulfilled when
Indeed, since B is a transversal 2-form (i.e., living in three spatial dimensions), we have B A 13 = 0.Consequently, B A e, J B = 0.Thus (0.52) solves (B.2.23) for any frlnctioll a = cu(z). Note that the ansatz (0.52) can be used even in the metric-free formulation of electrodynamics. In Maxwell-Lorentz electrodynamics, (0.52) further reduces to
"see the reviews of MoRntt and Marsh [24, 22, 23). '"ee the article of Frankel [9] and the experimental investigations of Tsebro and O~nel'yanovskii[30]. '"Tl~is could be true for the ball-like solutions of the vacuum Mnxwell equations of Cl~ubykaloand Espinoza [ 5 ] ;see also the literature cited there. 1 7 ~ l i iiss discussed in Marsh (231, for example.
Incidentally, this is the field equation of the so-called L'topologically massive clcctrodynamics" in three dimensions18 provided a is a constant. Specializing to axially symmetric configurations in Minkowski spacetime, we can easily find a solution of (0.53) for any choice of a . For example, in cylindrical coordinates (p, 4, z), for a ( p ) = 2 / [ a ( l + p2/a2)], we obtain
This solution describes a uniformly twisted flux tube. The constant parameter a determines the twist. Although da # 0, da A B = 0, which guarantees thr consistency of the solution. It is straightforward to read off from (0.54) the potential 1-form
where x is an arbitrary gauge function. There exist various topological numbers (or invariants) that evaluate the topological complexity of an electric and magnetic field configuration. The magnetic hclicity density in (B.3.17) provides an explicit example of such a number. Defined by the integral
the magnetic helicity h measures the "linkage" of the magnetic field lines. One can establish a direct relation of h to the classical Hopf invariant, which classifies the maps of a 3-sphere onto a 2-sphcrc, and can interpret it in terms of thc Gauss linking number. Turning again to the twisted flux-tube solution above, we see from (0.55) and (0.54) that in the exterior product of d with B only the contribution from tllc pure gauge survives: A A B = dx A B = ~ ( x B )As . a result, a nontrivial value of the hrlicity (0.56) can only be obtained by assuming a toroidal topology of space. This can be achieved by gluing two 2-dimensional cross sections a t some vn1uc.s tl ant1 22 of the third coordinate and, moreover, by assigning a nontrivial jump 6~ = x(z2) - ~ ( 2 to~ the ) gauge function. There is a close relation between magnetic interaction energy ant1 hclicity. For the force-free magnetic field, we have explicitly, P
P
where we used the Oersted-An~pBrelaw j = &Y and the ansatz (0.52). When cu is const,ant and fi can be neglected, then the energy is proportional to the magnetic l~elicit~.'" '"ee
Deser, Jackiw, and Templeton [6]
Another manifestation of topology in electl.odynamica1 systems is more of a quantum nature: the Aharonov-Bohm effect.20 If the wavc function of a11 electron surrounds an isolated magnetic flux such that it cannot penetrate it, then the electron wave resides in a region free of a magnetic ficltl B (but, not free of the covector potential A ) . Nevertheless, the phase of the wavc function is influenced by the enclosed magnetic flux. By interferoinctric experiincnts this flux can be measured. We shall now turn to the quasi-classical fields that built up the electric current 3-form.
Superconductivity: Remarks on Ginzburg-Landau theory Generalizing the classical Maxwell-London theory of superconductivity of 1935, Ginzburg and Landau achieved in 1950, by introducing a conlplex 3D scalar "order parameter" $, a quasi-classical static description of superconductivity for T = o . ~ The ' field $ describes the Cooper pairs (two bound electrons in a superconductor), the density of which is l$I2. The Ginzburg-Landau model is specified by the 3D energy functional
with q = 2e FLS the charge of a Cooper pair, li the Planck constant, ant1 y x 2 . Here denotes the coluplcx conjugate field. Tllc posU ( x )= - p x itive quant,ities m, P, y arc the parameters of the theory. By varying (0.58) witli respect to the ficlds A and $, onc finds the field equations, n;~mclyt,hc Oersted-Amphe law witli the superconducting current
+
iiiq ( $ ~ d $ - $ ~ d $ ) j =2 m.
q2 Id)12'd +m
and the Ginzburg-Landau equation. This nonrclativistic theory describes the static properties of s ~ ~ p e r c o n d ~ ~ c t o r s very well. It represents a good approximation to the exact theory of superconductivity for temperatures close to T = 0. One consequence of t,his theory is the Abrikosov lattice of magnetic flux lines, which we displayed in Fig. B.3.2. l g ~ o r on e thc electrotiynamics in multiply connected domains can be foul~din Mars11 [22, 231. T h e corresponding effects underline the physical importance o f tllc elrctromngnctic potential 1-form A. 20Sce the theoretical discussion by Aharonov and Rohm [I] and the first cxpcrimentnl findings by Chambers [4]. A rccent evaluation has becn given by Natnbll [25]. 2 1 ~ eTinkham e [29] and Zirnbauer [31], for example.
