Foundations of classical electrodynamics [1 ed.] 9780817642228, 0817642226

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Foundations of Classi al Ele trodynami s1 2 ;

Friedri h W. Hehl & Yuri N. Obukhov 01 June 2001 1 Keywords:

ele trodynami s, ele tromagnetism, axiomati s, exterior al ulus, lassi al eld theory, ( oupling to) gravitation, omputer algebra.{ The book is written in Ameri an English (or what the authors on eive as su h). All footnotes in the book will be olle ted at the end of ea h Part before the referen es. The image reated by Glatzmaier (Fig. B.3.4) is in olor, possibly also Fig. B.5.1 on the aspe ts of the ele tromagneti eld. The present format of the book should be hanged su h as to allow for a 1-line display of longer mathemati al formulas. 2 The book is written in Latex. Our master le is birk.tex. The dierent part of the books orrespond to the les partI.tex, partA.tex,

partB.tex,

partC.tex,

partD.tex,

partE.tex,

partO.tex. As of today, only a trun ated version is available of the outlook hapter on the le partO.trun .tex. The ompleted version will be handed in later as le partO.tex. { The gures are on separate les. Their names are presently given on the rst page of the text of ea h part of an outprint. For some of the gures we have extra les available with a higher resolution whi h are, however, not atta hed to the draft version of the book. Follow up of ommands on a Unix system: d birk, latex birk (2 times), makeindex birk, latex birk, dvips -f birk > birk.ps, gv birk.ps&.

Contents

Prefa e . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . 8 Five plus one axioms . . . . . . . . . . . . . . . . Topologi al approa h . . . . . . . . . . . . . . . . Ele tromagneti spa etime relation as fth axiom Ele trodynami s in matter and the sixth axiom . List of axioms . . . . . . . . . . . . . . . . . . . . A reminder: Ele trodynami s in 3-dimensional Eu lidean ve tor al ulus . . . . . . . . . . . On the literature . . . . . . . . . . . . . . . . . .

Referen es

8 10 11 13 13 13 16

19

A Mathemati s: Some exterior al ulus

26

Why exterior dierential forms?

27

A.1 Algebra

33

A.1.1 A real ve tor spa e   and its dual . . . . . . . . . 33 A.1.2 Tensors of type pq . . . . . . . . . . . . . . . . . 37

vi

Contents A.1.3 A generalization of tensors: geometri quantities A.1.4 Almost omplex stru ture . . . . . . . . . . . . . A.1.5 Exterior p-forms . . . . . . . . . . . . . . . . . . A.1.6 Exterior multipli ation . . . . . . . . . . . . . . A.1.7 Interior multipli ation of a ve tor with a form . . A.1.8 Volume elements on a ve tor spa e, densities, orientation . . . . . . . . . . . . . . . . . . . . . . A.1.9 Levi-Civita symbols and generalized Krone ker deltas . . . . . . . . . . . . . . . . . . . . . . . . A.1.10The spa e M 6 of two-forms in four dimensions . A.1.11Almost omplex stru ture on M 6 . . . . . . . . . A.1.12 Computer algebra . . . . . . . . . . . . . . . . .

A.2

Exterior al ulus

A.3

Integration on a manifold

38 40 41 43 46 47 51 55 60 63 77

A.2.1 Dierentiable manifolds . . . . . . . . . . . . . 78 A.2.2 Ve tor elds . . . . . . . . . . . . . . . . . . . . 82 A.2.3 One-form elds, dierential p-forms . . . . . . . 83 A.2.4 Images of ve tors and one-forms . . . . . . . . . 84 A.2.5 Volume forms and orientability . . . . . . . . . 87 A.2.6 Twisted forms . . . . . . . . . . . . . . . . . . 88 A.2.7 Exterior derivative . . . . . . . . . . . . . . . . . 90 A.2.8 Frame and oframe . . . . . . . . . . . . . . . . 93 A.2.9 Maps of manifolds: push-forward and pull-ba k 95 A.2.10 Lie derivative . . . . . . . . . . . . . . . . . . . 97 A.2.11Ex al , a Redu e pa kage . . . . . . . . . . . . . 104 A.2.12 Closed and exa t forms, de Rham ohomology groups . . . . . . . . . . . . . . . . . . . . . . . . 108 113

A.3.1 Integration of 0-forms and orientability of a manifold . . . . . . . . . . . . . . . . . . . . . . . . . 113 A.3.2 Integration of n-forms . . . . . . . . . . . . . . . 114 A.3.3 Integration of p-forms with 0 < p < n . . . . . . 116 A.3.4 Stokes's theorem . . . . . . . . . . . . . . . . . . 121 A.3.5 De Rham's theorems . . . . . . . . . . . . . . . 124

Referen es

131

Contents

B

vii

Axioms of lassi al ele trodynami s

136

B.1 Ele tri harge onservation

139

B.2 Lorentz for e density

153

B.3 Magneti ux onservation

161

B.1.1 Counting harges. Absolute and physi al dimension . . . . . . . . . . . . . . . . . . . . . . . . . 139 B.1.2 Spa etime and the rst axiom . . . . . . . . . . 145 B.1.3 Ele tromagneti ex itation . . . . . . . . . . . . 147 B.1.4 Time-spa e de omposition of the inhomogeneous Maxwell equation . . . . . . . . . . . . . . . . . . 148 B.2.1 Ele tromagneti eld strength . . . . . . . . . . 153 B.2.2 Se ond axiom relating me hani s and ele trodynami s . . . . . . . . . . . . . . . . 155 B.2.3 The rst three invariants of the ele tromagneti eld . . . . . . . . . . . . . . . . . . . . . . . . . 157 B.3.1 B.3.2 B.3.3 B.3.4

Third axiom . . . . . . . . . . . . . . . . Ele tromagneti potential . . . . . . . . . Abelian Chern-Simons and Kiehn 3-forms Measuring the ex itation . . . . . . . . .

. . . .

. . . .

. . . .

. 161 . 166 . 167 . 169

B.4 Basi lassi al ele trodynami s summarized, example 177

B.4.1 Integral version and Maxwell's equations . . . . 177 B.4.2 Jump onditions for ele tromagneti ex itation and eld strength . . . . . . . . . . . . . . . . . . 183 B.4.3 Arbitrary lo al non-inertial frame: Maxwell's equations in omponents . . . . . . . . . . . . . . . . . 184 B.4.4 Ele trodynami s in atland: 2-dimensional ele tron gas and quantum Hall ee t . . . . . . . . . 186

B.5Energy-momentum urrent and a tion

199

B.5.1 Fourth axiom: lo alization of energy-momentum 199 B.5.2Properties of energy-momentum, ele tri -magneti re ipro ity . . . . . . . . . . . . . . . . . . . . . . 202 B.5.3Time-spa e de omposition of energy-momentum . 212

viii

Contents B.5.4 A tion . . . . . . . . . . . . . . . . . . . . . . . 214 B.5.5 Coupling of the energy-momentum urrent to the oframe . . . . . . . . . . . . . . . . . . . . . 218 B.5.6 Maxwell's equations and the energy-momentum

urrent in Ex al . . . . . . . . . . . . . . . . . . 222 Referen es

227

C More mathemati s C.1

232

Linear onne tion

233

C.1.1 C.1.2 C.1.3

Covariant dierentiation of tensor elds . . . . . 234 Linear onne tion 1-forms . . . . . . . . . . . . . 236

Covariant dierentiation of a general geometri quantity . . . . . . . . . . . . . . . . . . . . . . . 239 C.1.4 Parallel transport . . . . . . . . . . . . . . . . . 240 C.1.5 Torsion and urvature . . . . . . . . . . . . . . 241 C.1.6 Cartan's geometri interpretation of torsion and

urvature . . . . . . . . . . . . . . . . . . . . . . 246 C.1.7 Covariant exterior derivative . . . . . . . . . . 248 C.1.8 The -forms o(a), onn1(a,b), torsion2(a), urv2(a,b)250

p

C.2

Metri

C.2.1 Metri ve tor spa es . . . . . . . . . . . . . . . C.2.2 Orthonormal, half-null, and null frames, the

oframe statement . . . . . . . . . . . . . . . . C.2.3 Metri volume 4-form . . . . . . . . . . . . . . C.2.4 Duality operator for 2-forms as a symmetri almost omplex stru ture on 6 . . . . . . . . . . C.2.5 From the duality operator to a triplet of omplex 2-forms . . . . . . . . . . . . . . . . . . . . . . . C.2.6 From the triplet of omplex 2-forms to a duality operator . . . . . . . . . . . . . . . . . . . . . . C.2.7 From a triplet of omplex 2-forms to the metri : S honberg-Urbantke formulas . . . . . . . . . . C.2.8 Hodge star and Ex al 's # . . . . . . . . . . . C.2.9 Manifold with a metri , Levi-Civita onne tion

M

253

. 254 . 256 . 260 . 262 . 264 . 266 . 269 . 271 . 275

Contents

ix

C.2.10Codierential and wave operator, also in Ex al 277 C.2.11 Nonmetri ity . . . . . . . . . . . . . . . . . . . 279 C.2.12 Post-Riemannian pie es of the onne tion . . . 281 C.2.13 Ex al again . . . . . . . . . . . . . . . . . . . . 285 Referen es

291

D The Maxwell-Lorentz spa etime relation D.1

Linearity between

surfa e

D.1.1 D.1.2 D.1.3 D.1.4

H

and

F

293

and quarti wave

Linearity . . . . . . . . . . . . . Extra ting the Abelian axion . . Fresnel equation . . . . . . . . . Analysis of the Fresnel equation

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

D.2

Ele tri -magneti re ipro ity swit hed on

D.3

Symmetry swit hed on additionally

. . . .

. . . .

295

. 295 . 298 . 300 . 305 311

D.2.1 Re ipro ity implies losure . . . . . . . . . . . . 311 D.2.2 Almost omplex stru ture . . . . . . . . . . . . . 313 D.2.3 Algebrai solution of the losure relation . . . . 314 D.3.1 Lagrangian and symmetry . . . . . . . . . . . D.3.2 Duality operator and metri . . . . . . . . . . D.3.3 Algebrai solution of the losure and symmetry relations . . . . . . . . . . . . . . . . . . . . . . D.3.4 From a quarti wave surfa e to the light one .

D.4

317

. 317 . 319 . 320 . 326

Extra ting the onformally invariant part of

the metri by an alternative method

333

D.4.1 Triplet of self-dual 2-forms and metri . . . . . 334 D.4.2 Maxwell-Lorentz spa etime relation and Minkowski spa etime . . . . . . . . . . . . . . . . . . . . . . 337 D.4.3 Hodge star operator and isotropy . . . . . . . . . 338 D.4.4 Covarian e properties . . . . . . . . . . . . . . 340

x

Contents

D.5

Fifth axiom

345

Referen es

347

E Ele trodynami s in va uum and in matter 352 E.1 Standard Maxwell{Lorentz theory in va uum

355

E.1.1 Maxwell-Lorentz equations, impedan e of the va uum . . . . . . . . . . . . . . . . . . . . . . . . . 355 E.1.2 A tion . . . . . . . . . . . . . . . . . . . . . . . 357 E.1.3 Foliation of a spa etime with a metri . Ee tive permeabilities . . . . . . . . . . . . . . . . . . . . 358 E.1.4 Symmetry of the energy-momentum urrent . . . 360

E.2 Ele tromagneti spa etime relations beyond lo ality and linearity

E.2.1 E.2.2 E.2.3 E.2.4 E.2.5 E.3

Keeping the rst four axioms xed

Mashhoon . . . . . . . . . . . . . Heisenberg-Euler . . . . . . . . . .

Born-Infeld . . . . . . . . . . . .

Plebanski . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

Ele trodynami s in matter, onstitutive law

E.3.1 E.3.2 E.3.3 E.3.4 E.3.5

Splitting of the urrent: Sixth axiom . Maxwell's equations in matter . . . . Linear onstitutive law . . . . . . . . Energy-momentum urrents in matter

Experiment of Walker & Walker . .

. . . . .

. . . . .

. . . . .

369

. . . . .

. 369 . 371 . 372 . 373 . 379

E.4.1 Laboratory and material foliation . . . . . . . E.4.2 Ele tromagneti eld in laboratory and material frames . . . . . . . . . . . . . . . . . . . . . . . E.4.3 Opti al metri from the onstitutive law . . . . E.4.4 Ele tromagneti eld generated in moving ontinua . . . . . . . . . . . . . . . . . . . . . . . .

. 383

E.4 Ele trodynami s of moving ontinua

. . . . .

363

. 363 . 364 . 365 . 366 . 367

383

. 387 . 391 . 392

Contents

xi

E.4.5 E.4.6 E.4.7 E.4.8 E.4.9

The experiments of Rontgen and Wilson & Wilson396 Non-inertial \rotating oordinates" . . . . . . . . 401 Rotating observer . . . . . . . . . . . . . . . . . 403 A

elerating observer . . . . . . . . . . . . . . . 405 The proper referen e frame of the noninertial observer (\noninertial frame") . . . . . . . . . . . 408 E.4.10 Universality of the Maxwell-Lorentz spa etime relation . . . . . . . . . . . . . . . . . . . . . . . 410

Referen es

413

Preliminary sket h version of Validity of las-

F

si al ele trodynami s, intera tion with gravity, outlook F.1

Classi al physi s

F.1.1 F.1.2 F.1.3 F.1.4

F.2

416

(preliminary)

Gravitational eld . . . . . . . . . . . . . . Classi al (1st quantized) Dira eld . . . . Topology and ele trodynami s . . . . . . . Remark on possible violations of Poin are varian e . . . . . . . . . . . . . . . . . . . .

Quantum physi s

(preliminary)

. . . . . . in. .

419

. 419 . 430 . 432 . 435 437

F.2.1 QED . . . . . . . . . . . . . . . . . . . . . . . . 437 F.2.2 Ele tro-weak uni ation . . . . . . . . . . . . . . 438 F.2.3 Quantum Chern-Simons and the QHE . . . . . . 440 Referen es

441

Index

444

Contents

1

le birk/partI.tex, with gures [I01 over.eps, I02 over.eps℄, 200106-01

Des ription of the book Ele tri and magneti phenomena are omnipresent in modern life. Their non-quantum aspe ts are su

essfully des ribed by

lassi al ele trodynami s (Maxwell's theory). In this book, whi h is an outgrowth of a physi s graduate ourse, the fundamental stru ture of lassi al ele trodynami s is presented in the form of six axioms: (1) ele tri harge onservation, (2) existen e of the Lorentz for e, (3) magneti ux onservation, (4) lo alization of ele tromagneti energy-momentum, (5) existen e of an ele tromagneti spa etime relation, and (6) splitting of the ele tri

urrent in material and external pie es. The rst four axioms are well-established. For their formulation an arbitrary 4-dimensional dierentiable manifold is required whi h allows for a foliation into 3-dimensional hypersurfa es. The fth axiom hara terizes the environment in whi h the ele tromagneti eld propagates, namely spa etime with or without gravitation. The relativisti des ription of su h general environments remains a resear h topi of onsiderable interest. In parti ular, it is only in this fth axiom that the metri tensor of spa etime makes its appearan e, thus oupling ele tromagnetism and gravitation. The operational interpretation of the physi al notions introdu ed is stressed throughout. In parti ular, the ele trodynami s of moving matter is developed ab initio. The tool for formulating the theory is the al ulus of exterior dierential forms whi h is explained in suÆ ient detail, in luding the orresponding omputer algebra programs. This book presents a fresh and original exposition of the foundations of lassi al ele trodynami s in the tradition of the so- alled metri -free approa h. The reader will win a new outlook on the interrelationship and the inner working of Maxwell's equations and their raison d'^etre.

2

Contents

Book Announ ement (old version of 1999):

Ele tri and magneti phenomena play an important role in the natural s ien es. They are des ribed by means of lassi al ele trodynami s, as formulated by Maxwell in 1864. As long as the ele tromagneti eld is not quantized, Maxwell's equations provide a orre t des ription of ele tromagnetism. In this work the foundations of lassi al ele trodynami s are displayed in a onsistent axiomati way. The presentation is based on the simple but far-rea hing axioms of ele tri harge

onservation, the existen e of the Lorentz for e, magneti ux

onservation, and the lo alization of energy-momentum. While the rst four axioms above are well-established, they are insuf ient to omplete the theory. The missing ingredient is a hara terization of the environment in whi h the ele tromagneti eld propagates by means of onstitutive relations. This environment is spa etime with or without a material medium and with or without gravitation ( urvature, perhaps torsion). The relativisti des ription of su h general environments remains a resear h topi of onsiderable interest. In parti ular, it is only in this last axiom that the spa etime metri tensor makes its appearan e, thus oupling ele tromagnetism and gravitation. The appropriate tool for this theory is the al ulus of exterior dierential forms, whi h is introdu ed before the axioms of ele trodynami s are formulated. The operational interpretation of the physi al notions introdu ed is stressed throughout. The book may be used for a ourse or seminar in theoreti al ele trodynami s by advan ed undergraduates and graduate students in mathemati s, physi s, and ele tri al engeneering. Its approa h to the fundamentals of lassi al ele trodynami s will also be of interest to resear hers and instru tors as well in the above-mentioned elds. ==================

Contents

3

B

E

Figure 1: Book over alternative 1: Faraday's indu tion law, with drawing program.

4

Contents

B

E

Figure 2: Book over alternative 2: Faraday's indu tion law, hand-drawn.

Contents

5

Prefa e In this book we will display the fundamental stru ture underlying lassi al ele trodynami s, i.e., the phenomenologi al theory of ele tri and magneti ee ts. The book an be used as supplementary reading during a graduate ourse in theoreti al ele trodynami s for physi s and mathemati s students and, perhaps, for some advan ed ele tri al engineering students. Our approa h rests on the metri -free integral formulation of the onservation laws of ele trodynami s in the tradition of F. Kottler (1922), E. Cartan (1923), and D. van Dantzig (1934), and we stress, in parti ular, the axiomati point of view. In this manner we are led to an understanding of why the Maxwell equations have their spe i form. We hope that our book an be seen in the lassi al tradition of the book by E.J. Post (1962) on the Formal stru ture of ele tromagneti s and of the hapter on Charge and magneti ux of the en y lopedi arti le on lassi al eld theories by C. Truesdell and R.A. Toupin (1960), in luding R.A. Toupin's Bressanone le tures (1965); for the exa t referen es see the end of the Introdu tion on page . The manner in whi h ele trodynami s is onventionally presented in physi s ourses a la R. Feynman (1962) or J.D. Ja kson (1999) is distin tively dierent, sin e it is based on a at spa etime manifold, i.e., on the (rigid) Poin are group, and on H.A. Lorentz's approa h (1916) to Maxwell's theory by means of his theory of ele trons. We believe that the approa h of this book is appropriate and, in our opinion, even superior for a good understanding of the stru ture of ele trodynami s as a lassi al eld theory. In parti ular, if gravity annot be negle ted, our framework allows for a smooth and trivial transition to the urved (and ontorted) spa etime of general relativisti eld theories. Mathemati ally, integrands in the onservation laws are represented by exterior dierential forms. Therefore exterior al ulus is the appropriate language in whi h ele trodynami s should be spelled out. A

ordingly, we will ex lusively use this formalism (even in our omputer algebra programs whi h we will introdu e in Se . A.1.12.). In an introdu tory Part A, and later in Part C,

6

Contents

we try to motivate and to supply the ne essary mathemati al framework. Readers who are familiar with this formalism may want to skip these parts. They ould start right away with the physi s in Part B and then turn to Part D and Part E. In Part B four axioms of lassi al ele trodynami s are formulated and the onsequen es derived. This general framework has to be ompleted by a spe i ele tromagneti spa etime relation as fth axiom. This will be done in Part D. The Maxwell-Lorentz approa h is then re overed under spe i onditions. In Part E, we will apply ele trodynami s to moving ontinua, inter alia, whi h requires a sixth axiom on the formulation of ele trodynami s inside matter. These notes grew out of a s ienti ollaboration with the late Dermott M Crea (University College Dublin). Mainly in Part A and Part C Dermott's handwriting an still be seen in numerous pla es. There are also some ontributions to `our' mathemati s from Wojtek Kop zynski (Warsaw University). At Cologne University in the summer term of 1991, Martin Zirnbauer started to tea h the theoreti al ele trodynami s ourse by using the al ulus of exterior dierential forms, and he wrote up su

essively improved notes to his ourse. One of the authors (FWH) also taught this ourse three times, partly based on M. Zirnbauer's notes. This in uen ed our way of presenting ele trodynami s (and, we believe, also his way). We are very grateful to M. Zirnbauer for many dis ussions. There are many olleagues and friends who helped us in riti ally reading parts of our book and who made suggestions for improvement or who ommuni ated to us their own ideas on ele trodynami s. We are very grateful to all of them: Carl Brans (New Orleans), David Hartley (Adelaide), Yakov Itin (Jerusalem), Martin Janssen (Cologne), Gerry Kaiser (Glen Allen, Virginia), R.M. Kiehn (formerly Houston), Attay Kovetz (Tel Aviv), Claus Lammerzahl (Konstanz/Dusseldorf), Bahram Mashhoon (Columbia, Missouri), E kehard Mielke (Mexi o City), E. Jan Post (Los Angeles), Dirk Putzfeld (Cologne), Guillermo Rubilar (Cologne), Yasha Shnir (Cologne), Andrzej Trautman (Warsaw), Wolfgang Weller (Leipzig), and others.

Contents

7

We are very obliged to Uwe Essmann (Stuttgart) and to Gary Glatzmaier (Santa Cruz, California) for providing beautiful and instru tive images. We are equally grateful to Peter S herer (Cologne) for his permission to reprint his three omi s on omputer algebra. Please let us know riti al remarks to our approa h or the dis overy of mistakes by surfa e or ele troni mail (hehlthp.unikoeln.de, yothp.uni-koeln.de). This proje t has been supported by the Alexander von HumboldtFoundation (Bonn), the German A ademi Ex hange Servi e DAAD, and the Volkswagen-Foundation (Hanover). We are very grateful for the unbureau rati help of these institutions. Cologne and Mos ow, July 2001 Friedri h W. Hehl & Yuri N. Obukhov

8

Contents

Introdu tion

Five plus one axioms In this book we will display the stru ture underlying lassi al ele trodynami s. For this purpose we will formulate six axioms: Conservation of ele tri harge ( rst axiom), existen e of the Lorentz for e (se ond axiom), onservation of magneti ux (third axiom), lo al energy-momentum distribution (fourth axiom), existen e of an ele tromagneti spa etime relation ( fth axiom), and, eventually, the splitting of the ele tri urrent in material and external pie es (sixth axiom). The axioms of the onservation of ele tri harge and magneti ux will be formulated as integral laws, whereas the axiom for the Lorentz for e is represented by a lo al expression basi ally de ning the ele tromagneti eld strength F = (E; B ) as for e per unit harge and thereby linking ele trodynami s to me hani s; here E is the ele tri and B the magneti eld strength. Also the energy-momentum distribution is spe i ed as a lo al law. The Maxwell-Lorentz spa etime relation, that we will use, is, as an axiom, not so unquestionable as the rst four axioms and non-lo al and non-linear alternatives will be mentioned. We want to stress the fundamental nature of the rst axiom. Ele tri harge onservation is experimentally rmly established. It is valid for single elementary parti le pro esses (like the -de ay, n p+ + e +  , for instan e, with n as neutron, p as proton, e as ele tron, and  as ele tron anti-neutrino). In other words, it is a mi ros opi law valid without any known ex eption. A

ordingly, it is basi to ele trodynami s to assume a new type of entity, alled ele tri harge, arrying positive or negative sign, with its own physi al dimension, independent of the

lassi al fundamental variables mass, length, and time. Furthermore, ele tri harge is onserved. In an age in whi h single ele trons and (anti-)protons are ounted and aught in traps, this law is so deeply ingrained in our thinking that its expli it formulation as a fundamental law (and not only as a onsequen e of Maxwell's equations) is often forgotten. We will show that this

!

Contents

9

rst axiom yields the inhomogeneous Maxwell equation together with a de nition of the ele tromagneti ex itation H = (H; D); here H is the magneti ex itation (`magneti eld') and D the ele tri ex itation (`diele tri displa ement'). The ex itation H is a mi ros opi eld of an analogous quality as the eld strength F . There exist operational de nitions of the ex itations D and H (via Maxwellian double plates or a ompensating super ondu ting wire, respe tively). The se ond axiom of the Lorentz for e, as mentioned above, leads to the notion of the eld strength and is thereby exhausted. Thus we need further axioms. The only onservation law whi h

an be naturally formulated in terms of the eld strength, is the onservation of magneti ux (lines). This third axiom has the homogeneous Maxwell equation as a onsequen e, that is, Faraday's indu tion law and the vanishing divergen e of the magneti eld strength. Magneti monopoles are alien to the stru ture of the axiomati s we are using. Moreover, with the help of these rst three axioms, we are led, not ompletely uniquely, however, to the ele tromagneti energy-momentum urrent (fourth axiom), subsuming the energy and momentum densities of the ele tromagneti eld and their orresponding uxes, and to the a tion of the ele tromagneti eld. In this way, the basi stru ture of ele trodynami s is set up in luding the omplete set of Maxwell's equations. For making this set of ele trodynami equations well{determined, we still have to add the fth axiom. Let us ome ba k to the magneti monopoles. In our axiomati framework, a lear asymmetry is built in between ele tri ity and magnetism in the sense of Oersted and Ampere that magneti ee ts are reated by moving ele tri harges. This asymmetry is hara teristi for and intrinsi to Maxwell's theory. Therefore the onservation of magneti ux and not that of magneti

harge is postulated as third axiom. If one spe ulated on a possible violation of the third axiom, i.e., introdu ed elementary magneti harges, so- alled magneti monopoles, then | in our framework | there would be no reason to believe any longer in ele tri harge onservation either. If the third axiom is vi-

10

Contents

olated, why should then the rst axiom, ele tri harge onservation, be untou hable? In other words, if ever a magneti monopole1 is found, our axiomati s has to be given up. Or, to formulate it more positively: Not long ago, He [20℄, Abbott et al. [1℄, and Kalb eis h et al. [29℄ determined experimentally new improved limits for the non-existen e of (Abelian or Dira ) magneti monopoles. This in reasing a

ura y in the ex lusion of magneti monopoles speaks in favor of the axiomati approa h in Part B.

Topologi al approa h Sin e the notion of metri is a ompli ated one, whi h requires measurements with lo ks and s ales, generally with

rigid

bod-

ies, whi h themselves are systems of great omplexity, it seems undesirable to take metri as fundamental, parti ularly for phenomena whi h are simpler and a tually independent of it.

E. Whittaker (1953) The distin tive feature of this type of axiomati approa h is that one only needs minimal assumptions about the stru ture of the spa etime in whi h these axioms are formulated. For the rst four axioms, a 4-dimensional dierentiable manifold is required whi h allows for a foliation into 3-dimensional hypersurfa es. Thus no onne tion and no metri are expli itly introdu ed in Parts A and B. This minimalisti topologi al type of approa h may appear

ontrived at a rst look. We should re ognize, however, that the metri of spa etime, in the framework of general relativity theory, represents the gravitational potential and, similarly, the

onne tion of spa etime (in the viable Einstein-Cartan theory of gravity, e.g.) is intri ately linked to gravitational properties of matter. We know that we really live in a urved and, perhaps, ontorted spa etime. Consequently our desire should be to formulate the foundations of ele trodynami s su h that the 1 Our arguments refer only to Abelian gauge theory. In non-Abelian gauge theories the situation is totally dierent. There monopoles are a must.

Contents

11

metri and the onne tion don't intervene or intervene only in the least possible way. When we know that the gravitational eld permeates all our laboratories in whi h we make experiments with ele tri ity, all the more we should take are that this ever present eld doesn't enter the formulation of the rst prin iples of ele trodynami s. In other words, a lear separation between pure ele trodynami ee ts and gravitational ee ts is desirable and an, indeed, be a hieved by means of the axiomati approa h to be presented in Part B. Eventually, in the spa etime relation, see Part D, the metri does enter. The power of the topologi al approa h is also learly indi ated by its ability to des ribe the phenomenology (at low frequen ies and large distan es) of the quantum Hall ee t su

essfully (not, however, its quantization, of ourse). Insofar as the ma ros opi aspe ts of the quantum Hall ee t an be approximately understood in terms of a 2-dimensional ele tron gas, we an start with (1+2)-dimensional ele trodynami s, the formulation of whi h is straightforward in our axiomati s. It is then a sheer nger exer ise to show that in this spe i ase of 1+2 dimensions there exists a linear onstitutive law that doesn't require a metri . As a onsequen e the a tion is metri -free, too. Thus the formulation of the quantum Hall ee t by means of a topologi al (Chern-Simons) Lagrangian is imminent in our way of looking at ele trodynami s.

Ele tromagneti spa etime relation as fth axiom Let us now turn to that domain where the metri does enter the 4-dimensional ele trodynami al formalism. When the Maxwellian stru ture, in luding the Lorentz for e and the a tion, is set up, it does not represent a on rete physi al theory yet. What is missing is the ele tromagneti spa etime relation linking the ex itation to the eld strength, i.e., D = D(E ; B ), H = H(B; E ), or, written 4-dimensionally, H = H (F ). Trying the simplest, we assume linearity between ex itation H and eld strength F , that is, H = (F ) , with the linear operator . Together with two more \te hni al" assumptions,

12

Contents

namely that  ful lls losure and symmetry { these properties will be dis ussed in detail in Part D { we will be able to derive the metri of spa etime from H = (F ) up to an arbitrary ( onformal) fa tor. A

ordingly, the light one stru ture of spa etime is a onsequen e of a linear ele tromagneti spa etime relation with the additional properties of losure and symmetry. In this sense, the light ones are derived from ele trodynami s: Ele trodynami s doesn't live in a preformed rigid Minkowskian spa etime. It rather has an arbitrary (1 + 3)-dimensional spa etime manifold as habitat whi h, as soon as a linear spa etime relation with losure and symmetry is supplied, will be equipped with lo al light ones everywhere. With the light ones it is possible to de ne the Hodge star operator ? that an map p-forms to (4 p)-forms and, in parti ular, H to F a

ording to H  ? F . Thus, in the end, that property of spa etime whi h des ribes its lo al ` onstitutive' stru ture, namely the metri , enters the formalism of ele trodynami s and makes it into a omplete theory. One merit of our approa h is that it doesn't matter whether it is the rigid, i.e., at Minkowskian metri as in spe ial relativity, or a ` exible' Riemannian metri eld whi h hanges from point to point a

ording to Einstein's eld equation as in general relativitisti gravity theory. In this way, the traditional dis ussion of how to translate ele trodynami s from spe ial to general relativity loses its sense: The Maxwell equations remain the same, i.e., the exterior derivatives (the ` ommas' in oordinate language) are kept and are not substituted by something ` ovariant' (the `semi olons'), and the spa etime relation H =  ? F looks the same ( is a suitable fa tor). However, the Hodge star `feels' the dieren e in referring either to a onstant or to a spa etime dependent metri , respe tively, see [46, 21℄. Our formalism an a

omodate generalizations of lassi al ele trodynami s simply by suitably modifying the fth axiom, while keeping the rst four axioms as indispensable. The Heisenberg-Euler and the Born-Infeld ele trodynami s are prime examples of su h possible modi ations. The spa etime relation be omes a non-linear, but still lo al expression.

Contents

13

Ele trodynami s in matter and the sixth axiom

Eventually, we have to fa e the problem of formulating ele trodynami s inside matter. We odify our orresponding approa h in the sixth axiom. The total ele tri urrent, entering as sour e the inhomogeneous Maxwell equation, is split into a onserved pie e arried by matter (`bound harge') and into an external manipulable pie e (`free harge'). In this way, following Truesdell & Toupin [57℄, see also the textbook of Kovetz [31℄, we an develop a onsistent theory of ele trodynami s in matter. For simple ases, we an amend the axioms by a linear onstitutive law. We believe that the onventional theory of ele trodynami s inside matter needs to be redesigned. In order to demonstrate the ee tiveness of our formalism, we apply it to the ele trodynami s of moving matter, thereby

oming ba k to the post-Maxwellian time of the 1880's when the relativisti version of Maxwell's theory had won momentum. In this ontext, we dis uss and analyse the experiments of RontgenEi henwald, Wilson & Wilson, and Walker & Walker. List of axioms

1. Conservation of ele tri harge: (B.1.17). 2. Lorentz for e density: (B.2.6). 3. Conservation of magneti ux: (B.3.1). 4. Lo alization of energy-momentum: (B.5.8). 5. Maxwell-Lorentz spa etime relation: (D.5.7). 6. Splitting of the ele tri urrent in a onserved matter pie e and an external pie e: (E.3.1) and (E.3.2). A reminder: Ele trodynami s in 3-dimensional Eu lidean ve tor al ulus

Before we will start to develop ele trodynami s in 4-dimensional spa etime in the framework of the al ulus of exterior dierential

14

Contents

forms, it may be useful to remind ourselves of ele trodynami s in terms of onventional 3-dimensional Eu lidean ve tor al ulus. We begin with the laws obeyed by ele tri harge and urrent. ~ = (Dx ; Dy ; Dz ) denotes the ele tri ex itation eld (hisIf D tori ally `diele tri displa ement') and  the ele tri harge density, then the integral version of the Gauss law, ` ux of D through any losed surfa e' equals `net harge inside', reads

Z

~ = ~
df D

Z

(1)

 dV ;

M1

V

V

~ as area and dV as volume element. The Oersted-Ampere with df ~ = (Hx ; Hy ; Hz ) (hislaw with the magneti ex itation eld H tori ally `magneti eld') and the ele tri urrent density ~j = (jx ; jy ; jz ) is a bit more involved be ause of the presen e of the ~ Maxwellian ele tri ex itation urrent:  The ` ir ulation of H d ~ through surfa e around any losed ontour' equals ` dt ux of D



spanned by ontour ' plus ` ux of ~j through surfa e' (t =time):

I S

0

~ = d  ~
dr H dt

Z

1 ~ A+ ~
df D

S

Z

~ : ~j
df

(2)

M2

S

~ is the ve torial line element. Here the dot
always deHere dr notes the 3-dimensional metri dependent s alar produ t, S denotes a 2-dimensional spatial surfa e, V a 3-dimensional spatial volume, and S and V the respe tive boundaries. Later we will re ognize that both, (1) and (2), an be derived from the harge

onservation law. The homogeneous Maxwell equations are formulated in terms ~ = (Ex ; Ey ; Ez ) and the magneti of the ele tri eld strength E ~ = (Bx ; By ; Bz ). They are de ned operationally eld strength B via the expression of the Lorentz for e F~ . An ele tri ally harged parti le with harge q and velo ity ~v experien es the for e ~ + ~v  B ~): F~ = q (E

(3)

M3

Contents

15

Here the ross  denotes the 3-dimensional ve tor produ t. Then Faraday's indu tion law in its integral version, namely` ir~ around any losed ontour' equals `minus d ux

ulation of E dt  ~ through surfa e spanned by ontour ' reads: of B

I S

0 1 Z ~A: ~ = d  B~
df E~
dr dt

(4)

M4

S

Note the minus sign on its right hand side whi h is hosen a

ording to the Lenz rule (following from energy onservation). ~ through any losed surfa e' equals Eventually, the ` ux of B `zero', that is, Z ~ = 0: B~
df (5)

M5

V

Also (4) and (5) are inherently related. Later we will nd the law of magneti ux onservation { and (4) and (5) just turn out to be onsequen es of it. Applying the Gauss and the Stokes theorems, the integral form of the Maxwell equations (1, 2) and (4, 5) an be transformed into their dierential versions: ~ = ; ~ D ~_ = ~j ; div D

url H (6)

~ = 0; div B

~_ = 0 :

url E~ + B

(7)

~ = Additionally, we have to spe ify the spa etime relations D ~ ~ ~ "0 E ; B = 0 H and possibly the onstitutive laws. This formulation of ele trodynami s by means of 3-dimensional Eu lidean ve tor al ulus represents only a preliminary version, sin e the 3-dimensional metri enters the s alar and the ve tor ~ produ ts and, in parti ular, the dierential operators div  r
~ ~ and url  r, with r as the nabla operator. In the Gauss law (1) or (6)1 , for instan e, only ounting pro edures enter, namely

ounting of elementary harges inside V { taking also are of

M6 M7

16

Contents

their sign, of ourse { and ounting of ux lines pier ing through a losed surfa e V . No length or time measurements and thus no metri is involved in su h type of pro esses, as will be des ribed in more detail below. Sin e similar arguments apply also to (5) or (7)1 , respe tively, it should be possible to remove the metri from the Maxwell equations altogether.

On the literature Basi ally not too mu h is new in our book. Probably Part D is the most original one. Most of the material an be found somewhere in the literature. What we do laim, however, is some originality in the ompleteness and in the appropriate arrangement of the material whi h is fundamental to the stru ture ele trodynami s is based on. Moreover, we try to stress the phenomena underlying the axioms hosen and the operational interpretation of the quantities introdu ed. The expli it derivation in Part D of the metri of spa etime from pre-metri ele trodynami s by means of linearity, re ipro ity, and symmetry, though onsidered earlier mainly by Toupin [55℄, S honberg [48℄, and Jad zyk [26℄, is new and rests on re ent results of Fukui, Gross, Rubilar, and the authors [38, 22, 37, 19, 47℄. Our main sour es are the works of Post [41, 42, 43, 44, 45℄, of Truesdell & Toupin [57℄, and of Toupin [55℄. Histori ally, the metri -free approa h to ele trodynami s, based on integral onservation laws, was pioneered by Kottler [30℄, E.Cartan [9℄, and van Dantzig [58℄, also the arti le of Einstein [15℄ and the books of Mie [35℄, Weyl [59℄, and Sommerfeld [52℄ should be onsulted on these matters, see also the re ent textbook of Kovetz [31℄. A des ription of the orresponding histori al development, with referen es to the original papers, an be found in Whittaker [60℄ and, up to about 1900, in the penetrating a

ount of Darrigol [11℄. The driving for es and the results of Maxwell in his resear h on ele trodynami s are vividly presented in Everitt's [16℄

on ise biography of Maxwell.

Contents

17

In our book, we will onsistently use exterior al ulus2 , in luding de Rham's odd (or twisted) dierential forms. Textbooks on ele trodynami s using exterior al ulus are s ar e. We know only of Ingarden & Jamiolkowski [24℄, in German of Meetz & Engl [34℄ and Zirnbauer [61℄ and, in Polish, of Jan ewi z [28℄, see also [27℄. However, as a mathemati al physi s' dis ipline, orresponding presentations an be found in Bamberg & Sternberg [4℄, in Thirring [54℄, and, as a short sket h, in Piron [39℄, see also [5, 40℄. Bamberg & Sternberg are parti ular easy to follow and present ele trodynami s in a very transparent way. That ele trodynami s in the framework of exterior al ulus is also in the s ope of ele tri al engineers, an be seen from Des hamps [13℄ and Baldomir & Hammond [3℄, e.g.. Presentations of exterior al ulus, partly together with appli ations in physi s and ele trodynami s, were given, amongst many others, by Burke [8℄, Choquet-Bruhat et al. [10℄, Edelen [14℄, and Frankel [18℄. For dierential geometry we refer to the

lassi s of de Rham [12℄ and S houten [49, 50℄ and to Trautman [56℄. What else did in uen e the writing of our notes? The axiomati s of Bopp [7℄ is dierent but related to ours. In the more mi rophysi al axiomati attempt of Lammerzahl et al., Maxwell's equations [32℄ (and the Dira equation [2℄) are dedu ed from dire t experien e with ele tromagneti (and matter) waves, inter alia. The lear separation of dierential, aÆne, and metri stru tures of spa etime is nowhere more pronoun ed than in S hrodinger's [51℄ `Spa e-time stru ture'. A further presentation of ele trodynami s in this spirit, somewhat similar to that of Post, has been given by Sta hel [53℄. Our (1+3)-de omposition of spa etime is based on the paper by Mielke & Wallner [36℄. Re ently, Hirst [23℄ has shown, mainly based on experien e with neutron s attering on magneti stru tures in solids, that magnetization is a mi ros opi quantity. This is in a

ord with

M

2 Baylis [6℄ also advo ates a geometri approa h, using Cliord algebras. In su h a framework, however, at least in the way Baylis does it, the metri of spa etime is introdu ed right from the beginning. In this sense, Baylis's Cliord algebra approa h is

omplementary to our metri -free ele trodynami s.

18

Contents

our axiomati s whi h yields the magneti ex itation H as mi ros opi quantity, quite analogously to the eld strength B , whereas in onventional texts M is only de ned as a ma ros opi average over mi ros opi ally u tuating magneti elds. Clearly, with H, also the ele tri ex itation D, i.e. the ele tromagneti ex itation H = fH; Dg altogether, ought to be a mi ros opi eld.

Referen es

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[1℄ B. Abbott et al. (D0 Collaboration), 529.

A new onstru tive axiomati s heme for the geometry of spa e-time In: Seman-

[2℄ J. Audrets h, C. L ammerzahl,

ti al Aspe ts of Spa e-Time Geometry. U. Majer, H.-J.

S hmidt, eds.. (BI Wissens haftsverlag: Mannheim, 1994) pp. 21-39.

Geometry and Ele tromagneti Systems (Clarendon Press: Oxford, 1996).

[3℄ D. Baldomir and P. Hammond,

[4℄ P. Bamberg and S. Sternberg,

Students of Physi s,

A Course in Mathemati s for

Vol. 2 (Cambridge University Press:

Cambridge, 1990).

Spa e-time models Phys. A ta 67 (1994)

[5℄ A.O. Barut, D.J. Moore and C. Piron,

from the ele tromagneti eld, Helv. 392-404.

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[6℄ W.E. Baylis, Ele trodynami s. A Modern proa h (Birkhauser: Boston, 1999). [7℄ F. Bopp, Prinzipien (1962) 45-52.

der Elektrodynamik,

Geometri ApZ. Physik

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[8℄ W.L. Burke, Applied Dierential Geometry (Cambridge University Press: Cambridge, 1985).  Cartan, [9℄ E.

On Manifolds with an AÆne Conne tion and the Theory of General Relativity, English translation of the Fren h original of 1923/24 (Bibliopolis: Napoli, 1986).

[10℄ Y. Choquet-Bruhat, C. DeWitt-Morette, and M. DillardBlei k, Analysis, Manifolds and Physi s, revised ed. (North-Holland: Amsterdam, 1982). [11℄ O. Darrigol, Ele trodynami s from Ampere to Einstein (Oxford University Press: New York, 2000). [12℄ G. de Rham, Dierentiable Manifolds { Forms, Currents, Harmoni Forms. Transl. from the Fren h original (Springer: Berlin, 1984). [13℄ G.A. Des hamps, Ele tromagneti s Pro . IEEE 69 (1981) 676-696. [14℄ D.G.B. Edelen, York, 1985).

and dierential forms,

Applied Exterior Cal ulus (Wiley: New

Eine neue formale Deutung der Maxwells hen Feldglei hungen der Elektrodynamik, Sitzungsber. Konigl. Preuss. Akad. Wiss. Berlin (1916) pp. 184-188; see also The

olle ted papers of Albert Einstein. Vol.6, A.J. Kox et al.,

[15℄ A. Einstein,

eds. (1996) pp. 263-269.

James Clerk Maxwell. Physi ist and Natural Philosopher (Charles Sribner's Sons: New York, 1975).

[16℄ C.W.F. Everitt,

Referen es

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[17℄ R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Le tures on Physi s, Vol. 2: Mainly Ele tromagnetism and Matter (Addison-Wesley: Reading, Mass., 1964). [18℄ T. Frankel, The Geometry of Physi s { An Introdu tion (Cambridge University Press: Cambridge, 1997). [19℄ A. Gross and G.F. Rubilar,

On

the

derivation

of

the

. Los Alamos

spa etime metri from linear ele trodynami s

Eprint Ar hive gr-q /0103016 (2001).

[20℄ Y.D. He, Sear h for a Dira magneti monopole in high energy nu leus-nu leus ollisions, Phys. Rev. Lett. 79 (1997) 3134-3137. [21℄ F.W. Hehl and Yu.N. Obukhov, How does the ele tromagneti eld ouple to gravity, in parti ular to metri , nonmetri ity, torsion, and urvature? In: Gyros, Clo ks, Interferometers...: Testing Relativisti Gravity in Spa e. C. Lammerzahl et al., eds.. Le ture Notes in Physi s Vol.562 (Springer: Berlin, 2001) pp. 479-504; see also Los Alamos Eprint Ar hive gr-q /0001010. [22℄ F.W. Hehl, Yu.N. Obukhov, and G.F. Rubilar, time metri from linear ele trodynami s II.

(Leipzig) 9 (2000) Spe ial issue, SI-71{SI-78.

Spa e-

Ann. Physik

[23℄ L.L. Hirst, The mi ros opi magnetization: on ept and appli ation, Rev. Mod. Phys. 69 (1997) 607-627. [24℄ R. Ingarden and A. Jamiolkowski, Classi al i s (Elsevier: Amsterdam, 1985). [25℄ J.D. Ja kson, Classi al New York, 1999). [26℄ A.Z. Jad zyk,

Ele trodynami s

Ele trodynam-

, 3rd ed. (Wiley:

Ele tromagneti permeability of the va uum

, Bull. A ad. Pol. S i., Ser. s i. phys. et astr. 27 (1979) 91-94.

and light- one stru ture

22

Referen es

[27℄ B. Jan ewi z, A variable metri ele trodynami s. The Coulomb and Biot-Savart laws in anisotropi media, Annals of Physi s (NY) 245 (1996) 227-274. [28℄ B. Jan ewi z, Wielkos i skierowane w elektrodynami e (in Polish). Dire ted Quantities in Ele trodynami s. (University of Wro law Press: Wro law, 2000); an English version is under preparation. [29℄ G.R. Kalb eis h, K.A. Milton, M.G. Strauss, L. Gamberg, E.H. Smith, W. Luo, Improved experimental limits on the produ tion of magneti monopoles, Phys. Rev. Lett. 85 (2000) 5292-5295. [30℄ F. Kottler, Maxwell's he Glei hungen und Metrik Sitzungsber. Akad. Wien IIa 131 (1922) 119-146. [31℄ A. Kovetz, Ele tromagneti Theory. (Oxford University Press: Oxford, 2000). [32℄ C. Lammerzahl and M.P. Haugan, On the interpretation of Mi helson-Morley experiments, Phys. Lett. A282 (2001) 223-229. [33℄ H.A. Lorentz, The Theory of Ele trons and its Appli ations to the Phenomena of Light and Radiant Heat. 2nd ed.. (Teubner: Leipzig, 1916). [34℄ K. Meetz and W.L. Engl, Elektromagnetis he Felder { Mathematis he und physikalis he Grundlagen, Anwendungen in Physik und Te hnik (Springer: Berlin, 1980). [35℄ G. Mie, Lehrbu h der Elektrizitat und des Magnetismus. 2nd ed. (Enke: Stuttgart 1941). [36℄ E.W. Mielke ad R.P. Wallner, Mass and spin of double dual solutions in Poin are gauge theory, Nuovo Cimento 101 (1988) 607-623, erratum B102 (1988) 555.

Referen es [37℄ Yu.N. Obukhov, T. Fukui, and G.F. Rubilar,

agation in linear ele trodynami s,

Phys. Rev.

23

Wave propD62 (2000)

044050, 5 pages.

Spa e-time metri from linear ele trodynami s, Phys. Lett. B458 (1999) 466-470.

[38℄ Yu.N. Obukhov and F.W. Hehl,

[39℄ C. Piron,

 Ele trodynamique et optique. Course

given by C.

Piron. Notes edited by E. Pittet (University of Geneva, 1975). [40℄ C. Piron and D.J. Moore, J. Phys.

New aspe ts of eld theory, Turk.

19 (1995) 202-216.

Formal Stru ture of Ele tromagneti s { General Covarian e and Ele tromagneti s (North Holland: Amster-

[41℄ E.J. Post,

dam, 1962, and Dover: Mineola, New York, 1997).

The onstitutive map and some of its rami ations, Annals of Physi s (NY) 71 (1972) 497-518.

[42℄ E.J. Post,

Kottler-Cartan-van Dantzig (KCD) and noninertial systems, Found. Phys. 9 (1979) 619-640.

[43℄ E.J. Post,

[44℄ E.J. Post,

Physi al dimension and ovarian e, Found. Phys.

12 (1982) 169-195.

Quantum Reprogramming { Ensembles and Single Systems: A Two-Tier Approa h to Quantum Me hani s

[45℄ E.J. Post,

(Kluwer: Dordre ht, 1995).

Maxwell's theory on a post-Riemannian spa etime and the equivalen e prin iple, Class. Quantum Grav. 14 (1997) 1347-1356.

[46℄ R.A. Puntigam, C. L ammerzahl and F.W. Hehl,

[47℄ G.F. Rubilar, Y.N. Obukhov, and F.W. Hehl,

metri from linear ele trodynami s. III

Spa etime

(2001) To be pub-

lished. [48℄ M. S h onberg,

Ele tromagnetism and gravitation, 1 (1971) 91-122.

Brasileira de Fisi a

Rivista

24

Referen es

Ri

i-Cal ulus,

[49℄ J.A. S houten,

2nd ed. (Springer: Berlin,

1954). [50℄ J.A.

S houten,

Tensor Analysis for Physi ists.

2nd

ed.

reprinted (Dover: Mineola, New York 1989). [51℄ E. S hr odinger,

Spa e-Time Stru ture (Cambridge Univer-

sity Press: Cambridge, 1954). [52℄ A. Sommerfeld,

Elektrodynamik.

Vorlesungen u  ber Theo-

retis he Physik, Band 3 (Dieteris h's he Verlagsbu hhandlung: Wiesbaden, 1948). [53℄ J. Sta hel,

tions,

The generally ovariant form of Maxwell's equa-

In: J.C. Maxwell, the Sesqui entennial Symposium.

M.S. Berger, ed. (Elsevier: Amsterdam, 1984) pp. 23-37.

Classi al Mathemati al Physi s { Dynami al Systems and Field Theories, 3rd ed. (Springer: New York,

[54℄ W. Thirring,

1997). [55℄ R.A.

Toupin,

Elasti ity and ele tro-magneti s,

Linear Continuum Theories,

in:

Non-

C.I.M.E. Conferen e,

Bres-

sanone, Italy 1965. C. Truesdell and G. Grioli oordinators. Pp.203-342. [56℄ A. Trautman,

Dierential Geometry for Physi ists,

Stony

Brook Le tures (Bibliopolis: Napoli, 1984). [57℄ C. Truesdell and R.A. Toupin,

ries,

The lassi al eld theo-

In: Handbu h der Physik, Vol. III/1, S. Fl ugge ed.