Classical (first quantized) Dirac field In classical clectrodynarnics, thc clectric current 3-form J is phenomcnologically specified; it cannot be resolved any further. In tlie Ginzburg-Landau theory, we exprcssctl tlie supcrconclucting current (0.59) in terms of the 3D scalar ordcr parameter ?j,. Quite generally, wc? know, however, that clectric charge is carried by the f ~ ~ n t l ; ~ m e npn,rticlos, tal narncly by leptons and quarks. They are both fcrmions of spin fi3/2. The electron, a lepton, is responsible for many everyday effects. Its field can be dcscribcd by means of a Dirac spinor, a half-integer rcprescnt,ation of the Lorcntz group. Thus, spinors can only he introduced after thc 5th axiom has been atloptcd. Then a metric is available and tlie Lorc~itz group surfaces. Strict,ly, t,hc sccond quaiit,izctl Dirnc t,licory governs the beliavior of the electron ant1 its intcract,ion with the? photon. However, if the cncrgics involved in iIn cxperimelit arc mr~clisri~allcrthan the mass of tlie rlect,ron, then tlic electron call bc a.pproximatcly viewed as a classical matter wave, i.c., as a first quantizcd Dirac wavc? function. An clect,ron microscope and its resolution may well bc dcscribetl in such a mnnncr; similarly, an electron intcrferomcter for sensing rot,nt,ion (Sagnac type of effect) tlocsn't need a more refined description. Let us then assume t,hat the Dirac matrices, referred to an orthonormal cofralne, arc given by
I-Icrc 0-0 = tli;ig(+l, -1, -1, -1). Then we can introduce tllc Dirac-algebravalrlcd 1-form y := y,, 19". Tlic Dirnc rquation, minillially c o ~ ~ p l ctod the elcctromiignetic field, reads
The Dirac adjoint is Lagrangian 4-fonn
With grmvity, o ~ i cmust gcncrtilizc the spinor covnriant derivative, which now sho111tlrcntl
-
where, as usual, GoP := iy["ypl, and roPis tlie Lcvi-Civita connection. In order t o be able t o forliiulate Einstein's gravitational ficld equation, one must syrnnict,rize the energy-monicntum current, D
D
a,
=
E n -Dp,,
(0.68)
1)
An ~ l t e r n a t i v eDirac coupling to gravity can he acliicvctl via tho Ein,.steinCartan theory of gravity.22 Wt, allow, generalizing the Levi-Civita connection of GR, for a metric compatible connection r e p = -roe carrying a torsion piece. Then the spill current of the Dirac field can be defined accortling t o
where ys := -irj,fi,,, ynyfiy~'y"/4!, and for the gravitational scct,or of the Einst,ein-Ct~rtan-Dirac-Maxwell system, wc find
:= *+yo. Tlic Dirac ecluation can be derived from the
T h r conscrvcd olcctric current turns o ~ t,o~ bc t
Tlic energy-li~onicntt~rii ctirrclit of tlir Dirac ficld, according t o the Lagrang(.Noctlicr procotl~lrc.tlcvclopcd in Scc. B.5.5, reiltls
J
r e p = dl, A ~ L ~ I ,
whcro 7,p = ( d L D / D 9 ) (i;inp/4) 9 is tlic canonical spin current 3-form of the Dirac ficltl. Then the complete Einstein-Dirac-Maxwell system, together with (E.1.22), (O.G8), (O.G5), ant1 (O.GG), reads
1
5 7""
Here D, := e ,
D
with
A
AT'
8rG
=
D
-rep
C3
This is n vial)lc altcrniitivc to the Einstein-Dirac-Maxwell system. It looks very symnietric. Note that the clcctroniagnetic field (since it is niassless) doesn't carry dynainic:~lspin. If we t a l e citlier t,lie Einstein-Dirac-Maxwell or tlic Einstcill-Cartan-DirncMaxwell syst,em, in any case wc can study the gravitational properties of the Dirnc clc~tron.~"ntl in both cases we recognize that there is no ncetl to change clcctrotlynamics. It is insrnsitivc to the gravitational niodel applied. Via tlic 5th xiom om tlic I-Iodgc star corresponds to a tlynarliic metric oheying the Einstein or t he Einstein-Cz~rtnii equat,ions.
D. Therefore tlie self-consistent Dirac-Maxwell syst,elii reads '?See Rlngojcvii. [2] or Grorlwalcl ct nl. [ I l l and refrronccs givm there. "'SCC
[ I d ] for
R
d i s c ~ ~ s s i oofr ~the gravit.atior~nlmoments of the Ilirac electron.
On the quantum Hall effect and the composite fermion In Sec. B.4.5 we presented a classical description of the quantum Hall effect (QHE). However, a true understanding of QHE is only possible when the qutmtum aspects of the phenomenon are carefully studied. We have no tools in our book to pursue this goal. What one 11% to do is to quantize the Lagrangian (R.4.72) by, for instance, path integral methods. In other words, QHE is described within the framework of an effective topological quantum field theory with a Chern-Simons action. The explanation of the quantization of the Hall conductance a~ is the ultimate goal achieved in these studies. The best thing we can do is to direct the interested reader to the corresponding l i t e r a t ~ r e . ~ ~ Incidentally, if one describes &HE for low lying Landau levels, then Jain's concept of a composite femnlon is very helpful: It consists of one electron and .S are attached to it.2"sn't that a very an even number of magnetic ~ ? I L T O I ~that clear additional indication of what the fundamental quantities arc in electrodynamics? Namely, clcctric charge (see first axiom) and magnetic flux (see third axiom).
On quantum electrodynamics The approach in our book is essentially classical. This is a legitimate assun~ption within the well known and rather wide limits of the idealized rcpr~scnt~ation of electric charges and currerit,~by means of particles and continuous media ant1 of clectromagnet,ic radiation by means of classical waves of clcctric aiitl magnetic fieltls. However, it is experi~nentallywell established that an electron, uncler crrtain conditions, may display wave properties, whereas clcctromagnctic ratliation must sometimes be described in terms of particles, i.e., photons. These facts arc. taken into account by quantum elect,rodynaniics (QED). The mathematical framework of QED is the scheme of second quantization. In this scheme, the electromagnetic (Maxwell) field A and the electron (Dirac) field ?1, are replaced by field operators acting in the Hilbcrt space of a11 quantlinl states. Physical processes are the11 described in terms of creation, annihilation, and propagation of the quanta of the electromagiictic and tlie spinor fields. The photon y is massless and has spin 1 (better: helicity I ) , in units of h, whereas the electron is massive and has spin 112. In QED, the electromagnetic potential 1-form A plays a funclamelltal role, representing the "generalized coordinate" with the excitation H being its canon-
I
2 4 ~ e Fr'riihlicli e and Pedrini [lo], e.g., nnd the literature cited therein. 2%ee Jnin [17, 181 and, in this general context, also Nnmbu [25].
+
ically conjugated "momentum." The gauge symmetry A --, A d x moves t o center stage: One can consistently build electrodynamics on the basis of the principle of local gauge invariance. The underlying symmetry group is the Abelian U(1) and A emerges naturally as the U(1)-gauge potential, which is geometrically interpreted as the connection on the principal U(1)-bundle of spacetime. The gauge freedom is responsible for the masslessness of a photon and, thus, ult,imately for the long range character of the electromagnetic interaction. QED describes correctly all quantum phenomena involving electrons and photons. It is experimentally very well verified; in fact, it is the most precisely tested physical theory. The most precise experiments are the proof of the predicted value of the anomalous magnetic moment of the electron and the observation of the relativistic shift of energy levels in atoms. The success of QED is to a great extent related to the smallness of its coupling constant ar = e2/(4mohc), which allows one to effectively use a perturbation approach.
On electroweak unification Since the 1960s, considerable progress ha.; been achieved in the construction of unified theories of physical interactions. In particular, QED is found to be naturally unified with the weak interaction. A typical example of a quantum process governed by the weak interaction is the radioactive decay of the neutron into a proton, an electron, and an antineutrino. The modern understanding of weak forces is based on the gauge approach. In the Glashow-Salam-Weinberg model of tlie electroweak interaction, the fundamental symmetry group SU(2) x U(1) gives rise to four gauge potentials as mediators of the electroweak interaction: the intermediate vector boson W (its positively or negatively charged versions W*), the neutral intermediate vector boson Z, and the photon y.The spin 1 particles W and Z become massive via the mechanism of spontaneous symmetry breaking of the gauge group, whereas y corresponds to the unbroken exact subgroup and remains massless. I11 the standard model of elementary particle physics, the group SU(2) x U(1) is enlarged to SU(3) x SU(2) x U(1) and the resulting theory represents a tinification of the electroweak and the strong forces. The color group SU(3) brings to life the eight additional gauge potential 1-forms Aa, a = 1 , . . . ,8, which are called the gluon potentials. In the standard model, one describes the interaction of quarks (fermionic constituents of the strongly interacting particlcs, the hadrons) and leptons (electron, muon, tauon, and their neutrinos) by means of the gluons A" and the intermediate gauge bosons W, 2 , and y. And classical electrodynamics is encapsulated in the y.
References
[I] Y. Ahnronov and D. Bohm, Significan.ce of elect~onragn~etic poten,tial.s i n the quantum tii.eor?y, Pl1.y~.Rev. 115 (1959) 485-491. [2] M. Blngojcvik, Gravitation and Gauge S?yrnrnetries (IOP Publisllilig: Bristol, UIi, 2002).
[4] R..G. Cliillnbcrs, Slrift of a n elcc21.on. intcrfe~rncr:pattern hy encloseti ir~c~gn,ctic ~ ~ I Lp11.y~. Z , Rev. Lett. 5 (1960) 3-5. (51 A.E. C h u b y k a nl nod A. Espinozn, Unusl~(alform,ntions of tlre f7.c~clr?ct7nrn,ngnetic field in vncuu,m, J. Phys. A35 (2002) 8043-8053.
[GI S. Dcscr, R. Jackiw, alltl S. Templeton, Tlrrce-dime~~,sion,al n~.c~ssi.vc gtrlrge theories, Pl1.y~.R.ev. Lctt. 48 (1982) 975--978; S. Dcser, R.. .J:tcltiw, alitl S. Tcinplct,on, Topolo,qicnlly m,assi~iegn~lgctheories, Aiiii. Phys. (NY) 140 (1982) 372-41 1 [Eir.o.tum., Ai~ii.PIIYS.( N Y ) 185 (1985) 40G].
[8] T. Frankol, Gmvitation,nl C ~ i ~ v n t An ~ r .Int,rotl~~ction ~. t,o Einst.cin's Theory (I;'lccmnn: Snn Fr:uicisco, 1979). [9] T. Rn.nltcl, Electric curren.ts i n rn~iltiplyconmected sprr,ce.q,~lltC~11. J . Thcor. Ph,ys. 27 (1988) 995--999.
394
References
[lo] J. Frblilicli and I3. Pcdri~ii,New applications of tlie chiral anomaly, in: "Mathematical Physics 2000JJ,Eds. A. Fokas, A. Grigoryan, T . Kibble, and B. Zcgarli~lslti(Impcrinl College Press: London, 2000), pp. 9-47; Eprint Archive hcp-t,li/0002195. [ l l ] F. Gronwnltl n~idF.W. I-Icl~l,On the G a ~ ~ gAsp~cts e of Gra~~lty, in Proc. of the 14th Colnsc of the Scl~oolof Cosmology and Gravitation on 'Qrlant urn Gravit,yl, licltl at Ericc, Italy, May 1995, P.G. Bcrgmann, V. cle Sabbata, a ~ l d13.-.I. Trc~tlcr,etls. (World Scientific: Singapore, 1996) pp. 148-198; Eprint Archive gr-qc/9602013. [12] F.W. Hchl, J. Lcmke, and E.W. Mielke, Two lectures on fermions and gravity~,in: Gcomcti:y and Tlicorctical Physics, Proc. of tlie Bad Honnef School 12-16 Feb 1990, J. Dchrus ant1 A.C. Hirshfcltl, cds. (Springer: Heidclherg, 1991) pp. 56-1110. [13] F.W. I-Iclil nntl A. Mzlcias, Mct~ic-afi11egauge theory/ of gravit?~II. Exact sol~~tion..~, I i ~ t J. . Adood. Phys. D8 (1999) 399-416. [14] F.W. Hclil, A. Mncins, E.W. Miclltc, and Yu.N. Obukliov, On the structure and the spin c11,rren.t~in. Dirac7.selectron theory, of tlre ene~:q?l-m.om.en,tl~n~ in: "On Einstcin 's Path", Essays in honor of E.Scliuckii~g,A. Harvcy, ed. (Springer: Ncw York, 1998) 257-274. [15] 14. Hcuslcr, Black Hole Uniqueness Theorems (Cambridge University Press: Cambritlgc, UI<, 1996). [ l G ] 14. I-Ior;~,Pla,s~n,c~.,s at Iligh ?Fm.pemt~~re a7~dDensityl: Applications arid I~iil)lic:;~(.iolis of L;\scxr-Pl;ls~iiziInteri~ct,ioll.Lcctrlro Nolcs in Pl~.ysicsm 1 (Springer: Berlin, 1991).