(Springer: Berlin, 1960) pp. 226-793.

The fundamental equations of ele tromagnetism, independent of metri al geometry, Pro . Cambridge

[58℄ D. van Dantzig,

Phil. So . [59℄ H. Weyl,

30 (1934) 421-427.

Raum, Zeit, Materie,

Vorlesungen u  ber Allge-

meine Relativit atstheorie, 8th ed. (Springer: Berlin, 1993). Engl. translation of the 4th ed.: New York, 1952).

Spa e-Time-Matter (Dover:

Referen es

25

[60℄ E. Whittaker, A History of the Theories of Aether and Ele tri ity. 2 volumes, reprinted (Humanities Press: New York,

1973). [61℄ M.R. Zirnbauer, Elektrodynamik. Tex-s ript July 1998 (Springer: Berlin, to be published).

Part A Mathemati s: Some exterior al ulus

26

Why exterior dierential forms?

le birk/partA.tex, with gures [A01ve t1.eps, A02ve t2.eps, A03ve t3.eps, A04orien.eps, A05hdis .ps, A06error.ps, A07nhaus.ps, A08re t1.ps, A09re t2.ps, A10re t3.ps, A11re t4.ps, A12imag1.eps, A13imag2.eps, A14imag3.eps, A15moebs.ps, A16 ow.ps,A17lie.ps, A18inout.ps, A19simp1.ps, A20simp2.ps, A21simp3.ps, A22simp4.ps, A23re t5.ps℄ 2001-06-01 In this Part A and later, in Part C, we shall be on erned with assembling the geometri on epts in the language of dierential forms that are needed to formulate a lassi al eld theory like ele trodynami s and/or the theory of gravitation. The basi geometri stru ture underlying su h a theory is that of a spa etime

ontinuum or, in mathemati al terms, a 4-dimensional dierentiable manifold X4 . The hara teristi s of the gravitational eld will be determined by the nature of the additional geometri stru tures that are superimposed on this `bare manifold' X4 . For instan e, in Einstein's general relativity theory (GR), the manifold is endowed with a metri together with a torsion-free, metri - ompatible onne tion: It is a 4-dimensional Riemannian spa etime V4 . The meaning of these terms will be explained in detail in what follows.

28

In Maxwell's theory of ele trodynami s, under most ir umstan es, gravity an safely be negle ted. Then the Riemannian spa etime be omes at, i.e. its urvature vanishes, and we have the (rigid) Minkowskian spa etime M4 of spe ial relativity theory (SR). Its spatial part is the ordinary 3-dimensional Eu lidean spa e R3 . However, and this is one of the messages of the book, for the fundamental axioms of ele trodynami s we don't need to take into a

ount the metri stru ture of spa etime and, even more so, we should not take it into a

ount. This helps to keep ele trodynami al stru tures leanly separated from the gravitational ones. This separation is parti ularly de isive for a proper understanding of the emergen e of the light one. On the one side, by its very de nition, it is an ele trodynami al on ept in that it determines the front of a propagating ele tromagneti disturban e; on the other hand it onstitutes the main ( onformally invariant) part of the metri tensor of spa etime and is as su h part of the gravitational potential of GR. This ompli ated interrelationship we will try to untangle in Part D. A entral role in the formulation of lassi al ele trodynami s adopted in the present work will be played by the onservation laws of ele tri harge and magneti ux. We will start from their integral formulation. A

ordingly, there is a ne essity for an adequate understanding of the on epts involved when one writes down an integral over some domain on a dierentiable manifold. Spe i ally, in the Eu lidean spa e R3 in Cartesian oordinates, one en ounters integrals like the ele tri tension (voltage) Z (Ex dx + Ey dy + Ez dz ) (8)

intC

C

evaluated along a (1-dimensional) urve C , the magneti ux Z (Bx dy dz + By dz dx + Bz dx dy ) (9) S

intflux

29 over a (2-dimensional) surfa e

Z

S,

and the (total) harge

 dx dy dz

(10)

int harge

V

V.

integrated over a (3-dimensional) volume

The fundamental result of lassi al integral al ulus is Stokes's theorem whi h relates an integral over the boundary of a region to one taken over the region itself. Familiar examples of this theorem are provided by the expressions

Z

Z  E

Ex dx + Ey dy + Ez dz ) =

(



S +

Ex z

Ez x



Ey z

z



y

S

dz dx +

Ey x

Ex y





dy dz



dx dy ;

(11)

stokes1

dx dy dz ;

(12)

stokes2

and

Z

(

Bx dy dz + By dz dx + Bz dx dy ) =

Z  B

V

x

x

V where

S

and

V

+

By y

+

are the boundaries of

Bz z S



and

V,

The right hand sides of these equations orrespond to

df~ and

R

V

div

~ dV , B

R

respe tively.

S

~
E

url

respe tively.

Consider the integral

Z

(x; y; z ) dx dy dz

(13)

intrho

(14)

uvw

and make a hange of variables:

x = x(u; v; w) ;

y

=

y (u; v; w ) ;

z

=

z (u; v; w) :

30 For simpli ity and only for the present purpose, let us suppose that the Ja obian determinant

x u  (x; y; z ) y := u  (u; v; w) z

positive.

v

w

v y

u

is

v z

w z

x

x

w y

(15)

ja obian

We obtain

Z

(x; y; z ) dx dy dz

=

Z

[x(u; v; w ); y (u; v; w ); z (u; v; w)℄

 (x; y; z ) du dv dw :  (u; v; w)

(16)

intrho2

This suggests that we should write

dx dy dz

=

 (x; y; z ) du dv dw  (u; v; w )

x u y = u z u

v z

w y w z

v

w

x

x

v y

du dv dw : (17)

If we set

x = y or x = z

or

y

=

z

ja obian2

the determinant has equal rows

and hen e vanishes. Also, an odd permutation of

x; y; z

hanges

the sign of the determinant while an even permutation leaves it un hanged. Hen e, we have

dx dx = 0 ; dx dy dz

It is this

=

dy dy

= 0

;

dz dz

dy dz dx = dz dx dy = dy dx dz = dx dz dy

=

= 0

;

(18)

dz dy dx :

(19)

alternating algebrai stru ture of integrands that gave

rise to the development of exterior algebra and al ulus whi h is be oming more and more re ognized as a powerful tool in mathemati al physi s. In general, an exterior

p-form

will be an

expression

!=

1

p!

!1

i ;:::ip

dx 1


dx p ; i

i

(20)

pform

31

where the omponents ! 1 p are ompletely antisymmetri in the indi es and i = 1; 2; 3. Furthermore, summation from 1 to 3 is understood over repeated indi es. Then, when translating (12) in exterior form al ulus, we re ognize B as a 2-form i ;:::i

m

Z

Z

B= V

1 B dx dx = 2! [ ℄ i

Z

j

k

ij

V

1  B ℄ dx dx dx = 3! [ k

i

Z

dB :

j

ij

V

V

(21)

Bform

Note that [ ℄ := ( )=2, similarly ( ) := ( + )=2, et . A

ordingly, in the E3 , we have the magneti eld B as 2-form and, as a look at (8) will show, the ele tri eld as 1-form; and the harge  in (10) turns out to be a 3-form. In the 4-dimensional Minkowski spa e M4 , the ele tri urrent J , like  in the E3 , is represented by a 3-form. Sin e the a tion fun tional of the ele tromagneti eld is de ned in terms of a 4-dimensional integral, the integrand, the Lagrangian L, is a 4-form. The oupling term in L of the urrent J to the potential A, namely  J ^ A, identi es A as 1-form. In the inhomogeneous Maxwell equation J = dH , the 3-form hara ter of J attributes to the ex itation H a 2-form. If, eventually, we exe ute a gauge transformation A ! A + df; F ! F , we meet a 0-form f . Consequently, a (gauge) eld theory, starting from a

onserved urrent 3-form, here the ele tri urrent J , generates in a straightforward way forms of all ranks p  4. We know from lassi al al ulus that if the Ja obian determinant (15) above has negative values, i.e. the two oordinate systems do not have the same orientation, then, in equations (16) and (17), the Ja obian determinant must be repla ed by its absolute value. In parti ular, instead of (17), we get the general formula ij

ij

ji

ij



ij

ji



 (x; y; z ) dx dy dz = du dv dw :  (u; v; w )

(22)

This behavior under a hange of oordinates is typi al of what is known as a density . We shall see that, if we wish to drop the requirement that all our oordinate systems should have the

ja obian3

32

same orientation, then densities be ome important and these latter are losely related to the twisted dierential forms whi h one has to introdu e { besides the ordinary dierential forms { sin e the ele tri urrent, e.g., is of su h a twisted type. In Chapter A.1, we rst onsider a ve tor spa e and its dual and study the algebrai aspe t of tensors and of geometri al quantities of a more general type. Then, we turn our attention to exterior forms and their algebra and to a orresponding omputer algebra program. Sin e the tangent spa e at every point of a dierentiable manifold is a linear ve tor spa e, we an asso iate an exterior algebra with ea h point and de ne dierentiable elds of exterior forms or, more on isely, dierential forms on the manifold. This is done in Chapter A.2, while Chapter A.3 deals then with integration on a manifold. It is important to note that in this Part A, we are dealing with the `bare manifold'. Linear onne tion and metri will be introdu ed not before Part C, after the basi axiomati s of ele trodynami s will have been laid down earlier in Part B.

A.1

Algebra

A.1.1

A real ve tor spa e and its dual

Our onsiderations are based on an n-dimensional real ve tor spa e V . One-forms are the elements of the dual ve tor spa e V  de ned as linear maps of the ve tor spa e V into the real numbers. The dual bases of V and V  transform re ipro ally to ea h other with respe t to the a tion of the linear group. The ve tors and 1-forms an be alternatively de ned by their omponents with a spe i ed transformation law. Let V be an n-dimensional real ve tor spa e. We an depi t a ve tor v V by an arrow. If we multiply v by a fa tor f , the ve tor has an f -fold size, see Fig. A.1.1. Ve tors are added a

ording to the parallelogram rule. A linear map ! : V R is

alled a 1-form on V . The set of all 1-forms on V an be given a stru ture of a ve tor spa e by de ning the sum of two arbitrary 1-forms ! and ',

2

(! + ')(v ) = ! (v ) + '(v ) ;

!

v

2 V;

(A.1.1)

sumforms

34

A.1.

Algebra

u 0.5 u

v

u+v 3.2 v

Figure A.1.1: Ve tors as arrows, their multipli ation by a fa tor, their addition (by the parallelogram rule). and the produ t of ! by a real number  2 R , (! )(v ) =  (! (v )) :

v 2 V:

(A.1.2)

prodforms

This ve tor spa e is denoted V  and alled dual of V . The dimension of V  is equal to the dimension of V . The identi ation V  = V holds for nite dimensional spa es. In a

ordan e with (A.1.1), (A.1.2), a 1-form an be represented by a pair of ordered hyperplanes, see Fig. A.1.2. The nearer the hyperplanes are to ea h other, the stronger is the 1-form. In Fig. A.1.3, the a tion of a 1-form on a ve tor is depi ted. Denote by e = (e1 ; : : : ; en ) a (ve tor) basis in V . An arbitrary ve tor v an be de omposed with respe t to su h a basis: v = v  e . Summation from 1 to n is understood over repeated indi es (Einstein's summation onvention). The n real numbers v  ;  = 1; : : : ; n, are alled omponents with respe t to the given basis. With a basis e of V we an asso iate its dual 1-form, or

ove tor basis, the so- alled obasis # = (#1 ; : : : ; #n ) of V  . It is determined by the relation

# (e ) = Æ  :

(A.1.3)

dualbasis

A.1.1

A real ve tor spa e and its dual

35

φ ω 3ω 1 -φ 2

Figure A.1.2: One-forms as two parallel hyperplanes (straight lines in

Here

Æ 

n = 2) with a dire tion, their multipli ation by a fa tor.

Æ 

6

is the Krone ker symbol with

= 0 for

 = .

The omponents

#

Æ 

= 1 for



=



and

! of a 1-form ! with respe t to the obasis

are then given by

)

! = ! #

! = ! (e ) :

=

A transformation from a basis

L 




0

e

2 GL n; R

prime' basis)

(

0

=

0 0 (e ; : : : ; e ) n

1

e of V

0

=

form omps

to another one (`alpha-

is des ribed by a matrix L :=

) (general linear real

e

(A.1.4)

n-dimensional group):

L  e : 0

(A.1.5)

basetrafo

(A.1.6)

dualtrafo

The orresponding obases are thus onne ted by

#

0







L  , i.e., L  L =
Æ : Symboli ally, we may also write e0 = L e and #0 = LT 1 #. Here T denotes the transpose of the matrix L. where

L 

L  # ; 0

=

0

is the inverse matrix to

0

0

0

36

A.1.

Algebra

v φ

u ω

v u w

v w

u

Figure A.1.3: A 1-form a ts on a ve tor. Here we have ! (u) = 1 ; ! (v )  2:3 ; (u)  0:3 ; (v )  4:4 ; (w)  2:1 : A 1-form

an be understood as a ma hine: You input a ve tor and the output is a number whi h an be read o from our images. Consequently, one an view a ve tor v 2 V as n ordered numbers v  whi h transform under a hange (A.1.5) of the basis as

v  = L  v  ; 0

0

(A.1.7)

ve trafo

whereas a 1-form ! 2 V  is des ribed by its omponents ! with the transformation law

! = L  ! : 0

0

(A.1.8)

The similarity of (A.1.7) to (A.1.6) and of (A.1.8) to (A.1.5) and the fa t that the two matri es in these formulas are ontragradient (i.e. inverse and transposed) to ea h other, explains the old-fashioned names for ve tors and 1-forms (or ove tors):

ontravariant and ovariant ve tors, respe tively. Nevertheless, one should be areful: (A.1.7) represents the transformation of n omponents of one ve tor, whereas (A.1.6) en rypts the transformation of n-dierent 1-forms.

omtrafo

A.1.2

A.1.2

Tensors of type

p q

Tensors of type [ ℄

 

37

p q

A tensor is a multilinear map of a produ t of ve tor and dual ve tor spa es into the real numbers. An alternative de nition of tensors spe i es the transformation law of their omponents with respe t to a

hange of the basis. The related on epts of a ve tor and a 1-form an be generalized to obje ts of higher rank. The prototype of su h an obje t is the stress p  tensor of ontinuum me hani s. A tensor T on V of type q is a multi-linear map

T : |V  
{z

 V }  |V p






{z q

V}

R:

!

(A.1.9)

deftensor

It an be des ribed as a geometri al quantity whose omponents with respe t to the obasis # and the basis e are given by


T 1 :::p 1 ::: q = T #1 ; : : : ; #p ; e 1 ; : : : ; e q :

(A.1.10)

tensor omps

The transformation law for tensor omponents an be dedu ed from (A.1.5) and (A.1.6):

T 1 :::p 1 ::: q = L1 1


Lp p L 1 1


L q q T 1 :::p 1 ::: q : 0

0

0

0

0

0

0

0

(A.1.11)

  If we have two tensors, T of type pq and S of type [rs ℄, we an

onstru t its tensor produ t { the tensor T S of type pq++sr de ned as

(T S )(!1; : : : ; !p+r ; v1 ; : : : ; vq+s ) = T (!1 ; : : : ; !p; v1 ; : : : ; vq ) S (!p+1; : : : ; !p+r ; vq+1 ; : : : ; vq+s ) ; (A.1.12) for any one-forms ! and  any ve tors v . p Tensors of type q whi h have the form v1


vp

!1


!q are alled de omposable. Ea h tensor is a linear

tentrafo

38

A.1.

Algebra

ombination of de omposable tensors. More pre isely, using the de nition of the omponents of T a

ording to (A.1.11), one an prove that T

= T 1 :::p 1 ::: q e1


ep # 1


# q :

(A.1.13)

de ompose

Therefore tensor produ ts of basis ve tors e and of basis 1forms # onstitute a basis of the ve tor spa e Vqp of tensors of type pq on V . Thus the dimension of this ve tor spa e is np+q . Elementary examples of the tensor spa es are given by the original ve tor spa e and its dual, V01 = V and V10 = V  . The Krone ker symbol Æ  is a tensor of type [11 ℄, as an be veri ed with the help of (A.1.11).

A

A.1.3

generalization of tensors: geometri

quantities

A geometri quantity is de ned by the a tion of the general linear group on a ertain set of elements. Important examples are tensor-valued forms, the orientation, and twisted tensors. In eld theory, tensors are not the only obje ts needed for the des ription of nature. Twisted forms or ve tor-valued forms, e.g., require a more general de nition. As we have seen above, there are two ways of dealing with tensors: either we an des ribe them as elements of the abstra t p+q tensor spa e Vqp or as omponents, i.e. elements of R n , whi h have a pres ribed transformation law. These observations an be generalized as follows: Let W be a set, and let  be a left a tion of the group GL(n; R ) in the set W , i.e. to ea h element L 2 GL(n; R ) we atta h a map L : W ! W in su h a way that L1

Æ

L2

= L1 L2

L1 ; L2 2 GL(n; R ):

(A.1.14)

Denote by P (V ) the spa e of all bases of V and onsider the Cartesian produ t W  P (V ). The formula (A.1.5) provides us

defrho

A.1.3

A generalization of tensors: geometri quantities

39

with a left a tion of GL(n; R ) in P (V ) whi h an be ompa tly written as e0 = Le ; and then L1 (L2 e) = (L1 L2 )e holds. Thus we

an de ne the left a tion of GL(n; R ) on the produ t W  P (V ):


(w; e) 7! L (w); Le :

(A.1.15)

An orbit of this a tion is alled an geometri quantity of type  on V . In other words, a geometri quantity of type  on V is an equivalen e lass [(!; e)℄. Two pairs (w; e) and (w0 ; e0 ) are equivalent if and only if there exists a matrix L 2 GL(n; R ) su h that w 0 =  (w ) and e0 = Le : (A.1.16) L

la tion

equiv

In many physi al appli ations, the set W is an N -dimensional ve tor spa e R N and it is required that the maps L are linear. In other words,  is a representation  : GL(n; R ) ! GL(N; R ) of the linear group GL(n; R ) in the ve tor spa e W by N  N matri es (L) = A B (L) 2 GL(N; R ), with A; B;


= 1; : : : ; N . Let us denote as eA the basis of the ve tor spa e W . Then, we

an represent the geometri quantity w = wA eA by means of its omponents wA with respe t to the basis. The a tion of the group GL(n; R ) in W results in a linear transformation (A.1.17) GQbasis eA ! eA = A B (L) eB ; and, a

ordingly, the omponents of the geometri quantity transform as w A ! w A = B A (L 1 ) w B : (A.1.18) GQ omp 0

0

0

0

Examples: 1)

2)

We an take W = Vqp and L = idW . The orresponding   geometri quantity is then a tensor of type pq .

p+q We an take W = R n and hoose  in su h a way that (A.1.16) will indu e (A.1.11) together with (A.1.5). p  This type of geometri quantity is also a tensor of type q . For instan e, if we take W = R n and either L = L or L = (LT ) 1 , then from (A.1.17) and (A.1.18) we get ve tors (A.1.7) or 1-forms (A.1.8), respe tively.

40

A.1.

3)

4)

5)

Algebra

The two examples above an be ombined. We an take W = p+q n R Vsr and  as in example 2). That means that we an

onsider obje ts with omponents T 1:::p 1::: q belonging to Vsr whi h transform a

ording to the rule (A.1.11). This mixture of two approa hes seems strange at rst sight, but it appears produ tive if, instead of Vsr , we take spa es of sforms sV . Su h tensor-valued forms turn out to be useful in dierential geometry and physi s. Let W = f+1; 1g and  = sgn(det L). This geometri quantity is an orientation in the ve tor spa e V . A frame e 2 P (V ) is said to have a positive orientation, if it forms a pair with +1 2 W . Ea h ve tor spa e has two dierent orientations. Combine the examples 1) and 4). Let W = Vqp and  = sgn(det L) idW . This geometri quantity is alled a twisted   p (or odd) tensor of type q on V . Parti ularly useful are twisted exterior forms sin e they an be integrated even on a manifold whi h is non-orientable. L

L

A.1.4

Almost omplex stru ture

An even-dimensional ve tor spa e V , with n = 2k, an be equipped with an additional stru ture whi h nds many interesting appli ations in ele trodynami s and in other physi al theories. We say that a real ve tor spa e V has an almost omplex stru ture if a tensor J of type [ ℄ is de ned on it whi h has the property J = 1: (A.1.19) With respe t to a hosen frame, this tensor is represented by the omponents J and the above ondition is then rewritten as (A.1.20) J J = Æ : 1

1 1

2

1 See Choquet et al. [4℄.

almost1

almost2

A.1.5

Exterior

p-forms

41

By means of the suitable hoi e of the basis e , the omplex stru ture an be brought into the anoni al form 



J =

0 Ik

Ik



0

:

(A.1.21)

almost3

Here Ik is the k-dimensional unity matrix with k = n=2. A.1.5

Exterior

p-forms

Exterior forms are totally antisymmetri

ovariant   0 tensors. Any tensor of type p de nes an exterior p-form by means of the alternating map involving the generalized Krone ker. As we saw at the beginning of Part A, exterior p-forms play a parti ular role as integrands in eld theory. We will now turn to their general de nition. Let on e more V be an n-dimensional linear ve tor spa e. An exterior p-form ! on V is a real-valued linear fun tion

! : V|

 V {z: : :  V} ! R p

(A.1.22)

fa tors

su h that

! (v1; : : : ; v ; : : : ; v ; : : : ; vp ) = ! (v1 ; : : : ; v ; : : : ; v ; : : : ; vp ) (A.1.23)

exform

for all v1 ; : : : ; vp 2 V and for all ; = 1; : : : ; p. In other words, ! is a ompletely antisymmetri tensor of type 0p . In terms of a basis e of V and the obasis # of V , the linear fun tion !

an be expressed as 

! = !1

1 p #






: : : #p ;

(A.1.24)

where ea h oeÆ ient !1 p := ! (e1 ; : : : ; ep ) is ompletely antisymmetri in all its indi es.




ex omp

42

A.1.

Algebra

The spa e of real-valued p-linear fun tions on V was denoted by Vp . Then, for any ' 2 Vp , with ' = '1


p #1 : : : #p ; (A.1.25) we an de ne a orresponding (alternating) exterior p-form Alt ' by Alt ' = ' 1


p #1 : : : #p : (A.1.26) Here we have 1 p (A.1.27) ' ' 1


p := Æ 11 ::: p! :::p 1


p with the generalized Krone ker delta 8 > +1 if ; : : : ; p is an even > > > > > permutation of  ; : : : ; p ; < 1 ::: p (A.1.28) Æ1 :::p = 1 if ; : : : ; p is an odd > > > permutation of  ; : : : ; p ; > > > : 0 otherwise ; where  ; : : : ; p are p dierent numbers from the set 1; : : : ; n. Provided '1 ;::: ;p is already antisymmetri in all its indi es, then (A.1.29) ' 1


p = '1 :::p :
The set of exterior p-forms on V forms an np -dimensional sub
n spa e of Vp whi h we denote by p V . Here p represent the binomial oeÆ ients. In parti ular, for p = 0 and p = 1, we shall have  V =R;  V = V: (A.1.30)
For n = 4, the dimensions of the spa es for p-forms are p = p p , or 0

0

[

℄

[

℄

lamb

altlamb

anti

1

1

1

delta

1

1

[

℄

anti2

0

0

1

lamb0

4

4!

!(4

)!

p = 0; 1; 2; 3; 4



respe tively, see Table A.1.5.

1; 4; 6; 4; 1 dimensions ; (A.1.31)

dimforms

A.1.6

Exterior multipli ation

43

Table A.1.1: Number of omponents of p-forms in 3 and 4 dimensions and examples from ele trodynami s:  ele tri harge and j ele tri urrent density, D ele tri and H magneti ex itation, E ele tri and B magneti eld, A ove tor potential, ' s alar potential, f gauge fun tion; furthermore, L is the Lagrangian, J = (; j ), H = (D; H), F = (E; B ), A = ('; A). A tually, the forms J and H are twisted forms, see Se . A.2.6. p-form

0-form 1-form 2-form 3-form 4-form 5-form

A.1.6

n

=3 1 3 3 1 0 0

examples

'; f ; E; j; ; B 

H

D

A

{ {

n

=4

examples

1 4 6 4 1 0

f A H; F J L

{

Exterior multipli ation

The exterior produ t de nes a (p + q )-form for every pair of p- and q -forms. The basis of the spa e of p-forms is then naturally onstru ted as the p-th exterior power of the 1-form basis. The exterior produ t

onverts the dire t sum of all forms into an algebra.

In order to be able to handle exterior forms, we have to de ne their multipli ation. The exterior produ t of the p 1-forms ! 1 ; : : : ; ! p 2 V  { taken in that order { is a p-form de ned by !1

^ : : : ^ !p := Æ1:::p:::p ! : : : !p ; 1

1

(A.1.32)

exprodu t

44

A.1.

Algebra

spoken as \omega-one wedge : : : wedge omega-p". It follows that for any set of ve tors v1; : : : ; vp 2 V , (!1 ^ : : : ^ !p)(v1; : : :

! 1 (v ) 1 ! 2 (v ) 1 ; vp ) = .. . p ! (v1 )

Given ! 2 p V , so that ! = !1 :::p #1 : : : #p ; we have

! 1 (v2 ) ! 2 (v2 )

... ... : ... ! p(v2 )


! p(vp ) (A.1.33)

with

! = ![1 :::p ℄ #1 : : : #p

![1 :::p ℄ = !1 :::p ;

p #1 : : : #p ; ! = 1 Æ 11::: p! :::p 1 ::: p and hen e, f. (20),

!=

1! # 1 ^ : : : ^ # p : p! 1 ::: p

Sin e, in addition, the 2 < : : : < p

n
p






!1(vp)


!2(vp)

exprodu t2

(A.1.34)

om1

(A.1.35)

om2

(A.1.36)

om3

p-forms f#1 ^ : : : ^ #p , 1  1
0, the interior produ t is a map : V  p V  ! p 1 V  (A.1.43) whi h is introdu ed as follows: Let v 2 V and  2 pV . Then (v ) 2 p 1 V  is de ned by (v )(u1; : : : ; up 1) := (v; u1; : : : ; up 1) ; (A.1.44) for all u1; : : : ; up 1 2 V . We speak it as \v in ". In the literature sometimes the interior produ t of v and  is alternatively abbreviated as iv . For p = 0, v  := 0 : (A.1.45) Note that, if p = 1, the de nition (A.1.44) implies (A.1.46) v  = (v ) : The following properties of interior multipli ation follow immediately from the de nitions (A.1.44), (A.1.45): 1) v ( + ) = v  + v [distributive law℄, [linearity in a ve tor℄, 2) (v + u)  = v  + u  [multipli ative law℄, 3) (av )  = a(v ) 4) v u  = u v  [anti ommutative law℄, 5) v ( ^ ! ) = (v ) ^ ! + ( 1)p  ^ (v ! ) [(anti)Leibniz rule℄,

definner

in0

in1

A.1.8

Volume elements on a ve tor

spa e, densities, orientation

where ; 2 pV ; ! 2 q V  , v; u 2 V , and a 2 R . Let e be a basis of V , # the obasis of V  , then, by (A.1.46), (A.1.47) e # = # (e ) = Æ : Hen e, if we apply the ve tor basis e to the p-form = 1 1 :::p #1 ^ : : : ^ #p ; (A.1.48) p! i.e. e , then the properties listed above yield e

=

1

(p 1)!

2 2 :::p #

^ : : : ^ #p :

(A.1.49)

If we multiply this formula by # , we nd the identity # ^ (e )=p : (A.1.50) A.1.8

Volume

47

indual

psi

inbasis

ontr

elements on a ve tor spa e,

densities, orientation

A volume element is a form of maximal rank. Thus, it has one nontrivial omponent. Under the a tion of the linear group, this omponent is a density of weight +1. Orientation is an equivalen e lass of volume forms related by a positive real fa tor. The hoi e of an orientation is equivalent to the sele tion of similarly oriented bases in V . The spa e nV  of exterior n-forms on an n dimensional ve tor spa e V is 1-dimensional and, for ! 2 nV , we have 1 != !1 :::n #1 ^ : : : ^ #n = !1:::n #1 ^ : : : ^ #n : (A.1.51) n! The nonzero elements of nV  are alled volume elements . Consider a linear transformation (A.1.5) of the basis e of V . The orresponding transformation of the obasis # of V  is given by (A.1.6).

exom

48

A.1.

Algebra

Let L := det #1

0

^

L0

: : : ^ #n

0




. Then

= L1 1 : : : Ln n #1 ^ : : : ^ #n
1 1 :::n # ^ : : : ^ #n : = L1 1 : : : Ln n Æ1:::n (A.1.52) 0

0

0

0

Hen e, for the volume element2 , we have: #1

0

^

: : : ^ #n

0

= det(L  ) #1 ^ : : : ^ #n : 0

(A.1.53)

volel

(A.1.54)

exom2

(A.1.55)

exom3

(A.1.56)

exom4

Sin e, in terms of the basis # , 0

!

= !1 :::n 0

#1

0

0

0 : : : ^ #n ;

^

it follows from (A.1.51) and (A.1.53) that !1:::n

= det(L  ) !1 :::n = L

1

0

0

0

!10 :::n0 ;

with L 1 = det(L  ), and, onversely, 0

!10 :::n0

= det(L ) !1:::n = 0

L !1:::n :

The geometri quantity with transformation law given by (A.1.56) is alled a s alar density. It an easily be generalized. The geometri quantity S with transformation law S

0 = det(L )w S 

(A.1.57)

0

deltrafo

is alled s alar density of weight w. The generalization to tensor densities of weight w in terms of omponents, see (A.1.11), reads T

 ::: 0

::: 0

= det(L )w L 0

0




L



0




T :::

::: ;

(A.1.58)

whereas twisted tensor densities of weight w on the right hand side pi k up an extra fa tor sgn det(L ). Let ! and ' be two arbitrary volume forms on V . We will say that volumes are equivalent, if a positive real number a > 0 exists, su h that ' = a !. This de nition divides the spa e nV  into the two equivalen e lasses of n-forms. One alls ea h of the 0

densityT

A.1.8

Volume elements on a ve tor spa e, densities, orientation

(a)

h1

e1 h2

h1

e2

(b)

e1 Figure A.1.4: Distinguishing orientation (a) on a line and (b) on a plane: The ve tor bases e and h are dierently oriented.

equivalen e lasses an orientation on V , be ause a spe i ation of a volume form uniquely determines a lass of oriented bases on V and onversely. This an be demonstrated as follows: On the intuitive level, in the ase of a straight line one speaks of an orientation by distinguishing between positive and negative dire tions, whereas in a plane one hooses positive and negative values of an angle, see Fig. A.1.4. The generalization of these intuitive ideas to an n-dimensional ve tor spa e is ontained in the on ept of orientation : one says that two bases e and h
 of V are similarly oriented if h = A e , with det A > 0. This is learly an equivalen e relation whi h divides the spa e of all bases of V into two lasses. An orientation of the ve tor spa e V is an equivalen e lass of ordered bases, and V is alled oriented ve tor spa e when a hoi e of orientation is made. 2 For the de nition of a volume element without the use of a metri , see Synge and S hild [32℄, and, in parti ular, Laurent [14℄.

49

50

A.1.

Algebra

Now we return to the volume elements in V . Given the volume ! , we an de ne the fun tion

o! (e) := sgn ! (e1 ; : : : ; en )

(A.1.59)

on the set of all bases of V . It has only two values: +1 and 1, and a

ordingly we have a division of the set of all bases into the two subsets. One lass is onstituted of the bases for whi h o! (e) = 1, and on the se ond lass we have o! (e) = 1. In ea h subset, the bases are similarly oriented. In order to show this, let us assume that a volume form (A.1.51) is hosen and xed, and let us take an arbitrary obasis # . The value of the volume form on the ve tors of the basis e reads ! (e1 ; : : : ; en ) = !1:::n (#1 ^


^#n )(e1 ; : : : ; en ) = !1:::n . Suppose this number is positive, then in a

ordan e with (A.1.59) we have o! (e) = +1. For a dierent basis h = A e
we ob
tain ! (h1 ; : : : ; hn) = det A ! (e1 ; : : : ; en ) = det A !1:::n . Consequently, if the basis h is
in the same subset as e , that is o! (h) = +1, then det A > 0, whi h means that the bases e
and h are similarly oriented. Conversely, assuming det A > 0 for any two bases h = A e , we nd that (A.1.59) holds true for both bases. Clearly, every volume form whi h is obtained by a \res aling" !1:::n ! '1:::n = a !1:::n with a positive fa tor a will de ne the same orientation fun tion (A.1.59): o! (e) = oa! (e). This yields the whole lass of equivalent volume forms whi h we introdu ed at the beginning of our dis ussion. The standard orientation of V for an arbitrary basis e is determined by the volume form #1 ^


^ #n with obasis # . A simple reordering of the ve tors (for example, an inter hange of the rst and the se ond leg) of a basis may hange the orientation.

orient

Levi-Civita symbols and generalized Krone ker deltas

A.1.9

A.1.9

Levi-Civita

51

symbols and generalized

Krone ker deltas The Levi-Civita symbols are numeri ally invariant quantities and lose relatives of the volume form. They an arise by applying the exterior produ t or the interior produ t

n-times,

^

respe tively. Levi-

Civita symbols are totally antisymmetri tensor densities, and their produ ts an be expressed in terms of the generalized Krone ker delta.

Volume forms provide a natural de nition of very important tensor densities, the Levi-Civita symbols. In order to des ribe them, let us hoose an arbitrary obasis # , and onsider the form of maximal rank ^ := #1 ^ : : : ^ #n : (A.1.60) We will all this an elementary volume. Re all that the transformation law of this form is given by (A.1.53), whi h means that ^ is the n-form density of the weight 1. By simple inspe tion it turns out that the wedge produ t #1 ^ : : : ^ #n (A.1.61) is either zero (when at least two of the wedge-fa tors are the same) or it is equal to ^ up to a sign. The latter holds when all the wedge-fa tors are dierent, and the sign is determined by the number of permutations whi h are needed for bringing the produ t (A.1.61) to the ordered form (A.1.60). This suggests a natural de nition of the obje t 1 :::n whi h has similar symmetry properties. That is, we de ne it by the relation (A.1.62) #1 ^ : : : ^ #n =: 1 :::n ^:

As one an immediately he k, the Levi-Civita symbol 1 :::n

an be expressed in terms of the generalized Krone ker symbol (A.1.28)3 :  ::: 1 :::n = Æ 11::: nn : (A.1.63) 3 See

Sokolniko [29℄.

elemvol

prodN

epsilon1

eps-del1

52

A.1.

Algebra

In parti ular, we see that the only nontrivial omponent is 1:::n = 1. With respe t to the hange of the basis, this quantity transforms as the [n0 ℄-valued 0-form density of the weight +1: 

1 :::n 0

0

= det(L ) L1 1 : : : Ln n 1 :::n : 0

0

0

(A.1.64)

Re alling the de nition of the determinant, we see that the omponents of the Levi-Civita symbol have the same numeri al values with respe t to all bases, 

1 :::n 0

0

= 1 :::n :

(A.1.65)

epsinv

They are +1; 1, or 0. Another fundamental antisymmetri obje t an be obtained from the elementary volume ^ with the help of the interior produ t operator. As we have learned from Se . A.1.7, the interior produ t of a ve tor with a p-form generates a (p 1)-form. Thus, starting with the elementary volume n-form, and using the ve tor of the basis e , we nd an (n 1)-form ^

:= e ^:

(A.1.66)

The transformation law of this obje t de nes it as a valued (n 1)-form density of the weight 1: ^ = det(L ) 0

0

1



L0 ^ :

epsA1

ove tor-

(A.1.67)

transeps

Applying on e more the interior produ t of the basis to (A.1.66), one obtains and (n 2)-form, et .. Thus, we an onstru t the

hain of forms: ^1 2

.. .

:= e2 ^1 ;

^1 :::n := en

:::

(A.1.68) e1 ^:

(A.1.69) (A.1.70)

The last obje t is a zero-form. The property 4) of the interior produ t for es all these epsilons to be totally antisymmetri in all their indi es. Similarly to (A.1.67),   we an verify that for p = 0; : : : ; n the obje t ^1 :::p is a 0p -valued (n p)-form

epsilon2

A.1.9

Levi-Civita symbols and generalized Krone ker deltas

53

density of the weight 1. These forms f^; ^; ^1 2 ; : : : ; ^1 :::n g, alternatively to f#; #1 ^ #2 ; : : : #1 ^ : : : #n g, an be used as a basis for arbitrary forms in the exterior algebra V . In parti ular, we nd that (A.1.70) is the [0n℄-valued 0-form density of the weight 1. This quantity is also alled the LeviCivita symbol be ause of its evident similarity to (A.1.62). Analogously to (A.1.63) we an express (A.1.70) in terms of the generalized Krone ker symbol (A.1.28): ^1 :::n

::: n : = Æ11 ::: n

(A.1.71)

eps-del2

Thus we nd again that the only nontrivial omponent is ^1:::n = +1. Note that despite the deep similarity, we annot identify the two Levi-Civita symbols in the absen e of the metri , hen e the dierent notation (with and without hat) is appropriate. It is worthwhile to derive a useful identity for the produ t of the two Levi-Civita symbols: 1 :::n ^ 1 ::: n

= ^1 :::n e n : : : e 1  = e n : : : e 1 (#1 ^ : : : ^ #n ) = (#1 ^ : : : ^ #n ) (e 1 ; : : : ; e n ) Æ 1 1  Æ n

= ...

1








 Æ n1






 Æ nn

...

... = Æ1 :::n : 1 ::: n

(A.1.72)

epseps

The whole derivation is based just on the use of the orresponding de nitions. Namely, we use (A.1.70) in the rst line, (A.1.62) in the se ond line, (A.1.44) in the third one, and nally (A.1.33) in the last line. This identity helps a lot in al ulations of the dierent ontra tions of the Levi-Civita symbols. For example, we easily obtain from (A.1.72): 1 :::p 1 ::: n p ^ 1 ::: p 1 ::: n p

= (n

1 :::p p)! Æ 1 ::: p :

(A.1.73)

Furthermore, let us take an integer q < p. The ontra tion of (A.1.72) over the (n q) indi es yields the same result (A.1.73) with p repla ed by q. Comparing the two ontra tions, we then

epseps1

54

A.1.

Algebra

dedu e for the generalized Krone kers: 1 :::q q+1 ::: p Æ 1 ::: q q+1 ::: p

=

(n (n

q )! 1 :::q Æ : p)! 1 ::: q

(A.1.74)

deldel

(A.1.75)

deldel1

In parti ular, we nd :::p Æ11::: p

=

(n

n!

p)!

:

Let us olle t for a ve tor spa e of 4 dimensions the de isive formulas for going down the p-form ladder by starting from the 4-form density ^ and arriving at the 0-form ^ Æ : ^ = e ^ = e ^ = e ^ Æ = eÆ

^ # ^ #Æ =3! ; ^ = ^ Æ # ^ #Æ =2! ; ^ = ^ Æ #

^

= ^ Æ #Æ ;

(A.1.76)

epsilonhats

^ :

Going up the ladder yields:

^ ^ ^ ^

# ^ Æ # ^  Æ # ^ # ^ 

= Æ ^ Æ ÆÆ ^  + Æ  ^ Æ = ÆÆ ^ + Æ  ^Æ + Æ  ^ Æ ; = Æ  ^ Æ  ^ ; = Æ  ^ :

Æ  ^  Æ ;

(A.1.77)

One an, with respe t to the ^-system, de ne a (pre-metri ) duality operator } whi h establishes an equivalen e between pforms and totally   antisymmetri tensor densities of weight +1 n p and of type 0 . In terms of the bases of the orresponding linear spa es, this operator is introdu ed as

}(^

1 :::p )

p := Æ 11::: :::p e 1


e p :

Consequently, given an arbitrary p-form spe t to the ^-basis as !

=

(n

1

p)!

!

! 1 :::n p ^1 :::n p ;

(A.1.78)

diamondef

expanded with re(A.1.79)

diamond1

A.1.10

The spa e

M

6 of two-forms in four dimensions

55

the map } de nes a tensor density by }

! :=

1

(n p)! !

1 :::n p e 1




en p :

(A.1.80)

For example, in n = 4 we have }^ = e and }^ = 1. Thus, every 3-form ' = ' ^ is mapped into a ve tor density }' = ' e , whereas a 4-form ! yields a s alar density }!.

A.1.10

The spa e dimensions

M

6

of two-forms in four

Ele tromagneti ex itation and eld strength are both 2-forms. On the 6-dimensional spa e of 2-forms, there exists a natural 6-metri , whi h is an important property of this spa e.

Let e be an arbitrary basis of V , with ; ; : : : = 0; 1; 2; 3. In later appli ations, the zeroth leg e0 an be related to the time

oordinate of spa etime, but this will not always be the ase (for the null symmetri basis (C.2.14), e.g., all the e 's have the same status with respe t to time). The three remaining legs will be denoted by ea , with a; b;


= 1; 2; 3. A

ordingly, the dual basis of V  is represented by # = (#0 ; #a ). In the linear spa e of 2-forms 2V  , every element an be de omposed a

ording to ' = 21 ' # ^ # . The basis # ^ #

onsists of 6 simple 2-forms. This 6-plet an be alternatively numbered by a olle tive index. A

ordingly, we enumerate the antisymmetri index pairs 01; 02; 03; 23; 31; 12 by upper ase let-

diamond2

56

A.1.

Algebra

ters I; J::: from 1 to 6: 0

BI =

=

B B B B B B 



B1 B2 B3 B4 B5 B6

1

0

C C C C C C A

B B B =B B B 

#0 ^ #1 #0 ^ #2 #0 ^ #3 #2 ^ #3 #3 ^ #1 #1 ^ #2

#0 ^ #a 1 b d # ^ #d 2^



=



1 C C C C C C A

a ^b



(A.1.81)

beh1

With the BI as basis (speak ` yrilli B' or `Beh'), we an set up a 6-dimensional ve tor spa e M 6 := 2 V  . This ve tor spa e will play an important role in our onsiderations in Parts D and E. The extra de omposition with respe t to a and ^b will be onvenient for re ognizing where the ele tri and where the magneti pie es of the eld are lo ated. We denote the elementary volume 3-form by ^ = #1 ^ #2 ^ #3 . Then ^a = ea ^ is the basis 2-form in the spa e spanned by the 3- oframe #a , see (A.1.66). This notation has been used in (A.1.81). Moreover, as usual, the 1-form basis then an be des ribed by ^ab = eb ^a . Some useful algebrai relations an be immediately derived: ^a ^ #b = Æab ^; ^ab ^ # = Æa ^b Æb ^a ; ^ab ^ ^ = ^ ^ab :

(A.1.82) (A.1.83) (A.1.84)

Correspondingly, taking into a

ount that #0 ^ ^ =: Vol is the elementary 4-volume in V , we nd a ^ b = 0; ^a ^ ^b = 0; ^a ^ b = Æab Vol;
^ab ^ ^ #d = Æa Æbd + Æb Æad Vol:

(A.1.85) (A.1.86) (A.1.87) (A.1.88)

bb0 ee0 epsbe

A.1.10

The spa e

M

6 of two-forms in four dimensions

57

Every 2-form, being an element of M 6 , an now be represented as ' = 'I BI by its 6 omponents with respe t to the basis (A.1.81). A 4-form ! , i.e., a form of the maximal rank in four dimensions, is expanded with respe t to the wedge produ ts of the B-basis as ! = 21 !IJ BI ^ BJ . The oeÆ ients !IJ form a symmetri 6  6 matrix sin e the wedge produ t between 2-forms is evidently ommutative. A 4-form has only one omponent. This simple observation enables us to introdu e a natural metri on the 6-dimensional spa e M 6 as the symmetri bilinear form "(!; ') := (! ^ ')(e0 ; e1 ; e2 ; e3 );

!; '

2 M 6;

(A.1.89)

6met

where e is a ve tor basis. Although the metri (A.1.89) apparently depends on the hoi e of the basis, the linear transformation e ! L  e indu es the pure res aling " ! det(L  ) ". Using the expansion of the 2-forms with respe t to the bive tor basis BI , the bilinear form (A.1.89) turns out to be 0

0

"(!; ') = !I 'J "IJ ;

0

"IJ

where

= (BI ^ BJ )(e0 ; e1 ; e2 ; e3 ): (A.1.90)

met6 o

A dire t inspe tion by using the de nition (A.1.81) and the identity (A.1.87) shows that the 6-metri omponents read expli itly "IJ

0 1  Here I3 := 0

1

=



0 I3

I3

0



:

(A.1.91)

0 0 1 0 A is the 3  3 unit matrix. Thus we see 0 0 1 that the metri (A.1.89) is always non-degenerate. Its signature is (+; +; +; ; ; ). Indeed, the eigenvalues  of the matrix (A.1.91) are de ned by the hara teristi equation det("IJ  Æ IJ ) = (2 1)3 = 0. The symmetry group whi h preserves the 6-metri (A.1.89) is isomorphi to O(3; 3). By onstru tion, the elements of (A.1.91) numeri ally oin ide with the omponents of the Levi-Civita symbol ijkl , see

Levi1

58

A.1.

Algebra

(A.1.63):

"IJ = IJ :=



0

I3

I3

0



:

(A.1.92)

Levi1a

Similarly, the ovariant Levi-Civita symbol  ^mnpq , see (A.1.71),

an be represented in 6D notation by the matrix

IK = ^KI := ^



0

I3

I3

0



:

(A.1.93)

Levi2

One an immediately prove by multiplying the matri es (A.1.91) and (A.1.93) that their produ t is equal the 6D-unity, in omplete agreement with (A.1.73). Thus, the Levi-Civita symbols

an be onsistently used for raising and lowering indi es in

Transformation of the

M 6 {basis

M 6.

!

is hanged, e e0 ? As we know, su h a hange is des ribed by the linear transfor-

What happens in

M 6 when the basis in V

mation (A.1.5). Then the obasis transforms in a

ordan e with (A.1.6):

#

=

L  # : 0

0

(A.1.94)

dualtrafo1

(A.1.95)

dualtrafo2

In the (1 + 3)-matrix form, this an be written as



#0 #a

  =

L0 0 Lb 0 L0 a Lb a



#0 #b 0



0

:

Correspondingly, the 2-form basis (A.1.81) transforms into a new bive tor basis

B

0

I

0 0 11 # ^# a =  # 0 ^# 2 A; 0

0

0

0

0

#0 0

^#

0

3

 =

a ^b 0

0



;

(A.1.96)

0 2 31 # ^# ^b =  # 3 ^ # 1 A : 0

0

0

0

0

#1 0

^#

0

2

beh2

A.1.10

The spa e

M

6 of two-forms in four dimensions

59

Substituting (A.1.94) into (A.1.81), we nd that the new and old 2-form bases are related by indu ed linear transformation 

a ^a



=



P a b W ab Zab Qa b



0b ^0b



(A.1.97)

;

B2B

where P a b = L0 0 Lb a L0 a Lb 0 ; Qb a = (det L d ) (L W ab = L 0 Ld a b d ; Zab = ^a d L0 Lb d :

1

)b a; (A.1.98) (A.1.99)

PQ MN

The 3  3 matrix (L 1 )ba is inverse to the 3  3 sub-blo k La b in (A.1.94). The inverse transformation is easily omputed: 

0a ^0a



= 1 det L





Qb a W ba Zba P b a

b ^b



;

(A.1.100)

invB2B

where the determinant of the transformation matrix (A.1.94) reads 



det L := det L = L0 0 La 0 (L 1)b aL0 b det L d: (A.1.101) 0

detLam

One an write an arbitrary linear transformation L 2 GL(4; R ) as a produ t L = L1 L2 L3

of three matri es of the form L1 L2 L3

= = =



 



1 Ub ; 0 Æba  1 0 ; V a Æa b

00 0 0 b a



:

(A.1.102)

LLL

(A.1.103)

sub1

(A.1.104)

sub2

(A.1.105)

sub3

Here V a ; Ub; 00; ba , with a; b = 1; 2; 3, des ribe 3+3+1+9 = 16 elements of an arbitrary linear transformation. The matri es fL3 g form the group R GL(3; R) whi h is a subgroup of

60

A.1.

Algebra

matri es fL1 g and f g in GL(4; R ). In the study of the ovarian e properties of various obje ts in M 6 , it is thus suÆ ient to onsider the three separate ases (A.1.103)-(A.1.105). Using (A.1.98) and (A.1.99), we nd for L = L1 P a b = Qb a = Æba ; W ab = ab U ; Zab = 0: (A.1.106) Similarly, for L = L2 we have P a b = Qb a = Æba ; W ab = 0; Zab = ^ab V ; (A.1.107) and for L = L3 P a b = 0 0 b a ; Qb a = (det )( 1 )b a ; W ab = Zab = 0: (A.1.108) GL(4; R ), whereas the sets of unimodular L2 evidently form two Abelian subgroups

A.1.11

Almost omplex stru ture on

M

ase1

ase2

ase3

6

An almost omplex stru ture on the spa e of 2-forms determines a splitting of the omplexi ation of M 6 into two invariant 3-dimensional subspa es.