[17] J.K. Jilin, Co~n,posite-fer111~i01~ npproach for the fractional quantum. Hall eflect, P11,ys. Rcv. Lctt. 63 (1989) 199-202. [18] J.K. .Jil.iii, Com.positefelrr~,iontlreomj of fractional quan.tum Hall eflect, Acta P1i.y~.Polon. I326 (1995) 2149-2166. 1191 H.E. I<noepfcl, Magnetzc Fields. A con~prclicnsivcthcoret,ical treatise for pract.ica1 use (Wilcy: Ncw York, 2000). [20] S.S. I
References
[23] G.E. Marsh, Topology in electromagnetics, in: Frontiers in Electromagneti c ~D.H. , Werner and R. Mittra, eds. (IEEE: New York, 2000) pp. 258-288. [24] H.K. Moffatt, Magnetic Fields Generated in Electrically Conducting Fluids (Cambridge Univ. Press: Cambridge, 1978). [25] Y. Nambu, The Aharonov-Bohm problem revisited, Nucl. Phys. B579 (2000) 590-616; Eprint Archive: hep-th/ 9810182. [26] R.A. Puntigam, C. Lammerzahl, and F.W. Hehl, Maxwell's theomj on a post-Riemannian spacetime and the equivalence principle, Class. Quantum Grav. 14 (1997) 1347-1356. [27] R.A. Puntigam, E. Schriifer, and F.W. Hehl, The use of computer algebra in Maxwell's theory. In Computer Algebra in Science and Engineering, J. Fleischcr et al., eds. (World Scientific: Singapore, 1995) pp. 195-211; Eprint Archive: gr-qc/9503023. [28] N. Straumann, The membrane model of black holes and applications. In: Black Holcs: Theory and Observation, F.W. Hehl, C. Kiefer, and R.J.K. Metzler, eds. (Springer: Berlin, 1998) pp. 111-156. [29] M. Tiakham, Introduction to Superconductivity, 2nd ed. (McGraw-Hill: New York, 1996). [30] V.I. Tsebro and O.E. Omel'yanovskii, Persistent currents and magnetic flux trapping in a multiply connected carbon nanotube structure, PliysicsUspeklii 43 (2000) 847-853 [the Russian version is contained in 170 (2000) 899--91 61. [31] M.R. Zirnbauer, Elektrodynamik. Tex-script July 1998 (Springer: Berlin, to bc published).
Author Index
Al~bottB., 3, 11 Abral~amM., 335, 371 Aharonov Y., 387, 393 Altltcrmnns E., 187 Alhrccht, J., 141, 188 Amoroso R., 189 Aritoci S., 335, 341, 371 Ashoori R.C., 109, 110, 187 Audrctscli J . , 8, 11 Avron J.E., 159, 187 B ~ l d o n ~D., i r 8, 11 I3ali:tn R., 242 Bnmbcrg P., 8, 11 Barrow J.D., 275, 310 Barut A.O., 8, 11, 303, 307 Baylis W.E., 8, 11 Bcltcnst.~inJ.D., 275, 307 Bergcr M.S., 15 Bcrgmann P.C., 394 I3lngojcvii: M., 215, 241, 389, 393 Bollm D., 387, 393 Bolot,ovsky B.M., 353, 371 Bopp F., 8, 11 Born M., 326, 371
Bossavit A., 8, 11 Bott R., 66, 103 Brans C.H., 244, 275, 307 Braun E., 155, 156, 187 Brevilt I., 335, 353, 371 Brodin G., 325, 371 Burkc W.L., 8, 11, 64, 66, 103 Cap F., 383, 393 CR[IOV~ R., ~ ~244, A 307 Carroll S.M., 252, 307 Cartan E ., v, 8, 12 Chaltrahorty T., 155, 187 Chambers R.G., 387, 393 Char B.W., 45, 103 Chicone C., 324, 372 Choquet-Bruhat Y., 8, 12, 29, 94, 103 Christensen S.M., 46, 104 Chubykalo A.E., 385, 393 Coll B., 215, 241 Cooper L., 253, 307 Crovini L.. 187 Darrigol O., 8, 12
Author Index
Debrus .I., 394 Dell J., 244, 307 cle Rliam G., 8, 12 Derrick G.H., 215, 241 de Sabbata V., 252, 310 Dcschamps G.A., 8, 12 Deser S., 386, 393 Devoret M.H., 110, 187, 188 DcWitt-Moretkc C., 8, 12, 29, 94, 103 Dickc R.H., 252, 275, 307 Dillnrd-Blcick M., 8, 12, 29, 94, 103
Geiger H., 311 Gibbons G.W., 326, 372 Gibbs J.M., 215, 241 Glatzmaier G.A., 133, 188, 190 Gonnet G.H., 45, 103 Gordon W., 354, 372 Grabert H., 110, 187 Grabmeicr J., 103 Grioli G., 15, 311 Gronwald F., 389, 394 Gross A,, 8, 12, 244, 308 Grozin A.G., 45, 103
Ebert G., 159, 187 Edelen D.G.B., 8, 12 Eichenwald A., 359, 372 Einstein A., 8, 12, 195, 233, 241, 376, 393 Engl W.L., 8, 13 Espinoza A,, 385, 393 Esslnailll U., 131, 187, 188 Euler H., 325, 372 Everitt C.W.F., 8, 12
Hadamard J., 263, 308 Hajdu J., 155, 188 Hammond P., 8, 11 Harnett G., 224, 242, 244, 308 Hartley D., 46, 103, 199, 242 Haugan M.P., 8, 13, 252, 308 Hc Y.D., 3, 12 Hearn A.C., 45, 103 Hehl F.W., 12-14,45, 104,105, 188, 241,254,308,309,325,366, 372, 373, 394, 395 Heilbron J.L., 109, 188 Heinicke C., 46, 104 Heisenberg W., 325, 372 Hernquist L., 325, 372 Heusler M., 381, 394 Hey1 J.S., 325, 372 Hirst L.L., 9, 13, 329, 372 Hoffnlann B., 310 Hora H., 383, 394 Huang K., 252, 308
Fastenrath U., 155, 188 Fauser B., 246, 307 Fcynman R.P., v, 12 Ficlcl G.B., 252, 307 Finkclstcin D., 215, 241 Flanders H., 8, 12 Flcischer J., 104 Flowers J.L., 302, 308 Fliiggc S., 15, 311 Fokas A., 188, 394 Frankel T . , 8, 12, 376, 385, 393 F'ranklin B., 109 Frohlich J., 159, 188, 390, 394 Fujiwara A,, 110, 188 Fukui T., 8, 14, 244, 267, 309 Gaillard M.K., 167, 188 Gnntmachcr F.R.., 275, 308 Garcclti J., 215, 241 Gaustcrer H., 190 Gcddes K.O., 45, 103
Iliev B.Z., 199, 242 Infeld L., 326, 371 Ingarden R., 8, 13, 140, 188 Isham C.J., 275, 308 Itin Y., 146, 181, 188, 238, 242 Ito N., 105 Itzykson C., 325, 372 Jackiw R., 386, 393 Jackson J.D., v, 13
J;tcohson T . , 244, 307 Jaclcayk A.Z., 8, 13, 244, 308 .Jain J.I<., 390, 394 Jnincs R.P., 372 .Jamiolltowski A., 8, 13, 140, 188 Jnncewicz B., 8, 13, 64, 104 .Jansscn M., 155, 188 .Jooss Ch., 141, 188 Jordan P., 275, 308 I
M., 45, 104
Macias A., 45, 105, 308, 379, 380, 394 Magllcijo J., 275, 310 Majcr U., 11 Mandclstam L.I., 267, 311 Marklund M., 325, 371 Marsh G.E., 135, 189, 385, 387, 30(1, 395 Mas!ilioon B., 112, 189, 324, 325, 372, 373 Maxwell J.C., 174 McCrea .J.D., 45, 47, 104, 105, 181, 188 h4cLcnaglinn R.G., 47, 104 Mcctz I<., 8, 13 Mi(%G., 8, 14 Miclkc E.W., 9, 14, 103, 167, 181, 188, 189, 389, 394 Miliirll L., 395, 341, 371 Minkowski H., 335, 373 Misncr C.W., 64, 104 Moffat,t H.K., 135, 189, 385, 395 Molir P.J., 302, 309 Molin;~A., 241 Moliagal~M.B., 45, 103 Mootly J.E., 252, 309 Moor(. D.,J., 8, 11, 14, 244, 310 h?or;llcs J.A., 215, 241 h411cncliU., 325, 373 Musgravc P., 47, 104 Nnml~r~ Y., 387, 390, 395 Ne'cman Y., 103, 181, 188 Nelsoli R.A., 110, 189 Ni W.-T., 252, 309, 366, 372 Nicvcs .J.F., 2G2, 309 Ol)~ikliovYII.N.,12-14, 104, 241, 244, 254, 267,308-310,325,332, 372, 373, 394 Ol)ukliov;t I.G., 47, 105 Omcllyanovsl
Author Index
Parltcr L., 46, 104 Parrott S., 8, 14 Pcccci R.D., 252, 310 Pcdrini B., 159, 188, 390, 394 Prrcs A., 144, 189, 244, 305, 310 Petlcy B.W., 302, 308 Picrcr J.F., 135, 189 Piptilbinrn P., 155, 187 Pirani F.A.E., 288, 310 Piron C., 8, 11, 14, 244, 310 Pitacvskii L.P., 267, 309 Pittct E., 14 Plcl>nliski J . , 327, 373 Ploog I<., 159, 187 Poll1 R.W., 138, 190, 315, 373 Poincclot P., 332, 373 Pollilcy D., 47, 104 Post E.J., v, 8, 14, 110, 159, 190, 245, 310 Probst C., 159, 187 Pantigam R.A., 14, 46, 104, 105, 379, 395 Qr~inri,H.R., 252, 310 Racxkn R., 303, 307 Rititll W., 155, 190 Ramachantlran G.N., 267, 310 Rnmnscslinn S., 267, 310 Raiind:~A.F., 135, 190 R;~sIiec~I D.A., 326, 372 R c l > o l ~ ~IM..J., n s 104 Richter T., 159, 190 Ridglry C.T., 353, 373 Robcrds J.B., 263, 308 Roberts P.H., 133, 188, 190 Rojo A.G., 141, 190 Roentgen W.C., 359, 373 Roqlir W.L., 104 Rovelli C., 215, 242 Ruhilar G.F., 8, 1 2 1 4 , 244,254,267, 271, 308-310 Rllticr H., 104
Sands M., 12 Sandvik H.B., 275, 310 Schcrcr P., vii Sctlild A., 34, 105, 288, 310 Schliitcr A., 125, 189 Schmidt H.-J., 11 Schonbcrg M., 7, 14, 224, 242, 244, 310 S c h o ~ ~ t cJ.A., n 8, 14, 15, 64, 104, 110, 190 Srhrijdingcr E., 8, 15 Schriifrr E., 46, 105, 379, 395 Scllwcigcrt C., 159, 188 Sciler R., 159, 190 Sikivic P., 252, 310 Sivaraln C., 252, 310 Slchodzinski W., 8, 15 Smit R.H.M., 157, 190 Socorro J., 45, 105 Sokolnikoff I.S., 37, 105 Soleng H.H., 47, 105 Sommcrfelcl A., 8, 15, 326, 373 Sowa A., 302, 311 Stachel .J., 8, 15 St,aoffrr D., 105 Stc~lmanG.E., 152, 190, 253, 307, 311 Stcnflo L., 325, 371 Strrnberg S., 8, 11 Stolyarov S.N., 353, 371 St,r;ttl~tlec.T., 275, 308 S t r a ~ ~ m n nN., n 383, 395 S t ~ ~ d U.M., cr 159, 188 Syngr J.L., 34, 105 Szivcssy G., 267, 311 Takahashi Y., 110, 188 Tamm I.E., 267, 311 Taylor B.N., 302, 309 Templeton S., 386, 393 Tert,ychniy S.I., 47, 105, 244, 310 Tllirring W., 8, 15 't Hooft G., 244, 308 Thorne I<.S., 64, 104 Tinkh:tm M., 387, 395
Author Index
Toupin R.A., v, 5, 7, 8, 15, 167, 190, 244, 245, 274, 311, 337, 368, 373 Toussaint M., 45, 105 TrAuble H., 131, 187, 188 Trautman A., 8, 15, 45, 104 Trueba J.L., 135, 190 Truesdell C., v, 5, 8, 15, 190, 245, 311, 337, 368, 373 Tsantilis E., 46, 105 Tscbro V.I., 385, 395 Tsuei C.C., 130, 190 Tu L.W., 66, 103 Turner M.S., 252, 308 Urbantkc H., 224, 242, 244, 311 Van Bladcl J., 152, 191, 353, 373 van Dantzig D., v, 8, 15 Vichweger O., 155, 188 von der Heycle P., 199, 242 von Klitzing K., 155, 159, 187, 191 Walcher J., 159, 188 Walker G., 341, 373 Walker G.B., 341, 373
Wallner R.P., 9, 14 Wang C., 244, 311 Watt S.M., 45, 103 Weinberg S., 252, 311 Weispfenning V., 103 Werner D.H., 189 Weyl H., 8, 15, 212, 242 Wheeler J.A., 64, 104 Whittaker E., 3, 8, 15, 163 Wilczek F., 252, 309, 311 Wilson H.A., 360, 373 Wilson M., 360, 373 Winkelmann V., 45, 105 Wise M.N., 167, 191 Wolfram S., 46, 105 Wright I?., 104 Write F., 45 Yoshioka D., 155, 191 Zabolitzky J.G., 105 Zhytnikov V.V., 47, 106 Zirnbauer M.R., vi, 8, 15, 140, 191, 387, 395 Zuber J.-B., 325, 372 Zumino B., 167, 188
Subject Index
The iden,tifiers (names of fi~nctions,etc.) of the computer algebrc7. system Reduce, including its Excalc package, are rliaplayed i n C A P I T A L letters.