Let us introdu e an almost omplex stru ture J on the M 6 . We re all that every tensor of type [11℄ represents a linear operator on a ve tor spa e. A

ordingly, if ' 2 M 6 , it is of type [01℄ and J(') an be de ned as a ontra tion. The result will be again an element of the M 6 . By de nition, J(J(')) = I6 ' or J2 = 1; (A.1.109) see (A.1.20). As a tensor of type [11℄, the operator J an be represented as a 6  6 matrix. Sin e the basis in M 6 is naturally split into 3 + 3 parts in (A.1.81), we an write it in terms of the set of four 3  3 matri es,

dualdef2

C ab Bab

almostIJ

JI = K



Aab Da b



:

(A.1.110)

A.1.11

Almost omplex stru ture on

M

6

61

Be ause of (A.1.109), the 3  3 blo ks A; B; C; D are onstrained by Aa B b + C a C b C a A b + Aa D b Ba C b + Da B b Ba A b + Da D b

= Æba ; = 0; = 0; = Æab :

(A.1.111)

almost lose

Complexi ation of the M 6 An almost omplex stru ture on the M 6 motivates a omplex generalization of the M 6 to the omplexi ed linear spa e M 6 (C ). The elements of the M 6 (C ) are the omplex 2-forms ! 2 M 6 (C ), i.e. their omponents !I in a de omposition ! = !I BI are omplex. Alternatively, one an onsider M 6 (C ) as a real 12-dimensional linear spa e spanned by the basis (BI ; iBI ), where i is the imaginary unit. We will denote by M 6 (C ) the omplex onjugate spa e. The same symmetri bilinear form as in (A.1.89) de nes also a natural metri in M 6 (C ). Note however, that now an orthogonal ( omplex) basis an be always introdu ed in M 6 (C ) so that "IJ = Æ IJ in that basis. In identally, one an de ne another s alar produ t on a omplex spa e M 6 (C ) by " (!; ') := (! ^ ')(e0 ; e1 ; e2 ; e3 ); 0

!; '

2 M 6:

(A.1.112)

6metri 1

The signi ant dieren e between these two metri s is that " assigns a real length to any omplex ve tor, whereas " de nes

omplex ve tor lengths. We will assume that the J operator is de ned in M 6 (C ) by the same formula J(! ) as in M 6 . In other words, J remains a real linear operator in M 6 (C ), i.e. for every omplex 2-form ! 2 M 6 (C ) one has J(! ) = J(! ). The eigenvalue problem for the operator J(! ) =  ! is meaningful only in the omplexi ed spa e M 6 (C ) be ause, in view of the property (A.1.109), the eigenvalues are  = i. Ea h of these two eigenvalues has multipli ity 3, whi h follows from the reality of J. Note that the 6  6 0

62

A.1.

Algebra

matrix of the

J operator has 6 eigenve tors,

but the number of

i

eigenve tors with eigenvalue + is equal to the number of eigen-

i be ause they are omplex onjugate

ve tors with eigenvalue to ea h other. Indeed, let

J(! ) = J(!) = i! .

J(! ) = i! , then the onjugation yields

Let us denote the 3-dimensional subspa es of

i

i by

orrespond to the eigenvalues + and

! 2 M 6 (C ) j J(! ) = i! ; (a)  M := ' 2 M 6 (C ) j J(') = i' ; (s)

M

:=



(s)

M

respe tively. Evidently,

M 6 (C )

whi h

(A.1.113)

subspa e

(A.1.114)

(a)

M . Therefore, we an restri t our

=

attention only to the self-dual subspa e. This will be assumed in our derivations from now on. A

ordingly, every form

! an be de omposed into a self-dual

and an anti-self-dual pie e4 ,

(s)

!=!

+

(a)

!;

(A.1.115)

selfantiself2

(A.1.116)

selfantiself1

with (s)

!

(a)

!

It an be he ked that

= =

1 2 1 2

(s)

!

[

iJ(! )℄ ;

! + iJ(! )℄ :

[

(s)

J( ! ) = +i !

and

(a)

(a)

J( ! ) = i ! .

4 A dis ussion of the use of self dual and anti-self dual 2-forms in general relativity

an be found in Kop zy nski and Trautman [13℄, e.g..

A.1.12

A.1.12

Computer algebra

63

Computer algebra

Also in ele trodynami s, resear h usually requires the appli ation of omputers. Besides numeri al methods and visualization te hniques, the manipulation of formulas by means of \ omputer algebra" systems is nearly a must. By no means are these methods on ned to pure algebra, also dierentiations and integrations, for example, an be exe uted with the help of omputer algebra tools. \If we do work on the foundations of lassi al ele trodynami s, we an dispense with omputer algebra," some true fundamentalists will laim. Is this really true? Well, later, as soon as we will analyze the Fresnel equation in Se . D.1.4, we ouldn't have done it to the extent we really did without using an eÆ ient omputer algebra system. Thus, our fundamentalist is well advised if she or he is going to learn some omputer algebra. A

ordingly, along with our introdu ing of some mathemati al tools in exterior al ulus, we will mention omputer algebra systems like Redu e 5 , Maple 6 , and Mathemati a 7 { and we will spe i ally explain of how to apply the Redu e pa kage Ex al 8 to the exterior forms immanent in ele trodynami s. In pra ti al work in solving problems by means of omputer algebra, it is our experien e that it is best to have a

ess to different omputer algebra systems. Even though in the ourse of time good features of one system \migrated" to other systems, still, for a ertain spe i ed purpose one system may be better suited than another one | and for dierent purposes these may 5 Hearn [8℄ reated this Lisp-based system. For introdu tions into Redu e, see Toussaint [36℄, Grozin [6℄, Ma Callum and Wright [15℄, or Winkelmann and Hehl [38℄; in the latter text you an learn of how to get hold of a Redu e system for your omputer. Redu e as applied to general-relativisti eld theories is des ribed, e.g., by M Crea [16℄ and by So orro et al. [28℄. In our presentation, we partly follow the le tures of Toussaint [36℄. 6 Maple, written in C, was reated by a group at the University of Waterloo, Canada. A good introdu tion is given by Char et al. [3℄. 7 Wolfram, see [39℄, reated the C-based Mathemati a software pa kage whi h is in very wide-spread use. 8 S hr ufer [25, 26℄ is the reator of that pa kage, f. also [27℄. Ex al is applied to Maxwell's theory by Puntigam et al. [22℄.

64

A.1.

Algebra

Figure A.1.5: \Here is the new Redu e-update on a hard disk."

be dierent systems. There does not exist as yet the optimal system for all purposes. Therefore, it is not only on rare o

asions that we have to feed the results of a al ulation by means of one system as input into another system. For omputations in ele trodynami s, relativity, and gravitation, we keep the three general-purpose omputer algebra systems Redu e, Maple, and Mathemati a. Other systems are available . Our workhorse for orresponding al ulations in exterior

al ulus is the Redu e-pa kage Ex al , but also in the MathTensor pa kage of Mathemati a exterior al ulus is implemented. For the manipulation of tensors we use the following pa kages: In 9

10

9 In

the review of Hartley [7℄ possible alternative systems are dis ussed, see also

Heini ke and Hehl [11℄.

10 Parker

tilis [37℄.

and Christensen [21℄ reated this pa kage; for a simple appli ation see Tsan-

A.1.12

Computer algebra

65

Redu e the library of M Crea11 and GRG 12 , in Maple GRTensorII 13, and in Mathemati a, besides MathTensor, the Cartan pa kage14 . Computer algebra systems are almost ex lusively intera tive systems nowadays. If one is installed in your omputer, you an

all the system usually by typing in its name or an abbreviation therefrom, i.e., `redu e', `maple', or `math', and then hitting the return key, or you have to li k the orresponding i on. In the

ase of `redu e', the system introdu es itself and issues a `1:'. It waits for your rst ommand. A ommand is a statement, usually some sort of expression, a part of a formula or a formula, followed by a terminator15. The latter is a semi olon ; if you want to see the answer of the system, otherwise a dollar sign $. Redu e is ase insensitive, i.e., the lower ase letter a is not distinguished from the upper ase letter A.

Formulating Redu e input As an input statement to Redu e, we type in a ertain legitimately formed expression. This means that, with the help of some operators, we ompose formulas a

ording to well-de ned rules. Most of the built-in operators of Redu e, like the arithmeti operators + (plus), - (minus), * (times), / (divided by), ** (to the power of)16 are self-explanatory. They are so- alled in x operators sin e the are positioned in between their arguments. By means of them we an onstru t ombined expressions of the type (x + y )2 or x3 sin x, whi h in Redu e read (x+y)**2 and x**3*sin x, respe tively. If the ommand 11 See M Crea's le tures [16℄. 12 The GRG system, reated by

Zhytnikov [40℄, and the GRGEC system of Terty hniy [35, 34, 20℄ grew from the same root; for an appli ation of GRGEC to the EinsteinMaxwell equations, see [33℄. 13 See the do umentation of Musgrave et al. [19℄. Maple appli ations to the EinsteinMaxwell system are overed in the le tures of M Lenaghan [17℄. 14 Soleng [30℄ is the reator of `Cartan'. 15 Has nothing to do with Arnold S hwarzenegger! 16 Usually one takes the ir um ex for exponentiation. However, in the Ex al pa kage this operator is rede ned and used as the wedge symbol for exterior multipli ation.

66

A.1.

Algebra

(x+y)**2;

is exe uted, you will get the expanded form x2 +2xy + y 2. There is a so- alled swit h exp in Redu e that is usually swit hed on. You an swit h it o by the ommand off exp;

Type in again (x+y)**2;

Now you will nd that Redu e doesn't do anything and gives the expression ba k as it re eived it. With on exp; you an go ba k to the original status. Using the swit hes is a typi al way to in uen e Redu e's way of how to evaluate an expression. A partial list of swit hes is

olle ted in a table on the next page. Let us give some more examples of expressions with in x operators: (u+v)*(y-x)/8 (a>b) and ( (greater than), < (less than). Widely used are also the in x operators neq

>=

18

% if a evaluates to an integer

If you want to display the truth value of a Boolean expression, use the if-statement, as in the following example: if

2**28 < 10**7 then write "less" else write "greater or equal";

Rudiments of evaluation A Redu e program is a follow-up of ommands. And the evaluation of the ommands may be onditioned by swit hes that we swit h on or o (also by a ommand). Let us look into the evaluation pro ess a bit loser. After a ommand has been sent to the omputer by hitting the return key, the whole ommand is evaluated. Ea h expression is evaluated from left to right, and the values obtained are ombined with the operators spe i ed. Sub{statements or sub{expressions existing within other expressions, like in

lear g,x$

72

A.1.

Algebra

a:=sin(g:=(x+7)**6);

os(n:=2)*df(x**10,x,n);

are always evaluated rst. In the rst ase, the value of (x+7)**6 is assigned to g, and then sin((x+7)**6) is assigned to a. Note that the value of a whole assignment statement is always the value of its right{hand{side. In the se ond ase, Redu e assigns 2 to n, then omputes df(x**10,x,2), and eventually returns 90*x**8* os(2) as the value of the whole statement. Note that both of these examples represents bad programming style, whi h should be avoided. One ex eption to the pro ess of evaluation exists for the assignment operator := . Usually, the arguments of an operator are evaluated before the operator is applied to its arguments. In an assignment statement, the left side of the assignment operator is not evaluated. Hen e

lear b, $ a:=b$ a:= $ a;

will not assign to b, but rather to a. The pro ess of evaluation in an assignment statement an be studied in the following examples:

lear h$ g:=1$ a:=(g+h)**3$ a; g:=7$ a;

% yields: (1+h)**3 % yields: (1+h)**3

After the se ond statement, the variable a hasn't the value (g+h)**3, but rather (1+h)**3. This doesn't hange by the fth statement either where a new value is assigned to g. As one will re ognize, a still has the value of (1+h)**3. If we want a to depend on g, then we must assign (g+h)**3 to a as long as g is still unbound:

A.1.12

lear g,h$ a:=(g+h)**3$ g:=1$ g:=7$ a;

Computer algebra

73

% all variables are still unbound % yields: (7+h)**3

Now a has the value of (7+h)**3 rather than (g+h)**3. Sometimes it is ne essary to remove the assigned value from a variable or an expression. This an be a hieved by using the operator lear as in

lear g,h$ a:=(g+h)**3$ g:=1$ a;

lear g$ a;

or by overwriting the old value by means of a new assignment statement:

lear b,u,v$ a:=(u+v)**2$ a:=a-v**2$ a; b:=b+1$ b;

The evaluation of a; results in the value u*(u+2*v), sin e (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+1; will, however, lead to a diÆ ulty: Sin e no value was previously assigned to b, the assignment repla es b literally with b+1 (whereas the previous a:=a-v**2 statement produ es the evaluation a:=(u+v)**2-v**2). The last evaluation b; will lead to an error or will even hang up the system, be ause b+1 is assigned to b. As soon as b is evaluated, Redu e returns b+1, whereby b still has the value b+1, and so on. Therefore the evaluation pro ess leads to an in nite loop. Hen e we should avoid su h re ursions.

74

A.1.

Algebra

In identally, if you want to nish a Redu e session, just type

in bye; After these glimpses on Redu e, we will turn to the real obje t of our interest.

Loading Ex al We load the Ex al pa kage by load_pa kage ex al $

The system will tell us that the operator ^ is rede ned, sin e it be ame the new wedge operator. Ex al is designed su h that the input to the omputer is the same as what would have been written down for a hand al ulation. For example, the statement f*x^y + u |(y^z^x) would be a legitimately built Ex al expression with ^ denoting the exterior and | (underline followed by a verti al bar) the interior produ t sign. Note that before the interior produ t sign | (spoken in) there must be a blank; the other blanks are optional. However, before Ex al an understand our intentions, we better de lare u to be a (tangential) ve tor tve tor u; f

to be a s alar (i.e. a zero-form), and

x; y; z

to be 1-forms:

pform f=0, x=1, y=1, z=1;

A variable that is not de lared to be a ve tor or a form is treated as a onstant; thus zero-forms must also be de lared. After our de larations, we an input our ommand f*x^y+u _|(y^z^x);

Of ourse, the system annot do mu h with this expression, but it expands the interior produ t. It also knows, of ourse, that u _|f;

A.1.12

Computer algebra

75

Figure A.1.6: \Catastrophi error," a Redu e error message. vanishes, that y ^ x = x ^ y , or that x ^ x = 0. If we want to

he k the rank of an expression, we an use exdegree (x^y);

This yields 2 for our example. Quite generally, Ex al an handle s alar-valued exterior forms, ve tors and operations between them, as well as non-s alar valued forms (indexed forms). Simple examples of indexed forms are the Krone ker delta Æ or the onne tion 1-form  of Se . C.1.2. Their de laration reads pform delta(a,b)=0, gamma1(a,b)=1;

The names of the indi es are arbitrary. Subsequently, in the program a lower index is marked by a minus sign, an upper index with a plus (or with nothing), i.e., Æ11 translates into delta(-1,1) et . Ex al is a good tool for studying dierential equations, for

al ulations in eld theory and general relativity or for su h simple things as al ulating the Lapla ian of a tensor eld for an arbitrarily given frame. Ex al is ompletely embedded in

76

A.1.

Algebra

Redu e. Thus, all features and fa ilities of Redu e are available in a al ulation. If we de lare the dimension of the underlying spa e by spa edim 4;

then pform a=2,b=3;

a^b;

yields 0. These are the fundamental ommands of Ex al for exterior algebra. As soon as we will have introdu ed exterior al ulus with frames and oframes, with ve tor elds and elds of forms { not to forget exterior and Lie dierentiation, we will ome ba k to Ex al and we will better appre iate its real power.

A.2

Exterior al ulus

Having developed the on epts involved in the exterior algebra asso iated with an n-dimensional linear ve tor spa e V , we now look at how this stru ture an be `lifted' onto an n-dimensional dierentiable manifold Xn or, for short, onto X . The pro edure for doing this is the same as for the transition from tensor algebra to tensor al ulus. At ea h point x of X there is an n-dimensional ve tor spa e Xx , the tangent ve tor spa e at x. We identify the spa e Xx with the ve tor spa e V onsidered in the previous hapter. Then, at ea h point x, the exterior algebra of forms is determined on V = Xx . However, in dierential geometry, one is on erned not so mu h with obje ts de ned at isolated points as with elds over the manifold X or over open sets U  X . A eld ! of p-forms on X is de ned by assigning a p-form to ea h point x of X and, if this assignment is performed in a smooth manner, we shall all the resulting eld of p-forms an exterior dierential p-form . For simpli ity we shall take `smooth' to mean C 1 , although in physi al appli ations the degree of dierentiability may be less.

78

A.2.1

A.2.

Exterior al ulus

Dierentiable manifolds A topologi al spa e be omes a dierentiable manifold when an atlas of oordinate harts is introdu ed in it. Coordinate transformations are smooth in the interse tions of the harts. The atlas is oriented when in all interse tions the Ja obians of the oordinate transformations are positive.

In order to des ribe more rigorously how elds are introdu ed on X , we have to re all some basi fa ts about manifolds. At the start, one needs a topologi al stru ture. To be spe i , we will normally assume that X is a onne ted, Hausdor, and para ompa t topologi al spa e. Topology on X is introdu ed by the olle tion of open sets T = fU  X j 2 I g whi h, by de nition, satisfy the three onditions: (i) both, the empty set  and the manifold itself X belong to that olle tion, ; X 2 T , S (ii) any union of open sets is again open, i.e., 2J U 2 T for any subset J 2 I , T (iii) any interse tion of a nite number of open sets is open, i.e., 2K U 2 T for any nite subset K 2 I . A topologi al spa eSX is onne ted, if one annot represent it T by the sum X = X1 X2 , with open X1;2 and X1 X2 = . Usually, for a spa etime manifold, one further requires a linear onne tedness whi h means that any two points of X an be onne ted by a ontinuous path. A topologi al spa e X is Hausdor when for any two points p1 6= p2 2 X one

an nd T open sets p1 2 U1  X , p2 2 U2  X su h that U1 U2 = . Hausdor's axiom forbids the `bran hed' manifolds of the sort depi ted in Fig. A.2.1. Finally, a onne ted Hausdor manifold is para ompa t when XS an be overed by a ountable number of open sets, i.e. X = 2K U for a ountable subset K 2 I . A dierentiable manifold is a topologi al spa e X plus a differentiable stru ture on it. The latter is de ned as follows: A

oordinate hart on X is a pair (U; ), where U 2 T is an open set and the map  : U ! R n is a homeomorphism (i.e., ontinuous with a ontinuous inverse map) of U onto an open subset of the arithmeti spa e of n-tuples R n . This map assigns n labels or

oordinates (p) = fx1 (p); : : : ; xn (p)g to any point p 2 U  X .

A.2.1 Dierentiable manifolds

79

(1/2) / 0

0

+

1

(1/2)

1

=

(1/2) 0

/

1/

(1/2) 1

Figure A.2.1: Non-Hausdor manifold: take two opies of the line segment f0; 1g, and identify (paste together) their left halves ex luding the points (1=2) and (1=2)0 . In the resulting manifold, the Hausdor axiom is violated for the pair of points (1=2) and (1=2)0 . Given T any two interse ting harts, (U ; ) and (U ;  ) with U U 6= , the map

f :=  Æ  1 : R n

! Rn

(A.2.1)

otra

is C 1 . The latter gives oordinate transformation in the interse tion of harts. The whole olle tion of harts f(US  ;  )j 2 I g is alled an atlas for every open overing of X = 2I U . The two atlases f(U ;  )g and f(V ;  )g are said to be ompatible if their union is again an atlas. Finally, the dierentiable stru ture on X is a maximal atlas A(X ) in the sense that its union with any atlas gives again A(X ). The atlas f(U ;  )g on X is said oriented, if all the transition fun tions (A.2.1) are orientation preserving, i.e., the orresponding Ja obian determinants are everywhere positive,





xi J (f ) = det > 0; y j

(A.2.2)

where  = fxi g and  = fy ig, with i; j = 1; : : : ; n. Then f = (x1 (y 1; : : : ; y n); : : : ; xn (y 1; : : : ; y n)). The dierentiable 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

J

80

A.2.

Exterior al ulus

D1111111111 C 0000000000 1111111111 0000000000 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111 0000000000 1111111111

A

B

Figure A.2.2: Re tangle ABCD in R 2 . of dierential forms. It is straightforward to provide examples of orientable and non-orientable manifolds. The following twodimensional manifolds an be easily onstru ted with the help of the ut and paste te hniques: Consider an R2 and ut out a re tangle ABCD, as shown in Fig. A.2.2. As su h, this is a ompa t two-dimensional manifold with boundary whi h is topologi ally equivalent to a dis . However, after gluing together the sides of this re tangle, one an onstru t a number of ompa t manifolds without a boundary. The rst example is obtained when we identify the opposite sides without twisting them, as shown in Fig. A.2.3. The resulting manifold is a two-dimensional torus T2 that is topologi ally equivalent to a sphere with one handle. This two-dimensional ompa t manifold is orientable. Another possibility is to glue the re tangle ABCD together after twisting both pairs of opposite sides, beforehand. This is shown in Fig. A.2.4. As a result, one obtains a real proje tive plane P2 whi h is represented by a sphere with a dis removed and the resulting hole is losed up by a \ ross- ap", i.e. by identifying its diametri ally opposite points (a more spe ta ular way is to say that a hole is losed by a Mobius strip). This two-dimensional manifold is also ompa t, but it is non-orientable. Finally, we an glue the re tangle ABCD with twisting one pair of opposite sides while mat hing the two other sides untwisted, as shown in Fig. A.2.5. The resulting

ompa t two-dimensional manifold is a famous Klein bottle K 2 . The Klein bottle annot be drawn in the R3 without self-interse tions. However, it is possible to understand it as a sphere with two dis s removed and the holes losed up with two \ ross- aps" (the Mobius strips). The Klein bottle is also a ompa t but non-orientable manifold. In general, one an prove that any onne ted ompa t two-dimensional manifold an be realized as a sphere with a ertain number of small dis s removed and a nite number of either handles or \ ross- aps" atta hed in order to lose the holes.

A.2.1 Dierentiable manifolds

D=A

C=A

= A

B=A Figure A.2.3: Torus

D=B

T2 .

C =A

= A

B Figure A.2.4: Real proje tive plane

P2 .

81

82

A.2.

D=A

Exterior al ulus

C =A

= B=A

A

Figure A.2.5: Klein bottle K 2 .

A.2.2

Ve tor elds Ve tor elds smoothly assign to ea h point of a manifold an element of the tangent spa e. The ommutator of two ve tor elds is a new ve tor eld.

Let us denote by C (X ) the algebra of dierentiable fun tions on X . A tangent ve tor u at a point x 2 X is de ned as an operator whi h maps C (X ) into R and satis es the ondition

8

( ) = f (x) u(g ) + g (x) u(f );

u fg

f; g

2

( )

C X :

(A.2.3)

A physi al motivation omes from the notion of velo ity. Indeed, let us onsider a smooth urve x(t) su h that 0  t  1 and x(0) = x. Then the dire tional derivative of a fun tion f 2 C (X ) along x(t) at x, ( )=

v f

( ( ))

df x t dt

t

=0

;

(A.2.4)

is a linear mapping v : C (X ) ! R satisfying (A.2.3). Choose a lo al oordinate system fxi g (i = 1; : : : ; n) on a oordinate neighborhood U 3 x. Then the dierential operator i := =xi , for ea h i, satis es (A.2.3). It an be demonstrated that the set of ve tors fi g, i = 1; : : : ; n provides a basis of the tangent

ve 1

A.2.3 One-form elds, dierential p-forms

83

spa e X at x 2 U . We will use Latin letters to label oordinate indi es. A mapping u whi h assigns a tangent ve tor u 2 X to ea h point x is alled a ve tor eld on the manifold X . If we onsider a smooth fun tion f (x) on X , then u(f ) := u (f ) is a fun tion on X . A ve tor eld is alled dierentiable when a fun tion u(f ) is dierentiable for any f 2 C (X ). In lo al oordinates fx g, a ve tor eld is des ribed u = u (x)  by its omponents u (x) whi h are smooth fun tions of oordinates. For every two ve tor elds u and v a ommutator [u; v ℄ is naturally de ned by x

x

x

x

i

i

i

i

[u; v ℄(f ) := u(v (f ))

v (u(f )):

(A.2.5)

omm-uv

This is again a ve tor eld. Please he k that the ondition (A.2.3) is satis ed. In lo al oordinates, be ause of u = u (x)  and v = v (x)  , we nd the omponents of the ommutator [u; v ℄ = [u; v ℄ (x)  as i

i

i

i

i

[u; v ℄ = u v i

A.2.3

i

j

v u :

i

j

;j

(A.2.6)

i

;j

omm-uv-i

One-form elds, dierential p-forms One-form elds assign to ea h point of a manifold an element of the dual tangent spa e. Dierential pforms are then de ned pointwise as the exterior produ ts of 1-form elds.

The dual ve tor spa e X  is alled otangent spa e at x. The elements of X  are 1-forms ! whi h map X into R . An exterior dierential 1-form ! is de ned on X if a 1-form ! 2 X  is assigned to ea h point x. A natural example is provided by the dierential df of a fun tion f whi h is de ned by x

x

x

x

8u 2 X ;

(df (u)) := u (f ); x

x

x

or, in lo al oordinates fx g,

x

(A.2.7)

dfun1

(A.2.8)

dfun2

i

df = f dx : i

;i

84

A.2.

Exterior al ulus

Obviously, the oordinate dierentials dx des ribe elds of 1forms whi h provide a basis of X  at ea h point. The bases f g and fdx g are dual to ea h other, i.e., dx ( ) = Æ . Repeating pointwise the onstru tions of the previous Se . A.2.2, we nd that a dierential p-form ! on U  X is expressible in the form 1 ! (x) dx 1 ^ : : : ^ dx p ; != (A.2.9) 1 p p! i

i

x

i

i

i

j

i

j

i

i :::i

where ! 1 p (x) = ![ 1 p℄(x) are dierentiable fun tions of the

oordinates. The spa e of dierential p-forms on X will be denoted by  (X ). In this ontext we may write 0(X ) = C (X ) for the set of dierentiable fun tions on X . i :::i

i :::i

p

A.2.4

Images of ve tors and one-forms

The image of a ve tor eld (an arrow) arises from the velo ity of a point moving along an arbitrary urve. The prototype of an image of a 1-form (a pair of ordered hyperplanes) emerges from dierential of a fun tion.

It seems worthwhile to provide simple pi tures for dierential 1-form elds and ve tor elds. The physi al prototype of a tangent ve tor v is a velo ity of a parti le moving along a given

urve. Therefore a ve tor an pi torially be represented by an arrow, see Fig. A.2.6. The prototype of a 1-form ! an be represented by the dierential df of a fun tion f : its omponents represent the gradient. Therefore, a suitable pi ture for a 1-form is given by two parallel hyperplanes, see Fig. A.2.6, whi h des ribe surfa es of onstant value of f . With an arrowhead it is indi ated in whi h dire tion the value of f is in reasing. The \stronger" the 1-form is, the loser those two planes are. In physi s, a generi 1-form is the wave 1-form de ned as the gradient of the phase (think of a de Broglie wave!), or the momentum 1-form, de ned as the gradient of the a tion fun tion.

expform

A.2.4

Images of ve tors and one-forms

v

.

85

ω

P.

P

(a) arbitrary vector v at a point P

(b) arbitrary 1-form at a point P

ω

Figure A.2.6: (a) Image of a ve tor (` ontravariant ve tor') at a point

P.

(b) Image of a 1-form (` ove tor' or ` ovariant ve tor')

P.

at a point

x3

e3=

3

e2=

.

x

2

.

.

x2

.

1

θ = dx 1

Figure A.2.7: Lo al oordinates (x

1

1

; x2 ; x3 )

at a point

dimensional manifold and the basis ve tors (e1 ; i

i

# = dx , i = 1; 2; 3, are supposed to #1 (e1 ) = 1; #1 (e2 ) = 0; #1 (e3 ) = 0, et ., i i a

ording to # (ej ) = Æj .

1-forms that

2

2

e1 = 1

P

θ = dx

θ = dx 3 3

P

of a 3-

e2 ; e3 ). The basis be also at P . Note i i.e. # is dual to ej

86

.

A.2.

v

Exterior al ulus

P.’

.

P

P ω

π

ω( v)=1

π( v)=2

v v

P. u

ρ 1 , ρ(u)=− 5 ρ( v)=− 2

Figure A.2.8: Two-dimensional images of 1-forms ! ,  ,  of different strengths at P . They are applied to a ertain ve tor v at the same point P and, in the ase of , also to the ve tor u.

To give a spe i example, let us onsider a three-dimensional manifold with lo al oordinates (x ; x ; x ). A hyperplane is then simply a two-dimensional plane. A lo al basis of ve tors is given by ei = i , with i = 1; 2; 3, whi h are tangent to the oordinate lines xi , respe tively. Similarly, a lo al basis of 1-forms #i = dxi is represented by the lo al planes whi h depi t surfa es of

onstant value of the relevant oordinate xi , see Fig. A.2.7. A 1-form is de ned in su h a way that, if applied to a ve tor, a number (s alar) pops out. The pi torial representation of su h a number is straightforward, see Fig. A.2.8: A straight line starting at P in the dire tion of the ve tor v disse ts the se ond hypersurfa e at P . The number ! (v ) is then the size of the ve tor v measured in terms of the segment P P used as a unit. 1

2

3

0

0

1

1 Pi tures

of ve tors and one-forms, and of many other geometri al quantities, an

be found in S houten [24℄, see, e.g., page 55. Also easily a

essible is Misner et al. [18℄ where in Chapter 4 a number of orresponding worked out examples and ni e pi tures are displayed. More re ent books with beautiful images of forms and their manipulation in lude those of Burke [2℄ and Jan ewi z [12℄.

A.2.5

Volume

A.2.5

An

Volume forms and orientability

87

forms and orientability

everywhere

non-vanishing

n-form

on

an

n-

dimensional manifold is alled a volume form.

Like in the ase of a ve tor spa e V , dierential forms of maximal rank on a manifold X are losely related to volume and orientation. Any everywhere non-vanishing n-form ! is alled a volume form on X . Obviously, it is determined by a single

omponent in every lo al oordinate hart (U ; ): !=

1 !i :::i (x) dxi1 ^ : : : ^ dxin = !1:::n (x) dx1 ^ : : : ^ dxn : n! 1 n (A.2.10)

A manifold X is orientable if and only if it has a global volume form. Indeed, given a volume form ! whi h, by de nition, does not vanish for any x, the fun tion !1:::n (x) is either everywhere positive or everywhere negative in U . If it is negative, we an simply repla e the lo al oordinate x1 by x1 = x1 . Then !1 2:::n (x) > 0. Thus, without any restri tion, we have positive oeÆ ient fun tions !1:::n (x) in all harts of the atlas f(U ; )g. In an interse tion of any two harts U U with lo al oordinates  = fxi g and  = fy ig, we have 0

0

T

! = !1:::n (y ) dy 1 ^ : : : ^ dy n = !1:::n (x) dx1 ^ : : : ^ dxn = !1:::n (x) J (f ) dy 1 ^ : : : ^ dy n: (A.2.11)

Thus !1:::n (y ) = !1:::n (x) J (f ). Sin e both !1:::n (x) and !1:::n (y ) are positive, we on lude that the Ja obian determinant J (f ) is also positive for all interse ting harts, f. (A.2.2). Hen e the atlas f(U ;  )g is oriented. Conversely, let the atlas f(U ;  )g be oriented, i.e., (A.2.2) holds true. Then in ea h hart (U ;  ) we have an evidently non-vanishing n-form ! () := dx1 ^ : : : ^ dxn . In the interse tions U U one nds ! () = J (f ) ! ( ). Let f g be a partition of unity subordinate to the overing fU g of X . Then we de ne a global n-form ! =   ! () . Sin e in the overlapping harts all non-trivial forms ! () are positive multiples of ea h other

T

P

volform

88

A.2.

P

Exterior al ulus

and  (p)  0,   (p) = 1 (i.e., all  annot vanish at any point p), we on lude that ! is a volume form. A.2.6

Twisted forms The dierential forms that an be de ned on a non-orientable manifold are alled twisted dierential forms. They are orientation-valued in terms of the

onventional dierential forms of Se . A.2.3.

\Sin e the integral of a dierential form on R n is not invariant under the whole group of dieomorphisms of R n , but only under the subgroup of orientation-preserving dieomorphisms, a dierential form annot be integrated over a nonorientable manifold. However, by modifying a dierential form we obtain something alled a density, whi h an be integrated over any manifold, orientable or not."2 Besides the onventional dierential forms, one an de ne slightly dierent obje ts whi h are alled twisted forms3 (or, sometimes, \odd forms", \impair forms", \densities" or \pseudoforms"). The twisted forms are ne essary for an appropriate representation of ertain physi al quantities, su h as the ele tri urrent density. Moreover, they are indispensable when one

onsiders the integration theory on manifolds and, in parti ular, on non-orientable manifolds. In Se . A.1.3, see Examples 4) and 5), a twisted form was de ned on a ve tor spa e V as a geometri quantity. Intuitively, a twisted form on the manifold X an be de ned as an \orientationvalued" onventional exterior form. Given an atlas f(U ;  )g, a twisted p-form is represented by a family of dierential p-forms 2 Bott & Tu [1℄ p.79. 3 \Twisted tensors were

introdu ed by Hermann Weyl....and de Rham... alled them tensors of odd kind... We ould make a good ase that the usual dierential forms are a tually the twisted ones, but the language is for ed on us by history. Twisted dierential forms are the natural representations for densities, and sometimes are a tually alled densities, whi h would be an ideal name were it not already in use in tensor analysis. I agonized over a notation for twisted tensors, say, a dierent typefa e. In the end I de ided against it..." William G. Burke [2℄, p. 183.

A.2.6

A

Twisted forms

89

A

ξ θ

B

B

U1

U2

U2

U1

U

U

U1

U2

Figure A.2.9: Mobius strip.

f!  g su h that in the interse tions U T U ( )

! ()

= sgnJ (f ) ! ( ):

(A.2.12)

Example: Consider the Mobius strip, a non-orientable two-

dimensional ompa t manifold with boundary, see Fig. A.2.9. It

an be easily realized by taking a re tangle f(;  ) 2 R 2 j0 <  < 2; 1 <  < 1g and gluing it together with one twist along verti al sides. The simplest atlas for the resulting manifold onsists of two harts (U1 ; 1 ) and (U2 ; 2). The open domains U1;2 are re tangles and they an be hosen as shown in Fig. A.2.9, with the evidently de ned lo al oordinate maps 1 = (x1 ; x2 ) and 2 = (y 1 ; y 2 ), where the rst oordinate runs along the re tanT gles and the se ond one a ross them. The interse tion U1 U2 is T T

omprised of two open sets, (U1 U2 )left and (U1 U2 )right . The transition fun tions f12 = 1 Æ 2 1 are f12 = fx1 = y 1T ; x2 = y 2 g T 1 1 2 2 in (U1 U2 )left and f12 = fx = y ; x = y g in (U1 U2 )right , so that J (f12 ) = 1 in these domains, respe tively. The 1-form ! = f! (1) = dx2 ; ! (2) = dx2 g is a twisted form on the M obius strip. In general, given a hart (U ;  ), both a usual and a twisted p-form is given by its omponents !i1 :::ip (x), see (A.2.9). With

twist

90

A.2.

Exterior al ulus

a hange of oordinates, the omponents of a twisted form, via (A.2.12), are transformed as !1

i :::i



sgn det x p (x) = y

i

j



y 1 y p ::: ! x 1 x p 1 j

j

i

i

j :::jp

(y): (A.2.13)

For a onventional p-form, the rst fa tor on the right-hand side, the sign of the Ja obian, is absent. Normally, in gravity and in eld theory one works on orientable manifolds with an oriented atlas hosen. Then the dieren e between ordinary and twisted obje ts disappears, be ause of (A.2.2). However, twisted forms are very important on nonorientable manifolds on whi h usual forms annot be integrated. A.2.7

Exterior derivative The exterior derivative maps a

p-form into a (p + 1)d2 = 0.

form. Its ru ial property is nilpoten y,

Denote the set of ve tor elds on X by X01. For 0-forms f 2 the dierential 1-form df is de ned by (A.2.7), (A.2.8), i.e., by df = f dx . We wish to extend this map d : 0 (X ) ! 1(X ) to a map d :  (X ) !  +1(X ): Ideally this should be performed in a oordinate-free way and we shall give su h a de nition at the end of this se tion. However, the de nition of exterior derivative of a p-form in terms of a oordinate basis is very transparent. Furthermore, it is a simple matter to prove that it is, in fa t, independent of the lo al oordinate system that is used. Starting with the expression (A.2.9) for a ! 2  (X ), we de ne d! 2  +1 (X ) by 0(X )

i

;i

p

p

p

p

d ! :=

1 d! p! 1

i :::ip

^ dx ^ : : : ^ dx p : i1

i

(A.2.14)

By (A.2.7), (A.2.8), the right-hand side of (A.2.14) is (1=p!) ! 1

i :::ip ;j

dx

j

^ dx ^ : : : ^ dx p : i1

i

(A.2.15)

exdf

A.2.7

Exterior derivative

91

Hen e, be ause of the antisymmetry of the exterior produ t, we may write 1 d ! = ![ 1 p ℄ dx ^ dx 1 ^ : : : ^ dx p : (A.2.16) p! j

i

i

i :::i ;j

domega

Under a oordinate transformation fx g ! fx g it is found that ![

i0 :::i0p ;j 0

Hen e, ![ 1 0

i :::i0p ;j 0

℄ dx

℄=

x 1 x 1 i

i

0




xx

+1

ip

i

i

![ 1

+1 ℄ :

0

(A.2.17)

i :::ip ;ip

j0

^ dx 1 ^ : : : ^ dx p = = ![ 1 p p+1℄ dx p+1 ^ dx 1 ^ : : : ^ dx p ; (A.2.18)

j

0

i

0

i

0

i

i

i

i :::i ;i

so that the exterior derivative, as de ned by (A.2.16), is independent of the oordinate system hosen. Proposition 1: The exterior derivative, as de ned by (A.2.16), is a map d :  (X ) !  +1(X ) (A.2.19) with the following properties: 1) d(! + ) = d! + d [linearity℄, 2) d(! ^ ) = d! ^  + ( 1) ! ^ d [(anti-)Leibniz rule℄, 3) df (u) = u(f ) [partial derivative for fun tions℄, 4) d(d! ) = 0 [nilpoten y℄. Here, !;  2  (X );  2  (X ); f 2 0(X ); u 2 X01 (X ). Proof. 1) and 3) are obvious from the de nition. Be ause of 1) and the distributive property of the exterior multipli ation, it is suÆ ient to prove 2) for ! and  of the `monomial' form: ! = f dx 1 ^ : : : ^ dx p ;  = h dx 1 ^ : : : ^ dx q : (A.2.20) p

p

d

p

p

i

q

i

j

j

monom

92

A.2.

Exterior al ulus

Thus ! ^  = fh dx 1 ^ : : : ^ dx p ^ dx 1 ^ : : : ^ dx q : i

i

j

(A.2.21)

j

monom2

Then d(! ^ ) = d(fh) ^ dx 1 ^ : : : ^ dx q = (f dh + h df ) ^ dx 1 ^ : : : ^ dx q = (df ^ dx 1 ^ : : : ^ dx p ) ^ (h dx 1 ^ : : : ^ dx q ) + ( 1) (f dx 1 ^ : : : ^ dx p ) ^ (dh ^ dx 1 ^ : : : ^ dx q ) = d! ^  + ( 1) ! ^ d : (A.2.22) To prove 4), we rst of all note that, for a fun tion f 2 0(X ), d(df ) = f [ ℄ dx ^ dx = 0 ; (A.2.23) ddo sin e partial derivatives ommute. For a p-form it is suÆ ient to onsider a monomial (A.2.24) mono ! = f dx 1 ^ : : : ^ dx p : Then d! = df ^ dx 1 ^ : : : ^ dx p ; (A.2.25) and repeated appli ation of property 2) and (A.2.23) yields the desired result d(d!) = 0. By linearity, this may be extended to a general p-form whi h is a linear ombination of terms like (A.2.24). i

j

i

j

i

i

p

j

i

j

i

j

j

p

j

i

; ij

i

i

i

i

Proposition 2: Invariant expression for the exterior derivative.

For ! 2  (X ), we an express d! in a oordinate-free manner as follows: p

X ( 1) u (!(u ; : : : ; ub ; : : : ; u )) d ! (u ; u ; : : : ; u ) = X ( 1) !([u ; u ℄; u ; : : : ; ub ; : : : ; ub ; : : : ; u ) ; + p

0 1

0

j

p

j

=0 + i

0

i 4 will vanish identi ally. Similarly, for the 1-form A, we have (A; dA; A ^ dA; dA ^ dA), and, if we mix both, then the sequen e (A ^ H; dA ^ H; A ^ dH ) emerges. Note that A ^ A  0 sin e A is a 1-form. In this way, we reate the new 3-forms K , see (B.3.20), and CA, see (B.3.16): By the same token, the four dierent invariants Ii of the ele tromagneti eld arise as 4-forms, see Table B.2.1 on page . B.3.4

Measuring the ex itation

The ele tri ex itation D an be operationally de ned by using the Gauss law (B.1.41). Put at some point P in free spa e

AJgauge

170

B.3.

Magneti ux onservation

Table B.3.1: Three-forms of the ele tromagneti eld in in 4D. 3-form

=A^H A =A^F

K C

J

dimension h

(h=q )2 q

un-/twisted

twisted untwisted untwisted

name

Kiehn Chern-Simons ele tri urrent

two small, thin, and ele tri ally ondu ting (metal) plates of arbitrary shape with insulating handles (\Maxwellian double plates"), see Fig.B.3.5. Suppose we hoose lo al oordinates (x1 ; x2 ; x3 ) in the neighborhood of P su h that P = (0; 0; 0). The ve tors a , with a = 1; 2; 3, span the tangent spa e at P . Press the plates together at P , and orient them in su h a way that their ommon boundary is given by the equation x3 = 0 (and thus (x1 ; x2 ) are the lo al oordinates on ea h plate's surfa e). Separate the plates and measure the net harge Q and the area S of that plate for whi h the ve tor 3 points outwards from its boundary surfa e. Determine experimentally limS !0 Q=S as well as possible. Then

Q D12 = Slim !0 S

:

(B.3.26)

dsurf1

Te hni ally, one an realize the limiting pro ess by onstru ting the plates in the form of parallelograms with the sides a 1 and a 2 . Then (B.3.26) is repla ed by

Q D12 = alim !0 2 a

;

(B.3.27)

dsurf1h

Repeat this measurement with the two other possible orientations of the plates (be areful with hoosing one of the two plates in a

ordan e with the orientation pres ription of above). Then, similarly, one nds D23 and D31 . Thus nally the ele tri ex itation is measured: D

=

1 a b 1 2 2 3 3 1 Dab dx ^ dx = D12 dx ^ dx + D23 dx ^ dx + D31 dx ^ dx : 2 (B.3.28) dsurf2

B.3.4

D =0

Measuring the ex itation

171

++ + ++ - - P + . - - insulating + + handles ++ - - -++ + + + + + external charges + ++ + +

Figure B.3.5: Measurement of D at a point P . The dimension of D is that of a harge, i.e. [D℄ = q , for its

omponents we have [Dab ℄ = q l 2 . Let us prove the orre tness of this pres ription. The double plates are assumed to be ideal ondu tors. Therefore the ele tri eld E inside the ondu tor has to vanish: E

inside

= 0:

(B.3.29)

Ein

A

ording to (B.2.6), the ele tri eld E is uniquely de ned. Thus the vanishing of E de nes a unique ele trodynami al state in the ondu tor. And for the ele tri ex itation D, guided by experien e,4 we assume that, in turn, it vanishes, too:

D inside = 0

:

(B.3.30)

This is a rudiment of the spa etime relation that links F and H and whi h makes the ex itation eld D unique. For that reason we had postponed this dis ussion until now, sin e the knowledge of the notion of the eld strength E is ne essary for it. 4 The only really appropriate dis ussion on the de nitions of D and E , whi h we know of, was given by Pohl [28℄.

Din

172

B.3.

Magneti ux onservation

By (B.3.30), we sele ted from the allowed lass (B.1.21) of the ex itations that D whi h will be measured by the double plates. In a more general setting, let us onsider the ele tromagneti ex itation D near a 2-dimensional boundary surfa e S  h whi h separates two parts of spa e lled with two dierent types of matter. Choose a point P 2 S . Introdu e lo al oordinates (x1 ; x2 ; x3) in the neighborhood of P in su h a way that P = (0; 0; 0), while (x1 ; x2) are the oordinates on S , see Fig.B.3.6. Let V be a 3-dimensional domain whi h is half (denoted V1) in one medium and half (V2) in another one, with the two halves V1;2 spanned by the triples of ve tors (1 ; 2; a3 ) and (1 ; 2; a3 ), respe tively. For de niteness, we assume that 3 points from medium 1 to medium 2. Now we onsider the Gauss law (B.1.41) in this domain. Integrate (B.1.41) over V , and take the limit a ! 0. The result is Z Z Z D(2) D(1) = alim (B.3.31) !0  ;
S12


S12

V

where D(1) and D(2) denote the values of the ele tri ex itations in the medium 1 and 2, respe tively, while
S12 is a pie e of S spanned by (1 ; 2 ). Despite the fa t that the volume V learly goes to zero, the right-hand side of (B.3.31) is nontrivial when there is a surfa e

harge exa tly on the boundary S . Mathemati ally, in lo al oordinates (x1; x2 ; x3 ), this an be des ribed by the Æ-fun tion stru ture of the harge density:  = 123 dx1 ^ dx2 ^ dx3 = Æ (x3 )  e12 dx1 ^ dx2 ^ dx3 : (B.3.32) Substituting (B.3.32) into (B.3.31), we then nd Z


S12

where e =

D(2)

Z

D(1) =


S12  e12 dx1 ^ dx2 is

harge density

difD1

Z


S12

 e;

(B.3.33)

the 2-form of the ele tri surfa e . Sin e P and
S12 are arbitrary, we on lude

delrho

difD2

B.3.4

3

Measuring the ex itation

173

x3

2

P

2

x2 1

V1 V2

x1

S

h -h

1

Figure B.3.6: Ele tri ex itation on the boundary between two media. that on the separating surfa e S between two media, the ele tri ex itation satis es

D

(2)

S

D

(1)

S

= e:

(B.3.34)

Returning to the measurement pro ess with plates, we have

D = 0 inside an ideal ondu tor, and (B.3.34) justi es the (1)

de nition of the ele tri ex itation as the harge density on the surfa e of the plate (B.3.26). Thereby we re ognize that the ele tri ex itation D is, by its very de nition, the ability to separate harges on (ideally ondu ting) double plates. A similar result holds for the magneti ex itation H. Analogously to the small 3-dimensional domain V , let us onsider a 2-dimensional domain  whi h is half ( ) in one medium and half ( ) in another one, with the two halves  spanned by the ve tors (l; a ) and (l; a ), respe tively. Here l = l  + l  is an arbitrary ve tor tangent to S at P . Integrating the Maxwell 1

2

1;2

3

3

1

1

2

2

DonS

174

B.3.

Magneti ux onservation

equation (B.1.40) over , and taking the limit a ! 0, we nd Z

H(2)

l

Z l

H(1) = alim !0

Z

(B.3.35)

j:



The se ond term on left-hand side of (B.1.40) has a zero limit, be ause D_ is ontinuous and nite in the domain of a loop whi h

ontra ts to zero. However, the right-hand side of (B.3.35) produ es a nontrivial result when there are surfa e urrents owing exa tly on the boundary surfa e S . Analogously to (B.3.32), this is des ribed by j = j13 dx1 ^ dx3 + j23 dx2 ^ dx3 = Æ (x3 ) e ja dxa ^ dx3 : (B.3.36) Substituting (B.3.36) into (B.3.35), we nd Z l

H(2)

Z

H(1) =

l

difH1

Z

ej;

delj

(B.3.37)

l

and, sin e l is arbitrary, eventually:

H(2) S H(1) S = ej :

(B.3.38)

Thus we see that on the boundary surfa e between the two media the magneti ex itation is dire tly related to the 1-form of the ele tri surfa e urrent density e j. To measure H, following a suggestion of M. Zirnbauer,5 we

an use the Meissner ee t. Be ause of this ee t, the magneti eld B is driven out from the super ondu ting state. Take a thin super ondu ting wire (sin e the wire is pretty old, perhaps around 10 K, taking is not to be understood too literally), and put it at the point P where you want to measure H (see Fig.B.3.7(a)). Be ause of the Meissner ee t, the magneti eld B and, if we assume again that B = 0 implies H = 0, the ex itation H is expelled from the super ondu ting region (apart from 5

See his le ture notes [40℄, ompare also Ingarden and Jamiolkowski [16℄.

HonS

B.3.4

Measuring the ex itation

175

H

H

A

P

.

L

.P

J H

H ind

ind

(a)

(b)

Figure B.3.7: Measurement of H at a point P . a thin surfa e layer of some 10 nm, where H an penetrate). A

ording to the Oersted-Ampere law (B.1.40) dH = j { we assume quasi-stationarity in order to be permitted to forget about D_ { the ompensation of H at P an be a hieved by surfa e urrents J owing around the super ondu ting wire. These indu ed surfa e urrents J are of su h a type that they generate an Hind whi h ompensates the H to be measured: Hind + H = 0. We have to hange the angular orientation su h that we eventually nd the maximal urrent Jmax . Then, if the x3 - oordinate line is hosen tangentially to the \maximal orientation", we have

Jmax H3 = Llim !0 L

;

(B.3.39)

max

where L is the length parallel to the wire axis transverse to whi h the urrent has been measured. A

ordingly, we nd

H = H3 3 (B.3.40) Clearly the dimension of H is that of an ele tri urrent: [H℄ = 1 SI 1 1 SI = and [Ha ℄ = = . dx

qt

A

ql

t

:

A=m

If you dislike this thought experiment, you an take a real small test oil at P , orient it suitably, and read o the orre-

H3

176

B.3.

Magneti ux onservation

sponding maximal urrent at a galvanometer as soon as the effe tive H vanishes (see Fig.B.3.7(b)). Multiply this urrent with the winding number of the oil and nd Jmax . Divide by the length of the test oil, and you are ba k to (B.3.39). Whether H is really ompensated for, you an he k with a magneti needle whi h, in the eld-free region, should be in an indierent equilibrium state. In this way, we an build up the ex itation H = D H ^ d as a measurable ele tromagneti quantity in its own right. The ex itation H , together with the eld strength F , we will all the \ele tromagneti eld".

B.4 Basi lassi al ele trodynami s summarized, example

B.4.1

Integral version and Maxwell's equations

We are now in a position to summarize the fundamental stru ture of ele trodynami s in a few lines. A

ording to (B.1.17), (B.2.6), and (B.3.1), the three axioms on a onne ted, Hausdor, para ompa t, and oriented spa etime read, for any C3 and C2 with C3 = 0 and C2 = 0: I I F) ^ J ; F = 0: (B.4.1) J = 0; f = (e C3

C2

The rst axiom reigns matter and its onserved ele tri harge, the se ond axiom links the notion of that harge and the on ept of a me hani al for e to an operational de nition of the ele tromagneti eld strength. The third axiom determines the

ux of the eld strength as sour e-free. In Part C we will learn that a metri of spa etime brings in the temporal and spatial distan e on epts and a linear onne tion the inertial guidan e eld (and thereby parallel displa ement). Sin e the metri of spa etime represents Einstein's gravitational potential (and the linear onne tion is also related to gravita-

3axioms

178

B.4.