I I
i I 1
I
2DEG 2-dimensional electron gas, 153 Abrlian Clienl 4-form, 134 Al~clianChern-Sirnons 3-form, 134, 1G2 Abraham force, 337, 339, 342, 345 Abriltosov lattice, 130, 387 accclcration, 364, 365 action principle, 178 Aharonov-Bolirn effect,, 387 alillost complcx st,ructure, 29, 43, 220 AND, 52 ANHOL2, 236 anti-Lenz rule, 146, 150, 177 anti-self-dual form, 45, 220 atlas, 59 oriented, 59, 65 autoparallel, 201 axioni fifth, 303 first, 115 forlrtli, 164 secontl, 123
sixth, 330 third, 129 axioms list of, 5 axion field, 252, 256 and light propagation, 270 vanishing, 303 axion part of constitutive tcilsor, 248 I~asis,24, see frame 77-forms, 226 i-forms, 38 CO-, 24 covector, 24 half-null, 214 null Coll-Morales, 238 Coll-Mornlcs, 2 16 Newman-Penrose, 215 orthonormal, 213 spacc of 2-f0r1ns, 40 transformatioli, 42 transfortnation law, 24 vector, 24 Betti numbers, 83
Subject Index Bianchi identity first, 208, 235 second, 208, 235 zeroth, 232, 235 boundary, 91, 97 map, 96
~
I
Cart an's displacement, 206, 233 structure equation first, 208 second, 208 zeroth, 232 chain, 91 singular, 96 charge, see electric charge CHRISTI, 236 Christoffcl symbols, 230 CLEAR, 54 closure relation, 218, 274, 275 cobasis, 24, see coframc codiffercnt,ial, 230 COFRAME, 81 axially symmet,ric, 214 half-null, 214 Newman-Penrose, 215 spherically symmetric Minkowski spacetime, 81 Ricmann spacetime, 82 coframc, 70, see cobasis accelerating and rotating, 366, 367 axially symmetric, 379 dimension of, 111 foliation compatible, 118 noninertial, 36G cohomology, 83 composite ferinion, 390 conductaace fundamental, 157 conductivity, 338 CONN1, 208, 236 conncct,ion, 196 1-fornls, 197 Riemannian, 229
transposed, 203 constitutive law anisotropic media, 338 inertial frame, 361 laboratory frame, 351 moving medium, 352 of Minkowski, 353 simple medium, 338 tensor of spacetime, 246 coordinate chart, 58 coordinates, 58 accelerating, 365 Cartesian, 20 comoving, 348 Lagrange, 348 noninertial, 366 rotating, 362 cosinological constant, 376 Coulomb potential, 378 covariant differential, 198 differentiation, 196 of geometric quantity, 200 exterior derivative, 207 Lie derivative, 208 covector, 24 CURR3, 144 current, see electric current CURV2, 208 curvature, 203 2-form, 203 geometrical meaning, 203 segmental, 209, 233 tensor, 204 cycle, 97 dlAlembertian, 231 de Rham coliomology groups, 85 complex, 97 map, 100 theorem, 100 first, 100
I
~ 1 I
i I
11
density, 22 tliff(:oinorphisni, 71 1-p~trnnlet,ergroup, 73, 180 diffcrentid~le manifold, 19, 57, 58 non-oricntablc, 59, G7 nonorientnble, G1 orientalIle, 59, 65 map, 71 strl~ctl~rc, 58 tlifferc?ntial,G2 forni, G2 tensor-valued, 28 twisted, 22, GG map, 71 t;wistccl form transformation law, 67 dilaton ficltl, 275 vanishing, 302 DILCURV2, 209 dimension, 110 absol~ltc,110 relative, 111 Dirac (?quation,388 Dir~lc-Maxwell syst,c:m, 388 distortion 1-form, 235 tlut~lityoperator, 219 closrlrc: condition, 274 Einstein 3-&1rm,227 Eilist,cin's ficltl oclr~ntion,376 Einst,cin's t,hcory of gravitation, 19, see gc>ncrnlroll~tivit~y (GR) Ei~istcin-Cnrt,t~~~ tlic?ory, 389 Einstein-Cart,nn--Dirnc-Maxwcll system, 359 Eillstein-Din~eh4;1vwcllsystenl, 389 EINSTEIN3, 227 clcctric clinrgc! bountl, 330 coi~scrvnt,io~i, 113 tlcnsit,y, 110
tlcnsity, 110 diincnsion of, 111 free, 330, 332 ~natcrial,332 constant, 316, see prrmitt ivity of v;Lculllll current, 112 3-form, 114 bountl, 330 tlcnsity, 112 tlinlensioii of, 114 free, 330 cxcit,atiol~ 2-form 'D, 119 tliincnsion of, 137 139 on a l~oti~ldnry, fic.ld strc~rigth1-form E , 123 line tension, 124 susc~ptibility,337 clrct,ric/inngn~ticreciprocity 3-fo1.111,167 of ciicrgy-mo111erit~1111 of spacctimr relation, 302 clt~ctromagnrtic rncrgy tlcnsity 3-forn~,175, 321, 333, 335
ciic.rgy-rnoinrntr~li~, 163, 332 (1 3)-dcconiposition, 175 Al~ral~am, 335, 341 canonical currcl~t,181 conservation law, 181, 31:) frer-charge., 333 ltinclnatir, 3-form. 1li.1 niatcrial, 334 Minkowski, 334, 335 syllllllctl~y,320 trnsor tlrnsity, 171 tracc. free., 167 clxcit.'I t 1011 ' 2-forin H, 1 IG tlimr~~sion of, 116 c x t c ~ i ~ 331 i~l, r, in m t ~ t t t ~330 ficltl. 141
+
Subject Index