Basi lassi al ele trodynami s summarized, example

tional properties), the three axioms (B.4.1) of ele trodynami s are not ontaminated by gravitational properties, in ontrast to what happens in the usual textbook approa h to ele trodynami s. A urved metri or a non- at linear onne tion do not ae t (B.4.1), sin e these geometri obje ts don't enter these axioms. Up to now, we ould do without a metri and without a onne tion. Yet we do have the basi Maxwellian stru ture already at our disposal. Consequently, the stru ture of ele trodynami s that emerged so far has nothing to do with Poin are or Lorentz

ovarian e. The transformations involved are dieomorphisms and frame transformations alone. If one desires to generalize spe ial relativity to general relativity theory or to the Einstein-Cartan theory of gravity (a viable alternative to Einstein's theory formulated in a non-Riemannian spa etime, see again Part C), then the Maxwellian stru ture in (B.4.1) is untou hed by it. As long as a 4-dimensional onne ted, Hausdor, para ompa t and oriented dierentiable manifold is used as spa etime, the axioms in (B.4.1) stay ovariant and remain the same. In parti ular, arbitrary frames, holonomi and anholonomi ones, an be used for the evaluation of (B.4.1). A

ording to (B.4.1), the more spe ialized dierential version of ele trodynami s (skipping the boundary onditions) reads as follows:

dJ = 0 ; f = (e F ) ^ J ; J = dH ;

dF = 0 F = dA :

(B.4.2)

maxeqns

This is the stru ture that we de ned by means of our axioms so far. But, in fa t, we know a bit more: Be ause of the existen e of ondu tors and super ondu tors, we an measure the ex itation H . Thus, even if H emerges as a kind of a potential for the ele tri urrent, it is more than that: It is measurable. This is in

lear ontrast to the potential A that is not measurable. Thus we have to take the gauge invarian e under the substitution

A = A +  ; with d = 0 ; 0

very seriously.

(B.4.3)

gauge

B.4.1

Integral version and Maxwell's equations

D

E

179

H

B

Figure B.4.1: Faraday-S houten pi tograms of the ele tromagneti eld in 3-dimensional spa e. The images of 1-forms are represented by two neighboring surfa es. The nearer the surfa es, the stronger the 1-form is. The 2-forms are pi tured as ux tubes. The thinner the tubes are, the stronger the ow is. The dieren e between a twisted and an untwisted form a

ounts for the two dierent types of 1- and 2-forms, respe tively.

180

B.4.

Basi lassi al ele trodynami s summarized, example

The system (B.4.2) an straightforwardly be translated into the Ex al language: We denote the ele tromagneti potential A by pot1. The `1' we wrote in order to remember better that the potential is a 1-form. The eld strength F is written as farad2, i.e., as the Faraday 2-form. The ex itation H will be named ex it2, the left hand sides of the homogeneous and the inhomogeneous Maxwell equations be alled maxhom3 and maxinh3, respe tively. Then we need the ele tri urrent density urr3 and the left hand side of the ontinuity equation ont4. For the rst axion, we have pform ont4=4, { urr3,maxinh3}=3, ex it2=2$

ont4 maxinh3

:= d urr3; := d ex it2;

and for the se ond and third axiom, pform for e4(a)=4, maxhom3=3, farad2=2, pot1=1$ % has to be pre eded by frame e$ % a oframe statement farad2 := d pot1; for e4(-a) := (e(-a) _|farad2)^ urr3; maxhom3 := d farad2;

These program bits and pie es, whi h look almost trivial, will be integrated into a omplete and exe utable Maxwell sample program after we will have learned about the energy-momentum distribution of the ele tromagneti eld and about its a tion. The physi al interpretation of the equations (B.4.2) an be found via the (1+3)-de omposition that we had derived earlier as J= H= F = A=

j ^ d +  ; H ^ d + D ; E ^ d + B ; ' d + A ;

(B.4.4) (B.4.5) (B.4.6) (B.4.7)

see (B.1.33), (B.1.36), (B.2.7), and (B.3.7), respe tively.

sumj sumh sumf suma

B.4.1

Integral version and Maxwell's equations

181

We rst on entrate on equations whi h ontain only measurable quantities, namely the Maxwell equations. For their (1+3)de omposition we found, see (B.1.40,B.1.41) and (B.3.13,B.3.14),

(

dH = J

( dF = 0

dD =  D_ = d H dB = 0 B_ = d E

j

(1 onstraint eq:) ; (3 time evol: eqs:) ;

(B.4.8)

evol1

(1 onstraint eq:) ; (3 time evol: eqs:) :

(B.4.9)

evol2

A

ordingly, we have 2  3 = 6 time evolution equations for the 2  6 = 12 variables (D; B; H; E ) of the ele tromagneti eld. Thus the Maxwellian stru ture in (B.4.2) is under-determined. We need, in addition, an ele tromagneti spa etime relation that expresses the ex itation H = (H; D) in terms of the eld strength F = (E; B ), i.e., H = H [F ℄. For lassi al ele trodynami s, this fun tional be omes the Maxwell-Lorentz spa etime relation that we will dis uss in Part D. Whereas, on the unquantized level, the Maxwellian stru ture in (B.4.1) or (B.4.2) is believed to be of universal validity, the spa etime relation is more of an ad ho nature and amenable to orre tions. The `va uum' an have dierent spa etime relations depending on whether we take it with or without va uum polarization. We will ome ba k to this question in Chapter E.2.

182

B.4.

Basi lassi al ele trodynami s summarized, example

Tables (put inside the

over of the book)

Table I. The ele tromagneti eld and its sour e Field  j

D H E B

name

math. obje t

independent related re e - absolute

omponents to tion dimension

ele tri twisted 123

harge 3-form ele tri twisted j23 ; j31 ; j12

urrent 2-form ele tri twisted D23 ; D31; D12 ex itation 2-form magneti twisted H1 ; H2; H3 ex itation 1-form ele tri untwisted E1; E2 ; E3 eld strength 1-form magneti untwisted B23 ; B31; B12 eld strength 2-form

volume



area

j

area

D

q

line

H

q=t

0 =t

area

B

0 = magneti ux

Field SI-unit of eld SI-unit of omponents of eld

D H E B

q=t

E

(C = oulomb, A = ampere, W b = weber, V = volt, T = tesla; m = meter, s = se ond. The units oersted and gauss are phased out and do not exist any longer in SI) C A = C=s C A = C=s Wb=s = V Wb

= ele tri

harge

line

Table II. SI-units of the ele tromagneti eld and its sour e

 j

q

C=m3 A=m2 = C=(sm2 ) C=m2 A=m = C=(sm) (! oersted) V =m = Wb=(sm) Wb=m2 = T (! gauss)

B.4.2 Jump onditions for ele tromagneti ex itation and eld strength

B.4.2

Jump onditions for ele tromagneti ex itation and eld strength

The equations (B.3.34) and (B.3.38) represent the so- alled jump (or ontinuity) onditions for the omponents of the ele tromagneti ex itation. In this se tion we will give a more onvenient fomulation of these onditions. Namely, let us onsider on a 3dimensional sli e h of spa etime, see Fig.B.1.3, an arbitrary 2-dimensional surfa e S , the points of whi h are de ned by the parametri equations xa = xa ( 1 ;  2 ) ; _

_

a = 1 ; 2; 3 :

(B.4.10)

Here  A = ( 1_ ;  2_ ) are the two parameters spe ifying the position on S . We will denote the orresponding indi es A; B; ::: = 1_ ; 2_ by a dot in order to distinguish them from the other indi es. We assume that this surfa e is not moving, i.e. its form and position are the same in every  = onst hypersurfa e. We introdu e the 1-form density  normal to the surfa e S ,   3 1 x x x2 x3 x2 x3 1 dx +  := _ _ 1 2_ 2 1_      1_  2_  1 2  x x x1 x2 + dx3 ;  1_  2_  2_  1_ 

x3 x1  2_  1_



dx2

(B.4.11)

nuS

(B.4.12)

tauS

and the two ve tors tangential to S , A :=

xa a ;  A

A = 1_ ; 2_ :

A

ordingly, we have the two onditions A  = 0. The surfa e S divides the whole sli e h into two halves. We denote them by the subs ripts (1) and (2) , respe tively. Then, by repeating the limiting pro ess for the integrals near S of above, we nd, instead of (B.3.34) and (B.3.38), the jump

onditions

D

(2)

A



H

D

(1)

(2)

H


e

(1)

S

^  = 0;

(B.4.13)

DonSnu

= 0:

(B.4.14)

HonStau

 e j

S

183

184

B.4.

Basi lassi al ele trodynami s summarized, example

Here, D(1) and H(1) are the ex itation forms in the rst half and D(2) and H(2) in the se ond half of the 3D spa e h . One an immediately verify that the analysis in Chap.B.3.4 of the operational determination of the ele tromagneti ex itation was arried out for the spe ial ase when the surfa e S was de ned by the equations x1 =  1_ ; x2 =  2_ ; x3 = 0 (then  = dx3 , and 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.4.13) and (B.4.14) sin e the lo al oordinates an always be hosen in su h a way that (B.4.13), (B.4.14) redu e to (B.3.34), (B.3.38). However, in pra ti al appli ations, the use of (B.4.13), (B.4.14) turns out to be more onvenient. In an analogous way, one an derive the jump onditions for the omponents of the ele tromagneti eld strength. Starting from the homogeneous Maxwell equations (B.3.13) and (B.3.14) and onsidering their integral form near S , we obtain B(2) A

B(1) E(2)




S

^  = 0;

(B.4.15)

BonSnu

= 0:

(B.4.16)

EonStau

E(1)




S

Similar as above, B(1) and E(1) are the eld strength forms in the rst half and E(2) and E(2) in the se ond half of the 3D spa e h . The homogeneous Maxwell equation does not ontain harge and urrent sour es. Thus, equations (B.4.15), (B.4.16) des ribe the ontinuity of the tangential pie e of the magneti eld and of the tangential part of the ele tri eld a ross the boundary between the two domains of spa e. In (B.4.13) and (B.4.14), the

omponents of the ele tromagneti ex itation are not ontinuous in general, with the surfa e harge and urrent densities e; ej de ning the orresponding dis ontinuities. B.4.3

Arbitrary lo al non-inertial frame: Maxwell's equations in omponents

In (B.4.8) and (B.4.9), we displayed the Maxwell equations in terms of geometri al obje ts in a oordinate and frame invari-

B.4.3

Arbitrary lo al non-inertial frame: Maxwell's equations in omponents

ant way. Sometimes it is ne essary, however, to introdu e lo ally arbitrary ( o-)frames of referen e that are non-inertial in general. Then the omponents of the ele tromagneti eld with respe t to a oframe # , the physi al omponents emerge and the Maxwell equations an be expressed in terms of these physi al

omponents. Let the urrent, the ex itation, and the eld strength be de omposed a

ording to J

H

=

=

1 

J # ^ # ^ # ; 3!

1  H # ^# ; 2 

F

=

1  F # ^# ; 2 

(B.4.17)

ur omp'

(B.4.18)

max omp'

respe tively. We substitute these expressions into the Maxwell equations. Then the oframe needs to be dierentiated. As a shorthand notation, we introdu e the anholonomi ity 2-form 1 C  := d# = 2 C  # ^ # , see (A.2.35). Then we straightforwardly nd: [ H ℄

Æ

C[ H ℄Æ

=

1 J ; 3 

[ F ℄

Æ

= 0: (B.4.19)

C[ F ℄Æ

maxanh

If we use the (metri -free) Levi-Civita tensor density  Æ ,

H := 2!1  Æ H Æ ;

J  := 3!1  Æ J Æ ;

(B.4.20)

maxlevi

then the ex itation and the urrent are represented as densities. A

ordingly, Maxwell's equations (B.4.2) in omponents read alternatively 

H + C  H = J  ;

[ F ℄

Æ

C[ F ℄Æ

= 0: (B.4.21)

The terms with the C 's emerge in non-inertial frames, i.e., they represent so- alled inertial terms. If we restri t ourselves to nat-

max omp

185

186

B.4.

Basi lassi al ele trodynami s summarized, example

ural (or oordinate) frames, then C = 0, and Maxwell's equations display their onventional form.1 This representation of ele trodynami s an be used in spe ial or in general relativity. If one desires to employ a laboratory frame of referen e, then this is the way to do it: The obje t of anholonomi ity in the lab frame has to be al ulated. By substituting it into (B.4.19) or (B.4.21), we nd the Maxwell equations in terms of the omponents F et . of the ele tromagneti eld quantities with respe t to the lab frame { and these are the quantities one observes in the laboratory. Therefore the F et . are alled physi al omponents of F et . As soon as one starts from (B.4.2), the derivation of the sets (B.4.19) or (B.4.21) is an elementary exer ise. Many dis ussions of the Maxwell equations within spe ial relativity in non-inertial frames ould be appre iable shortened by using this formalism.

B.4.4

Ele trodynami s in atland: 2-dimensional ele tron gas and quantum Hall ee t

Our formulation of ele trodynami s an be generalized straightforwardly to arbitrary dimensions n. If we assume again the

harge onservation law as rst axiom, then the rank of the ele tri urrent must be n 1. The for e density in me hani s, in a

ordan e with its de nition  L=xi within the Lagrange formalism, should remain a ove tor-valued n-form. Hen e we keep the se ond axiom in its original form. A

ordingly, the eld strength F is again a 2-form: I I F = 0: (B.4.22) F) ^ J ; J = 0; f = (e Cn

n

1

C2

This may seem like an a ademi exer ise. However, at least for = 3, there exists an appli ation: Sin e the middle of the 1960's, 1 For formulating ele trodynami s in a

elerated systems in terms of tensor analysis,

see J. Van Bladel [37℄. New experiments in rotating frames (with ring lasers) an be found in Stedman [35℄.

3axiomsn

B.4.4 Ele trodynami s in atland: 2-dimensional ele tron gas and quantum Hall ee t

y Ey

jy

ly

H jx Ex

Dy ρx y

Dx

Bxy

lx

x

Figure B.4.2: The arsenal of ele tromagneti quantities in atland: The sour es are the harge density xy and the urrent density (jx ; jy ), see (B.4.28). The ex itations (Dx ; Dy ) and H (twisted s alar) are somewhat unusual, see (B.4.29). The double plates for measuring D, e.g., be ome double wires. The magneti eld has only one independent omponent Bxy , whereas the ele tri eld has two, namely (Ex ; Ey ), see (B.4.30). experimentalists were able to reate a 2-dimensional ele tron gas (2DEG) in suitable transistors at suÆ iently low temperatures and to position the 2DEG in a strong external trasversal magneti eld. Under su h ir umstan es, the ele trons an only move in a plane transverse to B and one spa e dimension an be suppressed.

Ele trodynami s in 1 + 2 dimensions In ele trodynami s with 1 time and 2 spa e dimensions, we have from the rst and the third axiom, (2)

d J = 0;

(2)

(1)

J =dH;

(B.4.23)

first3

(B.4.24)

third3

and (2)

d F = 0;

(2)

(1)

F =d A;

187

188

B.4.

Basi lassi al ele trodynami s summarized, example

respe tively, where we indi ated the rank of the forms expli itly for better transparen y. Here, the remarkable feature is that eld strength F and urrent J arry the same rank; this is only possible for n = 3 spa etime dimensions. Moreover, the urrent J , the ex itation H , and the eld strength F all have the same number of independent omponents, namely 3. Now we (1 + 2)-de ompose the urrent and the ele tromagneti eld: (2)

twisted 2-form:

J =

(1)

twisted 1-form:

H= (2)

untwisted 2-form:

F =

j ^ d +  ;

(B.4.25)

zerj3

H d + D ; E ^ d + B :

(B.4.26)

zerh3

(B.4.27)

zerf3

A

ordingly, in the spa e of the 2DEG, we have

 = 12 dx1 ^ dx2 ; D = D1 dx1 + D2 dx2 ; B = B12 dx1 ^ dx2 :

j = j1 dx1 + j2 dx2 ; H; E =E1 dx1 + E2 dx2 ;

(B.4.28) (B.4.29) (B.4.30)

flatj flath flate

We re ognize the rather degenerate nature of su h a system. The magneti eld B , for example, has only 1 independent omponent B12 . Su h a on guration is depi ted in Fig. B.4.2. Charge onservation (B.4.23) in de omposed form and in omponents reads, d j + _ = 0 ;

1 j2

2 j1 + _ 12 = 0 :

(B.4.31)

harge3

The (1 + 2)-de omposed Maxwell equations look exa tly as in (B.4.8) and (B.4.9). We will also express them in omponents. We nd: dH

dD = ;

D_

1 D2

1 H

2 D1 = 12 ;

D_  H D_

=j;

2

= j1 ; 2 = j2 ; 1

(B.4.32) (B.4.33) (B.4.34)

3max1 3max2 3max3

and d E + B_ = 0 ; dB  0:

1 E2

2 E1 + B_ 12 = 0 ;

(B.4.35) (B.4.36)

3max4 3max5

B.4.4 Ele trodynami s in atland: 2-dimensional ele tron gas and quantum Hall ee t

We will assume in nite extension of atland. If that annot be assumed as a valid approximation, one has to allow for line

urrents at the boundary of atland (\edge urrents") in order to ful ll the Maxwell equations. In our formulation the Maxwell equations don't depend on the metri . Thus, instead of the Eu lidean plane, as in Fig. B.4.2, we

ould have drawn an arbitrary 2-dimensional manifold, a surfa e of a ylinder or of a sphere, e.g.. The Maxwell equations (B.4.32) to (B.4.36) would still be valid. Before we an apply this formalism to the quantum Hall ee t (QHE), we rst remind ourselves of the lassi al Hall ee t (of 1879).

Hall ee t

2

We onne t the two yz -fa es of a (semi-) ondu ting plate of volume lx  ly  lz with a battery, see Fig. B.4.3. A urrent I will ow and in the plate the urrent density jx . Transverse to the urrent, between the onta ts P and Q, there exists no voltage. However, if we apply a onstant magneti eld B along the z -axis, then the urrent j is de e ted by the Lorentz for e and the Hall voltage UH o

urs whi h, a

ording to experiment, turns out to be UH

= RH I = AH

BI lz

with

;

[AH ℄ =

l

3

q

:

(B.4.37)

Hallvoltage

RH is alled the Hall resistan e and AH the Hall onstant. We divide UH by ly . Be ause of Ey = UH =ly , we nd

Ey

= AH Bxy

Ix ly lz

= AH Bxy jx :

(B.4.38)

Let us stress that the lassi al Hall ee t is a volume (or bulk) ee t. It is to be des ribed in the framework of ordinary (1 + 3)-dimensional Maxwellian ele trodynami s. 2 See Landau-Lifshitz [21℄ pp.96-98 or Raith [29℄ p.502.

Hallvoltage'

189

190

B.4.

z

Basi lassi al ele trodynami s summarized, example

y

Bxy

Q ly jx

UH lz I

P

lx

x

Figure B.4.3: Hall ee t (s hemati ): The urrent density j in the ondu ting plate is ae ted by the external onstant magneti eld B (in the gure only symbolized by one arrow) su h as to reate the Hall voltage UH .

B.4.4 Ele trodynami s in atland: 2-dimensional ele tron gas and quantum Hall ee t

I

I gate source SiO2

Al

SiO2

Al

drain SiO2

Al n+

n+

p

UH

Si

z y

2DEG

x

Figure B.4.4: A Mosfet with a 2-dimensional ele tron gas (2DEG) layer between a semi ondu tor (Si) and an insulator (SiO ). Adapted from Braun [3℄. In 1980, with su h a transistor, von Klitzing et al. performed the original experiment on the QHE. 2

Quantum Hall ee t

3

A prerequisite for the dis overy, in 1980, of the QHE were the advan es in transistor te hnology. Sin e the 1960's one was able to assemble 2-dimensional ele tron gas layers in ertain types of transistors, su h as in a metal-oxide-semi ondu tor field ee t transistor (Mosfet), see a s hemati view of a Mosfet in Fig.B.4.4. The ele tron layer is only about 50 nanometers thi k, whereas its lateral extension may go up to the millimeter region. In the quantum Hall regime , we have very low temperatures (between 25 mK and 500 mK) and very high magneti elds (between 5 T and 15 T). Then the ondu ting ele trons of the spe imen, be ause of a (quantum me hani al) ex itation gap,

annot move in the z -dire tion, they are on ned to the xy 4

3 See,

for example, von Klitzing [38℄, Braun [3℄, Chakraborty and Pietil ainen [4℄,

Janssen et al. [17℄, and referen es given there.

4A

fairly detailed des ription an be found in Raith [29℄ pp.579-582, e.g..

191

192

B.4.

Basi lassi al ele trodynami s summarized, example

jx

jy

y

Uy Ux

UH

Q

Bxy

jx

x

P I

Figure B.4.5: S hemati view of a quantum Hall experiment with a 2-dimensional ele tron gas (2DEG). The urrent density x in the 2DEG is exposed to a strong transverse magneti eld xy . The Hall voltage H an be measured in the transverse dire tion to x between and . In the inset we denoted the longitudinal voltage with x and the transversal one with y (= H). j

B

U

j

P

Q

U

U

U

plane. Thus an almost ideal 2-dimensional ele tron gas (2DEG) is onstituted. The Hall ondu tan e (= 1/resistan e) exhibits very wellde ned plateaus at integral (and, in the fra tional QHE, at rational) multiples of the fundamental ondu tan e of 2 h SI = 1 (25 812 807 ), where is the elementary harge and h Plan k's

onstant. Therefore this ee t is instrumental in pre ision experiments for measuring, in onjun tion with the Josephson ee t, and h very a

urately. We will on entrate here on the integer QHE. e =

=

e

:

e

B.4.4 Ele trodynami s in atland: 2-dimensional ele tron gas and quantum Hall ee t

Turning to Fig.B.4.5, we onsider the re tangle in the xyplane with side lengths lx and ly , respe tively. The quantum Hall ee t is observed in su h a two-dimensional system of ele trons subje t to a strong uniform transverse magneti eld B~ . The on guration is similar to that of the lassi al Hall ee t (see Fig.B.4.3), but the system is ooled down to a uniform temperature of about of 0.1 K. The Hall resistan e RH is de ned by the ratio of the Hall voltage Uy and the ele tri urrent Ix in the x-dire tion: RH = Uy =Ix. Longituidinally, we have the ordinary dissipative Ohm resistan e RL = Ux=Ix. For xed values of magneti eld B and area harge density of ele trons ne e, the Hall resistan e RH is a onstant. Phenomenologi ally, the QHE

an be des ribed by means of a linear tensorial Ohm-Hall law as onstitutive relation. Thus, for an isotropi material, Ux = RL Ix RH Iy ; (B.4.39) Uy = RH Ix + RL Iy : (B.4.40) If we introdu e the ele tri eld Ex = Ux=lx ; Ey = Uy =ly and the 2D urrent densities jx = Ix=ly ; jy = Iy =lx, we nd ~ E

=  ~j or

~j

= E~

with

=

1

=

l

RL lxy RH

RH

!

RL llxy

(B.4.41) where , the spe i resistivity, and , the spe i ondu tivity, are represented by se ond rank tensors. With a lassi al ele tron model for the ondu tivity, the Hall resistan e an be al ulated to be RH = B=(ne e), where B is the only omponent of the magneti eld in 2-spa e. The magneti

ux quantum for an ele tron is h=e = 20 , with 0 SI = 2:07  10 15 Wb. With the fundamental resistan e we an rewrite the Hall resistan e with the dimensionless lling fa tor  as h n B h 1 R = =  ; where  := e e : (B.4.42) H

ne e

e2

resistan e

B

Observe that  1 measures the amount of magneti ux (in ux units) per ele tron.

; ve torQHE

filling

193

194

B.4.

Basi lassi al ele trodynami s summarized, example

Thus lassi ally, if we in rease the magneti eld, keeping ne (i.e., the gate voltage) xed, we would expe t a stri tly linear in reasing of the Hall resistan e. Surprisingly, however, we nd the following (see Fig.B.4.6): 1. The Hall R has plateaus at rational heights. The plateaus at integer height o

ur with a high a

ura y: R = (h=e )=i, for i = 1; 2 : : : (for holes we would get a negative sign). H

2

H

2. When (; R ) belongs to a plateau, the longitudinal resistan e, R , vanishes to a good approximation. H

L

In a

ordan e with these results, we hoose  su h that R is zero and  onstant at a plateau for one parti ular system, namely L

H

=R



0 1

H



1 0

=

or

H

This yields ~ j

=

H

~ E



with



:=



0 1 1 0

0 1 1 0



:

 ;

(B.4.43)

ondQHE

(B.4.44)

ondQHE1

a phenomenologi al law valid at low frequen ies and large distan es. We will see in the next subse tion that we an express this onstitutive law in a generally ovariant form. This shows that the laws governing the QHE are entirely independent of the parti ular geometry (metri ) under onsideration.

Covariant des ription of the phenomenology at the plateaus For a (1 + 2)-dimensional ovariant formulation , let us turn ba k to the rst and the third axioms in (B.4.23) and (B.4.24), ~ . Therefore respe tively. In (B.4.44), ~j is linearly related to E the simplest (1 + 2)- ovariant ansatz is 5

J 5 This

=

H F ;

[ ℄ = q

2

H

=h

= 1=resistan e :

(B.4.45)

an be found in the work of Fr ohli h et al. [11, 9, 10℄, see also Avron [2℄, Ri hter

and Seiler [33℄, and referen es given there.

QHE onstit

B.4.4 Ele trodynami s in atland: 2-dimensional ele tron gas and quantum Hall ee t R

10 kΩ 8 6 4

RH

2 0 0.8

RL

0.6 0.4 0.2

B

0 0

2

4

T

6

Figure B.4.6: The Hall resistan e RH at a temperature of 8 mK as a fun tion of the magneti eld B (adapted from Ebert et al. [6℄). The minus sign is hosen sin e we want to re over (B.4.44). The Hall resistan e H is a twisted s alar (or pseudo-s alar). Of

ourse, we an only hope that su h a simple ansatz is valid provided the Mosfet is isotropi in the plane of the Mosfet. Be ause of (B.4.23)1 and (B.4.24)1 , we nd quite generally that the Hall

ondu tan e must be onstant in spa e and time:

H = onst :

(B.4.46)

We should be aware that here happened something remarkable. We have not introdu ed a spa etime relation a la H  F , as is done in (1 + 3)-dimensional ele trodynami s, see Part D. Sin e, for n = 3, H and F have still the same number of independent omponents, this would be possible. However, it is the ansatz J  F that leads to a su

essful des ription of the phenomenology of the QHE. One an onsider (B.4.45) as a spa etime relation for the (1 + 2)-dimensional quantum Hall regime, see Fig. B.4.7. Of ourse, the integer or fra tional numbers of

sigmaH

195

el a

er im et ac

j

2+1 decomposition

field strength (untw.)

E

B

1+2 decomposition

Lorentz force coupl.

s rm fo 1-

s

rm

fo

2-

ρ

n

el. current (twisted)

Lorentz force coupl.

tio

sp

a el

ac et im er

sp

tio

n

Basi lassi al ele trodynami s summarized, example

magnetic

B.4.

electric

196

Figure B.4.7: Interrelationship between urrent and eld strength in the Chern-Simons ele trodynami s of the QHE: In horizontal dire tion, the 2-dimensional spa e part of a quantity is linked with its 1-dimensional time part to a 3-dimensional spa etime quantity. Lines in verti al dire tion onne t a pair of quantities whi h ontribute to the Lorentz for e density. And a diagonal line represents a spa etime relation.

B.4.4 Ele trodynami s in atland: 2-dimensional ele tron gas and quantum Hall ee t

annot be derived from a lassi al theory. Nevertheless, lassi al (1 + 2)-dimensional ele trodynami s immediately suggests a relation of the type (B.4.45), a relation whi h is free of any metri . In other words, the (1 + 2)-dimensional ele trodynami s of the QHE, as a lassi al theory, is metri -free. Eq.(B.4.45) transforms the Maxwell equations (B.4.23)2 and (B.4.24)2 to a

omplete system of partial dierential equations whi h an be integrated. At the same time, is also yields an expli it relation between H and F , namely

H

dH

=

H F

or

dH

=

H dA :

(B.4.47)

HF3

The last equation an be integrated. We nd H

=

(B.4.48)

H A ;

sin e the potential 1-form A is only de ned up to a gauge transformation anyway. The metri -free dierential relation (B.4.47)1 between ex itation H and eld strength F is ertainly not what we would have expe ted from lassi al Maxwellian ele trodynami s. It represents, for n = 3, a totally new type of (ChernSimons) ele trodynami s. We ompare (B.4.45) with (B.4.25) and (B.4.27) and nd j

= H E

and

=



H B :

(B.4.49)

QHE onstit12

In (B.4.44), the urrent is represented as ve tor. Therefore we introdu e the ve tor density ~j := }j by means of the diamond operator of (A.1.80), or, in omponents ~j a = ab ja =2. Then (a; b = 1; 2), ~j a = 1 ab jb = 2

H 

ab




Ea ;

(B.4.50)

that is, we re over (B.4.44) thereby verifying the ansatz (B.4.45). For the harge density ~ := }, we nd  ~ = H 

ab

Bab

with

1 ~ j

1

+ 2 ~j 2 + ~_ = 0 :

(B.4.51)

QHE onstit12''

197

198

B.4.

Basi lassi al ele trodynami s summarized, example

Preview: 3D Lagrangians

We will introdu e Lagrangians in Se . B.5.4. Nevertheless, let us on lude this se tion on ele trodynami s in atland with a few remarks on the appropriate Lagrangian for the QHE. The Lagrangian in (1 + 2)-dimensional ele trodynami s has to be a twisted 3-form with the dimension h of an a tion. Obviously, the rst invariant of (B.2.16) quali es, 1 1 J ^H = H ^dH : (B.4.52) I1 = F ^ H = H H But also the Chern-Simons 3-form of (B.3.16), if multiplied by the twisted s alar H , has the right hara teristi s, H CA = H A ^ F = H A ^ d A = A ^ d H = d(A ^ H ) F

(B.4.53) ^H = F ^ H +dK ;

lagr31

lagr32

with the Kiehn 2-form K = A ^ H . Therfore, both andidate Lagrangians are intimately linked, H CA = I1 + d K :

(B.4.54)

In other words, they yield the same Lagrangian sin e they only dier by an irrelevant exa t form. Let us de ne then the Lagrange 3-form 1 H ^ dH A ^dH : (B.4.55) L3D := 2H By means of its Euler-Lagrange equation,   L L 1 d + =0 or dH dA = 0;  (d H ) H H (B.4.56) we an re over the 3D spa etime relation (B.4.45) we started from. Alternatively, we an onsider J as external eld in the Lagrangian H A^ dA A^ J : (B.4.57) L3D := 2 A similar omputation as in (B.4.56) yields dire tly (B.4.45), q.e.d.. 0

link3

lagr3

lagr3'

lagr3''

B.5 Ele tromagneti energy-momentum

urrent and a tion

B.5.1

Fourth axiom: lo alization of energy-momentum

Minkowski's \greatest dis overy was that at any point in the ele tromagneti eld in va uo there exists a tensor of rank 2 of outstanding physi al importan e. ... ea h omponent of the tensor Epq has a physi al interpretation, whi h in every ase had been dis overed many years before Minkowski showed that these 16 omponents onstitute a tensor of rank 2. The tensor Epq is alled the energy tensor of the ele tromagneti eld."

Edmund Whittaker (1953) Let us onsider the Lorentz for e density f = (e F ) ^ J in (B.4.2). If we want to derive the energy-momentum law for ele trodynami s, we have to try to express f as an exa t form. Then energy-momentum is a kind of a generalized potential for the Lorentz for e density, namely f  d . For that purpose, we start from f. We substitute J = dH (inhomogeneous Maxwell equation) and subtra t out a term with H and F ex hanged and

200

B.5.

Energy-momentum urrent and a tion

multiplied by a onstant fa tor : a

f



=(



F

e

)^

dH

a

(

e



H

)^

(B.5.1)

dF :

inter1

Be ause of = 0 (homogeneous Maxwell equation), the subtra ted term vanishes. The fa tor will be left open for the moment. Note that we need a non-vanishing urrent 6= 0 for our derivation to be sensible. We partially integrate both terms in (B.5.1): dF

a

J



f

=





^(  ^ ( e

H

d e

H

d aF aF

) ^(  ) )+ ^ (  ) H

e

H

d e

F

F

(B.5.2)

:

inter2

The rst term has already the desired form. We re all the main formula for the Lie derivative of an arbitrary form , namely Le  = (  ) +  ( ) , see (A.2.51) This allows us to transform the se ond line of (B.5.2): d e

f



e

=





^(  ) ^(  ) ^ (Le ) + ^ (Le ) ^  ( ) ^  ( )

d aF

+

d

e

H

H

H

H

dH

H

aF aF

e

e

F

F

e

dF

:

(B.5.3)

inter3

(B.5.4)

inter4

The last line an be rewritten as +

ae





e

[ ^ F

[ ^ H

℄ (  )^ ℄+(  )^

dH

dF

a

e

e

F

H

dH

dF :

As 5-forms, the expressions in the square bra kets vanish. Two terms remain, and we nd 

f

=





^(  ) ^(  ) ^ (Le ) + ^ (Le ) (  )^ +(  )^

d aF

e

aF

a

Be ause of

dF

(1 + ) a

f

e

F

H

H

H

H

dH

e

e

F

F

H

= 0, the third line adds up to 

=

 d aF aF

(B.5.5)

dF :

 . A

ordingly,

af

^(  ) ^(  ) ^ (Le ) + ^ (Le ) e

H

H

H

H

e

inter3a



F

F

:

(B.5.6)

inter3b

B.5.1

Fourth axiom: lo alization of energy-momentum

201

Now we have to make up our mind about the hoi e of the fa tor . With = 1, the left hand side vanishes and we nd a mathemati al identity. A real onservation law is only obtained when, eventually, the se ond line vanishes. In other words, here we need an a posteriori argument, i.e., we have to take some information from experien e. For = 0, the se ond line does not vanish. However, for = 1, we an hope that the rst term in the se ond line ompensates the se ond term if somehow  . In fa t, under \ordinary ir umstan es", to be explored below, the two terms in the se ond line do ompensate ea h other for = 1. Therefore we postulate this hoi e and nd a

a

a

a

H

F

a



f

=(

e



F

) ^ = k + J

d

X

(B.5.7)

:

Here the kinemati energy-momentum 3-form of the ele tromagneti eld, a entral result of this se tion, reads k ^ (  )℄ (fourth axiom)  := 21 [ ^ (  ) (B.5.8) and the remaining for e density 4-form turns out to be 1( ^L (B.5.9) ^ Le ) e  := 2 The absolute dimension of k  as well as of b is . Our derivation of (B.5.7) doesn't lead to a unique de nition of k  . The addition of any losed 3-form would be possible, F

e

H

H

H

F

X

e

F

;

F

H

:

X

k

 := k + 0

;

Y

with

fSX

dY



=0

simax

Xal

h=l

;

(B.5.10)

nonunique

(B.5.11)

nonunique1

su h that f



= k + d

0

:

X

In parti ular,  ould be exa t:  = . The 2-form  has the same dimension as k  . It seems impossible to build up  ex lusively in terms of the quantities  in an algebrai way. Therefore,  = 0 appears to be the most natural hoi e. Y

Y

dZ

Z

Z

e ; H; F

Y

202

B.5.

Energy-momentum urrent and a tion

Thus, by the fourth axiom we postulate that k  in (B.5.8) represents the energy-momentum urrent that orre tly lo alizes the energy-momentum distribution of the ele tromagneti eld in spa etime. We all it the kinemati energy-momentum

urrent sin e we didn't nd it by a dynami prin iple, whi h we will not formulate before (B.5.78), but rather by means of some sort of kinemati arguments. The urrent k  an also be rewritten by applying the antiLeibniz rule for e either in the rst or the se ond term on the right hand side of (B.5.8). With the 4-form  := 1 F ^ H ; 2

(B.5.12)

lagr

(B.5.13)

simax'

we nd k

B.5.2

 = e  + F ^ (e H ) = e  H ^ (e F ) :

Properties of the energy-momentum

urrent, ele tri -magneti re ipro ity

 is tra efree The energy-momentum urrent k  is a 3-form. We an blow it up to a 4-form a

ording to # ^ k . Sin e it still has 16

omponents, we haven't lost any information. If we re all that for any p-form  we have # ^ (e ) = p, we immediately re ognize from (B.5.8) that k

# ^ k  = 0 ;

(B.5.14)

whi h amounts to one equation. This property | the vanishing of the \tra e" of k | is onne ted with the fa t that the ele tromagneti eld (the \photon") arries no mass and the theory is thus invariant under dilations. Why we all that the tra e of the energy-momentum will be ome lear below, see (B.5.29).

zerotra e

B.5.2 Properties of energy-momentum, ele tri -magneti re ipro ity

 is ele tri -magneti re ipro al Furthermore, we an observe another property of k  . It is remarkable how symmetri H and F enter (B.5.8). This was a hieved by our hoi e of a = 1. The energy-momentum urrent is ele tri -magneti re ipro al, i.e., it remains invariant under the transformation 1 H ; ) k ! k ; (B.5.15) H ! F ; F !   

203

k

duality1

with the twisted zero-form (pseudo-s alar fun tion)  =  (x) of dimension [ ℄ = [H ℄=[F ℄ = q2 =h. It should be stressed that in spite of k  being ele tri -magneti re ipro al, Maxwell's equations are not, dH = J ! dF + F ^ d = = J= ; (B.5.16) dF = 0 ! dH H ^ d = = 0 ; (B.5.17) not even for d  = 0, sin e we don't want to restri t ourselves to the free- eld ase with vanishing sour e J = 0. Eq.(B.5.15) expresses a ertain re ipro ity between ele tri and magneti ee ts with regard to their respe tive ontributions to the energy-momentum urrent of the eld. We all it ele tri -magneti re ipro ity.1 That this naming is appropriate,

an be seen from a (1+3)-de omposition. We re all the de ompositions of H and F in (B.4.5) and (B.4.6), respe tively. We substitute them in (B.5.15): H

F

! F ! 1 H

(

H ! E ; D ! B ; ( E ! 1 H ; B ! 1 D :

(B.5.18)

duality1a

(B.5.19)

duality1b

1 ...following Toupin [36℄ even if he introdu ed this notion in a somewhat more restri ted ontext. Maxwell spoke of the mutual embra e of ele tri ity and magnetism, see Wise [39℄. In the ase of a pres ribed metri , dis ussions of the orresponding Raini h \duality rotation" were given by Gaillard & Zumino [12℄ and by Mielke [26℄, amongst others. Note, however, that our transformation (B.5.15) is metri -free and thus of a dierent type.

3+1 decomposition

E

B

1+3 decomposition

oc ity

s rm

ip r ec .r s

rm

fo

2-

fo

+

+

1-

el .-m ag

r ip

field strength (untw.)

oc

H

ec

magnetic

.r

D

spacetime relation

ag .-m el

excitation (twisted)

ity

Energy-momentum urrent and a tion

electric

B.5.

spacetime relation

204

Figure B.5.1: Dierent aspe ts of the ele tromagneti eld: In horizontal dire tion, the 3-dimensional spa e part of a quantity is linked with its 1-dimensional time part to a 4-dimensional spa etime quantity. The energy-momentum urrent remains invariant under the ex hange of those quantities that are onne ted by a diagonal line. And a verti al line represents a spa etime relation between quantities that are anoni ally related like momentum and velo ity, see (B.5.73)1 .

B.5.2 Properties of energy-momentum, ele tri -magneti re ipro ity

Here it is learly visible that a magneti quantity is repla ed by an ele tri one and an ele tri quantity by a magneti one: ele tri ! magneti . In this sense, we an speak of an ele tri magneti re ipro ity in the expression for the energy-momentum

urrent  . Alternatively we an say that  ful lls ele tri magneti re ipro ity, it is ele tri -magneti re ipro al. Let us pause for a moment and wonder of how the notions \ele tri " and \magneti " are atta hed to ertain elds and whether there is a onventional element involved. By making experiments with a at's skin and a rod of amber, we an \liberate" what we all ele tri harges. In 3 dimensions, they are des ribed by the harge density . Set in motion, they produ e an ele tri urrent j . The ele tri harge is onserved ( rst axiom) and is linked, via the Gauss law d D = , to the ele tri ex itation D. Re urring to the Oersted experiment, it is lear that moving harges j indu e magneti ee ts, in a

ordan e with the Oersted-Ampere law d H D_ = j | also a onsequen e of the rst axiom. Hen e we an unanimously attribute the term magneti to the ex itation H. There is no room left for doubt about that. The se ond axiom links the ele tri harge density  to the eld strength E a

ording to (ea ) ^ E and the ele tri urrent j to the eld B a

ording to (ea j ) ^ B . Consequently, also for the eld strength F , there an be no other way than to label E as ele tri and B as magneti eld strength. These arguments imply that the substitutions H !  F as well as F ! H= both substitute an ele tri by a magneti eld and a magneti by an ele tri one, see (B.5.18) and (B.5.19). Be ause of the minus sign (that is, be ause of a = 1) that we found in (B.5.15) in analyzing the ele tromagneti energy-momentum urrent  , we annot speak of an equivalen e of ele tri and magneti elds, the expression re ipro ity is mu h more appropriate. Fundamentally, ele tri ity and magnetism enter into lassi al ele trodynami s in an a symmetri way.

205

206

B.5.

Energy-momentum urrent and a tion

Let us try to explain the ele tri -magneti re ipro ity by means of a simple example. In (B.5.52) we will show that the ele tri energy density reads uel = 12 E ^ D. If one wants to try to guess the orresponding expression for the magneti energy density umag , one substitutes for an ele tri a orresponding magneti quantity. However, the ele tri eld strength is a 1-form. One annot substitute it by the magneti eld strength B sin e that is a 2-form. Therefore one has to swit h to the magneti ex itation a

ording to E ! 1 H, with the 1-form H. The fun tion  is needed be ause of the dierent dimensions of E and H and sin e E is an untwisted and H a twisted form. Analogously, one substitutes D ! B thereby nding umg = 21 H ^ B . This is the orre t result, i.e., u = 21 (E ^ D + B ^ H), and we ould be happy. Naively, one would then postulate the invarian e of u under the substitution E ! 1 1  H ; D ! B ; B !  D ; H ! E . But, as a look at (B.4.5) and (B.4.6) will show, this annot be implemented in a ovariant way be ause of the minus sign in (B.4.5). One

ould re onsider the sign onvention for H in (B.4.5). However, as a matter of fa t, the relative sign between (B.4.5) and (B.4.6) is basi ally xed by Lenz's rule (the indu ed ele tromotive for e [measured in volt℄ is opposite in sign to the indu ing eld). Thus the relative minus sign is independent of onventions. How are we going to save our rule of thumb for extra ting the magneti energy from the ele tri one? Well, if we turn to the substitutions (B.5.18) and (B.5.19), i.e., if we introdu e two minus signs a

ording to E ! 1 H ; D ! B ; B ! 1 D ; H ! E , then u still remains invariant and we re over the ovariant rule (B.5.15). In other words, the naive approa h works up to two minus signs. Those we an supply by having insight into the ovariant version of ele trodynami s. A

ordingly, the ele tri -magneti re ipro ity is the one that we knew all the time { we just have to be areful with the sign. k

 expressed in terms of the omplex ele tromagneti

eld

We an understand the ele tri -magneti re ipro ity transformation as a ting on the olumn ve tor onsisting of H and F :    0   H 0 1 H : (B.5.20) = 0 F 1 0 F

olumn

In order to ompa tify this formula, we introdu e the omplex ele tromagneti eld 2-form 2 U 2 Even

:= H + i F

and

U

=H

i F ;

(B.5.21)

though we will introdu e the on ept of a metri only in Part C, it is ne essary to point out that the omplex ele tromagneti eld U , subsuming exitation and eld strength (see Fig. B.5.1), should arefully be distinguished from the omplex ele tromagneti eld strength introdu ed onventionally: Fb a := F ^0a + iab Fb , with F 0^a := g^0i gaj Fij . This an only be de ned after a metri has been introdu ed. Similarly, b a := H ^0a + iab Hb , with H ^0a := g^0i gaj Hij . for the ex itation we would then have H

omplex

B.5.2 Properties of energy-momentum, ele tri -magneti re ipro ity

207

with  denoting the onjugate omplex. Now the ele tri -magneti re ipro ity (B.5.20) translates into U0

=

U 0

iU ;

= i U :

(B.5.22)

This orresponds, in the omplex plane, where U lives, to a rotation by an angle of =2. We an resolve (B.5.21) with respe t to ex itation and eld strength: 1 i H = (U + U  ) ; F = ( U U ) : (B.5.23) 2 2 We dierentiate (B.5.21)1. Then the Maxwell equation for the

omplex eld turns out to be dU

+ (U 

U

) d 2 = J :

(B.5.24)

omplex1

omplex2

omplexmax

Clearly, if we hoose a onstant  , i.e., d = 0, the se ond term on the left hand side vanishes. The asymmetry between ele tri and magneti elds nds its expression in the fa t that the sour e term on the right hand side of (B.5.24) is a real quantity. If we substitute (B.5.23) into the energy-momentum urrent (B.5.8), we nd, after some algebra, k

 = 4i

  U

^ (e

U

)

U

^ (e

U



)

:

(B.5.25)

simax omplex

Now, a

ording to (B.5.22), ele tri -magneti re ipro ity of the energy-momentum urrent is manifest. If we exe ute su

essively two ele tri -magneti re ipro ity transformations, namely U ! U 0 ! U 00, then, as an be seen from (B.5.22) or (B.5.15), we nd a re e tion (a rotation of ), namely U 00 = U , i.e., U

!

U

or

(H !

H; F

!

)

F :

(B.5.26)

Only four ele tri -magneti re ipro ity transformations lead ba k to the identity. It should be stressed, however, that already one

2duality

208

B.5.

Energy-momentum urrent and a tion

ele tri -magneti re ipro ity transformation leaves k  invariant. It is now straightforward to formally extend the ele tri -magneti re ipro ity transformation (B.5.22) to U0

= e+i U ;

U 0

=e

i U  ;

(B.5.27)

omplex1a

with  = (x) as an arbitrary \rotation" angle. The energymomentum urrent k  is still invariant under this extended transformation, but in later appli ations only the sub ase of  = =2, treated above, will be of interest. Energy-momentum tensor density

k T

Sin e k is a 3-form, we an de ompose it either onventionally or with respe t to the basis 3-form ^ = e ^, with ^ = #^0 ^ ^ ^ ^ #1 ^ #2 ^ #3 , see (A.1.76): k = 1 k    k (B.5.28)   # ^ # ^ # =: T ^ : 3! The 2nd rank tensor density of weight 1, kT , is the Minkowski energy tensor density. We an resolve this equation with respe t to kT by exterior multipli ation with # . We re all # ^ ^ = Æ ^ and nd k T ^ = # ^ k  



(B.5.29)

or, with the new diamond operator } of (A.1.80), k T = } # ^ k 
= 1   k  (B.5.30)  :   3! Thereby we re ognize that # ^ k = 0, see (B.5.14), is equivalent to the vanishing of the tra e of the energy-momentum tensor density k T = 0. Thus kT as well as k  have 15 independent

omponents at this stage. Both quantities are equivalent. If we substitute (B.5.8) into (B.5.30), then we an express the energy-momentum tensor density in the omponents of H and

emtensor

emtensor1

emtensor1a

B.5.2 Properties of energy-momentum, ele tri -magneti re ipro ity

209

F as follows:3

k k

T = 41   (H F

F H ) :

(B.5.31)

emtensor2

T alternatively derived by means of tensor al ulus

We start with the Maxwell equations (B.4.21) in holonomi oordinates, i.e., in the natural frame e = Æi i : j Hij = J i ; [i Fjk℄ = 0 : (B.5.32) max omponents 1 klmn kl Hmn , see (B.4.20)1 . We substitute the Here H is de ned a

ording to H = 2  inhomogeneous Maxwell equation into the Lorentz for e density and integrate partially:   fi = Fij J j = Fij k Hjk = k Fij Hjk (k Fij ) Hjk : (B.5.33) lorentz The last term an be rewritten by means of the homogeneous Maxwell equation, i.e., k Fij = i Fjk + j Fki ; (B.5.34) or  
fi = k Fij Hjk + i Fjk + j Fki Hjk : (B.5.35) Again, we integrate partially. This time the two last terms:


fi = k Fij Hjk + i Fjk Hjk + j Fki Hjk Fjk i Hjk Fki j Hjk : (B.5.36) We olle t the rst three terms on the right hand side and substitute the left hand side of the inhomogeneous Maxwell equation into the last term:   fi = k Æik Fjl Hjl + 2Fij Hjk Fjk i Hjk Fik J k : (B.5.37) The last term represents the negative of a Lorentz for e density, see (B.5.33). Thus we nd   1 1 fi = k Æik Fjl Hjl + Fij Hjk F  Hjl : (B.5.38) 2 2 jl i The rst and the third term on the right hand side are of a related stru ture. We split the rst term into two equal pie es and dierentiate one pie e:   i
1h 1 i Fjl Hjl Fjl i Hjl : (B.5.39) fi = k Æik Fjl Hjl + Fij Hjk + 4 4 Introdu ing the kinemati energy-momentum tensor density k T j = 1 Æ j F Hkl Fik Hjk (B.5.40) i 4 i kl and the for e density h i (B.5.41) Xi := 41 i Fjk
Hjk Fjk i Hjk ; we nally have the desired result, fi = j k Tij + Xi ; (B.5.42) whi h is the omponent version of (B.5.7). By means of (B.4.20)1 , it is possible to transform (B.5.40) into (B.5.31). 3 We leave it to the readers to prove the formula derived rst by Minkowski in 1907.

k

T k T = 14 Æ k T 2 whi h was

aa bb

dd

ee

ee1

ff

gg hh

210

B.5.