ficltl strength 2-form F , 123 invaliants, 327 nlomt,l~tumdensity 3-foln~,175, 321, 339, 335 pot,ential 1-form A , 132 stress 2-folln, 175, 334, 335 ETAO, ETA1, ETA2, ETAS, 227 h l c r ch:tractcristic, 85, 101 Eulcr-L;~grangc cquat ion, 182 cvi~luation(Rcducc), 52 Excalc loatling, 54 EXCIT2, 144 vxcitation, src clcct ric, m:iglictic, c.lcctromagnc'tic expcrimclit of Jamc~s,342 of RontgCn, 358 of Walltcr & Walltc~,339 of Wilson k Wilson, 359 cxprc~ssio~i (Rcducc~), 50 ~ o o l c a l i52 , integer, 51 scnlitr, 51 ~ x t ico ~ dcl iviltivc. (Exc:tlc), 78 ])rol~(.rticxs, 68 difF(~rcl~tinl fol ni, (i2 tliff(\rcwtintioli, GX form, 29 closctl, 83 exact, 83 longitlitlinal, 117, 340 transversal, 117, 349 ~ ~ o d ~ 31 lct, (Excalc), 55 FARAD2, 144 FiIill'ild~y'~ ilitl~lctioli ~;Lw, 129 FDOMAIN, 79 ficltl strength, src c31cctric, niagnc.tic, clcctrolnagnrtic flux qttantuni, scc tnngl~cticflux quan-
fluxoitl, src liiaglictic flux quantum, 390 fol cc3-ft cc clcct~otlynamics,383 clcctl omtrgnctic field, 125 ~nttgnclicficltl, 125 FORCE4, 144 FRAME, 81 fr:~~li(l, 70, SCP basis accelerating and rotating, 367 diincllsion of, 111 foliation compatihlc, 118 holonoinic, 70 incrtial, 348 nonincrtial, 151, 3% normal, 199 F'rcsncl equation (oxtendcd), 267 (1 3)-tlcco~iiposition,270 functions (Rccluce) list of, 49
+
gauge ficlcl n~omcwtum,179 transformation, 22, 178 G a ~ ~law, s s 119, 136 gcllrr;tl r~litf.ivit~ (GR), 19, 211, 235 gconirtric q~li~lit,it,y, 27 Gi~izhurg-Lnndn~~ energy funct,ional, 387 Gratl-Shafrtu~ovequation, 384 IIall rc~sist,nncc,158, 1G0 Hcisenl)crg-Euler electrodynamics, 325 !cIotlge s t x , 225 in Excalc, 227 hotnology, 96 homotol>ic cqllivalcnce, 85 llypotl~csisof locality, 112, 348 idcal 2-tlilncnsional clectron gas, 157 colitluct,or, 137 suprrcontluctor, 140 IF, 52 int,cgr;rl, 87
Subject Index
of cxterior n-form, 89 of exterior p-form, 92 of twisted exterior n-form, 89 interior prodrlct, 33 (Excalc), 55 invariants of the elcctromagnetic field, 127 reciprocity transformation, 167 Jacobian determinant, 21, 65 jump conditions, 356 I<err-Newman solution, 379 Kiclin 3-form, 134, 135, 162 I
magnetic constant, 316, see permeability of vacuum excitation 1-form 3-1, 119 dimension of, 141 on a boundary, 140 field strength %form B, 123 flux, 124 and Aharonov-Bohm effect, 387 conservation, 129 flux quantum, 130 for electron, 158 half-integer, 130 helicity, 135, 386 monopole, 2 susceptibility, 337 magnetization, 330, 331, 353, 360 manifold, 57 metric-affine, 233 Riemann-Cartan, 232 Riemannian, 229 spacetime, 114 Weyl-Cartan, 233 matter current, 179 MAXENERGY3,185 MAXHOM3, 144 MAXINH3, 144 Maxwell sample program in Excalc, 184 Maxwell's equations, 144, 316 homogeneous, 132 in anholonomic coordinates, 152 in matter, 331 inhomogeneous, 116 Maxwell's theory axionis, 143 Mcaxwellian double plates, 136 Meissner cffect, 140 metric, 212 Minkowski, 211, 213, 295, 361 on space of 2-forms, 41, 219 optical, 336, 354 Riemannian, 229
Subject Index
Schonbcrg-Urbantket, 224 tensor, 212 fielcl, 229 three-spacc, 349 vector spacc, 212 NEQ, 52 Ncwton's gravitational constant, 376 NIL, 52 Nocthcr current, 180 NONMET1, 236 nonrnetricity, 232 1-form. 232 object of anholonomity, 71, 239 ohserver accelerating, 364 cornoving, 348 noninertial, 367 rotating, 363 Oersted-AnipBre law, 119, 140 Ohm-Ha11 law, 158 one-form, 23 image, 63 open set, 58 OPERATOR, 49 opcxrator (Reducc), 47 arithmcltic, 47 assignment, 49 infix, 47 logical ant1 rclational, 48 prcfix, 49 OR, 52 orientation, 28, 35, 65 inner, 87, 92 of a vector spacc, 35 outcr, 87, 93 stanclard, 36 parallel transport, 201 partial derivative (Excalc), 78 period, 99, 116, 120 permeability effective
magnetic, 319 of matter, 338 of vacuum, 316 tensor, 338 permittivity effective elcctric, 319 of matter, 338 of vacuum, 316 tensor, 338 PFORM, 55 Poincarh lemma, 83 polarization, 330, 331, 353, 360 POT1, 144 potential electric, 3-dim. scalar cp, 134 electromagnetic, 4-dim. 1-form A, 133 magnetic, 3-dim, covector A, 134 principal part of constitut,ivc tcnsor, 248, 256 projective plane, 60 homology groups, 99 pull-back map, 72
I I
l
QHE quantum Hall cffcct,, 155 Rcissiier-Nordstrom solution, 379 REPEAT, 52 Ricci 1-form, 204 Ricci identity, 208 RICCI1, 209 scalar density, 35 Sclionberg-Urk)antske formulas, 224 self-dual form, 45, 220 SIGNATURE, 81 simplex, 90 faces, 90 singular, 92 standard, 94 sillgular homology group, 96 skewon ficld, 254, 256, 304
1 1 1
Subject Indcx
ancl dissipation, 258 and light propagation, 270 spatially isotropic, 261 vanishing, 279, 302 sl
T, 52 Tamm-Rubilar tensor dcnsity, 267 tensor, 28 decomposable, 27 dcnsity, 35 transformation law, 26