Energy-momentum urrent and a tion

Preview: Covariant onservation law and vanishing extra for e density

b X

The Lorentz for e density f in (B.5.7) and the energy-momentum urrent k  in (B.5.8) are ovariant with respe t to frame and oordinate transformations. Nevertheless, ea h of the two terms on the right hand side of (B.5.7), namely d k  or X , are not ovariant by themselves. What an we do? For the rst three axioms of ele trodynami s, the spa etime arena is only required to be a (1+3)-de omposable 4-dimensional manifold. We annot be as e onomi al as this in general. Ordinarily a linear onne tion  on that manifold is needed. The linear onne tion will be the guiding eld that transports a ve tor, e.g., from one point of spa etime to a neighboring one. The onne tion will only be introdu ed in Part C. There, the

ovariant exterior dierential is de ned as D = d +  (L  ), see (C.1.64). With the help of this operator, a generally ovariant expression D k  an be onstru ted. Then (B.5.7) an be rewritten as f

= D k  + Xb ;

(B.5.43)

fSXgam

with the new supplementary for e density 1 = (H ^ Le F F ^ Le H ) ; (B.5.44) 2 whi h ontains the ovariant Lie derivative L = D  +  D, see (C.1.72). Note that the energy-momentum urrent k  remains the same, only the for e density X gets repla ed by Xb . It is remarkable, in (B.5.43) [or in (B.5.7)℄ the energy-momentum urrent an be de ned even if (B.5.43), as long as Xb 6= 0, doesn't represent a genuine onservation law. Only in this subse tion, we will use the linear onne tion and the ovariant exterior derivative, but not in the rest of this Part B. Then we will be able to show that the fourth axiom is exa tly what is needed for an appropriate and onsistent derivation of the onservation law for energy-momentum. Let us then exploit, as far as possible, the arbitrary linear onne tion  , b X

Xalgam

B.5.2 Properties of energy-momentum, ele tri -magneti re ipro ity

211

introdu ed above. As auxiliary quantities, atta hed to  , we need _ the torsion 2-form T  and the transposed onne tion 1form  :=  + e T , both to be introdu ed in Part C in (C.1.43) and (C.1.44), respe tively. Let us now go ba k to the extra for e density Xb of (B.5.44). What we need is the gauge ovariant Lie derivative of an arbitrary 2-form =  # ^ # =2 in terms of its omponents. Using the general formula (C.2.128) we have  1 _   Le = D   # ^ # ; (B.5.45) 2 _

_

_

where D := e D, with D as the exterior ovariant dierential with respe t to the transposed onne tion. Thus,  _ 1 _     b X = H D  F F D  H # ^ # ^ # ^ # ; 8 (B.5.46) or, sin e # ^ # ^ # ^ # =  ^ , we nd, as alternative to (B.5.44),  _  _ b = ^  H D F F D H : X (B.5.47) 8 This is as far as we an go with an arbitrary linear onne tion. Now it be omes obvious of how one ould a hieve the vanishing of Xb . Our four axioms don't make ele trodynami s a

omplete theory. What is missing is the ele tromagneti spa etime relation between ex itation H and eld strength F . Su h a fth axiom will be introdu ed in Chapter D.4. The starting point for arriving at su h an axiom will be the linear ansatz

H = 12  F ;

with

H = 21  H :

Substituted into (B.5.47), we have _  b = ^ D  F F : X 8

(B.5.48)

(B.5.49)

Xfinal

212

B.5.

Energy-momentum urrent and a tion

Thus, the extra for e density Xb will vanish provided  is

ovariantly onstant with respe t to the transposed onne tion of the underlying spa etime. We will ome ba k to this dis ussion in Se .E.1.4. B.5.3

Time-spa e de omposition of the energy-momentum urrent

Another theory of ele tri ity, whi h I prefer, denies a tion at a distan e and attributes ele tri a tion to tensions and pressures in an all-pervading medium, these stresses being the same in kind with those familiar to engineers, and the medium being identi al with that in whi h light is supposed to be propagated.

James Clerk Maxwell (1870) If we 1+3 de ompose the Lorentz for e and the energy-momentum urrent, then we arrive at the 3-dimensional version of the energy-momentum law of ele trodynami s in a rather dire t way. We re all that we work with a foliation- ompatible frame e , as spe i ed in (B.1.32), i.e. with e^0 = n, ea = a , together with the transversality ondition ea d = 0. Consider the de nition (B.5.8) of k . Substitute into it the (1+3)-de ompositions (B.4.5) and (B.4.6) of the ex itation H and the eld strength F , respe tively. Then, we obtain ^0 = u k a = pa k

d ^ s ; d ^ Sa ;

where we introdu ed the energy density 3-form 1 u := (E ^ D + B ^ H) ; 2 the energy ux density (or Poynting) 2-form s := E ^ H ;

(B.5.50) (B.5.51)

siga

(B.5.52)

maxener

(B.5.53)

poynting

sig0

B.5.3 Time-spa e de omposition of energy-momentum

213

the momentum density 3-form a

p

:= ^ ( B

D)

a

e

(B.5.54)

;

and the Maxwell stress (or momentum ux density) 2-form of the ele tromagneti eld 1 ) ^ D ( a D) ^ a := ( a 2 (B.5.55)  + ( a H) ^ ( a )^H e

S

E

e

e

B

E

e

B

maxmom

maxstress

:

A

ordingly, we an represent the s heme (B.5.50)-(B.5.51) in the form of a 4  4 matrix (for density we use the abbreviation d.): 



d. energy ux d. = ( ) = energy mom. d. mom. ux d.



k

u

s

a

S

p

a

 :

(B.5.56)

sigmatrix1

The entries of the rst olumn are 3-forms and those of the se ond olumn 2-forms. The absolute dimensions of the quantities emerging in the 4  4 matrix an be determined from their respe tive de nitions and the de ompositions (B.4.5) and (B.4.6): 



[ ℄ [℄ = [ a℄ [ a℄ u

p

s

S

 h

t l

1 1

t

( )



2

tl

(B.5.57)

:

1

The dimensions of their respe tive omponents read (here 1 2 3), ;

i; j; k

=

;





[ ijk℄ [ ij ℄ = [ ijk a ℄ [ ij a ℄ u

p

s

S

h tl

3



1 ( ) l=t

1

l=t

1

 :

(B.5.58)

This oin ides with the results from me hani s. A momentum 3

ux density, e.g., should have the dimension = 2 3= 2 = stress, in agreement with [ ij a℄ = ( 3 ) = energy 3 = SI stress = Pas al. Note that dimensionwise the energy ux density ij of the ele tromagneti eld equals its momentum density 2 ijk a times the square of a velo ity ( ) . pv=l

f =l

S

h= tl

s

p

sigmatrix1a

l=t

mv =l

=l

sigmatrix1b

214

B.5.

Energy-momentum urrent and a tion

Transve ting the Maxwell stress Sa , \familiar to engineers", with #a , we nd straightforwardly # a ^ Sa = u ;

(B.5.59)

stresstra e

whi h is the 3-dimensional version of (B.5.14). As soon as an ele tromagneti spa etime relation will be available, we an relate the energy ux density s ^ #a , whi h has the same number of independent omponents as the 2-form s, namely 3, to the momentum density pa. In Se t. E.1.4, for the Maxwell-Lorentz spa etime relation, we will prove in this way the symmetry of the energy-momentum urrent, see (E.1.31). Sin e the Lorentz for e density is longitudinal with respe t to n, i.e., fa = ?fa , the forms u, s, pa , and Sa are purely longitudinal, too. Eqs.(B.5.50)-(B.5.51) provide the de omposition of the energy-momentum 3-form into its `time' and `spa e' pie es. If we apply (B.1.26) to it, we nd for the exterior dierentials: d k ^0 = d ^ (u_ + d s) ; d k a = d ^ (p_a + d Sa ) :

(B.5.60) (B.5.61)

dsig0 dsiga

Combining all the results, we eventually obtain for the (1+3)de omposition of (B.5.7) the balan e equations for the ele tromagneti eld energy and momentum: k^0 = u_ + d s + (X^0 )? ; ka = p_a + d Sa + (Xa )? :

(B.5.62) (B.5.63)

Observe nally that all the formulas displayed in this se tion are independent of any metri and/or onne tion. B.5.4

A tion

Why did we postpone the dis ussion of the Lagrange formalism for so long even if we know that this formalism helps so mu h in an ee tive organization of eld-theoreti al stru tures? We

hose to base our axiomati s on the onservation laws of harge

k0 ka

B.5.4

A tion

215

and ux, inter alia, whi h are amenable to dire t experimental veri ation. And in the se ond axiom we used the on ept of a for e from me hani s that has also the appeal of being able to be grasped dire tly. A

ordingly, the proximity to experiment was one of our guiding prin iples in sele ting the axioms. Already via the se ond axiom the notion of a for e density i,

ame in. We know that this on ept, a

ording to i  has also a pla e in Lagrange formalism. When we \derived" the fourth axiom by trying to express the Lorentz for e density  as an exa t form    , we obviously moved already towards the Lagrange formalism. This be ame apparent in (B.5.12): the 4-form  is a possible Lagrangian. Still, we proposed the energymomentum urrent without appealing to a Lagrangian. This seemed to be more se ure be ause we ould avoid all the falla ies related to a not dire tly observable quantity like . We were led, pra ti ally in a unique fashion, to the fourth axiom (B.5.8). In any ase, after having formulated the integral and the differential versions of ele trodynami s in luding its energy-momentum distribution, we have enough understanding of its inner working as to be able to reformulate it in a Lagrangian form in a very straightforward way. As we dis ussed in Se . B.4.1, for the ompletion of ele trodynami s we need an ele tromagneti spa etime relation = [ ℄. This ould be a nonlo al and nonlinear fun tional in general, as we will dis uss in Chapter E.2. The eld variables in Maxwell's equations are and . Therefore, the Lagrange 4form of the ele tromagneti eld should depend on both of them: = ( [ ℄ )= [ ℄ (B.5.64) From a dimensional point of view, it is quite obvious what type of a tion we would expe t for the Maxwell eld. For the ex i1 tation we have [R ℄ = and for the eld strength [ ℄ = . A

ordingly,  ^ would qualify as a tion. And this is, indeed, what we will nd out eventually. Maxwell's equations are rst order in = [ ℄ and , respe tively. Sin e = , the eld strength itself is rst order in . We will take as eld variable. The Euler-Lagrange equaf

L=x

f

f

d

L

H

H F

H

V

H

V

F

H F ;F

V F

:

q

H

F

F

H

F

A

dA

A

hq

H F

F

F

216

B.5.

Energy-momentum urrent and a tion

tions of a variational prin iple with a Lagrangian of dierential order m are of dierential order 2m. The eld equations are assumed at most of 2nd dierential order in the eld variables A; . Therefore the Lagrangian is of 1st order in these elds. Consider an ele tri ally harged matter eld , whi h, for the time being, is assumed to be a p-form. The total Lagrangian of the system, a twisted 4-form, should onsist of a free eld part V of the ele tromagneti eld and a matter part Lmat , the latter of whi h des ribes the matter eld and its oupling to A: L = V + Lmat = V (A; dA; ) + Lmat (A; dA; ; d ; ) : (B.5.65) Here  are what we an all the stru tural elds. Their presen e, at this stage, is motivated by the te hni al reasons: in order to be able to onstru t a viable Lagrangian other than a trivial (\topologi al") F ^ F , one should have a tool whi h helps to do this. The role of su h a tool is played by . Here we do not spe ify the nature of the stru tural elds, but we will see below that as soon as the metri (and onne tion) are de ned on the spa etime, they will represent the elds  properly. We require V to be gauge invariant, that is ÆV = V (A + ÆA; d[A + ÆA℄; ) V (A; dA; ) = 0 ; (B.5.66) where ÆA = d! represents an in nitesimal gauge transformation of the type (B.3.9) for   !. We obtain ÆV

or

=0 = d! ^ V A

,

V A

= 0;

(B.5.67)

= V (dA; ) = V (F; ) : (B.5.68) Hen e the free eld Maxwell or gauge Lagrangian an depend on the potential A only via the eld strength F = dA. The matter Lagrangian should also be gauge-invariant. The a tion reads Z W = L: (B.5.69) V

4

lmatter

vinv

delV

vf

a tion

B.5.4

A tion

217

The eld equations for A are given by the stationary points of W under a variation Æ of A whi h ommutes with the exterior derivative by de nition, that is, Æd = dÆ , and vanishes at the boundary, i.e. ÆAj 4 = 0. Varying A yields  Z Z  L L + ÆdA ^ ÆA W = ÆA L = ÆA ^ = =

4 Z 

ÆA

4 Z ÆA

4

^

4

A

L

L ( 1)1 d

A

ÆL

^ ÆA +

dA



+d

dA

Z ÆA

 4



ÆA

:=

^ dA

L ; ^ dA

L A

+d



(B.5.70)

where the variational derivative of the 1-form

ording to ÆL

ÆA

L

L dA

A

is de ned a -

:

(B.5.71)

delL

(B.5.72)

gaugefield

Stationarity of W leads to the gauge eld equation ÆL ÆA

= 0:

Keeping in mind the inhomogeneous Maxwell eld equation in (B.4.2), we de ne the ex itation (\ eld momentum") onjugated to A and the matter urrent by H

=

V dA

=

V F

and

J

=

ÆLmat ÆA

;

(B.5.73)

fieldmom

(B.5.74)

maxin

respe tively. Then we re over, indeed: dH

=J:

The homogeneous Maxwell equation is a onsequen e of working with the potential A, sin e F = dA and dF = ddA = 0. In (B.5.74), we were also able to arrive at the inhomogeneous equation. The ex itation H and the urrent J are, however, only

218

B.5.

Energy-momentum urrent and a tion

impli itly given. As we an see from (B.5.73), only an expli it form of the Lagrangians V and Lmat promotes H and j to more than sheer pla eholders. On the other hand, it is very satisfying to re over the stru ture of Maxwell's theory already at su h an impli it level. Eq.(B.5.73)1 represents the as yet unknown spa etime relation of Maxwell's theory. Let us turn to the variational of the matter eld . Its variational derivative reads ÆL Æ

Sin e

L := L ( 1)p d d

does not depend on , we nd for the equation simply V

ÆLmat Æ

(B.5.75)

:

varpsi

matter eld

= 0:

(B.5.76)

delmat

All what is left to do now in this ontext, is to spe ify the spa etime relation (B.5.73)1 and thereby to transform the Maxwell and the matter Lagrangian into an expli it form. At this stage, the stru tural elds  are onsidered as nondynami al (\ba kground"), so we do not have equations of motion for them. B.5.5

Coupling of the energy-momentum

urrent to the oframe

In this se tion we will show that the anoni al de nition of the energy-momentum urrent (as a Noether urrent orresponding to spa etime translations) oin ides with its dynami de nition as a sour e of the gravitational eld that is represented by the

oframe { and both are losely related to the kinemati urrent of our fourth axiom. Let us assume that the intera tion of the ele tromagneti eld with gravity is `swit hed on'. On the Lagrangian level, it means that (B.5.68) should be repla ed by the Lagrangian V = V (# ; F ): (B.5.77)

vfvta

B.5.5

Coupling of the energy-momentum urrent to the oframe

From now on, the oframe assumes the role of the stru tural eld . In a standard way, the dynami (or Hilbert) energymomentum urrent for the oframe eld is de ned by   ÆV V V d  := Æ# = # + d  (d# ) : (B.5.78) The last term vanishes for the Lagrangian (B.5.77) under onsideration. As before, f. (B.5.73), the ele tromagneti eld momentum is de ned by

219

SigDyn

: (B.5.79) = V F The ru ial point is the ondition of general oordinate or diffeomorphism invarian e of the Lagrangian (B.5.77) of the intera ting ele tromagneti and oframe elds. The general variation of the Lagrangian reads H

ÆV

V V = Æ# ^ #  + ÆF ^ F =

Æ#

^ d 

ÆF

^H:

(B.5.80) If  is a ve tor eld generating an arbitrary one{parameter group Tt of dieomorphisms on X , the variation in (B.5.80) is des ribed by the Lie derivative, i.e., Æ = L =  d + d . Substituting this into the left-hand and right-hand sides of (B.5.80), we nd, after some rearrangements, the identity i h d ( V ) + ( # ) d  + ( F ) ^ H ( # ) d d + ( d# ) ^ d  ( F ) ^ dH = 0: (B.5.81) Sin e  is arbitrary, the rst and se ond lines are vanishing separately. Now, the nal step is to put  = e. Then (B.5.81) yields two identities. From the rst line of (B.5.81) we nd that the dynami urrent d , de ned in (B.5.78), an be identi ed with the anoni al (or Noether)energy-momentum urrent, i.e., d    with  := e V (e F ) ^ H : (B.5.82)

varL0

AB0

sig an

220

B.5.

Energy-momentum urrent and a tion

Thus we don't need to distinguish any longer between the dynami and the anoni al energy-momentum urrent. This fa t mat hes very well with the stru ture of the kinemati energymomentum urrent k of (B.5.8) or, better, of (B.5.13). We

ompare (B.5.82) with (B.5.13). If we hoose as Lagrangian V = F ^ H=2, then k  =  { and we an drop the k and the d from k and d  , respe tively. However, this hoi e of the Lagrangian will only be legitimized by our fth axiom in Chapter D.5. The se ond line of (B.5.81) yields the onservation law of energy-momentum: d 

= (e d# ) ^  + (e F ) ^ J;

(B.5.83)

onserv

where we have used the inhomogeneous Maxwell equation dH = The presen e of the rst term on the right-hand side guarantees the ovariant hara ter of that equation. It is easy to see that we an rewrite (B.5.83) in the equivalent form J.

e  D

= (e F ) ^ J:

(B.5.84)

by making use of the Riemannian ovariant exterior derivative e to be de ned in Part C. D Our dis ussion an even be made more general,4 going beyond a pure ele trodynami al theory. Namely, let us onsider a theory of the oframe # oupled to a generalized matter eld . Note that the latter may not be just one fun tion or an exterior form, but and arbitrary olle tion of forms of all possible ranks and/or exterior forms of type (0)

p

( )

, i.e. = ( U ; : : : ; A ; : : : ) (note that the range of indi es for the forms of dierent rank is, in general, also dierent, that is, e.g., U and A run over dierent ranges). We assume that the Lagrangian of su h a theory depends very generally on frame, matter eld, and its derivatives: L = L(# ; d# ; ; d ) : (B.5.85) Normally, the matter Lagrangian does not depend on the derivatives of the oframe, but we in lude d# for greater generality. [It is important to realize that the Lagrangian (B.5.85) indeed des ribes the most general theory: For example, the set an in lude p

( )

not only the true matter elds, des ribed, say, by a p-form A , but formally also other virtual gravitational potentials su h as the metri 0-form g , the onne tion 1-form  , as soon as they are de ned on X and are intera ting with ea h other℄. 4

See,e.g., Ref.[14℄.

Lvarpsi

B.5.5

Coupling of the energy-momentum urrent to the oframe

221

The basi assumption about the Lagrangian (B.5.85) is that L is a s alar-valued twisted n-form whi h is invariant under the spa etime dieomorphisms. This simple input has amazingly general onsequen e. The dynami energy-momentum urrent of matter will be de ned as in (B.5.78),  L  L ÆL d  := = + d : (B.5.86)     Æ# #  (d# ) Similarly, the variational derivative of the generalized matter reads ÆL = L ( 1)p d  (L ; (B.5.87) Æ d )

SigDyn'

VDpsi

where the sign fa tor ( 1)p orrelates with the relevant rank of a parti ular omponent in the the set of elds . This is a simple generalization of equation (B.5.75). A

ording to the Noether theorem, the onservation identities of the matter system result from the postulated invarian e of L under a lo al symmetry group. A tually, this is only true \weakly", i.e., provided the Euler{Lagrange equation (B.5.87) for the matter elds is satis ed. Here we onsider the onsequen es of the invarian e of L under the group dieomorphisms on the spa etime manifold. Let  be a ve tor eld generating an arbitrary one{parameter group Tt of dieomorphisms on X . In order to obtain a orresponding Noether identity from the invarian e of L under a one{parameter group of lo al translations Tt  T  Diff (4; R), it is important to re all that in nitesimally the a tion of a one{parameter group Tt on X is des ribed by the onventional Lie derivative (A.2.49) with respe t to a ve tor eld . Sin e we work with the elds whi h are exterior forms of various ranks, the most appropriate formula for the Lie derivative is (A.2.51), i.e. L =  d + d . The general variation of the Lagrangian (B.5.85) reads: L L L L  = Æ# ^ # (B.5.88)  + (Æd# ) ^ d# + Æ ^  + (Æd ) ^ d : For the variations generated by a one-parameter group of the ve tor eld  we have to substitute Æ = L in (B.5.88). This is straightforward and, after performing some `partial integrations', we nd h L L + ( d# ) ^ d# d( L) =d ( # )  # L L i + ( ) ^  + ( d ) ^ d ÆL ÆL  ( # )d Æ#  + ( d# ) ^ Æ# ÆL

ÆL p (B.5.89) + ( d ) ^ ÆÆL + ( 1) ( ) ^ d Æ : Now we rearrange the equation of above by olle ting terms under the exterior derivative separately. Then (B.5.89) an be written as A dB = 0 ; (B.5.90) where ÆL ÆL  A := ( # )d Æ#  + ( d# ) ^ Æ#

+ ( B

:= 

L

ÆL p ) ^ ÆÆL + ( 1) ( ) ^ d Æ ;

d

(

#

L ) # 

( ) ^ L

(

(

d#

varL1

A+dB

(B.5.91)

defA

(B.5.92)

defB

L ) ^ d# 

L ) ^ d :

d

varL

222

B.5.

Energy-momentum urrent and a tion

The fun tions A and B have the form A =   A ; B =   B : (B.5.93) Hen e, by (B.5.90),   (A dB ) d  ^ B = 0; (B.5.94)   where both  and d are pointwise arbitrary. Hen e we an on lude that B as well as A vanish: A = 0; and B = 0 : (B.5.95) Sin e the ve tor eld  is arbitrary, it is suÆ ient to repla e it via  ! e by the frame eld. Then, for B = 0, we obtain from (B.5.92) d

=

L + (e ) ^ L e L + (e d ) ^ d L L d  + (e d# ) ^ : d# d#

(B.5.96)

dyn urr

(B.5.97)

1stNoe

For A = 0, eq.(B.5.91) yields

d d   (e d# ) ^ d  + F ;

where

F := (e d ) ^ ÆÆL

( 1)p (e ) ^ d ÆÆL :

(B.5.98)

In fa t, when the matter Lagrangian is independent of the derivatives of the oframe eld, i.e., L=d# = 0, eq. (B.5.96) demonstrates the equality of the dynami energymomentum urrent, that is oupled to the oframe, with the anoni al one, the Noether

urrent of the translations. B.5.6

Maxwell's equations and the energy-momentum urrent in Ex al

It is a merit of exterior al ulus that ele trodynami s and, in parti ular, Maxwell's equations an be formulated in a very su

in t form. This translates into an equally on ise form of the

orresponding omputer programs in Ex al . The goal of better understanding the stru ture of ele trodynami s leads us, hand in hand, to a more transparent and a more ee tive way of omputer programming. In Ex al , as we mentioned in Se . A.2.11, the ele trodynami al quantities are evaluated with respe t to the oframe that is spe i ed in the program. If we put in a a

elerating and rotating oframe !  , e.g., then the ele tromagneti eld strength F in the program will be evaluated with respe t to this frame:

B.5.6

Maxwell's equations and the energy-momentum urrent in Ex al

F = F !  ^ ! =2. This is, of ourse, exa tly what we dis ussed

in Se . B.4.3 when we introdu ed arbitrary noninertial oframes. We autioned already our readers in Se . A.2.11 that we need to spe ify a oframe together with the metri . Thus, our Maxwell sample program proper, to be displayed below, is pre eded by

oframe and frame ommands. We pi k the spheri al oordinate system of Se . A.2.11 and require the spa etime to be Minkowskian, i.e., (r) = 1. Afterwards we spe ify the ele tromagneti potential A = A # , namely pot1. Sin e we haven't de ned a spe i problem so far, we leave its omponents aa0, aa1, aa2, aa3 open for the moment. Then we put in the pie es dis ussed subsequent to (B.4.2). In order to relate H and F , we have to make use of the fth axiom, only to be pinned down in eq.(D.5.7): % file mustermax.exi, 2001-05-31 load_pa kage ex al $ pform psi=0$ fdomain psi=psi(r)$

oframe o(0) = psi o(1) = (1/psi) o(2) = r o(3) = r * sin(theta) with signature (1,-1,-1,-1)$ frame e$ psi:=1;

* * * *

d d d d

% oframe defined t, r, theta, phi

% flat spa etime assumed

% start of Maxwell proper: unknown fun tions aa0, aa1... pform {aa0,aa1,aa2,aa3}=0, pot1=1, {farad2,ex it2}=2, {maxhom3,maxinh3}=3$ fdomain aa0=aa0(t,r,theta,phi),aa1=aa1(t,r,theta,phi), aa0=aa0(t,r,theta,phi),aa1=aa1(t,r,theta,phi)$ pot1

:= aa0*o(0) + aa1*o(1) + aa2*o(2) + aa3*o(3)$

223

224

B.5.

farad2 maxhom3 ex it2 maxinh3

:= := := :=

Energy-momentum urrent and a tion

d pot1; d farad2; lam * # farad2; d ex it2;

% spa etime relation, see % 5th axiom Eq.(D.5.7)

% Maxwell Lagrangian and energy-momentum urrent assigned pform lmax4=4, maxenergy3(a)=3$ lmax4 := -(1/2)*farad2^ex it2; maxenergy3(-a) := e(-a) _|lmax4 + (e(-a) _|farad2)^ex it2; % Use a blank before the interior produ t sign! end;

If this sample program is written onto a le with name muster(exi stands for ex al -input, the orresponding output le has the extension .exo), then this very le an be read into a Redu e session by the ommand in"mustermax.exi"; As a trivial test, you an spe ify the potential of a point harge sitting at the origin of our oordinate system:

max.exi

aa0 := -q/r;

aa1 := aa2 := aa3 := 0;

Determine its eld strength by farad2:=farad2; its ex itation by ex it2:=ex it2; and its energy-momentum distribution by maxenergy3(-a):maxenergy3(-a); you will nd the results you are familiar with. And, of ourse, you want to onvin e yourself that Maxwell's equations are ful lled by releasing the ommands maxhom3:=maxhom3; and maxinh3:=maxinh3; This sample program an be edited a

ording to the needs one has. Pres ribe a non-inertial oframe, i.e., an a

elerating and rotating oframe. Then you just have to edit the oframe ommand and an subsequently ompute the orresponding physi al omponents of an ele tromagneti quantity with respe t to that frame. Appli ations of this program in lude the ReissnerNordstrom and the Kerr-Newman solutions of general relativity; they represent the ele tromagneti and the gravitational elds

B.5.6

Maxwell's equations and the energy-momentum urrent in Ex al

of an ele tri ally harged mass of spheri al or axial symmetry, respe tively. They will be dis ussed in the outlook in the last part of the book.

225

226

B.5.

Energy-momentum urrent and a tion

Referen es

[1℄ R.C. Ashoori,

Ele trons in arti ial atoms,

Nature

379

(1996) 413-419.

[2℄ J.E. Avron, Quantum

Adiabati Quantum Transport.

Physi s,

Les

Hou hes

Session

In Mesos opi LXI

1994,

E.

Akkermans, G. Montambaux, J.L. Pi hard, and J. ZinnJustin, eds. (Elsevier: Amsterdam 1995) pp. 741-791.

[3℄ E. Braun,

The Quantum Hall Ee t.

In Metrology at the

Frontiers of Physi s and Te hnology, Pro . Intern. S hool of Physi s `Enri o Fermi' Course CX (1989), L. Crovini and T.J. Quinn, eds. (North Holland: Amsterdam, 1992) pp.211-257.

[4℄ T. Chakraborty and P. Pietil ainen,

fe ts, Fra tional and Integral,

The Quantum Hall Ef-

2nd ed. (Springer: Berlin,

1995).

[5℄ M.H.

Devoret

and

Charge Tunneling.

H.

Grabert,

Introdu tion to Single

In Single Charge Tunneling: Coulomb

Blo kade Phenomena in Nanostru tures, H. Grabert and

228

Referen es M.H. Devoret, eds. (Plenum Press: New York, 1992) pp. 1-19.

Magnetoquantumtransport on GaAs-Alx Ga1 x as heterostru tures at very low temperatures. Solid State Comm. 44 (1982) 95-

[6℄ G. Ebert, K. v. Klitzing, C. Probst, and K. Ploog,

98.

The dire t observation of individual ux lines in type II super ondu tors, Phys. Lett. 24A (1967) 526-527.

[7℄ U. Essmann and H. Tr auble,

The magneti stru ture of super ondu tors, S i. Ameri an 224 (Mar h 1971) 74-84.

[8℄ U. Essmann and H. Tr auble,

[9℄ J.

Fr ohli h,

anomaly.

B.

Los

Pedrini, Alamos

New appli ations of the hrial Eprint

Ar hive

hep-th/0002195

(2000) 39 pages.

Universality in quantum Hall systems: Coset onstru tion of in ompressible states. Los Alamos Eprint Ar hive ond-

[10℄ J. Fr ohli h, B. Pedrini, C. S hweigert, J. Wal her,

mat/0002330 (2000) 36 pages.

Gauge invarian e and urrent algebra in nonrelativisti many-body theory, Rev. Mod. Phys. 65 (1993) 733-802.

[11℄ J. Fr ohli h and U.M. Studer,

Duality rotations for interB193 (1981) 221-244.

[12℄ M.K. Gaillard and B. Zumino,

a ting elds, Nu l. Phys.

Rotation and magnetism of Earth's inner ore, S ien e 274 (1996) 1887-1891.

[13℄ G.A.

Glatzmaier and

P.H. Roberts,

[14℄ F.W. Hehl, J.D. M Crea, E.W. Mielke, and Y. Ne'eman,

Metri -AÆne Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invarian e. Phys. Rep. 258 (1995) 1{171.

Referen es

229

[15℄ J.L. Heilbron, Ele tri ity in the 17th and 18th Centuries. A Study of Early Modern Physi s (University of California Press: Berkeley, 1979). [16℄ R. Ingarden and A. Jamiolkowski, Classi al Ele trodynami s (Elsevier: Amsterdam, 1985). [17℄ M. Janssen, O. Viehweger, U. Fastenrath, and J. Hajdu,

Introdu tion to the Theory of the Integer Quantum Hall Ee t (VCH: Weinheim, Germany, 1994).

[18℄ R.M. Kiehn, Periods on manifolds, quantization, and gauge, J. Math. Phys. 18 (1977) 614-624. [19℄ R.M. Kiehn, The photon spin and other topologi al features of lassi al ele tromagnetism (10 pages). In Gravitation and Cosmology: From the Hubble Radius to the Plan k S ale. R. Amoroso et al., eds. (Kluver: Dordre ht, Netherlands, in press 2001). [20℄ R.M. Kiehn and J.F. Pier e, Intrinsi Transport Theorem, Phys. Fluids 12 (1969) 1941-1943. [21℄ L.D. Landau and E.M. Lifshitz, Ele trodynami s of Continuous Media. Volume 8 of Course of Theor. Physi s. Transl. from the Russian (Pergamon Press: Oxford, 1960). [22℄ R. Lust and A. S hluter, Kraftfreie Magnetfelder, Z. Astrophysik 34 (1954) 263-282. [23℄ G.E. Marsh, For e-Free Magneti Fields: Solutions, topology and appli ations (World S ienti : Singapore, 1996). [24℄ G.E. Marsh, Topology in ele tromagneti s. Chapter 6 of Frontiers in Ele tromagneti s. D.H. Werner, R. Mittra, eds. (IEEE Press: New York, 2000) pp. 258-288. [25℄ B. Mashhoon, The hypothesis of lo ality in relativisti phyi s, Phys. Lett. A145 (1990) 147-153.

230

Referen es

[26℄ E.W. Mielke, Geometrodynami s of Gauge Fields | On the geometry of Yang-Mills and gravitational gauge theories (Akademie{Verlag: Berlin 1987) Se .V.1. [27℄ H.K. Moatt,

Magneti Field Generation in Ele tri ally

Fluids (Cambridge University Press: Cambridge, England, 1978).

Condu ting

[28℄ R.W. Pohl, Elektrizitatslehre, 21st ed. (Springer: Berlin, 1975) pp. 27/28, see also earlier editions. [29℄ W. Raith,

Bergmann-S haefer, Lehrbu h der Experimen-

, 8th ed. (de Gruyter:

talphysik, Vol.2, Elektromagnetismus

Berlin, 1999).

[30℄ A.F. Ra~nada, Topologi al (1992) 1621-1641. [31℄ A.F. Ra~nada, (1992) 70-76.

ele tromagnetism, J. Phys.

On the magneti heli ity, Eur. J.

[32℄ J.L. Trueba, A.F. Ra~nada, The Eur. J. Phys. 17 (1996) 141-144. [33℄ T. Ri hter and R. Seiler,

A25

Phys. 13

ele tromagneti heli ity,

Geometri properties of trans-

In Geometry and Quantum Physi s. Pro . 38th S hladming Conferen e, H. Gausterer et al., eds. Le ture Notes in Physi s 543 (2000) 275-310.

port in quantum Hall systems.

[34℄ P.H. Roberts and G.A. Glatzmaier, Geodynamo theory and simulations, Rev. Mod. Phys. 72 (2000) 1081-1123. [35℄ G.E. Stedman, Ring-laser tests of fundamental physi s and geophysi s, Rept. Prog. Phys. 60 (1997) 615-688. [36℄ R.A. Toupin,

, in:

Elasti ity and ele tro-magneti s

Linear Continuum Theories,

Non-

Conferen e, Bres-

. C. Truesdell and G. Grioli oordinators.

sanone, Italy 1965

Pp.203-342.

C.I.M.E.

Referen es [37℄ J. Van Bladel,

231

Relativity and Engineering, Springer Series

in Ele trophysi s Vol.15 (Springer: Berlin, 1984). [38℄ K. von Klitzing, The

58 (1986) 519-531.

quantized Hall ee t, Rev. Mod. Phys.

The mutual embra e of ele tri ity and magnetism, S ien e 203 (1979) 1310-1318.

[39℄ M.N. Wise,

[40℄ M.R. Zirnbauer,

Elektrodynamik.

(Springer: Berlin, to be published).

Tex-s ript July 1998

Part C More mathemati s

232

C.1

Linear onne tion

le birk/partC.tex, with gures [C01tors.ps, C02 urv.eps, C03 art.eps, C04tetra.eps, C05m ube.eps℄ 2001-06-01

\...the essential a hievement of general relativity, namely to over ome `rigid' spa e (ie the inertial frame), is only indire tly

onne ted with the introdu tion of a Riemannian metri . The dire tly relevant on eptual element is the `displa ement eld' ( ), whi h expresses the in nitesimal displa ement of ve tors. It is this whi h repla es the parallelism of spatially arbitrarily separated ve tors xed by the inertial frame (ie the equality of

orresponding omponents) by an in nitesimal operation. This makes it possible to onstru t tensors by dierentiation and hen e to dispense with the introdu tion of `rigid' spa e (the inertial frame). In the fa e of this, it seems to be of se ondary importan e in some sense that some parti ular eld an be dedu ed from a Riemannian metri ..." l ik

A. Einstein (1955 April 04)

1

1 Translation

by F. Gronwald, D. Hartley, and F.W. Hehl from the German original:

See Prefa e in [6℄.

234

C.1.

C.1.1

Linear onne tion

Covariant dierentiation of tensor elds The hange of s alar fun tions f along a ve tor u is des ribed by the dire tional derivative u f . The generalization of this notion from s alars f to tensors T is provided by the ovariant dierentiation ru T .

When al ulating a dire tional derivative of a fun tion f (x) along a ve tor eld u, one has to know the values of f (x) at dierent points on the integral lines of u. With the standard de nition whi h involves taking a limit of the separation between points, the dire tional derivative reads u f

:= u

= df (u) f (x)

df

= ui(x)

i x

in lo al oordinates fxi g:

(C.1.1)

Obviously, (C.1.1) des ribes a map u : T00 (X )  T01 (X ) ! 0 0 0 T0 (X ) of s alar elds, i.e., tensors of type [0 ℄, again into T0 (X ) = C (X ). This map has simple properties: 1) R -linearity, 2) C (X )linearity with respe t to u, i.e., gu+hv f = gu f + hv f , 3) additivity u (f + g ) = u f + u g . In order to generalize   the dire tional derivative to arbitrary tensor elds of type pq , one needs a re ipe of how to ompare tensor quantities at two dierent points of a manifold. This is provided by an additional stru ture on X , alled the linear

onne tion. The linear onne tion or, equivalently, the ovariant dierentiation is ne essary in order to formulate dierential equations for various physi al elds like, for instan e, the Einstein equation for the gravitational eld or the Navier-Stokes equation of hydrodynami s. In line with the dire tional derivative, a ovariant dierentiation r is de ned as an smooth R -linear map r : Tqp(X )  T01(X ) ! Tqp(X ) (C.1.2) 1 whi h to any ve tor eld u 2 T0 (X ) and to any tensor eld  p p T 2 Tq (X ) of type q assigns a tensor eld ru T 2 Tqp (X ) of   type pq that satis es the following properties:

dire tU

CDdef

C.1.1 Covariant dierentiation of tensor elds 1)

C (X )-linearity with respe t to u,

rf u+gv T = f ruT + grv T ; 2)

df ;

(C.1.4)

odiff2

(C.1.5)

odiff3

(C.1.6)

odiff4

(C.1.7)

odiff5

the Leibniz rule with respe t to tensor produ t,

ru (T S ) = ruT S + T ruS ; 5)

odiff1

for a s alar eld f a dire tional derivative is re overed,

ruf = u 4)

(C.1.3)

additivity with respe t to T ,

ru(T + S ) = ruT + ruS ; 3)

235

the Leibniz rule with respe t to interior produ t,

ru(v

! ) = (ru v ) ! + v

ru! ;

for all ve tor elds u; v , all fun tions f; g , all tensor elds T; S , and all forms ! . The Lie derivative Lu , de ned in Se . A.2.10, is also a map p Tq (X )  T01(X ) ! Tqp(X ). The properties of ovariant dierentiation 3)-6) are the same as those of the Lie derivative, f. (C.1.4)(C.1.7) with (A.2.56), (A.2.53), (A.2.58) and (A.2.55), respe tively. The property (C.1.3) is however somewhat dierent than the orresponding property of the Lie derivative (A.2.57) whi h

an be visualized, e.g., by substituting u ! fu in (A.2.57). This dieren e re e ts the fa t that the de nition of Lu T at a given point x 2 X makes use of u in the neighborhood of this point, whereas in order to de ne ru T at x one has to know only the value of u at x.

236

C.1.

C.1.2

Linear onne tion

Linear onne tion 1-forms

The dieren e between the ovariant and the partial dierentiation of a tensor eld is determined by the linear onne tion 1-forms . They show up in the a tion of r on a frame  and supply a onstru tive realization of the ovariant dierentiation r of an arbitrary tensor eld. i

u

j

i

u

Consider the hart U1 with the lo al oordinates x whi h

ontains a point x 2 U1  X . Take the natural basis  at a x. The ovariant dierentiation r of the ve tors  with respe t to an arbitrary ve tor eld u = u  reads i

i

u

i

k

ru i

=u

k

k

(C.1.8)

rk i :

Here we used the property (C.1.3). These n ve tor elds an be de omposed with respe t to the oordinate frame  : i

ru i

:=

i

=

ki

j

(u)  :

(C.1.9)

onn1

(C.1.10)

onn2

j

The onne tion 1-forms j i

j

k

dx

an be read o with their omponents . Suppose a dierent hart U2 with the lo al

oordinates x interse ts with U1 , T and the point x 2 U1 U2 belongs to an interse tion of the two harts. Then the onne tion 1-forms satisfy the onsisten y

ondition ki i

0

i

j

0

(x ) = 0

i

x

i0

x

j

x

0

j j

x

i

(x) +

j

x

0

k

x

j

0

k

d

x

i0

x

(C.1.11)

oord on

T

everywhere in the interse tion of the lo al harts U1 U2 . Now, making use of the properties 1)-5),   one an des ribe a

ovariant dierentiation of an arbitrary tensor eld p q

T

=T

i1 :::i

p

j1 :::j

j1 jq q i1


ip dx


dx

(C.1.12)

arbtensor

C.1.2

Linear onne tion 1-forms

237

in terms   of the onne tion 1-forms. Namely, we have expli itly the pq tensor eld

(DT i1 :::ip j1 :::jq )i1


ip dxj1


dxjq ; (C.1.13) where the ovariant dierential of the natural tensor omponents is introdu ed by D T i1 :::ip j1 :::jq := d (T i1 :::ip j1 :::jq ) + k i1 T ki2 :::ip j1 :::jq +


+ k ip T i1 :::ip 1 k j1 :::jq

ru T = u

j1

k T i1 :::ip

kj2 :::jq

:::

jq

k T i1 :::ip

oderiv

j1 :::jq 1 k :

(C.1.14)

oderiv2

The step in (C.1.9) an be generalized to an arbitrary frame

e . Its ovariant dierentiation with respe t to a ve tor eld u

reads

rue =



(u)e

(C.1.15)

;

onne t

with the orresponding linear onne tion 1-forms  . In 4 dimensions, we have 16 one-forms  at our disposal. The omponents of the onne tion 1-forms with respe t to the oframe # are given by

re e =  e : In terms of a lo al oordinate system fxi g, 







=

=



i

#

dxi

or

where

i



=



( ) : i

(C.1.16)

omps

(C.1.17)

hol omps

The 1-forms  are not a new independent obje t: sin e an arbitrary frame may be de omposed with respe t to the oordinate frame, we nd, with the help of (A.2.30) and (A.2.31), the simple relation 



= ej

i

j ei



+ ei d ei  :

(C.1.18)

gamgam

Under a hange of the frame whi h is des ribed by a linear transformation e0 = L0  e ;

(C.1.19)

hframe

238

C.1.

Linear onne tion

the onne tion 1-forms transform in a non-tensorial way, 

0



0

= L  L 0

0





+ L dL :

(C.1.20)

0

0

trafo on

For an in nitesimal linear transformation, L = Æ + " , Æe

= e

e

0

= " e ;

(C.1.21)

hframe1

the onne tion one-form hanges as Æ 

= D" = d" +



"

 " :

(C.1.22)

Although the transformation law (C.1.20) is inhomogeneous, we

annot, on an open set U , a hieve  = 0 in general; it an be only done if the urvature (to be introdu ed later) vanishes in U . At a given point x , however, we an always hoose the rst derivatives of L  ontained in dL  in su h a way that  (x ) = 0. A frame e , su h that 0

0

0

0

0

0

0

0

 (x0 )

=0

at a given point x

0

(C.1.23)

;

normframe

will be alled normal at x . A normal frame is given up to transformations L  whose rst derivatives vanish at x . This freedom may be used, and we an always hoose a oordinate system fxig su h that the frame e(x) in (C.1.23) is also `trivialized': 0

0

e

0

= Æi i

at a given point x

0

:

(C.1.24)

norm oo

(C.1.25)

trivial

Summing up , for a trivialized frame we have 2

(e ;



 ) = (Æi i ; 0)

at a given point x

0

:

Despite the fa t that the normal frame looks like a oordinate frame [in the sense, e.g., that (C.1.24) shows apparently that the tangent ve tors i of the oordinate frame oin ide with the ve tors of e basis at x ℄, one annot, in general, introdu e new lo al oordinates in the neighborhood of x in whi h (C.1.23) is ful lled. 0

0

2 See

Hartley [3℄ and Iliev [4℄.

C.1.3 Covariant dierentiation of a general geometri quantity

C.1.3

239

Covariant dierentiation of a general geometri quantity What is the ovariant dierential of a tensor density, for example?

Let us now onsider the ovariant dierentiation of a general geometri quantity that was introdu ed in Se . A.1.3. As earlier in Se . A.2.10, we will treat a geometri quantity w as a set of smooth elds wA on X . These elds are the omponents of w = wA eA with respe t to a frame eA 2 W = R N in the spa e of a -representation of the group GL(n; R) of lo al linear frame transformations (C.1.19). The transformation (C.1.19) of a frame of spa etime a ts on the geometri quantity of type  by the means of the lo al generalization of (A.1.18), wA

! wA

0

=

0 B A (L 1 ) w B ;

(C.1.26)

or, in the in nitesimal ase (C.1.21), Æ wA

=

" B A  w B :

(C.1.27)

geomtrafo

The generator matrix B A  was introdu ed in (A.2.67) when we dis ussed the Lie derivative of geometri quantities of type . A ovariant dierentiation for geometri quantities of type  is introdu ed as a natural generalization of the map (C.1.2) with all the properties 1)-6) preserved: the ovariant dierentiation for a W -frame reads

rueA = AB 

 (u)eB ;

(C.1.28)

onne t2

whereas for an arbitrary geometri quantity w = wA eA of type  eqs. (C.1.13), (C.1.14) are repla ed by

rw = u

(DwA) eA ; with

Dw A

:= dwA + B A 

B  w :

(C.1.29)

The general formula (C.1.29) is onsistent with the ovariant derivative of usual tensor elds when the latter are treated as a

oderiv3

240

C.1.

Linear onne tion

geometri quantity of a spe ial kind; one an ompare this with the examples 2), 3) in Se . A.1.3. Two simple appli ations of this general te hnique are in order. As a rst one, we re all that the Levi-Civita symbols have the same values with respe t to all frames, see (A.1.65). This means that they are the geometri quantities of the type  = id (identity transformation) or, plainly speaking, Æ1 :::n = 0. Comparing this with (C.1.27) and (C.1.28), we on lude that D 1 :::n = 0 (C.1.30) for an arbitrary onne tion. As a se ond example, let us take a s alar density S of weight w . This geometri quantity is des ribed by the transformation law (A.1.57) or, in the in nitesimal form, by Æ S = w "  S : (C.1.31) Comparing with (C.1.27), we nd  = w Æ  , and hen e (C.1.29) yields D S = dS w  S : (C.1.32) If we generalize this to a tensor density ; T , we nd  D T = d T


+  T   T +


w   T : (C.1.33)



























































C.1.4

Deps0

Parallel transport

By means of the ovariant dierentiation ru , a tensor an be parallelly transported along a urve on a manifold. This provides a onvenient tool for omparing values of the tensor eld at dierent points of the manifold.

A onne tion enables us to de ne parallel transport of a tensor along a urve. A dierentiable urve  on X is a smooth map  : (a; b)!X t 7!x(t) ; (C.1.34)

density

Torsion and urvature

C.1.5

241

where (a; b) is an interval in R. In lo al oordinates fx g, the tangent ve tor to the urve  = fx (t)g is u = dx =dt. A tensor eld T is said to be parallelly transported along  if i

i

i

i

r T =0

(C.1.35)

u

nabla0

along  . Taking into a

ount (C.1.14), we get for T d T dt

i1 :::ip




j1 :::jq

x(t) +

dx dt

l lk

i1

(x(t)) T

ki2 :::ip

j1 :::jq

(x(t)) + : : : !

ljq

k

(x(t)) T

i1 :::ip

j1 :::jq

1k

(x(t)) = 0 : (C.1.36)

paralT

If 0 2 (a; b) and T ( (0)) is given, then there exists (at least lo ally) a unique solution T ( (t)) of this linear ordinary dierential equation for t2 (a; b). Therefore, we have a linear map  of tensors of type at the point  (0) to tensors of type at the point  (t). Taking T = u, we obtain a dierential equation for autoparallels:

r u=0 u

C.1.5

or

p

p

q

q

d2 x + dt2 i

Torsion and

i jk

(x(t))

dx dx = 0: (C.1.37) dt dt j

k

urvature

Torsion and urvature both measure the deviation from the at spa etime geometry (of spe ial relativity). When both of them are zero, one an globally de ne the trivialized frame (C.1.25) all over the spa etime manifold.

We now want to asso iate with a onne tion two tensor elds: torsion and urvature. As we have seen above, the onne tion form an always be transformed to zero at one given point. However, torsion and urvature will in general give a non-vanishing

hara terization of the onne tion at this point.

autopar

Linear onne tion

uR R

vu

u||R T(u,v)

[u,v]

∆

C.1.

∆

242

uv

vQ|| v Q

vP

uP

P

Q

Figure C.1.1: On the geometri al interpretation of torsion: a

losure failure of in nitesimal displa ements.This is a s hemati view. Note that R and Q are in nitesimally near to P . The torsion of a onne tion r is a map T that assigns to ea h pair of ve tor elds u and v a ve tor eld T (u; v ) by (

T u; v

) := ru v

rv

u

[u; v ℄ :

(C.1.38)

The ommutator [u; v ℄ has been de ned in (A.2.5). One an straightforwardly verify the tensor hara ter of T (u; v ) so that its value at any given point is determined by the values of u and v at that point. Fig. C.1.1 illustrates s hemati ally the geometri al meaning of torsion: Choose two ve tors v and u at a point P 2 X . Transfer u parallelly along v to the point R and likewise v along u to the point Q. If the resulting parallelogram is broken, i.e., if it has a gap or a losure failure, then the onne tion arries a torsion. Su h situations o

ur in the ontinuum theory of dislo ations. 3

3 See

Kr oner [5℄ and referen es given there.

deftorsion

C.1.5

Torsion and urvature

243

Sin e the torsion T (u; v ) is a ve tor eld, one an expand it with respe t to a lo al frame, T (u; v ) = T  (u; v ) e :

(C.1.39)

torexp

By onstru tion, the oeÆ ients T  (u; v ) of this expansion are are 2-forms, i.e., fun tions whi h assign real numbers to every pair of the ve tor elds u; v . Taking the ve tors of a frame, T (e ; e ) = T  (e ; e ) e = T  e ;

we an read o the oeÆ ients of this form:

(C.1.40)

ve tor-valued torsion 2-

1 1 (C.1.41) 2 2 The expli it form of these oeÆ ients is obtained dire tly from the de nition (C.1.38) whi h we evaluate with respe t to a frame e , T  = T  # ^ # = Tij  dxi ^ dxj :

T  (e ; e ) e = re e

re e

tor oef

[e ; e ℄ e :

tor2form

(C.1.42)

The rst two terms on the right-hand side bring in the onne tion oeÆ ients (C.1.16), whereas the ommutator an be rewritten in terms of the obje t of anholonomity, see (A.2.36). A

ordingly, we nd for the omponents of the torsion T  =









+ C  :

(C.1.43)

tor omps

The torsion 2-form T  allows to de ne the 1-form e T . If added to the onne tion 1-form  , it is again a onne tion. We all it the transposed onne tion _ 

:=





+ e T = =




+ T  #
 + C # ;





(C.1.44)

sin e in a natural frame, i.e. for C = 0, the indi es of the _

omponents of  are transposed with respe t to that of  .

trans onn

244

C.1.

Linear onne tion

u(1) u(0)

.