twisted, 28 topological invariant, 84, 99 structure, 58 topology, 58 torsion, 201 2-form, 202 geometrical meaning, 201 tensor, 203 TORSION2, 208 torus, GO homology groups, 98 TRATOR2, 209 triangulation, 97 triplet of 2-forms completeness, 222 self-dual, 221 TVECTOR. 55 vacuum impedance, 248, 303, 315 variables (Reduce) unbound, 51 variational derivative, 179 vector, 23 contravariant, 25 covariant, 25 field, 62 integral curve, 74 of foliation, 117 position, 205 image, 63 length, 212 null, 213 spacc, 23 affinc, 205 dual, 23 oriented, 35 spacelike, 213 tangent, 61 timelikc, 213 transversal, 93 velocit,y angular, 362 mean matcrial, 348 of light, 316
Subject Index
one-form, 352 relative, 349 volume, 34 elcmcnt, 34 clcmcntary, 37 form, 64, 65 twisted 4-form, 217 wave operator, 231, 316 wave surface
quadratic, 287 quartic, 267 WAVETOCOFRAMEl, 231 wedge operator, 31 Weyl covector, 232 one-form, 232 Weyl's segmental (or dilational) curvature 2-form, 209 WHILE, 52
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5
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COLLET/ECKMANN. Iterated Maps on the Interval as Dynamical Systems ISBN 3-7643-3510-6 J A F F E ~ A U BVortices ES. and Monopoles, Structure of Static Gauge Thcorics ISBN 3-7643-3025-2 MANIN.Mathematics and Physics ISBN 3-7643-3027-9 ATW~~D/BJ~RKEN/BR~D~KY/STROYNOWSKI. Lectures on Lepton Nucleon Scattering and Quantum Chromodynamics ISBN 3-7643-3079- 1 DITA/GEORGE~CU/PURICE. Gauge Theories: Fundamental Interactions and Rigorous Results ISBN 3-7643-3095-3
FRAMFTON/GLASHOW/VAN DAM. Third Workshop on Grand Unification, 1982 ISBN 3-7643-3105-4 FROHLICH. Scaling and Self-similarity in Physics: Renormalization in Statistical Mechanics and Dynamics ISBN 3-7643-3168-2 M I L T O N ~ ~ A MWorkshop U E L . on Non-Perturbative Quantum Chromodynamics ISBN 3-7643-3127-5 LANGACKERISTEINHARDT/WELDON. Fourth Workshop on Grand Unification ISBN 3-7643-3169-0 FRITZIJAFFFJSZASZ. Statistical Physics and Dynamical Systems: Rigorous Results ISBN 3-7643-3300-6 C E A ~ ~ E ~ C ~ / C ~ ~ T A C H E / G ECritical O R G EPhenomena: SCU. 1983 Brasov School Conference ISBN 3-7643-3289-1 PIGUETISIROI~D. Renormalized Supersymmetry: The Perturbation Theory of N=l Supersymmetric Theories in Flat Space-Time ISBN 3-7643-3346-4 HARAISORCLYK. Functional Integration, Geometry and Strings: Proceedings of the XXV Karpacz Winter School of Theoretical Physics ISBN 3-7643-2387-6 S MIRNOV . Renormalization and Asymptotic Expansions ISBN 3-7643-2640-9 Leznov/Saveliev. Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems ISBN 3-7643-2615-8 MASLOV. The Complex WKB Method for Nonlinear Equations I: Linear Theory ISBN 3-7643-5088-1 BAYLIS. Electrodynamics: A Modern Geometric Approach ISBN 0-8 176-4025-8 ABLAMOWIC~~~A Clifford U S E RAlgebras . and their Applications in Mathematical Physics, Volume 1: Algebra and Physics ISBN 0-8176-4182-3 RYAN~SPR~~RIG. Clifford Algebras and their Applications in Mathematical Physics, Volume 2: Clifford Analysis ISBN 0-8176-4183-1 S TOLLMANN. Caught by Disorder: Bound States in Random Media ISBN 0-8 176-42 10-2 PETTERSLEVINEIWAMRSGANSS. Singularity Theory and Gravitational Lensing ISBN 0-8176-3668-4 CERCIGNANI. The Relativistic Boltzmann Equation: Theory and Applications ISBN 3-7643-6693-1 K A S H I W A R ~ ~MathPhys I W A . Odyssey 2001: Integrable Models and Beyond-In Honor of narry M. McCoy ISBN 0-8176-4260-9 CNOPS. An Introduction to Dirac Operators on Manifolds ISBN 0-8 176-4298-6 KLAINERMANINICOL~. The Evolution Problem in General Relativity ISBN 0-8176-4254-4 ~ L A N C H A R D / B R ~Mathematical ~ N I N G . Methods in Physics ISBN 0-8176-4228-5
WILLIAMS. Topics in Quantum Mechanics ISBN 0-8176-4311-7 OBOLASHVILI. Higher Order Partial Differential Equations in Clifford Analysis ISBN 0-8176-4286-2 CORDANI. The Kepler Problem: Group Theoretical Apects, Regularization and Quantization, with Applications to the Study of Perturbations ISBN 3-7643-6902-7 DUPLANTIER/RIVASSEAU. P o i n c d Seminar 2002: Vacuum Energy-Renormalization ISBN 3-7643-0579-7 RAKOTOMANANA. A Geometrical Approach to Thermomechanics of Dissipating Continua ISBN 0-8176-4283-8 TORRES DEL CASTILLO. 3-D Spinors, Spin-Weighted Functions and their Applications ISBN 0-8176-3249-2 HEHL/OBUKHOV. Foundations of Classical Electrodynamics: Charge, Flux, and Metric ISBN 3-7643-4222-6