P

Figure C.1.2: On the geometri al interpretation of urvature: parallel transport of a ve tor around a losed loop. The urvature of a onne tion r is a map that assigns to ea h pair of ve tor elds u and v a linear transformation R(u; v ) : Xx ! Xx of the tangent spa e at an arbitrary point x by

R(u; v )w := ru rv w

rv ru w r u;v w : [

℄

(C.1.45)

def urv

One an verify the tensor hara ter of the urvature so that the value of R(u; v )w at any given point depends only on the values of ve tor elds u, v , and w at that point. Analogously to torsion, we an expand the four ve tor elds R(u; v )e with respe t to a lo al ve tor frame,

R(u; v ) e = R (u; v ) e :

(C.1.46)

urvexp

Thereby the urvature 2-form R is de ned. Similarly to (C.1.41), we may express these 2-forms with respe t to a oframe # or a

oordinate frame dxi as follows: 1 1 R = R # ^ # = Rij dxi ^ dxj : (C.1.47) urv2form 2 2 Taking (C.1.46) with respe t to a frame, one nds the omponents of the urvature 2-form,

R(e ; e )e = R e and R(i ; j )e = Rij e :

(C.1.48)

C.1.5

Torsion and urvature

245

Thus, by (C.1.45), R =  

  +   









+ C   (C.1.49)

urv2

and Rij = i j

j i + i j 

j



i

:

(C.1.50)

urv omp

Here we introdu ed the abbreviation  := ei  i . The urvature 2-form R an be ontra ted by means of the frame e . In this way we nd the Ri

i 1-form Ri  := e R = Ri  # :

(C.1.51)

ri

i1

(C.1.52)

ri

i2

Using (C.1.47), we immediately nd Ri  = R  :

The geometri al meaning of the urvature is revealed when we

onsider a parallel transport of a ve tor along a losed urve in X , see Fig. C.1.2. Let  : fxi (t)g; 0  t  1; be a smooth urve whi h starts and ends at a point P = xi (0) = xi (1) [in other words,  is a 1- y le℄. Taking a ve tor u(0) at P and transporting it parallelly along  [whi h te hni ally redu es to the solution of a dierential equation (C.1.36)℄, one nds at the return point xi (1) a ve tor u(1) whi h diers from u(0) . The dieren e is determined by the urvature,
u

=

u

(1)

u

(0)

=

Z

R  u ;

(C.1.53)

S

where S is the two-dimensional surfa e whi h is bounded by  , i.e.,  = S . When the urvature is zero everywhere in X , we all su h a manifold a at aÆne spa e with torsion (or a teleparallelism spa etime). If the torsion vanishes additionally, we speak of a

at aÆne spa e. AÆne means that the onne tion r is still there and it allows to ompare tensors at dierent points. Clearly, the urvature is

int urv

246

C.1.

Linear onne tion

vanishing for an everywhere trivial onne tion form  = 0. However, as we know, the omponents  depend on the frame eld and, in general, if we have urvature, it may happen to be impossible to hoose frames e in su h a way that  = 0 on the whole X . On a at aÆne spa e, the omponents of a ve tor do not hange after a parallel transport around a losed loop, in other words, parallel transport is integrable. Curvature is a measure of the deviation from the at ase. C.1.6

Cartan's geometri interpretation of torsion and urvature

On a manifold with a linear onne tion, the notion of a position ve tor

an be de ned along a urve. If we transport the position ve tor around an in nitesimal losed loop, it is subje t to a translation and a linear transformation. The translation reveals the torsion and the linear transformation the urvature of the manifold.

Let us start, following Cartan, with the at aÆne spa e, in whi h a onne tion r has zero torsion and zero urvature. In su h a manifold, we may de ne an aÆne position ve tor eld (or radius ve tor eld) r as one that satis es the equation

r r=v

(C.1.54)

v

affinetrans

for all ve tor elds v. With respe t to a lo al oordinates fx g, r = r i , and (C.1.54) is a system of sixteen partial dierential equations for the four fun tions ri (x), namely i

i

D rj = dxi ; or i rj (x) + ik j (x) rk (x) = Æij : (C.1.55) In at aÆne spa e, a oordinate basis an always be hosen, at least within one hart, in su h a way that ik j = 0, and then the equation (C.1.54) or (C.1.55) is simply j ri = Æji . The solution is ri = xi + Ai , where Ai a onstant ve tor, so that ri is, indeed, the position (or radius) ve tor of xi with respe t to an origin xi = Ai . In an aÆne spa etime, the integrability ondition for (C.1.54) is rv rw r rw rv r r v;w r = rv w rw v [v; w℄ ; (C.1.56) or R(v; w)r T (v; w) = 0 ; (C.1.57) for all ve tor elds v; w. Hen e a suÆ ient ondition for the existen e of global radius ve tor elds r is R(v; w) = 0 and T (v; w) = 0 (C.1.58) for arbitrary v; w, i.e. the vanishing urvature and torsion. In a general manifold, when torsion and urvature are non{zero, the position ve tor eld does not exist on X . Nevertheless, it is possible to de ne a position ve tor along a

urve. This obje t turns out to be extremely useful for revealing the geometri al meaning of torsion and urvature. [

℄

affinetrans1

C.1.6

Cartan's geometri interpretation of torsion and urvature

x1

247

r+∆ r p’ r+∆ p ∆p r 2 x p

Figure C.1.3: Cartan's position ve tor on manifold with urvature and torsion: AÆne transport of a ve tor r around an in nitesimal loop. Here p and p0 denote the initial and nal positions of the position ve tor in the aÆne tangent spa e. Let p 2 X be an arbitrary point, and  = fxi (t)g; t  0, be a smooth urve whi h starts at p, i.e. xi (0) = xip . We now atta h at p an aÆne tangent spa e, or, to put it dierently, we will onsider the tangent spa e Xp at p as an n-dimensional aÆne ve tor spa e. Re all that in an aÆne ve tor spa e, ea h element (ve tor) is given by its origin and its omponents to some xed basis. We will onstru t Cartan's position ve tor as a map whi h assigns to ea h point of a urve  a ve tor in the aÆne tangent spa e Xp . Geometri ally su h a onstru tion an be onveniently understood as a generalized development map whi h is de ned by `rolling' the tangent spa e along a given urve. The de ning equation of the position ve tor is again (C.1.54), but this time v is not an arbitrary ve tor eld but tangent to the urve under onsideration, i.e. v = (dxi =dt)i . Substituting this into (C.1.54), we get h i dxi dr  (t) = ei (x(t)) : (C.1.59) i  (x(t)) r (t) dt

dt

With the growing of t, one `moves' along the urve  and the fun tions r (t) always des ribe the omponents of the position ve tor in the aÆne tangent spa e at the xed original point p. Thus, for example, a displa ement along the urve from xi to xi + d i yields the hange of the position ve tor dr 

= ei  (x) d i

i  (x)r (x) d i :

(C.1.60)

hangeu

Following Cartan, we may interpret this equation as telling us that the position ve tor map onsists of a translation ei d i and a linear transformation i  d i r in the aÆne tangent spa e at p. Let us now onsider a losed urve , i.e. su h that xi (1) = xip , see Fig. C.1.3. Then, on integrating around , it is found that the total hange in r is given by
r  =

Z

S

(T 

R  r )

= (Tij 

Rij  r )

Z

S

dxi

^ dxj ;

(C.1.61)

total hange

248

C.1.

Linear onne tion

where S is the two{dimensional in nitesimal surfa e en losed by . Thus, in going around the in nitesimal losed loop , the position ve tor r in the aÆne tangent spa e at p undergoes a translation and a linear transformation, of the same order of magnitude as the area of S . The translation is determined by the torsion Z
p = T  ; (C.1.62) S

whereas the linear transformation is determined by the urvature [it is instru tive to

ompare this with the hange of a ve tor under the parallel transport (C.1.53)℄. C.1.7

Covariant exterior derivative If an exterior form is generalized to a tensor-valued exterior form, then the usual de nition of the exterior derivative an be naturally extended to the ovariant exterior derivative. Covariant exterior derivatives of torsion and urvature are involved in the two Bian hi identities.

Torsion 2-form (C.1.39), and (C.1.41) and urvature 2-form (C.1.46), and (C.1.47) are examples of tensor-valued p-forms, that is, of generalized geometri quantities. For su h obje ts we need to introdu e the notion of ovariant exterior derivative whi h shares the properties of a ovariant derivative of a geometri quantity and of an exterior derivative of a s alar-valued form. Let 'A be an arbitrary p-form of type . It an be written as sum of de omposable p-forms of type , namely 'A = wA ! where wA is a s alar of type  and ! a usual exterior p-form. For su h a form we de ne D'A

:= (rwA) ! + wA d!

(C.1.63)

and extend this de nition by R-linearity to arbitrary p-forms of type . Using (C.1.29), it is straightforward to obtain the general formula D'A

= d'A + B A 





^ 'B

(C.1.64)

Drhomega

Covariant

C.1.7

exterior derivative

249

and to prove that D satis es the Leibniz rule D('A ^ B ) = D'A ^ B + (

1)p 'A ^ D

B;

(C.1.65)

where p is the degree of 'A . Unlike the usual exterior derivative, whi h satis es dd = 0, the ovariant exterior derivative is no longer nilpotent: DD'A

= B A  R ^ 'B :

(C.1.66)

ri

ident

The simplest proof makes use of the normal frame (C.1.23) in whi h D'A = d'A , R = d  . We hoose a normal frame and dierentiate (C.1.64). Sin e the resulting formula is an equality of two (p + 2)-forms of type , it holds in an arbitrary frame. The relation (C.1.66) is alled the Ri

i identity. Now we an appre iable simplify all al ulations involving frame, onne tion, urvature, and torsion. At rst, noti ing that the oframe # is a 1-form of the ve tor type, we re over the torsion 2-form (C.1.39), (C.1.41) as a ovariant exterior derivative T

= D# = d# +





^ # :

(C.1.67)

stru ture1

This equation is often alled the rst (Cartan) stru ture equation. Applying (C.1.66), we obtain the 1st Bian hi identity: DT

= R  ^ # :

(C.1.68)

bian hi1

Analogously, after re ognizing the urvature 2-form as a generalized 2-form of tensor type [11 ℄, we immediately rewrite (C.1.46), (C.1.47) as R

=d





+





^

  ;

(C.1.69)

stru ture2

whi h is alled the se ond (Cartan) stru ture equation. Using again the tri k with the normal frame, we obtain the 2nd Bian hi identity: DR

=0 :

(C.1.70)

bian hi2

250

C.1.

Linear onne tion

Finally, we an link up the notions of Lie derivative and ovariant derivative. For de omposable p-forms of type , 'A = wA ! , where wA is a s alar of type  and ! a p-form, we de ne the

ovariant Lie derivative as Lu 'A := (ru wA) ! + wALu !

(C.1.71)

and extend this de nition by R-linearity to arbitrary p-forms of type . It is an interesting exer ise to show that Lu 'A = u D'A + D(u 'A ) : C.1.8

(C.1.72)

The p-forms o(a), onn1(a,b), torsion2(a),

urv2(a,b)

We ome ba k to our Ex al programming. We put n = 4. On ea h dierential manifold, we an spe ify an arbitrary oframe eld # , in Ex al o(a). Ex al is made familiar with o(a) by means of the oframe statement as des ribed in Se . A.2.11. Moreover, sin e we introdu ed a linear onne tion 1-form  , we do the same in Ex al with pform onn1(a,b)=1; then it is straightforward to implement the torsion and urvature 2-forms T  and R by means of the stru ture equations (C.1.67) and (C.1.69), respe tively: pform torsion2(a)=2, urv2(a,b)=2; % pre eded by oframe ommand torsion2(a):=d o(a) + onn1(-b,a)^o(b);

urv2(-a,b):=d onn1(-a,b) + onn1(-a, )^ onn1(- ,b);

In Ex al , the tra e of the torsion T := e T  reads e(-a) j torsion2(a), and the orresponding tra e part of the torsion T  = 13 # ^ T be omes pform trator2(a); trator2(a) :=o(a)^(e(-a) _|torsion2(a));

The Ri

i 1-form is en oded as

ovariantLie

C.1.8

The p-forms o(a), onn1(a,b), torsion2(a), urv2(a,b)

251

pform ri

i1(a)=1; ri

i1(-a) :=e(-b) _| urv2(-a,b);

Weyl's (purely non-Riemannian) dilational (or segmental) urvature 2-form 41 Æ R is the other generally ovariant ontra tion of the urvature. We have pform delta(a,b)=0, dil urv2(a,b)=2; delta(-0,0:=delta(-1,1):=delta-3,3):=delta(-1,1):=1; dil urv2(-a,b):=delta(-a,b)* urv2(- , )/4;

These are the quantities whi h play a role in a 4-dimensional dierential manifold with a pres ribed onne tion. The orresponding Ex al expressions de ned here an be put into an exe utable Ex al program. However, rst we want to get a

ess to a possible metri of this manifold.

252

C.1.

Linear onne tion

C.2

Metri

Although in our axiomati dis ussion of ele trodynami s in Part B we adhered to the onne tion-free and metri -free point of view, the notions of onne tion and metri are unavoidable in the end. In the previous Chapter C.1, we gave the fundamentals of the geometry of a manifolds equipped with a linear onne tion. Here we dis uss the metri . In Spe ial Relativity theory (SR) and in the orresponding

lassi al eld theory in at spa etime, the Lorentzian metri enters as a fundamental absolute element. In parti ular, all physi al parti les are de ned in terms of representations of the Poin are (or inhomogeneous Lorentz) group whi h has a metri built in from the very beginning. In General Relativity theory (GR), the metri eld is upgraded to the status of a gravitational potential. In parti ular, the Einstein eld equation is formulated in terms of a Riemannian metri with Lorentz signature arrying on its right-hand side the symmetri (Hilbert or metri ) energy-momentum tensor as a material sour e. The physi al signi an e of the spa etime metri lies in the fa t that it determines intervals between

254

C.2.

Metri

events in spa etime ds2 , and, furthermore, establishes the ausal stru ture of spa etime. It is important to realize that the two geometri al stru tures | onne tion and metri | a priori are absolutely independent from ea h other. Modern data onvin ingly demonstrate the validity of Riemannian geometry and Einstein's GR on ma ros opi s ales where mass (energy-momentum) of matter alone determines the stru ture of spa etime. However, at high energies, the properties of matter are signi antly dierent, with additional spa etime related hara teristi s, su h as spin and s ale harge oming into play. Correspondingly, one an expe t that the geometri stru ture of spa etime on small distan es may deviate from Riemannian geometry. \In the dilemma whether one should as ribe to the world primarily a metri or an aÆne stru ture, the best point of view may be the neutral one whi h treats the g 's as well as the 's as independent state quantities. Then the two sets of equations, whi h link them together, be ome laws of nature without attributing a preferential status as de nitions to one or the other half." 1

C.2.1

Metri ve tor spa es

A metri tensor introdu es the length of a ve tor and an angle between every two ve tors. The omponents of the metri are de ned by the values of the s alar produ ts of the basis ve tors.

Let us onsider a linear ve tor spa e V . It is alled a metri ve tor spa e if on V a s alar produ t is de ned as a bilinear 1 \In dem Dilemma, ob man der Welt urspr ungli h eine metris he oder eine aÆne Struktur zus hreiben soll, ist viellei ht der beste Standpunkt der neutrale, der sowohl die g wie die als unabhangige Zustandsgroen behandelt. Dann werden die beiden Satze von Glei hungen, wel he sie verbinden, zu Naturgesetzen ohne da die eine oder andere Halfte als De nitionen eine bevorzugte Stellung bekommen." H. Weyl: 50 Jahre Relativit atstheorie [9℄, our translation.

C.2.1

Metri ve tor spa es

255

symmetri and non-degenerate map

g:

V

 V ! R:

(C.2.1)

In other words, a s alar produ t is introdu ed by a metri tensor g of type [02 ℄ whi h is symmetri , i.e. g(u; v ) = g(v; u) for all u; v 2 V , and non-degenerate in the sense that g(u; v ) = 0 holds for all v if and only if u = 0. The real number

g(u; u)

(C.2.2)

is alled a length of a ve tor u. The metri g de nes a anoni al isomorphism of the ve tor spa e and its dual,

g~ :

V

! V ;

(C.2.3)

isoVV

(C.2.4)

gtilde

where the 1-form g~ (u), if applied to a ve tor v , yields

g~ (u) (v ) := g(u; v );

for all v 2 V

:

Alternatively, we may write g~ (u) = g(u;
). In terms of a basis e of V and the dual basis # of

g = g # # ;

with

g

:= g(e ; e ) = g  :

V ,

(C.2.5)

met oeff

Thus the isomorphism (C.2.3) is te hni ally redu ed to the verti al motion of indi es,

g~ (e ) (e ) = g(e ; e ) = g = g Æ = g # (e ) : (C.2.6) A

ordingly, the basis ve tors de ne the 1-forms via

g~ (e ) = g # =: # :

(C.2.7)

gtildinv

Under a hange of the basis (A.1.5), the metri oeÆ ients g transform a

ording to (A.1.11). Re all that a symmetri matrix an always be brought into a diagonal form by a linear transformation. A basis for whi h g

= diag (1; : : : ; 1; 1; : : : ; 1)

(C.2.8)

orthomet

256

C.2.

Metri

is alled orthonormal. We will mainly be interested in 4-dimensional spa etime. Its tangent ve tor spa e at ea h event is Minkowskian. Therefore, from now on let us take V to be a 4dimensional Minkowskian ve tor spa e, unless spe i ed otherwise. The omponents of the metri tensor with respe t to an orthonormal basis are then given by 

g

=



o

01 B0 := B 0 0

C.2.2

0 1 0 0

0 0 1 0

0 0 0 1

1 CC A

:

Orthonormal, half-null, and null

(C.2.9)

oij

frames,

the oframe statement

A null ve tor has zero length. A set of null ve tors is in many ases a onvenient tool for the onstru tion of a spe ial basis in a Minkowski ve tor spa e.

The Minkowski metri has many interesting `fa es' whi h we will mention here only brie y. Traditionally, in relativity theory the ve tors of an orthonormal basis are labeled by 0; 1; 2; 3, thus underlining the fundamental dieren e between e0 , whi h has a positive length g00 = g (e0 ; e0 ) = 1, and ea , a = 1; 2; 3, whi h have negative length gaa = g(ea ; ea ) = 1. In general, a ve tors u 2 V is

alled time-like if g(u; u) < 0, spa e-like if g(u; u) > 0, and null if g(u; u) = 0. In Ex al one spe i es the oframe as the primary quantity. If we use Cartesian oordinates, an orthonormal oframe and frame in Minkowski spa e read, respe tively,

oframe o(0) o(1) o(2) o(3) metri g frame e;

= = = = =

d t , d x , d y , d z with o(0)*o(0)-o(1)*o(1)-o(2)*o(2)-o(3)*o(3);

C.2.2

Orthonormal, half-null, and null frames, the oframe statement

257

The blank between d and t et . is ne essary! Note that the phrase with metri g=o(0)*o(0)-o(1)*o(1)-o(2)*o(2)-o(3)*o(3); in this ase of a diagonal metri , an also be abbreviated by with signature 1,-1,-1,-1;

We re all that, in Ex al , a spe i spheri ally symmetri

oframe in a 4-dimensional Riemannian spa etime with Lorentzian signature has already been de ned in Se . B.5.6 in our Maxwell sample program. In general relativity, for the gravitational eld of a mass m and angular momentum per unit mass a, one has axially symmetri metri s with oframes like pform rr=0, delsqrt=0, ffsqrt=0$ fdomain rr=rr(rho,theta),delsqrt=delsqrt(rho), ffsqrt=ffsqrt(theta)$

oframe o(0)=(delsqrt/rr)*(d t-(a0*sin(theta)**2)*d phi), o(1)=(ffsqrt/rr)*sin(theta)*(a0*d t-(rho**2+a0**2)*d phi), o(2)=(rr/ffsqrt)*d theta, o(3)=(rr/delsqrt)*d rho with signature 1,-1,-1,-1$

Here rr, delsqrt, ffsqrt are fun tions to be determined by the Einstein equation. This is an example of oframe that is a bit more involved. Starting from an orthonormal basis e with respe t to whi h the metri has the standard form (C.2.9), we an build a new frame e = (l; n; e2 ; e3 ) by the linear transformation: 1 1 l = p (e0 + e1 ); n = p (e0 e1 ); (C.2.10) 2 2 and e2 = e2 ; e3 = e3 . The rst two ve tors of the new frame are null: g(l; l) = g(n; n) = 0. Correspondingly, the metri in this half-null basis reads 0

0

0

g



00 B1  := B 0

=h

1 0 0 0 0

0 0 1 0

0 0 0 1

1 CC A

:

(C.2.11)

halfnull

258

C.2.

Metri

In Ex al , again with Cartesian oordinates, we nd

oframe h(0) h(1) h(2) h(3) metri hh

= (d t+d x)/sqrt(2), = (d t-d x)/sqrt(2), = d y, = d z with = h(0)*h(1)+h(1)*h(0)-h(2)*h(2)-h(3)*h(3);

You an onvin e yourself by displayframe; and on nero; hh(-a,-b); that all has been understood by Ex al orre tly. Following Newman & Penrose, we an further onstru t two more null ve tors as the omplex linear ombinations of e2 and e3 : m

p1

=

2

(e2 + i e3 );

m

=

p1

2

(e2

)

i e3 :

(C.2.12)

Here i is the imaginary unit, and overbar means the omplex

onjugation. This transformation leads to the Minkowski metri in a null (Newman-Penrose) basis e = (l; n; m; m):



g

=n

00 B1  := B 0

1 0 0 0 0

0

0 0 0 1

0 0 1 0

1 CC A

(C.2.13)

NPnull

Su h a basis is onvenient for investigating the properties of gravitational and ele tromagneti waves. In Ex al , we have

oframe n(0) n(1) n(2) n(3) metri nn

= = = = =

(d t + 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(0)*n(1)+n(1)*n(0)-n(2)*n(3)-n(3)*n(2);

In the Newman & Penrose frame, we have two real null legs, namely l and n, and two omplex ones, m and m. It may be surprising to learn that it is also possible to de ne the null symmetri frame of D. Finkelstein whi h onsists of four real null

C.2.2

Orthonormal, half-null, and null frames, the oframe statement

259

ve tors. We start from an orthonormal basis e , with g(e ; e ) = o , and de ne the new basis f a

ording to

p

f0 = ( 3 e0 + e1 + e2 + e3 )=2; p f1 = ( 3 e0 + e1 e2 e3 )=2; p f2 = ( 3 e0 e1 + e2 e3 )=2; p f3 = ( 3 e0 e1 e2 + e3 )=2:

(C.2.14)

f2e

Sin e g(f ; f ) = 0 for all , the null symmetri frame onsists solely of real non-orthogonal null-ve tors. The metri with respe t to this frame reads

g

00 B1 = f := B 1

1 0 1 1 1

1 1 0 1

1 1 1 0

1 CC : A

(C.2.15)

The metri (C.2.15) looks ompletely symmetri in all its omponents: Seemingly the time oordinate is not preferred in any sense. Nevertheless (C.2.15) is a truly Lorentzian metri . Its determinant is 3 and the eigenvalues are readily omputed to be 3;

1;

1;

1;

(C.2.16)

whi h shows that the metri (C.2.15) has, indeed, the orre t signature. There is a beautiful geometri al interpretation of the four null legs of the Finkelstein frame. In a Minkowski spa etime, let us

onsider the three-dimensional spa elike hypersurfa e whi h is spanned by (e1 ; e2 ; e3 ). The four points whi h are de ned by the spatial parts of the Finkelstein basis ve tors (C.2.14), with

oordinates A = (1; 1; 1), B = (1; 1; 1), C = ( 1; 1; 1), and D = ( 1; 1; 1), form a perfe t tetrahedron in the 3-subspa e. Thepverti es A, B , C , and D lie at the same distan es of equal to 3 from the origin O = (0; 0; 0) (and orrespondingly all p sides of this tetrahedron have equal length, namely 8). If we

finkmat

260

C.2.

Metri

D

O

A

C

B Figure C.2.1: Tetrahedron whi h de nes Finkelstein basis

p

now send, at the moment t = 0, a light pulse from the origin O , it rea hes all four verti es of the tetrahedron at t = 3.

Thus four light rays provide the operational de nition for the light-like Finkelstein basis (C.2.14). C.2.3

Metri volume 4-form Given a metri , a orresponding orthonormal oframe determines a metri volume 4-form on every ve tor spa e.

Let # be an orthonormal oframe in the four-dimensional Minkowski ve tor spa e (V  ; g). Let us de ne the produ t  = #0

^ #1 ^ #2 ^ #3 :

(C.2.17)

If # is another orthonormal oframe in (V  ; g), then 0

# = L  # ; 0

(C.2.18)

0

where the transformation matrix is (pseudo)orthogonal, i.e., o L  L = o 0

0

0

0




and L := det L  = 1 : (C.2.19) 0

eta4

C.2.3

Metri volume 4-form

261

Therefore, under this hange of the basis, we have 00

#

^ #1 ^ #2 ^ #3 0

0

0

= L #0 ^ #1 ^ #2 ^ #3 :

(C.2.20)

In other words, the de nition (C.2.17) gives us a unique (i.e. basis independent) twisted volume 4-form of the Minkowski ve tor spa e. Alternatively, one may onsider two non-twisted volume 4-forms separately for ea h orientation. If # is an arbitrary (not ne essarily orthonormal) frame, then we have # = L  # and { sin e  is twisted { 0

0

0



= jLj #0

0

^ #1 ^ #2 ^ #3 : 0

0

(C.2.21)

0

On the other hand e = L  e . Thus, from the tensor transformation g = L  L o , we obtain jLj2 = det (g ). Hen e the twisted volume element with respe t to the frame e reads 0

0

0

0

0

0

0

0

0



=

q

det (g ) #0 0

0

0

^ #1 ^ #2 ^ #3 : 0

0

0

(C.2.22)

eta

Dropping the primes in (C.2.22), we may write, for any basis # , the metri volume 4-form as 

:=

1 

Æ  # ^# ^# ^# 4!  Æ

;

(C.2.23)

eta1

where the twisted antisymmetri tensor  Æ of type [04 ℄ is de ned by  Æ

:=

q

det (g ) ^ Æ ;

(C.2.24)

and ^ Æ is the Levi-Civita permutation symbol with ^0123 = +1. If we raise the indi es in the usual way, we nd the ontravariant omponents as 

 Æ

= g  g  g  g Æ  =

p

1  Æ : (C.2.25)  det (g )

It follows from (C.2.25) that #



^ # ^ # ^ #Æ =



 Æ  :

(C.2.26)

eta2

262

C.2.4

C.2.

Metri

Duality operator for 2-forms as a symmetri almost omplex stru ture on

M

6

The duality operator grows out of the almost omplex stru ture

J

on the

M6

when

J

is additionally self-

adjoint with respe t to the natural metri on the

M 6.

Let us now turn again to the spa e M 6 of the 2-forms in four dimensions whi h we dis ussed in Se . A.1.10. When an almost omplex stru ture J is introdu ed in M 6 in su h a way that it is additionally symmetri , i.e. self-adjoint, with respe t to the natural metri " (A.1.89) on M 6 , then a linear operator on 2-forms # : 2 V 

! 2 V  ;

(C.2.27)

dualdef1

de ned by means of # = J, is alled the duality operator on M 6 . In view of (A.1.109), the duality operator satis es ## = 1:

(C.2.28)

This will be alled the losure relation of the duality operator. The self-adjointness with respe t to the 6-metri (A.1.89) means "(# !; ') = "(!; # ')

(C.2.29)

for all !; ' 2 M 6 . The expli it a tion of the duality operator on the basis 2forms reads # # ^ #
= 1 #  (# ^ # ) ; (C.2.30) 2 where the elements of the matrix #  are, by de nition, the

omponents of the almost omplex stru ture in M 6 . In the 6dimensional notation introdu ed in Se . A.1.10, the duality operator is thus des ribed by # B K = #I K B I :

(C.2.31)

symsharp

sharp2B

sharp4

C.2.4

Duality operator for 2-forms as a symmetri almost omplex stru ture on

Expressed in terms of omponents, the metri reads "(!; ') = "IJ !I 'J , see (A.1.90). Therefore, the self-adjointness an also be written as "IJ #I K !K 'J = "IJ #J K !I 'K , or

o IJ



o JI

=

o IJ

for



:= "IK #K J :

(C.2.32)

hisym

By onstru tion, the omponents of the duality matrix an be expressed in terms of the 3  3 blo ks imported from (A.1.110), #I = K



C ab Bab



Aab C ba

(C.2.33)

:

sharpIJ

Here the omponents are onstrained by the self-adjointness (C.2.29), Aab

= Aba ;

Bab

= Bba ;

C aa

= 0;

(C.2.34)

sharp5

as ompared to the almost omplex stru ture (in parti ular, the D -blo k is expressed in terms of C ). Besides that, the algebrai

ondition (A.1.111) is repla ed by the losure relation Aa B b

+ C a C b =

Æba ;

C (a Ab)

= 0;

= 0: (C.2.35)

C (a Bb)

sharp lose

The existen e of the duality operator has an immediate onsequen e for the omplexi ed spa e M 6 (C ) of 2-forms. As we saw in Se . A.1.11, the almost omplex stru ture provides for a splitting of the M 6 (C ) into the 2 three-dimensional subspa es

orresponding to the i eigenvalues of J. Now we an say even more: These two subspa es are orthogonal to ea h other in the sense of the natural 6-metri (A.1.89): "(!; ') = 0;

for all

!

(s)

(a)

2 M; ' 2 M:

(C.2.36)

The proof is straightforward: "(!; ') = i"(!; #') = i"(# !; ') = "(!; '), where we used the de nitions (A.1.113) and the symmetry property (C.2.29).

ort

M6

263

264

C.2.

Metri

The 6-metri " indu es a metri on the three-dimensional subspa es (A.1.113), turning them into the omplex Eu lidean 3spa es. The symmetry group, whi h preserves the indu ed 3-

(s)

(a)

metri on M (and on M ), is SO(3; C ). This is a group-theoreti origin of the re onstru tion of the spa etime metri from the duality operator: the Lorentz group, being isomorphi to SO(3; C ), is en oded in the stru ture of the self-dual (or, equivalently, antiself-dual) omplex 2-forms on X . The signi an e of the duality operator will be ome learer in the next se tions where we will expli itly demonstrate that # enables us to onstru t a Lorentzian metri on spa etime. C.2.5

From the duality operator to a triplet of

omplex 2-forms Every duality operator on

M6

determines a triplet

of omplex 2-forms that satisfy ertain ompleteness

onditions.

Suppose the duality operator (C.2.27) is de ned in the M 6 with the losure property (A.1.109) and the self-adjointness (C.2.29). Its a tion 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 two-form basis (A.1.81), eq.(C.2.31) is rewritten, with the help of (C.2.33), as

#

   ^

C A B CT

  ^

;

(C.2.37)

dualB

# a = C a b b + Aab ^b ; # ^a = Bab b + C b a ^b :

(C.2.38)

dualuv

=

or, in terms of its matrix elements,

By means of the duality operator # (as well as with J of the almost omplex stru ture), one an de ompose any 2-form into

C.2.5

From the duality operator to a triplet of omplex 2-forms

a self-dual and an basis, this reads,

anti-self-dual

265

part. In terms of the 2-form

(s) (a) BI = B I + B I ;

(C.2.39)

where we de ne

(s)I 1 I B := (B 2 (a)I 1 I B := (B + 2

i #BI );

(C.2.40)

s b

i #BI ):

(C.2.41)

a b

(C.2.42)

sa b

Here i is the imaginary unit. One an he k that

(s) # (s) B I = +i B I ; (a) # (a) B I = i B I:

Thus the 6-dimensional spa e of omplex 2-forms M 6 (C ) de omposes into two 3-dimensional invariant subspa es whi h orrespond to the two eigenvalues i of the duality operator. In order to onstru t the bases of these subspa es, we have to inspe t the 3+3 representation (A.1.81). One has, using (A.1.81) in (C.2.40), the two sets of the self-dual 2-forms,

(s)a

1 = ( a 2 (s) 1  a = (^a 2

i # a );

(C.2.43)

i #^a );

(C.2.44)

and similarly for the anti-self-dual forms. With the help of (C.2.38), we nd expli itly:

(s)a

1 a (Æ 2 b (s) 1  a = (Æab 2

=



iC a b ) b

iAab ^b ;

iC b a ) ^b

iBab b :



(C.2.45)

Sbet

(C.2.46)

Sgam

Sin e the invariant subspa e of the self-dual forms is 3-dimensional, these two triplets of self-dual forms annot be independent

266

C.2.

Metri

from ea h other. Indeed, let us multiply (C.2.45) with B a and (C.2.46) with C a . Then the sum of the resulting relation, with the help of the losure property (C.2.35), yields (s)

Ba a

(s)

a = (iÆ

C a )  a :

(C.2.47)

This shows that these two triplets are linearly dependent. For the non-degenerate (s)

a

B -matrix,

(s)

one an express

a

in terms of

expli itly.

Let us ompute the exterior produ ts of the triplets (C.2.45) and (C.2.46): (s)

(s)

a ^ b = a^ b =

(s)

(s)

1 2 1 2

iAab + C (a Ab) iBab + C (a Bb)





Vol = Vol =

i 2

i

2

Aab Vol;

(C.2.48)

Bab Vol:

(C.2.49)

We used here the algebra (A.1.85)-(A.1.87) and the losure property (C.2.35). Furthermore, we have (s)

(s)

a ^ b =

1

C (a Ab) Vol = 0;

(C.2.50)

 a ^  b = C (a Bb) Vol = 0:

(C.2.51)

(s)

(s)

2 1

2

Here the overbar denotes omplex onjugate obje ts.

C.2.6

From the triplet of omplex 2-forms to a duality operator Every triplet of omplex 2-forms with the ompleteness property determine a duality operator on

M 6.

Both (C.2.45) and (C.2.46) are parti ular representations of the following general stru ture: Given is a triplet of self-dual (a)

omplex 2-forms S su h that

S (a) ^ S (b) = 2i Gab Vol; S (a) ^ S (b) = 0;

(C.2.52)

SSG

(C.2.53)

SOS

C.2.6

From the triplet of omplex 2-forms to a duality operator

267

where the matrix Gab is real and non-degenerate. Overbar denotes omplex onjugate obje ts. Equivalently, one an rewrite (C.2.52) as

S (a) ^ S (b) =

1 ab G G d S ( ) ^ S (d) ; 3

(C.2.54)

where Gab denotes the matrix inverse to Gab . We will all (C.2.52), (C.2.53) the ompleteness onditions for a triplet of 2-forms. In the previous se tion we saw that every duality operator de nes a triplet of self-dual omplex 2-forms. Here we show that the onverse is also true: Let S (a) be an arbitrary triplet of omplex 2-forms whi h satis es the ompleteness onditions (C.2.52), (C.2.53). Then they determine a duality operator in M 6. Expanding the arbitrary 2-forms with respe t to the basis (A.1.81), we an write

S (a) = M a b b + N ab ^b :

(C.2.55)

matS

In view of (A.1.85)-(A.1.87), we nd that (C.2.52) imposes an algebrai onstraint on the matrix omponents,

M a N b + M b N a = 2Gab :

(C.2.56)

SSG0

Introdu ing the real variables

M ab = V ab + i U a b;

N ab = X ab + iY ab ;

(C.2.57)

one an de ompose (C.2.56) into the two real equations:

V (a X b) U (a Y b) = 0; V (a Y b) + U (a X b) = Gab :

(C.2.58) (C.2.59)

SSG1 SSG2

Analogously, (C.2.53) yields another pair of the real matrix equations:

V (a X b) + U (a Y b) = 0; V [a Y b℄ U [a X b℄ = 0:

(C.2.60) (C.2.61)

SOS1 SOS2

268

C.2.

Metri

Combining (C.2.58) and (C.2.60), we nd

V (a X b) = 0;

U (a Y b) = 0;

(C.2.62)

VWXY1

(C.2.63)

VWXY2

whereas the sum of (C.2.59) and (C.2.61) gives

V a Y b + U b X a = Gab :

We an ount the number of independent degrees of freedom. The total number of variables is 36 (= 4  9 unknown omponents of the matri es V; U; X; Y ). They are subje t to 21 onstraints (=6 + 6 + 9) imposed by the equations (C.2.62) and (C.2.63). Thus, in general, 15 degrees are left over for the unknown matri es in (C.2.55). In order to see how the triplet is related to the duality operator, let us de ne a new basis in the M 6 by means of the linear transformation



a ^a 0

0

  =

V ab Uab

X ab Ya b



b ^b



;

(C.2.64)

B2B-S

where Uab := Ga U b and Ya b := Ga Y b . The new basis elements satisfy the same algebrai onditions

a ^ b = 0; 0

0

^a ^ ^b = 0; 0

0

^a ^ b = Æab Vol 0

0

(C.2.65)

as those in (A.1.85)-(A.1.87). The proof follows dire tly from (C.2.62) and (C.2.63). Interestingly, the transformation (C.2.64) is always invertible, with the inverse given by



a ^a

  =

Yb a X ba Uba V b a



b ^b 0

0



:

(C.2.66)

B2B-I

The dire t he k again involves only the ompleteness onditions (C.2.62) and (C.2.63). The original triplet (C.2.55), with respe t to the new basis (C.2.64), then redu es to

S (a) = a 0

iGab ^b : 0

(C.2.67)

matSnew

C.2.7

From a triplet of omplex 2-forms to the metri : S h onberg-Urbantke formulas

Now we are prepared to introdu e the duality operator. We de ne it by simply postulating that its a tion on the triplet amounts to a mere multipli ation by the imaginary unit, i.e.,

#S (a) = i S (a)

hen e




# S (a) =

i S (a) :

(C.2.68)

From (C.2.67) we have a = (S (a) + S (a) )=2 and ^a = iGab (S (a) S (a) )=2. Then, immediately, we nd the a tion of the duality operator on the new 2-form basis: 0

0

#



a ^a 0

0





=

0 Gab Gab 0



b ^b 0

0



:

(C.2.69)

newdual

We an now re onstru t the original basis in (C.2.37). We use (C.2.66) and (C.2.64) and nd the 3  3 matri es

Aab = G d Y a Y bd B ab = G d U a U d b C a b = G d U b Y da




X a X bd ;
V aV db ;
V b X da :

(C.2.70) (C.2.71) (C.2.72)

As a onsequen e, every triplet of omplex 2-forms, whi h satisfy the ompleteness onditions (C.2.52) and (C.2.53), de nes a duality operator with the losure and the symmetry properties. C.2.7

From a triplet of omplex 2-forms to the metri : S h onberg-Urbantke formulas The triplet of omplex 2-forms is a building material for the metri of spa etime. The Lorentzian metri , up to a s ale fa tor, an be onstru ted from the triplet of 2-forms by means of the S h onberg-Urbantke formulas.

The importan e of the duality operator # and the orresponding triplet of 2-forms lies in the fa t that they determine a Lorentzian metri on 4-spa e. Let us formulate this result. Let a triplet of two-forms S (a) be given on V whi h satisfy (C.2.52) with some symmetri regular

A3S B3S C3S

269

270

C.2.

Metri

matrix G. Then the Lorentzian metri of spa etime is re overed with the help of the S honberg-Urbantke formulas: 16 p (a) (b) ( ) det g gij = det G klmn ^ab Sik Slm Snj ; (C.2.73) 3 1 ( ) (d) det g = klmn G d Skl Smn : (C.2.74) 24

p

p

urbantke1 urbantke2

Here, klmn is the Levi-Civita symbol, and Sij(a) are the omponents of the basis two-forms with respe t to the lo al oordinates fxig, i.e., S (a) = 12 Sij(a) dxi ^ dxj . Despite the appearan e of the Levi-Civita symbols in (C.2.73)-(C.2.74), these expressions are tensorial. A rigorous proof of the fa t that on a 4-dimensional ve tor spa e V every three omplex two-forms S (a) , whi h satisfy the

ompleteness ondition (C.2.52), de ne a (pseudo-)Riemannian metri will not be presented here2 . The metri is, in general,

omplex, however it is real when (C.2.53) is ful lled. Note that the sign in the algebrai equations (A.1.109) and (C.2.35) is important. If, instead of the minus, there appeared a plus, then the resulting spa etime metri would have a Riemannian (Eu lidean) signature (+1; +1; +1; +1) or a mixed one (+1; +1; 1; 1). As a omment to the S honberg-Urbantke me hanism, we would like to mention the lo al isomorphism of the three following omplex Lie groups (symple ti , spe ial linear, and orthogonal): Sp(1; C )

 SL(2; C )  SO(3; C ):

(C.2.75)

The easiest way to see this is to analyze the orresponding Lie algebras. At rst, re all that the orthogonal algebra so(3; C )

onsists of all skew-symmetri 3  3 matri es with omplex elements: 0 q3 q2 q3 0 q1 : a= (C.2.76) q2 q1 0

0 

1 A

2 See, however, S h onberg [7℄, Urbantke [8℄, or Harnett [2℄.

so3mat

C.2.8 Hodge star and Ex al 's #

271

Here q1 ; q2 ; q3 is a triplet of omplex numbers. Sin e the linear algebra sl(2; C ) onsists of all tra eless 2  2

omplex matri es, its arbitrary element an be written as

ea = 21



iq3

iq1

q2

iq1 + q2 iq3



:

(C.2.77)

sl2mat

Assuming that the three omplex parameters q1; q2 ; q3 in (C.2.77) are the same as in (C.2.76), we obtain a map so(3; C ) ! sl(2; C ) whi h is obviously an isomorphism. It is straightforward to he k, for example, that the ommutator [a; b℄ of any two matri es of the form (C.2.76) is mapped into the ommutator [ea; eb℄ of the

orresponding matri es (C.2.77). The symple ti algebra sp(1; C ) onsistsof all 2 2 matri es 0 1 . One an ea whi h satisfy ea s + s eaT = 0, with s := 1 0 easily verify that any matrix (C.2.77) satis es this relation, thus proving the isomorphism sp(1; C ) = sl(2; C ).

C.2.8 Hodge star and Ex al 's # In a metri ve tor spa e, the Hodge operator establishes an map between p-forms and (n p)-forms. Besides the #-basis for exterior forms we an de ne  -basis whi h is Hodge dual to the #-basis.

Consider an n-dimensional ve tor spa e with metri g. Usually (when an orientation is xed), the Hodge star is de ned as a linear map ? : pV  ! n pV , su h that for an arbitrary p-form ! and for an arbitrary 1-form ' it satis es ?

(! ^ ') = g~ 1(')

?

!:

(C.2.78)

The formula (C.2.78) redu es the de nition of a Hodge dual for an arbitrary p-form to the de nition of a 4-form dual to a number ?1. Usually, ? 1 is taken as the ordinary (untwisted) volume form and su h a pro edure distinguishes a ertain orientation

hodge2

272

C.2.

Metri

in V . We hange this onvention and require instead the usual de nition that ? : p V  ! twisted n pV  ; (C.2.79) and vi e versa ? : twisted p V  ! n p V  : (C.2.80) A

ordingly, we put ? 1 equal to the twisted volume form: ?1 =  : (C.2.81) Let us now restri t our attention to the 4-dimensional Minkowski ve tor spa e. We an use (C.2.78) to de ne Hodge dual for an arbitrary p-form. Take at rst ! = 1 and ' = # as oframe 1-form. Then, with (C.2.4), (C.2.81), and (C.2.7), we nd ? # = g ~ 1(# ) ? 1 = ge  = 1  Æ # ^ # ^ #Æ ; 3! (C.2.82) where we used (A.1.49) in order to ompute the interior produ t e  for (C.2.23). We an now go on in a re ursive way. Choosing in (C.2.78) ! = # and again ' = # , we obtain: ? # ^ #
= e ? # : (C.2.83) Here e := g e . Substituting (C.2.82) into (C.2.83) and again using (A.1.49) to evaluate the interior produ t, we nd su

essively the formulas

Æ  ? # = e  = 1   Æ # ^ # ^ # =:  ; 3! (C.2.84)
1 ? # ^ #
= e e  =   Æ # ^ #Æ =:   ; 2! (C.2.85)  

? # ^ # ^ # = e e e  =   Æ #Æ =:   ; (C.2.86) ? # ^ # ^ # ^ #Æ
=   Æ : (C.2.87)

hodge3

hodge4

hodge5

hodge6

hodge9

hodge10

hodge11

hodge12 hodge13

C.2.8 Hodge star and Ex al 's #

273

The newly de ned -system of p-forms, p = 0; : : : ; 4, n o     Æ ;  ;  ;  ;  (C.2.88)

onstitutes, along with the usual #-system, o n 1; # ; # ^ # ; # ^ # ^ # ; # ^ # ^ # ^ #Æ ; (C.2.89)

etasystem

varthetasystem

a new metri dependent basis for the exterior algebra over the Minkowski ve tor spa e. This onstru tion an evidently be generalized to an n-dimensional ase. In the n-dimensional Minkowski spa e, ?? !

= ( 1)p n (

p)+1 !;

for ! 2 pV :

(C.2.90)

Thus, for n = 4, we have ?? ! = ! for exterior forms of odd degree, p = 1; 3, and ??! = ! for forms of even degree, p = 0; 2; 4. From the de nition (C.2.78), we an read o the rules ? ( +

)=

? + ?

and

? (a )

= a ? ;

(C.2.91)

for a 2 R and ; 2 pV . These linearity properties together with (C.2.84)-(C.2.87) enable one to al ulate the Hodge star of any exterior form. The implementation of these stru tures in Ex al is simple. By means of the oframe statement, the metri is put in. Ex al provides the operator # as Hodge star. In ele trodynami s the most prominent role of the Hodge star operator is that it maps, up to a dimensonful fa tor , the eld strength F into the ex itation H , namely H =  ?F , as we will see in the fth axiom (D.5.7). Therefore, in Ex al we simply have ex it2 := lam * # farad2; this spa etime relation is all we need in order to make the Maxwell equations to a omplete system. We used this Ex al ommand already in our Maxwell sample program of Se . B.5.6. As a further example, we study the ele tromagneti energymomentum urrent. We re all that we onstru ted in Se . B.5.2

hodge14

274

C.2.

Metri

in Eq.(B.5.30) the ele tromagneti energy-momentum tensor density in terms of the energy-momentum 3-form. The orresponding tensor we now nd, instead of with } rather by means of the star operator ? . Thus, % defs. of o(a) and maxenergy3(a) pre ede this de laration pform maxenergy0(a,b)=0; maxenergy0(-a,b):= #(o(b)^maxenergy3(-a));

Or, to turn to the -system of (C.2.88). It an be programmed as follows: pform eta0(a,b, ,d)=0,eta1(a,b, )=1, eta2(a,b)=2,eta3(a)=3,eta4=4$ eta4 eta3(a) eta2(a,b) eta1(a,b, ) eta0(a,b, ,d)

:= := := := :=

# 1$ e(a) e(b) e( ) e(d)

_|eta4$ _|eta3(a)$ _|eta2(a,b)$ _|eta1(a,b, )$

We ould de ne  alternatively as eta4:=o(0)^o(1)^o(2)^o(3)$ see (C.2.17). With these tools, the Einstein 3-form now emerges simply as pform einstein3(a)=3; einstein3(-a):=(1/2)*eta1(-a,-b,- )^ urv2(b, );

A

ordingly, exterior al ulus and the Ex al pa kage are really of equal power.

From a metri to the duality operator Let us assume that a metri is introdu ed in a 4-dimensional ve tor spa e V . Then, the Hodge star map (C.2.79), (C.2.80) is de ned for p-forms. Restri ting our attention to 2-forms, we nd that the Hodge star maps M 6 = 2V  into itself. The restri tion #

=?

2 V 

(C.2.92)

sharphodge

C.2.9

Manifold with a metri , Levi-Civita onne tion

275

is obviously a duality operator in M 6 . Re alling the de nitions of Se . A.1.11, we an straightforwardly verify that both, the

losure (A.1.109) and the symmetry (C.2.29), are ful lled by (C.2.92). The matrix of the duality operator is given by g

#  =   :=  g [g  ℄ :

(C.2.93)

sharp3

(C.2.94)

sparpIJ0

Alternatively, in 6-dimensional notation, it reads g

#I J = I J :=

g

!

g

C

A

B

C

g

g

;

T

where a straightforward al ulation yields the 3  3 blo ks g

ab

=

B ab

=

A g

g

a

C b

C.2.9

=

p

g g

1p 4 p 2

00

g

ab

g

0

g g

ad




(C.2.95)

Ahodge

g g

(C.2.96)

Bhodge

a 0d
 : b d

(C.2.97)

Chodge

g

;

de ef
^ a d ^bef ;

e df

g g g g

0a 0b

g g

Manifold with a metri , Levi-Civita

onne tion

The metri on a manifold is introdu ed pointwise as a smooth s alar produ t on the tangent spa es. The Levi-Civita (or Riemannian) onne tion is a unique linear onne tion with vanishing torsion and ovariantly onstant metri .

Let Xn be an n-dimensional dierentiable manifold. We say that a metri is de ned on Xn , if a metri tensor g is smoothly assigned to the tangent ve tor spa es Xx at ea h point x. In terms of the oframe eld, g = g (x) # # ;

where

g (x)

= g(e ; e )

(C.2.98)

is a smooth tensor eld in every lo al oordinate hart. The manifold with a metri stru ture de ned on it is alled a (pseudo)Riemannian manifold, denoted Vn = (Xn ; g). Usually, a metri

metlo al

276

C.2.

Metri

(and, orrespondingly, a manifold) is alled Riemannian, if the g (x) is positive de nite for all x. However, in order to simplify formulations, we will omit the `pseudo' and all Riemannian also metri s with a Lorentzian signature. The metri brings a whole bun h of related obje ts on a manifold Vn . First of all, a metri volume n-form emerges on a Vn : For every oframe eld # = (#^1 ; : : : ; #n^ ) it is de ned by 

:=

p

p = The world

1   1 :::n # 1 ^


^ # n n! (C.2.99) 1 = i1 :::in dxi1 ^


^ dxin : n! (C.2.100)

det g #^1 ^


^ #nb = det gkl dx1 ^


^ dxn metri tensor gij (x)

= i1 :::in (x) =

= ei  (x) ej (x) g (x)

p

ei1

etaNdx

with omponents (C.2.101)

is de ned in every lo al oordinate hart fxi g. The prin ipal dieren e between g and gij is that the former always an be `gauged away' by the suitable hoi e of the frame eld e . One an hoose an orthonormal frame eld, e.g., in whi h g has the diagonal form (C.2.8) independent of lo al oordinates. However, it is impossible in general to hoose the oordinates fxig in su h a way that gij is onstant everywhere on the Vn. The Levi-Civita tensor densities in (C.2.99), (C.2.100) are introdu ed by 1 :::n (x)

etaN

det g 1 :::n ;

1 : : : e n  in 1 :::n

=

p

det gkl i1 :::in ; (C.2.102)

with the numeri al permutation symbol hosen as 1:::n = +1. Hats over numeri al indi es help to distinguish omponents with respe t to lo al frames form omponents with respe t to oordinate frames. The next relative of the metri is the Hodge star operator. It is naturally introdu ed on a Vn pointwise with the help of formulas derived in Se . C.2.8.

metworld

C.2.10 Codierential and wave operator, also in Ex al

277

Finally, the most far-rea hing and non-trivial onsequen e of the metri g is the existen e on a Vn of a spe ial ovariant diere whi h is usually alled a Riemannian or Levi-Civita entiation r

onne tion. We will use tilde to denote this onne tion and any obje ts or operators onstru ted from it. As was shown in Se . C.1.1, a ovariant dierentiation is de ned on a manifold as soon as in every lo al hart the onne tion 1-forms i j are given whi h obey the onsisten y ondition (C.1.11). The Riemannian onne tion is de ned, in ea h lo al oordinate system, by the Christoel symbols eki j : ei j = eki j dxk ; eki j := 1 g jl (i gkl + k gil l gik ) : 2 (C.2.103)

Chris

e has some speThe Levi-Civita (or Riemannian) onne tion r

ial properties whi h distinguish it from other ovariant dierentiations. It has vanishing torsion. This is trivially seen from (C.1.43) and (C.1.18) if we noti e that the Christoel symbols (C.2.103) are symmetri in its lower indi es. Moreover, the ovariant exterior derivative of the metri with respe t to the LeviCivita onne tion vanishes identi ally: e  = dg

Dg

e  g

e  g = dg

2 e ( ) = 0; (C.2.104)

see (C.2.134). A onne tion for whi h the ovariant derivative of the metri is zero, is alled metri - ompatible.

C.2.10

Codierential and wave operator, also in Ex al By means of the Hodge star operator, we an de ne the odierential whi h is adjoint to the exterior differential d with respe t to the s alar produ t on exterior forms. This yields dire tly a wave operator.

Consider the spa e p(X ) of all smooth exterior p-forms on X . The Hodge operator makes it possible to de ne a natural

zeroDg

278

C.2.

Metri

s alar produ t on this fun tional spa e: Z

(!; ') :=

! ^ ';

!; ' 2  (X ): (C.2.105)

for all

?

p

p-s al

X

Then the odierential operator dy an be introdu ed as an adjoint to the exterior dierential d with respe t to the s alar produ t (C.2.105), (!; dy') := (d!; '):

(C.2.106)

By onstru tion, the odierential maps p-forms into (p 1)forms ( ontrary to the exterior dierential whi h in reases the rank of a form by one). Using the property (C.2.90) of the Hodge operator, one an verify that in an n-dimensional Lorentzian spa e the odierential on p-forms is given expli itly by dy = ( 1)

n(p

1) ?

(C.2.107)

d : ?

ddagger

The Leibniz rule for d was used at an intermediate step, and the boundary integral is vanishing due to the proper behavior of the forms at in nity. A

ordingly, in n = 4 we have dy = d for all forms. In lo al oordinates, the odierential of an arbitrary p-form ' = ' 1 p dx 1 ^


^ dx p reads: ?

1

p!

i :::i

dy ' =

i

?

i

1 r e (p 1)! ' j

j i1 :::ip

1

dx 1 ^


^ dx p 1 : i

i

(C.2.108)

e is the ovariant dierentiation for the Levi-Civita onHere r ne tion (C.2.103). The odierential is nilpotent

dy dy = 0

(C.2.109)

whi h follows dire tly from (C.2.107), (C.2.90) and the nilpoten y property d = 0 of the exterior dierential (A.2.19). The operator d (resp., dy) in reases (resp., de reases) the rank of a 2

ddaggerlo

C.2.11

Nonmetri ity

279

form by 1. Hen e, the ombinations ddy and dy d both map pforms into p-forms. However, these operators are not self-adjoint with respe t to the s alar produ t (C.2.105). The symmetrized se ond order dierential operator := dy d + d dy

(C.2.110)

d'Alembertian

is alled the wave operator (or d'Alembertian, also alled Lapla eBeltrami operator). It is, by onstru tion, self-adjoint with respe t to (C.2.105). On one o

asion, we had to he k whether the wave operator, if applied to a ertain oframe eld, vanishes, i.e., #a =? 0. Ex al ould help. After the oframe statement spe ifying the appropriate value for the oframe, we de ned a suitable 1-form: pform waveto oframe1(a)=1$ waveto oframe1(a):= d(#(d(#o(a)))) + #(d(#(d o(a))));

The emerging expression we had to treat with swit hes and suitable substitutions, but the quite messy omputation of the wave operator was given to the ma hine. C.2.11

Nonmetri ity Let a onne tion and a metri be de ned independently on the same spa etime manifold. Then the nonmetri ity is a measure of the in ompatibility between metri and onne tion.

Let us onsider the general ase when on a manifold Xn metri and onne tion are de ned independently. Su h a manifold is denoted (Xn ; r; g) and is alled a metri -aÆne spa etime. Sin e the metri is a tensor eld of type [02 ℄, its ovariant dierentiation yields a type [03 ℄ tensor eld whi h is alled nonmetri ity: Q(u; v; w) := g(ruv; w) + g(v; ruw)

ufg(v; w)g ; (C.2.111)

nonmeti ity

280

C.2.

Metri

for all ve tor elds u; v; w. Nonmetri ity measures the extent to whi h a onne tion r is in ompatible with the metri g. Metri - ompatibility (also alled metri ity) Q(u; v; w) = 0 implies the onservation of lengths and angles under parallel transport. A manifold whi h is endowed with a metri and a metri

ompatible onne tion is said to be a Riemann{Cartan manifold (or a Un ). In general, the Riemann-Cartan manifold has a non-vanishing torsion. When the latter is zero, we re over the Riemannian manifold des ribed in Se . C.2.9. Similarly to the 2-forms of torsion (C.1.39), (C.1.41) and urvature (C.1.46), (C.1.47), we de ne the nonmetri ity 1-form Q = Q  #

(C.2.112)

by Q  = Q (e ) = Q(e ; e ; e ) :

(C.2.113)

nonexp

Sin e u(g ) = dg (u), equation (C.2.111) is equivalent to Q = dg +



+



= Dg ;

(C.2.114)

stru ture0

where  := g  . If the g are onstants, then it follows from (C.2.114) that Q = 2 ( ) . Hen e, in a Un , where Q = 0, we have antisymmetri onne tion one-forms 

=

(C.2.115)

 ;

provided the g are onstants, i.e., with respe t to orthonormal

oframe elds, e.g. We shall refer to (C.2.114) as the 0th Cartan stru tural relation and shall all the expression obtained as its exterior derivative, DQ = dQ



^ Q



^ Q = 2 R  (

)

;

(C.2.116)

the 0th Bian hi identity. The proof of (C.2.116) makes use of the Ri

i identity (C.1.66).

bian hi0

C.2.12

Post-Riemannian pie es of the onne tion

281

It is onvenient to separate the tra e part of the nonmetri ity from its tra eless pie e. Let us de ne a Weyl 1-form (or Weyl

ove tor) by := 1   (C.2.117) The fa tor 1 is onventional. Then the nonmetri ity is de omposed into its deviator % and its tra e a

ording to (C.2.118)  = % +  The tra e of the urvature, whi h is alled the segmental urvature, an be expressed in terms of the Weyl 1-form:

(C.2.119)

= 2 The physi al importan e of the Weyl 1-form is related to the fa t that, during parallel transport, the ontribution of the Weyl 1-form does not ae t the light one, whereas lengths of non-null ve tors are merely s aled with some (path-dependent) fa tor. A spa e with % = 0 is alled a Weyl-Cartan spa e n. In this latter ase, the position ve tor (C.1.61) hanges a

ording to Z 
= (  1   ) (C.2.120) Q

n

Q

g

:

weyl1form

=n

Q

Q

Q

Qg

n

R

:

dQ:

Q

Y

r

T

S

n

R

r

R

[

℄

r

;

if it is Cartan displa ed over a losed loop whi h en ir les . The rst urvature term indu es a dilation, while the se ond one is a pure rotation. S

C.2.12

QQ

Post-Riemannian pie es of the onne tion An arbitrary linear onne tion an always be split into the Levi-Civita onne tion plus a post-Riemannian tensorial pie e alled the distortion. The latter depends on torsion and nonmetri ity. Correspondingly, all the geometri obje ts and operators an be systemati ally de omposed into Riemannian and postRiemannian parts.

artandilat

282

C.2.

Metri

\...the question whether this [spa etime℄ ontinuum is Eu lidean or stru tured a

ording to the Riemannian s heme or still otherwise is a genuine physi al question whi h has to be answered by experien e rather than being a mere onvention to be hosen on the basis of expedien y." The geometri al properties of an arbitrary metri -aÆne spa etime are des ribed by the 2-forms of urvature R and torsion  and by the 1-form of nonmetri ity Q . Parti ular values T  of these fundamental obje ts spe ify dierent geometries whi h may be realized on a spa etime manifold. Physi ally, one an think of a number of `phase transitions' whi h the spa etime geometry undergoes at dierent energies (or distan e s ales). Correspondingly, it is onvenient to study parti ular realizations of geometri al stru tures within the framework of several spe i gravitational models. The overview of these models and of the relevant geometries is given in Fig. C.2.2. The most general gravitational model { metri -aÆne gravity (MAG) { employs the (Ln ; g ) geometry in whi h all three main obje ts, urvature, torsion, and nonmetri ity are non-trivial. Su h a geometry ould be realized at extremely small distan es (high energies) when the hypermomentum urrent of matter elds plays a entral role. Other gravitational models and the relevant geometries appear as spe ial ases when one or several main geometri al obje ts are ompletely or partly \swit hed o". The Z geometry is hara terized by T  = 0 and was used in the uni ed eld theory of Eddington and in so- alled SKY-gravity (theories of Stephenson-Kilmister-Yang). Swit hing o the tra eless nonmetri ity, Q = 0, yields the Weyl-Cartan spa e Y (with torsion) or standard Weyl theory W (with T  = 0). Furthermore, swit hing o the nonmetri ity ompletely, one re overs the Riemann-Cartan geometry U whi h is the arena of Poin are gauge (PG) gravity in whi h the spin urrent of matter, besides its energy-momentum urrent, is an additional sour e of the gravitational eld. The Riemannian geometry V (with 3

4

%

4

4

4

4

3 A.

Einstein:

Geometrie und Erfahrung [1℄, our translation.

C.2.12

Post-Riemannian pie es of the onne tion

283

Rα β

R=0 T=0 Q=0

L4,g

Q=0

Y

MAG

Z4

Eddington SKY

4 WeylCartan

U4 PG

W

4 Weyl

V4 GR

P

+ 4 Teleparallelism +Q

T

α

P

4 Teleparallelism

Qαβ ?4 M4 SR

Figure C.2.2: MAGi ube: Classi ation of geometries and gravity theories in the three \dimensions" (R; T ; Q).

284

C.2.

Metri

Q = 0 and T  = 0) des ribes, via Einstein's General Rela-

tivity (GR), gravitational ee ts on a ma ros opi s ale when the energy-momentum urrent is the only sour e of gravity. Finally, when the urvature is zero, R = 0, one obtains the Weitzenbo k spa e P4 and the teleparallelism theory of gravity (when Q = 0) or a generalized teleparallelism theory in the spa etime with nontrivial torsion and nonmetri ity P4+ . The Minkowski spa etime M4 with vanishing Q = 0; T  = 0, R = 0 underlies Spe ial Relativity (SR) theory. The relations between the dierent theories and geometries are given in Fig. C.2.2 by means of arrows of dierent type whi h spe ify whi h obje t is swit hed o. In a metri -aÆne spa e, urvature, torsion, and nonmetri ity satisfy the three Bian hi identities (C.2.116), (C.1.68), and (C.1.70): DQ = 2R( ) ; DT  = R  ^ # ; DR = 0:

0th Bian hi identity (C.2.121) 1st Bian hi identity (C.2.122) 2nd Bian hi identity (C.2.123)

Bia0 Bia1 Bia2

In pra ti al al ulations, it is important to know exa tly the number of geometri al and physi al variables and their algebrai properties (e.g., symmetries, orthogonality relations, et .). These aspe ts an be lari ed with the help of two types of de ompositions. A linear onne tion an always be de omposed into Riemannian and post-Riemannian parts,



= e  + N  ;

(C.2.124)

where the distortion 1-form N is expressed in terms of torsion and nonmetri ity as follows: 1 1 N = e[ T ℄ + (e e T ) # + (e[ Q ℄ ) # + Q : 2 2 (C.2.125) The distortion \measures" a deviation of a parti ular geometry from the purely Riemannian one. As a by-produ t of the de omposition (C.2.124) we verify that a metri - ompatible onne tion

de om

N

C.2.13

Ex al again

285

without torsion is unique: it is the Levi-Civita onne tion. Nonmetri ity and torsion an easily be re overed from the distortion, namely Q = 2 N( ) ;

T  = N  ^ # :

(C.2.126)

distorsion2

If we olle t then the information we have on the splitting of a

onne tion into Riemannian and non-Riemannian pie es, then, in terms of the metri g , the oframe # , the anholonomity C  , the torsion T  , and the nonmetri ity Q , we have the highly symmetri master formula 

1 (e e C )# (Vn ) 2  1 T ℄ + (e e T )# (Un ) 2 (Ln ; g ) : (C.2.127) masternonriem

1 = dg + (e[ dg ℄ )# + e[ C ℄ 2 e[

1 + Q + (e[ Q ℄ )# : 2

The rst line of this formula refers to the Riemannian part of the onne tion; together with the se ond line a Riemann-Cartan geometry is en ompassed; and only the third line makes the

onne tion a really independent quantity. Note that lo ally the torsion T an be mimi ked by the negative of the anholonomity C and the nonmetri ity Q by the exterior derivative dg of the metri . In a 4-dimensional metri -aÆne spa e, the urvature has 96

omponents, torsion 24, and nonmetri ity 40. In order to make the work with all these variable manageable, one usually de omposes all geometri al quantities in irredu ible pie es with respe t to the Lorentz group. C.2.13

Ex al again

Ex al has a ommodity: If a oframe o(a) and a metri g are pres ribed, it al ulates the Riemannian pie e of the onne tion e  on demand. One just has to issue the ommand

286

C.2.

Metri

riemann onx hris1; then hris1 (one ould take any other

name) is, without further de laration, a 1-form with the index stru ture hris1(a,b). Sin e we use S houten's onventions in this book, we have to rede ne S hrufer's riemann onx a

ording to hris1(a,-b):= hris1(-b,a); If you distrust Ex al , you ould also ompute your own Riemannian onne tion a

ording to (C.2.127), i.e., pform anhol2(a)=2, hrist1(a,b)=1$ anhol2(a) := d o(a)$

hrist1(-a,-b):= (1/2)*d g(-a,-b) +(1/2)*((e(-a)_|(d g(-b,- )))-(e(-b)_|(d g(-a,- ))))^o( ) +(1/2)*( e(-a)_|anhol2(-b) - e(-b)_|anhol2(-a)) -(1/2)*( e(-a)_|(e(-b)_|anhol2(- )))^o( )$

But I an assure you that Ex al does its job orre tly. In any ase, with hris1(a,b) or with hrist1(a,b) you an equally well ompute the distortion 1-form and the nonmetri ity 1-form in terms of oframe o(a), metri g, and onne tion

onn1(a,b). The al ulation would run as follows:

oframe o(0)=...; % input 1 frame e; riemann onx hris1; hris1(a,b):= hris1(b,a); pform onn1(a,b)=1, distor1(a,b)=1, ,nonmet1(a,b)=1$

onn1(0,0):=...; % input 2 distor1(a,b):= onn1(a,b)- hris1(a,b); nonmet1(a,b):=distor1(a,b)-distor1(b,a);

and one ould ontinue in this line and ompute torsion and

urvature a

ording to pform torsion2(a)=2, urv2(a,b)=2; torsion2(a):=d o(a)+ onn1(-b,a)^o(b);

urv2(-a,b):=d onn1(-a,b)- onn1(-a, )^ onn1(- ,b);

C.2.13

Ex al again

287

All other relevant geoemtri al quantities an be derived thereform.

288

C.2.

Metri

Problems Problem C.1.

Che k the geometri al interpretation of torsion given in Fig. C.1.1 by dire t al ulation using the de nition of the parallel transport. Problem C.2.

Prove (C.1.53) in the in nitesimal ase, approximating the urve  by a small parallelogram. Problem C.3.

Prove the following relations involving the transposed onne tion: _ _ 1. T  = T  , i.e. T  = D # ; _

2. D  Æ = 0; 3. Le # = e T ; 4. The ovariant Lie derivative of an arbitrary p-form =  :::p # ^


^ #p =p!: _  Le = p1! D  :::p # ^


^ #p ; (C.2.128) 1

1

1

1

Problem C.4.

1. Find a transformation matrix from an orthonormal basis to the half-null frame in whi h metri has the form (C.2.11). 2. Find a transformation matrix from an orthonormal basis to Newman-Penrose null frame in whi h metri has the form (C.2.13). 3. Find a linear transformation e = L f whi h brings the Finkelstein basis f ba k to an orthonormal frame e .

ovarLIE

C.2.13

Solution:

0 1=p3 p2 0 p p B 1 1 = 3 0 p2 L = B  p

Ex al again

289

1

1 1C C 2 1 A : (C.2.129) ttfnn 2 1=p3 p0 1= 3 2 0 1 Che k that (C.2.14) is inverse to (C.2.129), up to a Lorentz transformation. 4. Show that the matrix (C.2.129) is represented as a produ t L = S R of the two matri es: 0 1 1 1 11 0 p3 0 0 0 1 B 0 1 0 0 CC ; R = 1 BB 1 1 1 1 CC S=B  0 0 1 0A 2 1 1 1 1 A 1 1 1 1 0 0 0 1 (C.2.130) The matrix S just s ales the time oordinate and a short

al ulation with Redu e shows that R is an element of SO (4). Problem C.5.

1. Prove that for ; 2 pV  ?

^ = ? ^ :

2. If  2 pV , show that # ^ (e ) = p  ; ? ( ^ # ) = e  

? :

(C.2.131) (C.2.132) (C.2.133)

3. Show that in 4-dimensional spa e (a) # ^  = Æ   ; (b) # ^  = Æ   Æ   ; ( ) # ^  Æ = ÆÆ  + Æ  Æ + Æ   Æ ; (d) #  Æ = Æ  Æ ÆÆ   + Æ   Æ Æ   Æ : Hint: # ^  = 0 be ause it is a 5-form in a 4-dimensional spa e.

moveast

ephi east

290

C.2.

Metri

Problem C.6.

1. Prove that Christoel symbols de ne a ovariant dierentiation by he king the transformation law (C.1.11) for the one-forms (C.2.103). 2. Show that with respe t to an arbitrary lo al frame eld, the Riemannian onne tion form reads, f. (C.1.18): e  = ej ei j ei  + ei d ei  1  = g dg + (e dg  e dg ) # 2   1 + e C e C (e e C ) # ; 2 (C.2.134)

where C  = d# is the anholonomity 2-form, see (A.2.35).

nonChris

Referen es

Referen es Part C [1℄ A.

Einstein,

Geometrie und Erfahrung,

Sitzungsber.

Preuss. Akad. Wiss. (1921) pp. 123-130.

The bive tor Cliord algebra and the geometry of Hodge dual operators, J. Phys. A25 (1992) 5649-5662.

[2℄ G. Harnett,

Normal frames for non-Riemannian onne tions, Class. Quantum Grav. 12 (1995) L103{L105.

[3℄ D. Hartley,

[4℄ B.Z. Iliev, Normal

frames and the validity of the equivalen e prin iple. 3. The ase along smooth maps with separable points of sel nterse tion, J. Phys. A31 (1998) 1287-1296.

[5℄ E. Kr oner,

Continuum theory of defe ts, in: Physi s of De-

fe ts, Les Hou hes, Session XXXV, 1980, R. Balian et al.,

eds. (North-Holland: Amsterdam, 1981) pp. 215-315. [6℄ M. Pantaleo, ed.,

Cinquant'anni di Relativita 1905{1955

(Edizioni Giuntine and Sansoni Editore: Firenze, 1955).

292

Referen es

Ele tromagnetism and gravitation, Brasileira de Fisi a 1 (1971) 91-122.

[7℄ M. S h onberg,

Rivista

A quasi-metri asso iated with SU (2) YangMills eld, A ta Phys. Austria a Suppl. XIX (1978) 875-

[8℄ H. Urbantke, 816.

50 Jahre Relativitatstheorie, sens haften 38 (1950) 73-83.

[9℄ H.

Weyl,

Die

Naturwis-

Part D The Maxwell-Lorentz spa etime relation

293

294

les birk/partD.tex and gures [D01 ones.eps℄ 2001-06-01

So far, the Maxwell equations (B.4.8) and (B.4.9) represent an underdetermined system of partial dierential equations of rst order for the ex itation H and the eld strength F . In order to redu e the number of independent variables, we have to set up a relation between H and F , H = H (F ) :

We will all this the ele tromagneti spa etime relation. Therefore we an omplete ele trodynami s, formulated in Part B up to now metri - and onne tion-free, by introdu ing a suitable spa etime relation as fth axiom. The simplest hoi e is, of ourse, a linear relation H   Æ F , with the \ onstitutive" tensor . This yields eventually onventional Maxwell-Lorentz ele trodynami s. It is remarkable that this linear relation, if supplemented merely by a losure property (basi ally  Æ   1) and a symmetry of  (namely   T ), indu es a light one at ea h point of spa etime. In other words, we are able to derive the onformally invariant part of the metri of spa etime from the existen e of a linear  together with its losure property and its symmetry.4 Alternatively, one ould simply assume the existen e of a (pseudo-) Riemannian metri g of signature (+1; 1; 1; 1) on the spa etime manifold. In both ases, the Hodge star operator ? is available and ordinary ele trodynami s an be re overed via the spa etime relation H  ? F .

4 This type of ideas goes ba k to Toupin [35℄ and S h onberg [30℄, see also Urbantke [37℄ and Jad zyk [12℄. Wang [38℄ gave a revised presentation of Toupin's results. A forerunner was Peres [23℄, see in this ontext also the more re ent papers by Piron and Moore [26℄. For new and re ent results, see [20℄. It was re ognized by Brans [1℄ that, within general relativity, it is possible to de ne a duality operator in mu h the same way as we will present it below, see (D.3.11), and that from this duality operator the metri an be re overed. Subsequently numerous authors dis ussed su h stru tures in the framework of general relativity theory, see, e.g., Capovilla, Ja obson, and Dell [2℄, 't Hooft [11℄, Harnett [8, 9℄, Obukhov and Terty hniy [21℄, and the referen es given there. In the present Part D, we will also use freely the results of Gross and Rubilar [6, 28℄.

D.1

Linearity between H and F and quarti wave surfa e

We assume a linear spa etime relation between ex itation H and eld strength F en ompassing 36 independent omponents of the

onstitutive tensor. As a onsequen e, the wave ve tors of ele tromagneti va uum waves lie on quarti surfa es. This is unphysi al and requires additional onstraints. D.1.1

Linearity

The ele tromagneti spa etime relation1 expreses the ex itation in terms of the eld strength . Both are elements of the spa e of 2-forms 2 . However, is twisted and is unwisted. Thus one an formulate the spa etime relation as = ( ) (D.1.1) where  : 2 ! 2 (D.1.2) H

F

X

H

F

H

F

X

;

X

1 Post [27℄ named it onstitutive map in luding also the onstitutive relation for matter, see (E.3.16), Truesdell & Toupin [36℄, Toupin [35℄, and Kovetz [15℄ use the term aether relations.

oper1

oper2

296

D.1.

Linearity between

H and F and quarti wave surfa e

is an invertible operator that maps an untwisted 2-form in a twisted 2-form and vi e versa. The most important ase is that of a linear law between the 2-forms H and F . A

ordingly, the operator (D.1.2) is required to be linear, i.e., for all a; b 2 0X and ; 2 2X we have (a  + b ) = a () + b ( ) :

(D.1.3)

For physi al appli ations, it may be useful to present our linear operator  in a more expli it form. Be ause of its linearity, it is suÆ ient to know the a tion of  on the basis 2-forms. The orresponding mathemati al preliminaries were outlined in Se . A.1.10. A hoi e of the natural oframe #i = dxi yields the spe i 2-form basis BI of (A.1.81). The operator  a ts on the 2-form basis dxk ^ dxl (= BI in the equivalent bive tor language) and maps them in twisted 2-forms the latter of whi h we an again de ompose:

 dxk ^ dxl = 1 ij kl dxi ^ dxj or  BK = I K BI : 2 (D.1.4) Now, we de ompose the 2-forms in (D.1.1): 1 H = Hij dxi ^ dxj or H = HI BI : (D.1.5) 2 Substituting (D.1.5), together with the similar expansion for F , and making use of (D.1.4), we nd 1 Hij = ij kl Fkl or HI = I K FK : (D.1.6) 2 Thus a linear spa etime relation postulates the existen e of 6  6 fun tions ij kl with ij kl =

ji kl =

ij lk :

(D.1.7)

In the bive tor notation, these 36 independent omponents are arranged into a 6  6 matrix I K .

linear0

oper5

oper3

hiHF

oper6

D.1.1

Linearity

297

The hoi e of the lo al oordinates is unimportant. In a dierent oordinate system, the linear law preserves its form due to the tensorial transformation properties of ij kl . Alternatively, instead of the lo al oordinates, one may hoose an arbitrary (anholonomi ) oframe # = ei dxi and may then de ompose the two-forms H and F with respe t to it a

ording to H = H # ^ # =2 and F = F # ^ # =2. Then, if we redo the

al ulations of above, we nd 1 H =  Æ F Æ with  Æ = ei  ej ek el Æ ij kl : 2 (D.1.8) Here we used also the omponents of the frame e = ek  k . As we re all from Se . A.1.10, the Levi-Civita symbols (A.1.91) and (A.1.93) an be used as a \metri " for raising and lowering (pairs of) indi es. We de ne 1 IK := IM M K or ijkl = ijmn mn kl (D.1.9) 2 and, onversely, 1 (D.1.10) I K = ^IM MK or ij kl = ^ijmn mnkl : 2 The 36 fun tions ij kl(; x) as well as the ijkl(; x) depend on time  and on spa e x in general. Be ause of the orresponding properties of the Levi-Civita symbol, the ijkl represent an (untwisted) tensor density of weight +1. Ex itation H and eld strength F are measurable quantities. The fun tions ij kl (or ijkl) are \quotients" of H and F . Thus they rst of all arry the dimension [℄ = [℄ = q2 =h = q=0 = SI (q=t)=(0=t) = urrent/voltage = 1/resistan e = 1= = S (for Siemens), but, moreover, they are measurable, too. Two invariants of , a linear and a quadrati one, play a leading role: The twisted s alar 1 1  := ij ij = ^ijkl [ijkl℄ (D.1.11) 12 4!

hiHFanh

raise

lower

invariant1

298

D.1.

Linearity between

H and F and quarti wave surfa e

and the true s alar

2 := =

1 kl ij   4! ij kl 1 ^ijkl ^mnpq ijmn pqkl : 96

(D.1.12)

invariant2

In later appli ations we will see that it always ful lls 2 > 0. Note that [℄ = [℄ = 1/resistan e. It is as if spa etime arried an intrinsi resistan e or, the inverse of it, an intrinsi impedan e ( ommonly alled \wave resistan e of the va uum" or \va uum impedan e"). One ould also build up invariants of order p a

ording to the pattern I1 I2 I2 I3 : : : Ip I1 , with p = 1; 2; 3; 4 : : : , but there seems no need to do so. D.1.2

Extra ting the Abelian axion

Right now (still without a metri ), we an split o the totally antisymmetri part of ijkl a

ording to

ijkl = eijkl +  ijkl ;

with

e[ijkl℄ = 0 :

(D.1.13)

split

The invariant (D.1.11) shows up in the se ond term as a pseudos alar fun tion  = (; x). Thus the linearity ansatz (D.1.6) eventually reads 1 4

Hij = ^ijmn emnkl Fkl +  Fij ;

(D.1.14)

linear

e mnkl = e nmkl = e mnlk and e [mnkl℄ = 0 :

(D.1.15)

hisymm

with Besides the Abelian axion eld , we have 35 independent fun tions. It is remarkable that the pseudo-s alar axion eld  enters here as a quantity that does not interfere at all with the rst four axioms of ele trodynami s. Already at the pre-metri level,

D.1.2

Extra ting the Abelian axion

299

su h a eld emerges as a not unnatural ompanion of the ele tromagneti eld. Hen e a possible axion eld has a high degree of universality | after all, it arises, in the framework of our axiomati approa h, even before the metri eld (Einstein's gravitational potential) omes into being. Pseudo-s alars are also alled axial s alars.2 So far, our axial s alar (x) is some kind of universal permittivity/permeability eld. If one adds a kineti term of the - eld to the purely ele tromagneti Lagrangian, then (x) be omes propagating and one an all it legitimately an Abelian3 axion. The orresponding hypotheti al parti le4 has spin = 0 and parity = 1. The split (D.1.13) ee tively introdu es the linear operator e : 2X ! 2X whi h a ts in the spa e of two-forms similarly to . Patterned after (D.1.4), its a tion on the B's reads
1 e dxi ^ dxj = ekl ij dxk ^ dxl or e (BI ) = e K I BK : 2 (D.1.16)

Here the linear operator matrix is evidently de ned by 1 kl ij := ^klmn e mnij or eI K := ^IM eMK : e (D.1.17) 2 Formulated in the 6D spa e of 2-forms, (D.1.14) then reads HI = I K FK = ^IM eMK FK +  FI ;

sharp1

sharp2

(D.1.18)

hiHF1

(D.1.19)

also

where ^MK eMK = eK K = 0 :

2 For a dis ussion of a possible primordial osmologi al heli ity, see G.B. Field, S.M. Carroll [4℄; also magneti heli ity, that we addressed earlier in (B.3.17), is mentioned therein. 3 In ontrast to the axions related to non-Abelian gauge theories, see Pe

ei and Quinn [22℄, Weinberg [39℄, Wil zek and Moody [40, 17℄ and the reviews in Kolb and Turner [14℄ and Sikivie [31℄. 4 Ni [18, 19℄ was the rst to introdu e su h an axion eld  in the ontext of the

oupling of ele tromagnetism to gravity, see also deSabbata & Sivaram [29℄ and the referen es given there.

300

D.1.

Linearity between

H and F and quarti wave surfa e

Summing up, the linear spa etime relation (D.1.6), see also (D.1.14), an be written as H = (e + ) F ;

Tr e = 0 :

(D.1.20)

lin1

Hen e the Maxwell equations in this shorthand notation read 



d (e + )F = J ;

dF = 0 :

(D.1.21)

Maxsharp

We an also exe ute the dierentiation in the inhomogeneous equation and substitute the homogeneous one. Then we nd for the inhomogeneous Maxwell equation d e (F ) + d ^ F = J :

(D.1.22)

As yet, the Abelian axion has not been found experimentally. In parti ular, it ouldn't be tra ed in ring laser experiments.5 D.1.3

Fresnel equation

As soon as the onstitutive law is spe i ed, ele trodynami s be omes a predi tive theory and one an study various of its physi al ee ts, su h as the propagation of ele tromagneti disturban es and, in parti ular, the phenomenon of wave propagation in va uum. In the theory of partial dierential equations, the propagation of disturban es is des ribed by the Hadamard dis ontinuities of solutions a ross a hara teristi hypersurfa e S , the wave front.6 One an lo ally de ne S by the equation (xi ) = onst. The Hadamard dis ontinuity of any fun tion F (x) a ross the hypersurfa e S is determined as the dieren e [F ℄ (x) := F (x+) F (x ), where x := "lim (x  ") are points on the opposite sides !0 of S 3 x. We all [F ℄ (x) the jump of the fun tion F at x. 5 See Cooper & Stedman [3℄ and Stedman [33℄ for a systemati and extended series of experiments. 6 The orresponding theory was developped in detail by Hadamard [7℄ and Li hnerowi z [16℄, e.g..

Maxsharp'

D.1.3

Fresnel equation

301

An ordinary ele tromagneti wave is a solution of Maxwell's va uum equations dH = 0 and dF = 0 for whi h the derivatives of H and F are dis ontinuous a ross the wave front hypersurfa e S . In terms of H and F , we have on the hara teristi hypersurfa e S [H ℄ = 0 ; [dH ℄ = q ^ h ; [F ℄ = 0 ; [dF ℄ = q ^ f :

(D.1.23) (D.1.24)

this that

Here the 2-forms h; f des ribe the jumps (dis ontinuities) of the dierentials of the ele tromagneti eld a ross S , and the wave- ove tor normal to the front is given by q := d  :

(D.1.25)

Equations (D.1.23) and (D.1.24) represent the Hadamard geometri al ompatibility onditions. If we use Maxwell's va uum equations dH = 0 and dF = 0, then (D.1.23) and (D.1.24) yield q ^h = 0;

q ^f = 0:

(D.1.26)

these

We an say something more about the jump h, if we apply the spa etime relation (D.1.20). Provided the omponents of the linear operator  (that is, of e and ) are ontinuous a ross S , we nd by dierentiating (D.1.20) and using (D.1.23) and (D.1.24), [dH ℄ = [d e (F )℄ +  [dF ℄ = q ^ (e (f ) +  f ) = q ^ h : (D.1.27) A

ordingly, the jump equations (D.1.26) an be put into the form q ^ h = q ^ e (f ) = 0 ;

q ^f = 0:

(D.1.28)

Note that the axion drops out ompletely, i.e., e arries 35 independent omponents. If we multiply the two equations of (D.1.28) by q, both vanish identi ally. Hen e only 3+3 equations turn out to be independent.

those

302

D.1.

Linearity between

H and F and quarti wave surfa e

Spa etime omponents

We an rewrite these equations in spa etime omponents: q

^ e (f ) = qm dxm ^ 21 fij e (dxi ^ dxj ) =

1 ij m k l q[m e kl℄ fij dx ^ dx ^ dx = 0; 4

(D.1.29)

and q

^ f = 21 q[i fjk℄ dxi ^ dxj ^ dxk = 0

(D.1.30)

As a onsequen e, we nd 

ijkl

qj hkl

= e ijkl qj fkl = 0 ;



ijkl

qj fkl

= 0:

(D.1.31)

4Dwave

If the onstitutive tensor density e ijkl is pres ribed, the set in (D.1.31) represents homogeneous algebrai equations for the 2forms fij . Components in the spa e of 2-forms

e (f ) and f in (D.1.28) with reAlternatively, we an de ompose  spe t to the BI -basis of the M 6 . Then (D.1.28) an be rewritten as q

^ hI BI = q ^ eI K fK BI = 0 ;

q

^ fI BI = 0 :

(D.1.32)

Before we start to exploit these equations, it will be onvenient to make in this pi ture the presen e of ele tri and magneti pie es more pronoun ed. For this reason, we need to re all, on the one hand, the (1+3)-de ompositions (B.4.5) and (B.4.6) of the ex itations and the eld strengths, and to ompare these, on the other hand, with the the de ompositions of H and F with respe t to the BI basis: H F

= =

BI = I FI B =

HI

H ^ d + D ; E ^ d + B :

(D.1.33) (D.1.34)

those1

D.1.3

Fresnel equation

303

Sin e every longitudinal (spatial) 1-form an be de omposed with respe t to the natural oframe dxa , whereas every 2-form

an be onveniently expanded with respe t to the -dual 2-form basis ^a , we have (a; b;


= 1; 2; 3)

H = Ha dxa ;

E = Ea dxa ;

(D.1.35)

and

D = D ^ ;

B = B ^ :

(D.1.36)

We substitute this into (D.1.33), (D.1.34) and nd

H = HI BI = F = FI BI =

Ha a + Db ^b ;

Ea a + B b ^b :

(D.1.37) (D.1.38)

omp1

omp2

e (F ), in omponents HI = Then the spa etime relation H =  K eI FK , reads 

Ha Da



=

Ceb a Beab e ba Aeab D

!

Eb Bb



:

(D.1.39)

CR

For the jumps h and f , we nd analogous equations. Instead of (D.1.37) and (D.1.38), we have

h = ha a + da ^a ; f = ea a + ba ^a ; and the equation derived from (D.1.39) is



ha da



=

! b e e C a Bab eba Aeab D

eb bb

(D.1.40) (D.1.41)



:

(D.1.42)

dish disf

CR1

Now we turn to (D.1.32). We insert the wave ove tor in its expanded form

q = q0 d + qa dxa

(D.1.43)

wavesplit

304

D.1.

H and F and quarti wave surfa e

Linearity between

and use (D.1.40) and (D.1.41): a

q0 d



ab

= 0; a qa d = 0; qb h

a

+ ab qb e = 0; a qa b = 0: q0 b

(D.1.44) (D.1.45)

geo2 geo3

In this system, whi h is a omponent version of (D.1.26), only the 6 equations (D.1.44) are independent. This has already been foreseen above. Assuming that q0 6= 0, one nds that the equations (D.1.45) are trivially satis ed if one substitutes (D.1.44) into them. Note that the hara teristi s with q0 = 0 do not have an intrinsi meaning for the evolution equations, sin e they obviously depend on the hoi e of the oordinates. We an now substitute da and ha from (D.1.42) into (D.1.44)1 : q0

   ab d d ab a b e e e e =  qb C ed + B d b A eb + Db b



Then we eliminate bb by means of (D.1.44)2 :

h

e q0 A 2

ab

q0 q





a d

ed C b



e d a + q qd a e bdf Beef + b d D

(D.1.46)

i

= 0: (D.1.47)

3Dwave

eb

algeq

This homogeneous linear algebrai equation for eb has a nontrivial solution provided the determinant W := det W of its

oeÆ ient matrix W

ab

:= q02 Aeab + q0 Y ab + Z ab

vanishes, where Y Z



b

e a + b d D

:=

ab

:= a e bdf Be d qe qf :



a d

e C

ab



qd ;

(D.1.48)

Fresnelmat

(D.1.49)

Ydef

(D.1.50)

Zdef

We have then, with the de nition of the determinant for a 3  3 matrix, W = det W = 61 ^ab ^def W adW beW f = 0 : (D.1.51) This is the Fresnel equation that is of entral importan e in any wave propagation analysis. It determines the geometry of the wave normals in terms of the 35 independent onstitutive e B; e C; eD e.

omponents of the 3  3 matri es A;

deteq0

D.1.4

D.1.4

Analysis of the Fresnel equation

305

Analysis of the Fresnel equation

The following properties are evident from the de nitions above: Z

ab

= 0;

qb

Z

ab

qa

= 0;

Y

ab

qa qb

= 0:

(D.1.52)

YZprop

Immediate inspe tion shows that the determinant reads: 1 det Ae + q05 ^ab ^def Aead Aebe Y f 2   1 ae ebe e f ad be e f 4 + q0 ^ab ^def Z A A + Y Y A 2   3 + q0 det Y + ^ab ^def Z ad Aebe Y f

W =q

6 0

  1 + q02 ^ab ^def Z ad Y be Y f + Z ad Z be Ae f 2 1 (D.1.53) + q0 ^ab ^def Z ad Z be Y f + det Z: 2 We have det Z = 0 sin e the matrix Z has null eigenve tors. Hen e, the last term drops out ompletely. Furthermore, by means of (D.1.50), we nd

^ab ^def Z ad = Bebe q qf

ebf q qe B

e e qb qf B

e f qb qe : +B (D.1.54)

determinant

epsepsZ

If we multiply this equation by Z be Y f , we re over the term linear in q0 of the last line in (D.1.53). Then, by using (D.1.52), we straightforwardly re ognize that it vanishes identi ally. Consequently, we have veri ed that the determinant W fa torizes into a produ t of q02 and a 4th order polynomial:

W =q

2 0

4

q0 M

+ q03 qa M a + q02 qa qb M ab
(D.1.55) +q0 qa qb q M ab + qa qb q qd M ab d = 0 :

Thus, the Fresnel equation des ribes ultimately a 4th order surfa e7 and not one of 6th order, as it appears from (D.1.53). 7 Numeri al evaluations and orresponding plots of Fresnel wave surfa es for a onstitutive tensor in luding opti al a tivity and Faraday rotation have been presented by Kiehn et al.[13℄.

detlin

306

D.1.

Linearity between

H and F and quarti wave surfa e

By using (D.1.54), (D.1.49), (D.1.50) and (D.1.52), we nd for the dierent pie es in (D.1.53):

1 ^ ^ Z ad Y be Y f = Bebf q qe Y be Y f 2 ab def e e Ced g Bhf ; (D.1.56) = qa qb q qd efa ghb D 1 ad be e f 1 e ^ab ^def Z Z A = Bbe q qf Z be Ae f 2 2 1 = qa qb q qd efa ghb Beeg Bfh Ae d : (D.1.57) 2

Furthermore,

^ab ^def Z

ad

e Y A be

f

h   e e f + Bedf Cef e = qa qb q dea Aeb Befd D i e f Ceb e Bedf Aef D e e b Befd : (D.1.58) A

Eventually, a somewhat lengthy al ulation yields:

1 det Y = ^ab ^def Y ad Y be Y f 6   e f b Ce e Cef d + Ceb f D e e D edf : = qa qb q dea D

(D.1.59)

D.1.4

Analysis of the Fresnel equation

307

As a result, we an nalize the omputation of the oeÆ ients of the Fresnel equation. We nd expli itly,

M M M

a

ab

:= det Ae ;



e := ^ Ae Ae Ce + Ae Ae D ; (D.1.61) h i 1 e( ) e 2 2 e e e e e (C ) + (D ) (C + D )(C + D ) := A 2 e )(C e( A e) +A e( D e )) + (Ce + D ab

e

d

d d

e C

ab)

de(

h

df

+ Ce := ( j

f

a

e A

(a

e A

e C

ab)

e

f



j Be

eg

b)

b)d

e C e A

d

d

e d ; B

jj D e

a e

a

f

e A



e A

f

e +D 1 2

d



d)

e A

b)f

e C

b)

f a

b)

ef h B



ed D

b)

(D.1.62)

ma2

(D.1.63)

ma3

(D.1.64)

ma4

e



e e C

(a

i

f

e C

;

d

f

eh D

d)

 :

e B; e C; e D e . We The M 's are all ubi in the omponents of A; de ned the M 's as ompletely symmetri expressions, that is, M 1 p = M ( 1 p ) , p = 2; 3; 4, sin e only these pie es ontribute to the Fresnel equation. Sin e

a totally symmetri ten+ 1 1+ sor of rank p has = independent omponents, 1 the M 's altogether arry 1 + 3 + 6 + 10 + 15 = 35 independent

omponents. Sin e q0 6= 0, one an delete the rst fa tor in (D.1.55). Thus we nd nally that the wave ove tor q lies, in general, on a 4th order (or quarti ) surfa e. Of ourse, a quarti wave surfa e for light propagation is unphysi al, at least at the present epo h of the universe. It is dierent from the 2nd order stru ture of the light one, whi h arises from the quarti surfa e only under parti ular ir umstan es. Further below we will demona :::a

a :::a

n

p

p

ma1

b

a

e D ed D

e A

e C

f

ed D

gh b



ee D

ab)

d

d

ma0

d

(a

e A

e A

b)

b

e A

b)

d



d

d

e A

ee D

a

a ef

ef d B 

+ Be

ab d

(a





+ Ae( := 

d

d



e

e

d

ab

e C

M

ba

e

ab

d

b d

d

M

(D.1.60)



n

p

n

i

308

D.1.

Linearity between

H and F and quarti wave surfa e

strate whi h type of onstraint an enfor es su h a redu tion. Nevertheless, if one wanted to generalize the light one stru ture for the very early universe, for example, then there would be no need to turn to nonlinear ele trodynami s (see Chapter E.2); also a linear spa etime relation an support nontrivial generalizations of the onventional (2nd order) light one to a quarti wave surfa e. Consider the expression in the parenthesis of the Fresnel equation (D.1.55). If we re all that the wave ove tor splits a

ording to (D.1.43), then this expression an be rewritten in a 4dimensional invariant form, 8

G

ijkl

qi qj qk ql

= 0;

i; j;




= 0; 1; 2; 3 :

(D.1.65)

Fresnel

The totally symmetri fourth order tensor density G ijkl of weight +1 has 35 independent omponents, exa tly the same number as those of the M 's. We an express G ijkl omponentwise in terms of the M 's. Start with 4 q 's. Then we nd su

essively, by omparing (D.1.55) with (D.1.65), 0

1 a 1 ab M ; G = M ab ; 4 6 1 ab G = M ab ; G ab d = M ab d : (D.1.66) 4 In the end, we have expressed the 35 independent omponents of G ijkl in terms of the 35 independent omponents of the M 's; and the M 's an be expressend in terms of the 35 independent e B; e C; e D e . These onstitute the tensor 

omponents of A; eij kl , see (D.1.39), whi h, a

ording to (D.1.17), is equivalent to e ijkl . The on lusion from this ba ktra ing pro ess is that G ijkl should be expressible in terms of e ijkl . The ubi Tamm-Rubilar formula is the answer: G

0000

=M;

G

000

a

=

00

0

G

8 Earlier,

ijkl

:=

1 ^ ^  e mnr i e j jpsjk e l qtu : 4! mnpq rstu (

)

(D.1.67)

the relation between the fourth- and the se ond-order wave geometry was

studied by Tamm [34℄ for a spe ial ase of a linear onstitutive law. He also introdu ed a `fourth-order metri ' of the type of our

G

tensor density, see (D.1.67).

ompare

G4

D.1.4

Analysis of the Fresnel equation

309

Here the total symmetrization is extended only over the four indi es i; j; k; l with all the dummy (or dead) summation indi es ex luded. Although eq. (D.1.67) an be veri ed by means of

omputer algebra, its ovariant analyti derivation remains an interesting and diÆ ult problem. Our readers are invited to solve it. By de nition, G ijkl (e) does not depend on the axion. We saw that in our analysis a la Hadamard the axion drops out, see the remark after (D.1.28). It is onsistent with that result that G ijkl () = G ijkl (e +  ) = G ijkl (e), as an be seen by dire t

al ulation.

310

D.1.

Linearity between

H and F and quarti wave surfa e

D.2 Ele tri -magneti re ipro ity swit hed on

The linear spa etime relation of the last hapter is required to obey ele tri -magneti re ipro ity. This implies an almost omplex stru ture on spa etime thereby redu ing the onstitutive tensor

e from 35 to 18 independent omponents.

D.2.1

Re ipro ity implies losure

The linear spa etime relation leaves us still with 35+1 independent omponents of the tensor . Clearly we need a new method to onstrain  in some way. An obvious hoi e is to require ele tri -magneti re ipro ity for (D.1.20). We have dis overed ele tri -magneti re ipro ity as a property of the energymomentum urrent k  of the ele tromagneti eld. Why should't we apply it to (D.1.20), too? The ele tri -magneti re ipro ity transformation (B.5.15) H

! F ;

F

! 1 H ;

(D.2.1)

duality2

312

D.2.

Ele tri -magneti re ipro ity swit hed on

an alternatively be written as



H F



If

W :=

!







0

1

0





0

1



0





H F

  =

; then W

1

=

F 1 H 



0 1 



:

(D.2.2)

0





duality2a

:

(D.2.3)

duality2b

Let us perform an ele tri -magneti re ipro ity transformation in (D.1.20). By de nition, the re ipro ity transformation ome . Then we nd mutes with the linear operator 

F = (e + )



H 



or

e (H ) =  2 F

 H : (D.2.4)

duality3

e to (D.1.20). On the other hand, we an also apply the operator  e ommutes with 0-forms, we get Be ause  e (H ) = (e e +  e ) F :

(D.2.5)

lin3

If we postulate ele tri -magneti re ipro ity of the linear law (D.1.20), then, as a omparison of (D.2.5) and (D.2.4) shows, we have to assume additionally

e e =  2 16 ;

 = 0:

(D.2.6)

e e =  2 16 the losure relation1 sin e applying the We all  e twi e, we ome ba k, up to a negative fun tion, to the operator  ij identity 16 (= Ækl , in omponents). In this sense, the operation

loses. At the same time, (D.2.6) tells us that the spa etime relation is not ele tri -magneti re ipro al for an arbitrary transformation fun tion  (like the energy-momentum urrent is). The line is based on the measurable omponents e ij kl . If ear operator  1 Toupin [35℄, for  2 = onst, just alled the losure relation \ele tri and magneti re ipro ity".

lose1

D.2.2

Almost omplex stru ture

313

applied twi e, as in (D.2.6), there must not emerge an arbitrary fun tion. In other words, we an solve (D.2.6)1 for  by taking its tra e, 1 1 e e) = Tr (  ekl ij  eij kl = 2 ; (D.2.7) 2 = 6 24 with the quadrati invariant 2 of (D.1.12). D.2.2

zetasquare

Almost omplex stru ture

Now we an fa torize the \ onstitutive" matrix (D.1.13) by means of the dimensionfull fun tion  a

ording to  e ijkl

o

=:   ijkl :

(D.2.8)

ir le

o

Here [℄ = 1/resistan e, as we already know, and  ijkl is a dimensionless tensor with the same symmetries as displayed in (D.1.15). This ee tively de nes a new operator J via

e =:  J :

(D.2.9)

lose1a'

For J the losure reads JJ =

(D.2.10)

16 :

lose1a

As we will see, the minus sign is very de isive: It will eventually yield the Lorentzian signature of the metri of spa etime. With the losure relation (D.2.10), the operator J is an almost omplex stru ture on the spa e M 6 of 2-forms, as dis ussed in Se . A.1.11. If we apply J to the 2-form basis, see (D.1.16), we nd J(BI ) =

1 

e (BI ) =

1 

 K I (B K ) e

= JK I B K :

(D.2.11)

Jde omp

A omparison with (D.1.39) allows to express J in terms of the

onstitutive fun tions a

ording to JI

K




=

1 

eab C eab B

eab A

eab D

!



=:

C ab Bab

Aab Da b

 :

(D.2.12)

Jmatrix

314

D.2.

Ele tri -magneti re ipro ity swit hed on

Be ause of (D.2.10), the 3  3 blo ks A; B; C; D are onstrained by + C 2 = 13 ; C A + AD = 0; BC + DB = 0; 2 BA + D = 13 :

(D.2.13) (D.2.14) (D.2.15) (D.2.16)

AB

D.2.3

almost lose1m almost lose2m almost lose3m almost lose4m

Algebrai solution of the losure relation

We are able to solve this losure relation. Assume that det B 6= 0. Consider (D.2.15). Then we an make an ansatz for the matrix C , namely 1

=B

C

K:

(D.2.17)

CDKK

(D.2.18)

KK'

We substitute this into (D.2.15) and nd D

=

1

KB

:

Next, we straightforwardly solve (D.2.13) with respe t to A: A

=

1

B

B

1

K


2

1

B

(D.2.19)

:

ABK

Now we turn to (D.2.14). Eqs.(D.2.17) and (D.2.19) yield, CA

=

AD

=B

B

1

1

KB

KB

1

1

+

B B

1

1

K

K


3


3

1

B

B

1

:

;

(D.2.20) (D.2.21)

Thus, we on lude that (D.2.14) is satis ed. Finally, (D.2.14) is left for onsideration. From (D.2.19) and (D.2.17)2 , we obtain: BA D

2

= 1 =

KB

KB


1 2

:


1 2

;

(D.2.22) (D.2.23)

Thus, in view of (D.3.28), the equation (D.2.16) is also ful lled.

CAAD

D.2.3

Algebrai solution of the losure relation

315

Summing up, we have derived the general solution of the losure system (D.2.13)-(D.2.16). It reads,
2

A = B 1 B 1K B 1; C = B 1 K; D = KB 1 ;

(D.2.24) (D.2.25) (D.2.26)

or, if we put this in 6  6 matrix form,

J=



C A B D





=

B 1K B

[1 + (B 1 K )2 ℄ B KB 1

1



summary ompA summary ompC summary ompD

:

(D.2.27)

summarymB1

By squaring J, one an dire tly verify the losure relation (D.2.10). The arbitrary matri es B and K parametrize the solution whi h thus has 2  9 = 18 independent degrees of freedom. Clearly, it would be possible to substitute K via K = BC , but the matrix K will turn out to be parti ularly useful in Chap. D.3. If we measure the elements of the 3  3 matri es e B; e C; e D e of  e and thereby, a

ording to (D.2.12), also those A; of A; B; C; D, then losure is only guaranteed provided the relations (D.2.24) to (D.2.26) are ful lled. Therefore, losure has a well-de ned operational meaning. If we assume that det A 6= 0, then an analogous derivation leads to

J=



C A B D



=

1 b KA 1 2 b A 1 [1 + (KA )℄

!

A : b A 1K (D.2.28)

Before turning to the dierent sub ases, a word to the physi s of all of this appears to be in order. We know from experiments in va uum that D  "0 E and H  0 B . If we ompare this with the spa etime relation (D.1.39), then we re ognize that "0 is related the 3  3 matrix Ae and 0 to the 3  3 matrix Be . Therefore it is safe to assume that det A 6= 0 and det B 6= 0. From this pra ti al point of view the other sub ases don't seem to be of mu h interest.

summarymA1

316

D.2.

Ele tri -magneti re ipro ity swit hed on

We ould substitute the matri es A; C; D of (D.2.24) to (D.2.26) into the M 's (D.1.60) to (D.1.64) of the Fresnel equation and

ould dis uss the orresponding onsequen es. However, we get a mu h more de isive restru turing of the Fresnel equation if we employ, in addition, the symmetry assumption whi h we will now turn to.

D.3

Symmetry swit hed on additionally

Having found an almost omplex stru ture on spa etime by linearity and ele tri -magneti re ipro ity, how an we re over a spa etime metri eventually? By requiring symmetry of the onstitutive tensor thereby redu ing its independent omponents to 9. Thus, a light one is de ned at ea h point of spa etime. D.3.1

Lagrangian and symmetry

Besides linearity and ele tri -magneti re ipro ity of the ele tromagneti spa etime relation, we require the operator

symmetri ,

() ^

=

 ^ (

);

for arbitrary twisted or untwisted 2-forms lations are valid for

e and J.

 to be

(D.3.1)

 and

. Similar re-

This an be motivated in that we usually assume that a Lagrange 4-form exists for a fundamental theory, as we dis ussed in Se . B.5.4. If we do this in the ontext of our linear spa etime relation, then, be ause of

H

=

V=F ,

the Lagrangian must

symm

318

D.3.

Symmetry swit hed on additionally

be quadrati in F . Thus we nd 1 1 V= H ^F = H F dxi ^ dxj ^ dxp ^ dxq 2 8 ij pq 1 ^ mnkl Fkl Fpq dxi ^ dxj ^ dxp ^ dxq : (D.3.2) = 32 ijmn We rewrite the exterior produ ts with the Levi-Civita symbol: 1 ijkl V =  Fij Fkl dx0 ^ dx1 ^ dx2 ^ dx3 : (D.3.3) 8 The omponents of the eld strength F enter in a symmetri way. Therefore, without loss of generality, we an impose the symmetry ondition

ijkl = klij

IK = KI

or

(D.3.4)

sym

redu ing them to 21 independent fun tions at this stage. This

ondition is equivalent to (D.3.1). Thus we split  into 36 = 21 + 15 independent omponents and require the vanishing of 15 of them. The split (D.1.13) of the Abelian axion obviously does not disturb the symmetry property. Thus, 36 = 20 + 1 + 15 and the tensor e ijkl has the same algebrai symmetries and the same number of 20 independent omponents as a urvature tensor in a 4-dimensional Riemannian spa etime. Of ourse, the losure relation further restri ts the remaining 20 omponents. In the language of M 6 spa e, the 6  6 matrix eIK is symmetri , and it an be represented with the help of 3  3 matri es e B; e Ce as A;

e

IK

= e

KI

=

Be CeT Ce Ae

!

; Ae = AeT ; Be = Be T : (D.3.5)

A

ordingly, for the fun tions in (D.1.18), we nd

I = ^IM  K

MK

=

=



0 13 13 0

 + Ce Ae Be  + CeT



! =

Be

 + CeT  + Ce Ae Ce Ae Be CeT

xixi1

!

!

+  16 : (D.3.6)

xi0

D.3.2

Duality operator and metri

319

Note that we have D = C T in this ase. The spa etime relation (D.1.20) with the symmetry (D.3.1)

H = (e + ) F ; with (F ) ^ F = F ^ (F ) ;

(D.3.7)

lin1a

(D.3.8)

axlagrangian

leads to the Lagrangian

V=

1 [F 2

^ e (F ) +  F ^ F ℄ :

Hen e, as a look at (D.1.12) will show, the oupling term to the axion reads 1  F ^ F =  E ^ B ^ d ; (D.3.9) 2 with the ele tri eld strenght 1-form E and the magneti 2form B in 3D. This term an also be rewritten as an exa t form plus a supplementary term: 1 F 2

^F =d



1 F 2



^A

1 F 2

^ A ^ d :

(D.3.10)

axlagrangian1

For the spe ial ase of  = onst, we are left with a pure surfa e term. D.3.2

Duality operator and metri

A

ordingly, it is lear that the existen e of a Lagrangian enfor es the Onsager type of symmetry (D.3.1). We don't know whether the reverse is also true. What we do know from Se . C.2.4, however, is that, in the framework of an almost omplex stru ture of (D.2.9), the symmetry (D.3.1) introdu es a duality operator. Therefore, we de ne a duality operator a

ording to

# := J = 1  e ; with J() ^ =  ^ J( ) :



(D.3.11)

dualityJ

Then, we also have self-adjointness with respe t to the 6-metri ,

"('; #!) = "(#'; !):

(D.3.12)

selfadj

320

D.3.

Symmetry swit hed on additionally

By means of (D.3.11) and (D.2.10), ele tri -magneti re ipro ity and symmetry of the linear ansatz eventually lead to the spa etime relation and its inverse, namely H

=  #F

and

F

=

1#



H;

(D.3.13)

olle tHF

whi h will be investigated in the next hapter. We an onsider the duality operator # also from another point of view. It is our desire to des ribe eventually empty spa etime with su h a linear ansatz. Therefore o ijkl we have to redu e the  number of independent fun tions somehow. The only onstants with even parity are the Krone ker deltas Æij . Obviously o the Æij 's are of no help here in spe ifying the  ijkl 's, sin e they

arry also lower indi es whi h annot be absorbed in a nono trivial way in order to reate the 4 upper indi es of  ijkl . Re ognizing that in the framework of ele trodynami s in matter a similar linear ansatz an des ribe anisotropi media, we need a o

ondition in order to guarantee isotropy. A \square" of  will do the job, 1 o klrs o mnpq   ^rsmn ^ pqij = 8

kl Æij ;

(D.3.14)

ij with Ærs as a generalized Krone ker delta. This equation, together the linearity ansatz and the symmetry, represents our fth axiom for spa etime. Alternatively, it an also be written as 1 # mn #mn kl = Æijkl : (D.3.15) 2 ij

D.3.3

Algebrai solution of the losure

and

symmetry relations

In addition to the almost omplex stru ture J, we found the symmetry of the onstitutive matrix (D.3.5). As a onsequen e,

lose3

lose4

D.3.3

Algebrai solution of the losure and symmetry relations

321

the onstraints (D.2.13) to (D.2.16), follwoing from the losure relation, pi k up the additional properties A

= AT ;

B

= BT;

D

= CT :

(D.3.16)

algsym

Then they redu e to Aa B b

+ C a C b = C (a Ab) = 0 C (a Bb) = 0

Æba

or or or

+ C 2 = 13 ; (D.3.17) CA + AC T = 0; (D.3.18) T BC + C B = 0: (D.3.19) AB

alg1 alg2 alg3

Naturally, we would like to resolve these algebrai onstraints.

Preliminary analysis Being a solution of the system (D.3.16)-(D.3.19), the matrix C e = 0, we has very spe i properties. First of all, be ause of Tr  have Tr C +Tr D = 0. With the symmetry assumption, D = C T . Thus, TrC = 0. More generally, the tra es of all odd powers of the matrix C are zero: TrC = Tr(C 3 ) = Tr(C 5 ) =


= 0:

(D.3.20)

tra eC

Indeed, multiplying (D.3.17) by C and taking the tra e, we nd Tr(ABC ) + Tr(C 3 ) = 0. On the other hand, if we transpose (D.3.17) and multiply the result by C T , then the tra e yields Tr(BAC T ) + Tr(C 3 ) = 0. of the two last equations
The sum T 3 reads Tr A(BC + C B ) + 2Tr(C ) = 0. In view of (D.3.19), we then on lude that Tr(C 3 ) = 0. The same line of arguments yields generalizations to the higher odd powers. It follows from (D.3.20) that the matrix C is always degenerate, det C = 0:

(D.3.21)

Indeed, re all that the determinant of an arbitary 3  3 matrix M (= Mb a ) reads 1 det M = ^ab a b Ma a Mb b M 6  1 = (TrM )3 3TrM Tr(M 2 ) + 2Tr(M 3 ) : (D.3.22) 6 0

detC

0 0

0

0

0

detdef

322

D.3.

Symmetry swit hed on additionally

Be ause of (D.3.20), we an immediately read o (D.3.21). Let us now analyse the determinants of A and B . When the term C 2 is moved from the left-hand side of (D.3.17) to the right-hand side, a dire t omputation of the determinant yields: 

det A det B =

det(1 + C ) = 2

Tr(C 2 ) 1+ 2

2 :

(D.3.23)

detABC

We used the formula (D.3.22) and the properties (D.3.20) and (D.3.21) to evaluate the right-hand side. A

ordingly, the matri es A and B annot be both positive de nite. Moreover, when at least one of them is degenerate, we nd that ne essarily Tr(C 2 ) = 2.

General regular solution Let us onsider the ase when both A and B are regular matri es, i.e., det A 6= 0; det B 6= 0. The general solution has been given in (D.2.27) and (D.2.28). Together with the symmetries (D.3.16), the general solution of (D.3.16) to (D.3.19) an be presented in one of the following two equivalent forms. B-representation: 

A C

= 1 + (B = B 1 K;

1

K

)2

 B

1

;

(D.3.24) (D.3.25)

BrepA BrepC

where K = K T . The two arbitrary matri es B (symmetri ) and K (antisymmetri ) des ribe the 6+3=9 degrees of freedom of the general solution. A-representation: B

=

C

bA =K

A

1 1

h

;

b A 1 )2 1 + (K

i ;

(D.3.26)

ArepB

(D.3.27)

ArepC

b = K b T . In this ase, the 6+3=9 degrees of freedom of where K the general solution are en oded in the matri es A (symmetri ) b (antisymmetri ). and K

D.3.3

Algebrai solution of the losure and symmetry relations

323

The transition between the two representations is established with the help of the relation

b =B

K

1

(D.3.28)

K A:

One an readily he k that (D.3.24), (D.3.25) and (D.3.26), (D.3.27) are really the solution of the losure and symmetry relations. Indeed, sin e the matrix B is non-degenerate, we nd that the ansatz C = B 1 K solves (D.3.19) provided K + K T = 0. Then from (D.3.17) we obtain the matrix A in the form (D.3.24). Finally, from (D.3.24), (D.3.25) we get CA

=

B

1

KB

1

B

1

KB

1

KB

1

KB

1

:

(D.3.29)

The right-hand side is obviously antisymmetri (i.e., the sign is

hanged under transposition). Hen e equation (D.3.18) is satis ed identi ally. If, instead, we start from a non-degenerate A, then the analb A 1 solves (D.3.18), whereas B , be ause ogous ansatz C = K of (D.3.17), is found to be in the form of (D.3.26). This time, equation (D.3.19) is full lled be ause of (D.3.26) and (D.3.27). In the ase when both A and B are non-degenerate, the formulas (D.3.24), (D.3.25) and (D.3.26), (D.3.27) are merely two alternative representations of the same solution. By using (D.3.28), one an re ast (D.3.24) and (D.3.25) into (D.3.26) and (D.3.27), and vi e versa. However, if det A = 0 and det B 6= 0, then the B-representation (D.3.24), (D.3.25) an be used for the solution of the problem. In the opposite ase, i.e., for det A 6= 0 and det B = 0, we turn to the A-representation (D.3.26) and (D.3.27). In these ases the equivalen e of both sets, mediated via (D.3.28), is removed. The totally degenerate ase will be treated in the next subse tion. It will be useful to write the regular solution exli itly in omponents. As usual, we denote the omponents of the matri es as ab a B = Bab ; A = A ; C = C b , and the omponents of the inverse matri es as (B 1 ) = B ab and (A 1 ) = Aab . We introdu e the b =K b ab . Then the antisymmetri matri es by K = Kab and K

omponent version of the B -representation (D.3.24), (D.3.25)

KK

324

D.3.

Symmetry swit hed on additionally

reads: A

ab

a

C b

=

B

=

B

ab ab

B

+

= B a K b

am

d

Km B Kdn B

1 (k 2 B ab det B

= B a ^ bd k d =

1 det B



a d

nb

a

b

k k );

B b kd :

(D.3.30)

Aab

(D.3.31)

Cab

Here we introdu ed k a := 21 ab Kb and ka := Bab k b , moreover, k 2 := ka k a . Analogously, the A-representation (D.3.26), (D.3.27) reads: Bab

a

C b

=

Aab

=

Aab

b Aam K +

b a A b =K

m

b A d K

1 2 (b k Aab det A

= Ab a d b kd =

dn

Anb

b ka b k b );

1 d ^ bd Aa b k ; det A

(D.3.32)

Bab

(D.3.33)

Cab1

1 b b , bka := Aab b kb , and b k 2 := b ka b ka . where b ka := 2 ^ab K

Degenerate solution Besides the regular ase of above, the losure and symmetry relations also admit a degenerate ase when all the matri es are singular, i.e. det A = det B = 0:

(D.3.34)

Re all that we always have det C = 0, see (D.3.21). We will not give here a detailed analysis of the degenerate ase be ause, in a ertain sense to be explained below, it redu es to the regular solution. Nevertheless, let us outline the main steps whi h yield the expli it onstru tion of the degenerate solution. The basi tool for this will be the use of the \gauge" freedom of the system (D.3.16)-(D.3.19) whi h is obviously invariant under

zeroABdet

D.3.3

Algebrai solution of the losure and symmetry relations

the a tion of the general linear group by

GL(3; R )

3 Lb a de ned

! L a Ldb A d ; ! (L 1 )a (L 1)b d B d; ! L a (L 1)b d C d :

Aab Bab a

C b

325

(D.3.35)

gl3ABC

This transformation does not hange the determinants of the matri es and hen e, by means of (D.3.35), the degenerate solutions are mapped again into degenerate ones. We an use the freedom (D.3.35) in order to simplify the onstru tion of the singular solutions. The vanishing of a determinant det B = 0 means that the algebrai rank of the matrix B is less than 3 (and the same for A). A rather lengthly analysis then shows that the system (D.3.16) to (D.3.19) does not have real solutions when the matri es A or B have rank 2. As a result, we have to admit that both, A and B , arry rank 1. Then dire t inspe tion shows that the degenerate matri es A and B an be represented in the general form Aab

=

vavb;

Bab

= ua ub :

(D.3.36)

ABdeg

We substitute (D.3.36) into (D.3.17). This yields, for the square of the C matrix, C a C b = Æba + (v u ) v a ub . Taking into a

ount the onstraint Tr(C 2 ) = 2, whi h arises from (D.3.23), we nd the general stru ture of C 2 as C a C b

=

Æba

+ v a ub ;

with

v u

= 1:

(D.3.37)

If we multiply (D.3.37)1 with ua and v b , respe tively, we nd that ua and v b are eigenve tors of C 2 with eigenvalues zero; this is ne essary for the validity of the equations (D.3.18) and (D.3.19). It remains to nd the matrix C as the square root of (D.3.37)1. Although this is a rather tedious task, one an solve it with the help of the linear transformations (D.3.35). It is always possible

Csquare

326

D.3.

Symmetry swit hed on additionally

to use (D.3.35) and to bring the olumn v a and the row ua into the spe i form

011 oa = 0 A

v

1

;

o

ua

= (0; 0; 1) :

(D.3.38)

va0

(D.3.39)

Cdeg

Then (D.3.37) an be solved expli itly and yields o C

0 a b =

0 1 0 1 0 1 0 0 0

1 A

:

Summarizing, the general degenerate solution is given by the o o matri es A; B of (D.3.36), with v a = Lb a vb , ua = (L 1 )a b ub , o and the matrix C a b = L a (L 1 )b d C d . An arbitrary matrix a Lb 2 GL(3; R ) embodies the 9 degrees of freedom of this solution.

D.3.4

From a quarti wave surfa e to the light one

After losure and symmetry have been taken are of in the last se tion and the expli it form of the matrix J been determined, we an ome ba k to the Fresnel equation (D.1.55) and its M oeÆ ients (D.1.60) to (D.1.64). The latter an now be al ulated. The regular and the degenerate solutions should be onsidered separately. Let us begin with the regular ase.

Regular ase Starting from (D.3.30)-(D.3.31) or from (D.3.32)-(D.3.33), dire t al ulation yields for the oeÆ ients of the Fresnel equation

D.3.4

From a quarti wave surfa e to the light one

327

(D.1.60)-(D.1.64): 



2 2 1 k M = 1 det B det B = det A;   2 k 1 a a 4k 1 M = det B det B = 4bka ;   2 1 k ab a b ab M = 4k k + 2 B 1 det B det B 6 ba bb k k ; = 2Aab + det A ab M = 4 B (ab k )   4 1 ba bb b (ab b ) ; = A k + k k k det A det A ab d M = (det B ) B (ab B d)

=

1 det A

A

(ab

A

d)

(D.3.40)

rB1

(D.3.41)

rA1

(D.3.42)

rB2

(D.3.43)

rA2

(D.3.44)

rB3

(D.3.45)

rA3

(D.3.46)

rB4

(D.3.47)

rA4

(D.3.48)

rB5

2A(ab bk bkd) bka bkb bk bkd : + det A det A (D.3.49)

rA5

!

Here every M is des ribed by two lines, the rst one displaying the expression in terms of the B -representation whereas the se ond line refer to the A-representation. Substituting all this into the general Fresnel equation (D.1.55), we nd

W= = =

2 q0

"

det B 2 q0

det A 2 q0

 2 q0

1

" 2 q0

k

2



det B

det A + 2q0 qa bka

qi qj g

ij


2

= 0:

#2

2q0 qa ka qa qb

qa qb

ab

A

(det B )B ab a kb b k b

! #2

det A (D.3.50)

Here g ij is the symmetri tensor eld. Let, for de nitness, det B > 0. Hen e, det A < 0. Then from (D.3.50) we read o the om-

Wgij

328

D.3.

Symmetry swit hed on additionally

ponents g

ij

1 =p det B =



1

1 det A

p

(det B )

1

ka

det A b ka

B d k k d

(det B ) B ab (D.3.51)

gijBup

(D.3.52)

gijAup

!

b kb

Aab



kb

+ (det A) 1 b ka b kb

One an prove that this tensor has Lorentz signature. Hen e it

an be understood as the metri of spa etime. Thus, from our general analysis, we indeed re over the null or light- one stru ture qi q i = qi qj g ij = 0 for the propagation of ele tromagneti waves: Provided the linear spa etime relation satis es losure and symmetry, the quarti surfa e in (D.1.55) redu es to the light one for the metri g ij . A visualization of the

orresponding onformal manifold is given in Fig. D.3.4.

Degenerate ase The same on lusion is also true for the degenerate ase. Substituting (D.3.36) into (D.1.61)-(D.1.64), and using (D.3.37)(D.3.39), we nd a = 0; M = 0; ab a b M = 4v v ; ab M = 4ueC (a f v b  )ef ; ab d M = ue C (a f bjef ug C j h d)gh :

(D.3.53) (D.3.54) (D.3.55) (D.3.56)

M

Inserting this into (D.1.55), we obtain

W=

2

q0



2q0 qa v a + qa qb ue C a f bef

2

=

2

q0 qi qj g

ij


2

= 0: (D.3.57)

This time the tensor eld g ij is des ribed by g

ij



=

0

va

vb

u C (a d b) d



:

(D.3.58)

This tensor is nondegenerate and it has Lorentzian signature.

gijDEG

D.3.4

From a quarti wave surfa e to the light one

329

Figure D.3.1: Null ones tted together to form a onformal manifold, see Pirani and S hild [25℄.

330

D.3.

Symmetry swit hed on additionally

Is losure also ne essary for the light one?

The losure property is suÆ ient for the light one to exist. Is it also a ne essary ondition for this? We an demonstrate some eviden e in favor of the onje ture that the losure relation is not only a suÆ ient, but also a ne essary ondition for the redu tion of the quarti geometry (D.1.55) to the light one. For the spe i ase when the matrix C = 0, we will be able to prove also ne essity. Putting C a b = 0, we nd from (D.1.61)-(D.1.64) that M a = 0 and M ab = 0, whereas

M ab = B d (Aab A d Aa Abd ); M ab d = (det B ) A(ab B d) : Consequently, (D.1.55) redu es to

W =q

2 0




det A q04 + q02 + det B  = 0;

(D.3.59) (D.3.60) (D.3.61)

where  := Aab qa qb , := B ab qa qb , and := M ab qa qb . Assuming that the last equation des ribes a light one, one on ludes that the roots for q02 should oin ide. Thus ne essarily

2 = 4 det A det B  :

(D.3.62)

abg

Let us write (det A det B ) = s j det A det B j, with s = sign(det A det B ). Then (D.3.62) yields 2

j det A det B j  = s ;

p

2

j det A det B j = 1 ;

p

(D.3.63)

where  is an arbitrary s alar fa tor. Re alling the de nitions of ; ; , we then nd

Aab = s 2 B ab :

(D.3.64)

AB

Consequently, M = det A = s 6 = det B and M ab = 24 B ab . Therefore one veri es that

q W = sdet q B 2 2 0

2 2 0

+ s qa qb B ab det B


2

= 0:

(D.3.65)

quad

D.3.4

From a quarti wave surfa e to the light one

331

For s = 1, we immediately re ognize that the quadrati form in (D.3.65) an have either the (+ ) signature, or (+ + + ). Similarly, for s = 1, the signature is either (+ + ++), or (+ + ). Therefore, the Fresnel equation des ribes the orre t (hyperboli ) light one stru ture only in the ase s = 1. Finally, one an verify that the above solutions satis es ij mn mn kl

= s2 Æijkl ;

(D.3.66)

whi h reprodu es the losure relation (D.2.6)1 for s = 1.

332

D.3.

Symmetry swit hed on additionally

D.4

Extra ting the onformally invariant part of the metri by an alternative method

The dis ussion above of the wave propagation in linear ele trodynami s shows that the losure and symmetry relations enable us to obtain the spa etime metri from the spa etime relation. Here we will present an alternative onstru tion of the Lorentzian metri from the onstitutive oeÆ ients. As above, the ru ial point will be a well-known mathemati al fa t. Here it is the one-to-one orresponden e between the duality operators # and the onformal lasses of the metri s of spa etime. Hen e, if we take as fth axiom the linearity ondition (D.1.6) together with

losure (D.2.6) and symmetry (D.3.5), then we an onstru t, up to a onformal fa tor, the metri of spa etime from the omponents of the duality operator #. In this sense, the metri is a derived on ept from the ele trodynami spa etime relation. At the enter of this derivation is the triplet of (anti)self-dual 2-forms whi h provides the basis of the (anti)self-dual subspa e of the omplexi ed spa e of all 2-forms 6 (C ).

M

334

D.4.1

D.4.

Extra ting the onformally invariant part of the metri by an alternative method

Triplet of

self-dual 2-forms and metri

In Se . C.2.5 we saw that the basis of the self-dual 2-forms an be des ribed either by (C.2.45) or by (C.2.46). These triplets are linearly dependent, and one an use any of them for the a tual omputation. For example, one an take as the funda(s)

mental triplet S (a) = a , with Gab = Aab =4. Alternatively, we (s)

an work with S (a) = B ab  b , with Gab = B ab =4. Both hoi es yield equivalent results if A and B are non-degenerate. For definiteness, let us hoose the se ond option. Then from (C.2.46), we have in the B -representation expli itly (s) 1 S (a) = Sij(a) dxi ^ dxj = B ab  b 2 1 i dx0 ^ dxa + i(det B ) 1 kb dxb ^ dxa (D.4.1) = 2  1 + B ab ^b d dx ^ dxd : 2 Here x0 =  and xa are the spatial oordinates, with a; b; ;


= 1; 2; 3. A useful al ulational tool is provided by a set of 1-forms Si(a) := i S (a) = Sij(a) dxj . With their help, we an read o the ontra tions needed in the S honberg-Urbantke formulas (C.2.73)-(C.2.74) from the exterior produ ts 1 (a) (b) ( ) k Si(a) ^ S (b) ^ Sj( ) = Sik Slm Snj dx ^ dxl ^ dxm ^ dxn 2 1 klmn (a) (b) ( ) Sik Slm Snj Vol: (D.4.2) =  2 Here, as usual, Vol = dx0 ^ dx1 ^ dx2 ^ dx3 . From (D.4.1) we have i a S0(a) = dx ; (D.4.3) 2 i dx0 Æba + e B a ^ bd dxd Sb(a) = 2  + (det B ) 1 k dx Æba (det B ) 1 kb dxa : (D.4.4)

Sforms1

SwSwS

Triplet of self-dual 2-forms and metri

D.4.1

335

By a rather lengthy al ulation we nd Gab S

(a)

^ab S0(a) ^ S (b)

^S b ^S

( )

=

( ) 0

=

^ab Sn(a) ^ S (b) ^ S0( ) = (a) ^ab Sm ^ S (b) ^ Sn( ) =

6i Vol; 3i Vol; 4 3i (det B ) 1 kn Vol; 4 3i  (det B ) 1 Bmn 4  + (det B ) 2 km kn Vol:

(D.4.5)

det

(D.4.6)

S00

(D.4.7)

Sn0

(D.4.8)

Smn

We use (D.4.5)-(D.4.8) together with (D.4.2). Then, the S honberg-Urbantke formulas (C.2.73)-(C.2.74), for the omponents of the spa etime metri , yield





1 det B kb gij = p : (D.4.9) ka Bab + (det B ) 1 ka kb det B One

an ndpthe determinant of this expression and verify that p det g = 1, in a

ordan e with (C.2.74). (i det g ) = If, instead of the B -representation (D.3.30), (D.3.31) of the solution of the losure and symmetry relations, we start from the A-representation (D.3.32), (D.3.33), then the S honbergUrbantke formulas yield an alternative form of the spa etime metri , gij

=

p

1 det A

1

(det A)

1

bka

A d b k b kd

bkb

gijB

!

:

(det A) Aab (D.4.10)

The triplet of the self-dual 2-forms S (a) is de ned up to an arbitrary s alar fa tor: By multiplying them with an arbitrary (in general omplex) fun tion h(x), one preserves the ompleteness

ondition (C.2.52), (C.2.53). Correspondingly, the determinant of the metri will be res aled by a fa tor h4 , whereas the metri itself will be res aled by a fa tor h. In other words, the whole pro edure de nes a onformal lass of metri rather than a metri itself. Clearly, one an always hoose the onformal fa tor h su h as to eliminate the rst fa tor in (D.4.9).

gijA

336

D.4.

Extra ting the onformally invariant part of the metri by an alternative method

For the metri (D.4.9), the spa etime interval expli itly reads:

"



1 ds2 = p det B d det B

ka dxa det B

2

#

Bab dxa dxb :

(D.4.11)

gij1

However, as just dis ussed, a onformal fa tor is irrelevant. Therefore we may limit ourselves to the interval 2



ds = det B d 0

ka dxa det B

2

Bab dxa dxb :

(D.4.12)

If we onsider the 3  3 matrix B , we an distinguish four dierent ases with the signatures (+ + +), ( + +), ( +), and ( ), respe tively. Let us demonstrate that all these

ases lead to a Lorentzian signature: For (+ + +), the determinant be omes positive and the expression Bab dxa dxb is positive de nite. Thus, we an read o the Lorentzian signature immediately:  will be the time oordinate, the xa 's the three spatial oordinates. For ( + +), the determinant be omes negative and the expression Bab dxa dxb is inde nite. Therefore the square of one dx

oordinate dierential, say (dx1 )2 , arries a positive sign and an be identi ed as the time oordinate, whereas  an be identi ed as spa e oordinate. For ( +), the determinant be omes positive again and the expression Bab dxa dxb is inde nite with signature (+ + ). Therefore x3 is the time oordinate in this ase. For ( ), the determinant be omes negative and the expression Bab dxa dxb is positive de nite. In other words, this

orresponds to the ase (+ + +) with an overall sign hange, q.e.d.. A

ordingly, the stru ture in (D.4.12) is rather robust and the quantity ka doesn't in uen e the Lorentzian signature of (D.4.12). It is quite satisfa tory to note that the S honberg-Urbantke formalism produ es the same Lorentzian metri that we re overed earlier in Se . D.3.4 when dis ussing the redu tion of the

Yur2

D.4.2

Maxwell-Lorentz spa etime relation and Minkowski spa etime

337

fourth order wave surfa e to the light one. In that approa h, sin e the wave ve tor is a 1-form, we obtained the ontravariant

omponents (D.3.51) and (D.3.52) of the metri whi h are just the inverse to (D.4.9) and (D.4.10), respe tively. However, our new \redu tion" method produ es a spa etime metri also for the degenerate solution of the losure relation, see (D.3.58). The S honberg-Urbantke formulas are inappli able to the degenerate solutions. In this sense, the redu tion method is further rea hing. D.4.2

Maxwell-Lorentz spa etime relation and Minkowski spa etime

Let us take a parti ular example for the spa etime relation. We assume that we are in a suitable frame su h that we measure 1 1 1 H01 = F23 ; H02 = F31 ; H03 = F12 ; 0 0 0 (D.4.13) H23 = "0 F01 ; H31 = "0 F02 ; H12 = "0 F03 : This set is known as the Maxwell-Lorentz spa etime relation in the frame hosen. The onstitutive oeÆ ients (D.1.6), for the example dis ussed, are independent of the spa etime oordinates and are given by (D.4.13) in terms of a pair of fundamental

onstants "0; 0, with (a; b; = 1; 2; 3)  H   H=l2  q2 l Am 1 m SI ["0℄ = F ab = F=(tl) = h t = = ; (D.4.14) Vs s 0

F   2  h l SI V m F=l m [0℄ = Hab = H= = = =

: (D.4.15) 2 (tl) q t As s 0 We an read o ij kl from (D.4.13). Then a dire t omputation shows that the quadrati invariant (D.2.7) is equal to SI 1= . Consequently, the dual2 = "0 =0 , with [℄ = q 2 =h = ity operator # of (D.3.11), as de ned by the spa etime relation (D.4.13), is given by the 6  6 matrix  1 13  0 K

; (D.4.16) #I = 1 0 3

exCR

epsmu'

ex1

338

D.4.

Extra ting the onformally invariant part of the metri by an alternative method

p

f. (D.1.6). Here we denoted := 1= "0 0 , with [ ℄ = velo ity. Thus we have the matri es A = B 1 , B = 13 , whereas C = 0. One then immediately nds the metri (D.4.9), (D.4.10) in the form: 0

gij

0 1 0 0

2

B 0

p1 B 

=

0 0

0 0 1 0

1

0 0C C: 0A 1

(D.4.17)

This is the standard Minkowski metri in orthonormal oordinates. D.4.3

Hodge star operator and isotropy

The inverse of (D.4.9) is given by g

ij

1 =p det B



1

(det B )

1

ka

kb

k2

(det B )B ab

 :

(D.4.18)

With the help of (D.4.9) and (D.4.18), we an de ne the orresponding Hodge star operator ? atta hed to the metri extra ted by means of the S honberg-Urbantke formalism. Its a tion on a 2-form, say F , is des ribed by (C.2.92), or expli itly, by ?

Fij

:=

p

2

g

^ijkl g

km g ln F : mn

(D.4.19)

Su h a Hodge duality operator, see (C.2.33), has the spa etime g g relation matrix ij kl = ^ijmn  mnkl =2 with g

 ijkl g

 ijkl

=

p

ik jl

g g g

g

jk g il
:

(D.4.20)

is invariant under onformal transformations gij ! this takes are that only 9 of the possible 10 g

omponents of the metri an ever enter  ijkl . g o We an ompare ijkl with the original ijkl of the linear spa etime relation. For this purpose, we have to substitute the metri

Note that

e(x) gij ;

metri inv

D.4.3

Hodge star operator and isotropy

339

omponents (D.4.9) into the A; B; C blo ks (C.2.95)-(C.2.97) of the Hodge duality matrix (C.2.94). Then inspe tion of (D.3.30)(D.3.33) demonstrates the exa t oin iden e g

A ab = Aab ;

g

g

C ab = C ab:

B ab = Bab ; g

(D.4.21)

o

the metri extra ted allows us to write the original duality operator # as Hodge star operator asso iated to that metri . Therefore, the

Summing up, it turns out that  IJ =  IJ , i.e.,

original linear onstitutive tensor (D.1.13) an then be nally written as

ijkl =  (x)

p g gik gjl gjk gil
+ (x) ijkl :

(D.4.22)

de omp1

This representation naturally suggests to interprete  (x) as a s alar eld of a dilaton type.1 Given a metri , we an de ne the notion of lo al isotropy. Let i 1 :::ip T be the ontravariant oordinate omponents of a tensor eld and T 1 :::p := ei1 1


eip p T i1 :::ip its frame omponents with respe t to an orthonormal frame e = ei  i . A tensor is said to be lo ally isotropi at a given point, if its frame omponents are invariant under a Lorentz rotation of the orthonormal frame. Similar onsiderations extend to tensor densities. There are only two geometri al obje ts whi h are numeri ally invariant under (lo al) Lorentz transformations: the Minkowski metri o = diag(+1; 1; 1; 1) and the Levi-Civita tensor density  Æ . Thus

T  Æ = (x) p g g g Æ g gÆ




+ '(x)  Æ

(D.4.23)

iso

is the most general lo ally isotropi ontravariant fourth rank tensor density of weight +1 with the symmetries T  Æ = T  Æ = T  Æ = T Æ . Here  and ' are s alar and pseudo-s alar elds, respe tively. A

ordingly, in view of (D.4.22), we have proved that the

onstitutive tensor (D.1.13) with the losure property (D.3.14) is lo ally isotropi with respe t to the metri (D.4.9). 1 In the low-energy string models this fa tor is usually written as  (x) = e b(x) , with a onstant b and the dilation eld (x).

340

D.4.

Extra ting the onformally invariant part of the metri by an alternative method

Covarian e properties

D.4.4

The dis ussion above was on ned to a xed oordinate system. However, one may ask: What happens when the oordinates are

hanged? Or, more generally, when a lo al ( o)frame is transformed? Up to now, we onsidered only a holonomi oframe # = Æi dxi , but in physi ally important ases one often needs to go to nonholonomi oframes. In this se tion we will study the

ovarian e properties of the S honberg-Urbantke onstru tion. The behavior of the metri (C.2.73)-(C.2.74) under oordinate and frame transformations is by no means obvious. Although the fundamental ompleteness relation (C.2.52), (C.2.53) looks

ovariant, one should re all that the index (a) of the self-dual S forms omes, in a non- ovariant manner, from the 3 + 3 split of the basis BI . A transformation of the spa etime oframe (A.1.94) (a) a ts on both types of indi es in the oeÆ ients S whi h enter the S honberg-Urbantke formulas (C.2.73)-(C.2.74). Thus, the determination of the new omponents of the metri with respe t to transformed frame be omes a nontrivial problem. For the sake of generality, we will onsider an arbitrary linear transformation (A.1.94) of the oframe whi h in ludes the holonomi oordinate transformation as a parti ular ase. With respe t to the original oframe # = Æi dxi , the duality operator is des ribed by the omponents of the spa etime matrix #  = Æi Æ j Æk Æl #ij kl . A

ordingly, the omponents of the extra ted spa etime metri read g = Æi Æ j gij . Be ause of the tensorial nature of the de nition of the duality operator (D.1.17), a linear transformation (A.1.94) of the basis, # = L  # , yields the orresponding transformation of the duality omponents, 0

0

#   = L  L L  L  #  : 0

0

0

0

0

0

0

0

(D.4.24)

Re all that L  is the inverse of L  . We will now demonstrate that the S honberg-Urbantke onstru tion is ompletely ovariant and that the extra ted metri 0

0

dualtrafo3

D.4.4

Covarian e properties

341

(C.2.73) transforms as

1

g 00 0 = (det L) 2 L0  L 0 g

(D.4.25)

gijL

under the linear transformation (A.1.94), (D.4.24).

The proof is te hni ally simple and straightforward, but it is somewhat lengthy. We have prepared the ne essary tools in Se . A.1.10. Combining now the matrix equations (A.1.97), (A.1.100), and (C.2.37), we nd that, with respe t to the new oframe, the duality operator



#

0 ^0



=



C 0 A0 B0 C 0 T

is des ribed by the new matrix omponents   T  0  1 C A0 C Q WT = T T T 0 0 P B B C det L Z



A CT

0 ^0





P W Z Q



:

(D.4.26)

dualB1

(D.4.27)

new oeff

This is the dire t matrix remake of the tensorial version (D.4.24). The matri es P; Q; W; Z are des ribed in (A.1.98) and (A.1.99). Expli itly, eq. (D.4.27) yields





A0 = (det L) 1 QT AQ + W T BW + QT CW + W T C T Q ;

  B 0 = (det L) 1 P T BP + Z T AZ + Z T CP + P T C T Z ;   C 0 = (det L) 1 QT CP + W T C T Z + QT AZ + W T BP :

(D.4.28)

A1

(D.4.29)

B1

(D.4.30)

C1

It is onvenient to onsider the three sub ases of the linear transformation (A.1.103)(A.1.105), be ause their produ t (A.1.102) des ribes a general linear tranformation. For L = L1 , we have (A.1.106) with only the matrix W ab = ab U being nontrivial. Then (D.4.28)-(D.4.30) yields A0 = A + W T BW + CW + W T C T ;

B 0 = B; (D.4.31) C 0 = C + W T B: Comparing this with the B -representation (D.3.24) and (D.3.25), we see that the trans-

formation (D.4.31) means a mere shift of the antisymmetri matrix: K 0 = K + BW T B; or; equivalently; k0 a = ka (det B ) Ua : Substituting this into (D.4.9), we nd the transformed metri :



(D.4.32)

ABC1

K ase1



1 0 (det B ) Ub (det B ) Ua ka Ub kb Ua + (det B ) Ua Ub : (D.4.33) gij2 det B Comparing with (A.1.103), we get a parti ular ase of (D.4.25): g0 0 0 = L0  L 0 g with L = L1 : (D.4.34) gij1L When L = L2 , the matrix Zab = ^ab V des ribes the ase (A.1.107). Then, from (D.4.28)-(D.4.30), we nd:

g0  = g + p

A0 = A; B 0 = B + Z T A Z + Z T C + C T Z; C 0 = C + AZ:

(D.4.35)

ABC2

342

D.4.

Extra ting the onformally invariant part of the metri by an alternative method

Contrary to the rst ase, it is now more onvenient to pro eed in the A-representation. From (D.3.26) and (D.3.27), we then see that the transformation (D.4.35) means a mere shift of the antisymmetri matrix 0 = bka (det A) Aab V b : b0 = K b + AZA; or; equivalently; b K ka (D.4.36) Substituting this into (D.4.10), we obtain the transformed metri : 1  2bk V (det A) A d V V d (det B ) Abd V d  : (D.4.37) g 0 = g + p (det A) Aa V 0 det A We ompare with (A.1.104) and re over a sub ase of (D.4.25), with L = L2 : (D.4.38) g 0  = L  L g Finally, for L = L3 we have (A.1.108). Then (D.4.28)-(D.4.30) redu e to det   1 A( 1 )T ; A0 = 0 0 0 0 T (D.4.39) B0 = det   B ; C 0 =  1 C : For the antisymmetri matrix K , this yields: 0 0 T K ; or; equivalently; k0 a =  0 ( 1 ) a kb : (D.4.40) K0 = 0 b det  As we saw above, the analysis of the ase (A.1.106) was easier in the B -representation, whereas the A-representation was more suitable for the treatment of the ase (A.1.107). However, the last ase (A.1.108), (D.4.39) looks the same in both pi tures. For de niteness, let us hoose the B -representation. A new and nontrivial feature of the present

ase is that the transformation L3 is not unimodular, det L3 = 0 0 det  6= 1. Re all that det L1 = det L2 = 1. As a onsequen e, one should arefully study the behavior of the determinant of the metri de ned by the se ond Urbantke formula (C.2.74). From (A.1.100) we have the transformation of the self-dual basis (C.2.46): (s)0 1  b (s)  :  a= (D.4.41) det  a b Hen e, for the S -forms (D.4.1), we nd (a) (D.4.42) = 1 0 ( 1 )b a S (b) ; S0 0 and, onsequently, (a) B 0 ab S 0 ^ S 0 (b) = det1L Bab S (a) ^ S (b) : (D.4.43) 3 Using this in (C.2.74), one nally proves the invarian e of the determinant: p p det g0 = det g: (D.4.44) Taking this result into a

ount when substituting (D.4.39) and (D.4.40) into (D.4.9), one obtains the transformed metri   1 (0 0 )2 det B 0 0 b d k d   : g 0 = p 0

d 1 0 a k a b B d + (det B ) k kd det B det L3 (D.4.45) Thus, the metri transforms as a tensor density, 1 L L g g 0 = p with L = L3 : (D.4.46) det L   0

0

0

0

0

0

0

0

K ase2

gij2a

gij2L

ABC3

K ase3

invdet

gij3

gij3L

D.4.4

Covarian e properties

343

This transformation law is ompletely onsistent with the invarian e of the determinant (D.4.44). Turning now to the ase of a general linear transformation, one an use the fa torization (A.1.102) and perform the three transformations (A.1.103)-(A.1.105) one after another. This yields subsequently (D.4.34), (D.4.38), and (D.4.46). The nal result is then represented in (D.4.25) with an arbitrary transformation matrix L.

Redu ing degenerate ase to the regular one Using the formalism above, we an show that the degenerate solutions of the losure relation, A; B; C , an always be transformed into the regular on gurations. Indeed, let us take the degenerate solution (D.3.36), (D.3.38), (D.3.39), whi h is expli itly given by the three matri es o

A=

01 0

0 1 0 0 1 0 1

00 1 A ; Bo =  0

0 0 0 0 0 0 1

0 1 A ; Co = 

1 A

0 1 0 1 0 1 : 0 0 0 (D.4.47)

ABCdeg

Consider a simple transformation of the oframe # = L  # by means of the L = L1 matrix (A.1.103) with Ua = (0; 0; 1). This indu es the transformation of the 2-form basis (A.1.97) where the matrix W of (A.1.106) reads expli itly 0

0

00 W ab =  0 0

0 0 0 1 1 0

1 A:

(D.4.48)

Wdeg

The other matri es are P = Q = I3 and Z = 0. Then, using (D.4.31), we nd the transformed spa etime relation matri es as o0

A =

01 0

0 0 1 0 0 0 1

1 00 o0 A; B =  0

0 0 0 0 0 0 1

1 0 o0 A; C = 

1 A

0 1 0 1 0 0 : 0 0 0 (D.4.49) ABCdeg1

With respe t to the transformed basis, as we immediately see, o the onstitutive matri es be ome non-degenerate, det A0 6= 0, and we an use the A-representation to des ribe this on guration and to onstru t the orresponding metri of spa etime.

344

D.4.

Extra ting the onformally invariant part of the metri by an alternative method

Evidently, the general degenerate solution an also be redu ed to the regular ase. Then the linear transformation above, with L = L1 , should be supplemented by the appropriate GL(3; R ) transformation (D.3.35). We thus on lude that the separate treatment of the degenerate ase is in fa t not ne essary: The degenera y (D.3.34) merely re e ts an unfortunate hoi e of the frame whi h an easily be removed by a linear transformation.

D.5

Fifth axiom

Summarizing the ontent of Part D, we an now give a lear formulation of the fth axiom. In Maxwell{Lorentz ele trodynami s, the 2-forms of the ele tromagneti ex itation H and the eld strength F are related by the universal linear law 1 (D.5.1) H = (F ) or Hij = ij kl Fkl ; 2 with the linear operator

(a  + b ) = a () + b ( ) ;

5 hiHF

(D.5.2)

this operator ful lls losure

2 =

2 16

or

ij mn mn kl

=

kl ; 2 Æij

(D.5.3)

5 lose

and symmetry

() ^ =  ^ ( ) or

ijmn mn kl

= klmn mn ij : (D.5.4)

Linearity, losure,1 and symmetry provide a unique light one stru ture for the propagation of ele tromagneti waves, see Fig. 1 With (D.5.3) and the understanding that H and F live also in the M , we ould 1 6 H= 16 F . For the square-root of su h a negative

write (D.5.1)1 symboli ally as

p

unit matrix, see Gantma her [5℄ pp.214 et seq.

5sym

346

D.5.

Fifth axiom

D.3.4. As a result, the spa etime metri with the orre t Lorentzian signature is, up to a onformal fa tor, re onstru ted from the

onstitutive oeÆ ients ij kl. The Maxwell{Lorentz ele trodynami s is spe i ed, among other viable models, by a vanishing axion eld 1 (no axion) ; (D.5.5)  = Tr  = 0 6 and a onstant universal dilation fa tor 1 Tr(2 ) = onst (no dilaton) : 2 = (D.5.6) 6 We ould have ex luded the axion eld alternatively by insisting on ele tri -magneti re ipro ity of the linear law in the rst pla e instead of only assuming losure, as in (D.5.3). Then, making use of the metri extra ted and of the orresponding Hodge star operator ?, the Maxwell-Lorentz spa etime relation an be written as H =  ?F

( fth axiom) ;

or, in omponents, as

p  Hij = ^ijmn g g mk g nl 2




g nk g ml Fmn = ^ij kl

(D.5.7)

onstva

p gF : kl

(D.5.8)

onstva '

Here  is a universal onstant with the dimension of an impedan e. Its value is r "0 1 : =  (D.5.9) 0 377

A

ordingly, the experimentally well-established Maxwell-Lorentz ele trodynami s is distinguished from other models by linearity, losure, symmetry, no axion, and no dilaton. Generalizations are obvious. We will dis uss nonlinearity and nonlo ality in Chap. E.2.

Referen es

Complex 2-forms representation of the Einstein equations: The Petrov Type III solutions, J. Math. Phys. 12 (1971) 1616-1619.

[1℄ C.H. Brans,

[2℄ R. Capovilla, T. Ja obson, and J. Dell,

without the metri , Phys.

Rev. Lett.

[3℄ L. Cooper and G.E. Stedman,

lasers, Phys.

Lett.

General relativity

63 (1989) 2325-2328.

Axion dete tion by ring

B357 (1995) 464-468.

Cosmologi al magneti elds from primordial heli ity, Phys. Rev. D62 (2000) 103008, 5

[4℄ G.B. Field and S.M. Carroll, pages. [5℄ F.R. Gantma her, Theorie

(VEB

Matrizenre hnung.

Deuts her

Verlag

Teil I: Allgemeine

der

Wissens haften:

Berlin, 1958).

On the derivation of the spa etime metri from linear ele trodynami s, Phys. Lett. A

[6℄ A. Gross and G.F. Rubilar,

(2001) in print; Los Alamos Eprint Ar hive gr-q /0103016.

348

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Le ons sur la propagation des ondes et les equations de l'hydrodynamique (Hermann: Paris, 1903).

[7℄ J. Hadamard,

[8℄ G. Harnett,

Metri s and dual operators, J. Math. Phys. 32

(1991) 84-91.

The bive tor Cliord algebra and the geometry of Hodge dual operators, J. Phys. A25 (1992) 5649-5662.

[9℄ G. Harnett,

On the experimental foundations of the Maxwell equations, Ann. Physik (Leipzig) 9

[10℄ M. Haugan and C. L ammerzahl,

(2000) Spe ial Issue, SI-119{SI-124.

A hiral alternative to the vierbein eld in general relativity, Nu l. Phys. B357 (1991) 211-221.

[11℄ G. 't Hooft,

Ele tromagneti permeability of the va uum and light- one stru ture, Bull. A ad. Pol. S i., Ser. s i. phys.

[12℄ A.Z. Jad zyk,

et astr. 27 (1979) 91-94.

Parity and time-reversal symmetry breaking, singular solutions, and Fresnel surfa es, Phys. Rev. A43 (1991) 5665-5671.

[13℄ R.M. Kiehn, G.P. Kiehn, and J.B. Roberds,

[14℄ E.W. Kolb and M.S. Turner,

The Early Universe (Addison-

Wesley: Redwood, 1990) Chapter 10: Axions.

[15℄ A.

Kovetz,

Ele tromagneti Theory

(Oxford

University

Press: Oxford, 2000).

[16℄ A.

Li hnerowi z,

physi s,

Relativity theory and mathemati al

\Astro si a e osmologia gravitazione quanti e relativita", Centenario di Einstein (Giunti Barbera: in:

Firenze, 1979.)

[17℄ J.E. Moody and F. Wil zek,

Rev. D30

[18℄ W.-T.

Physi s,

New ma ros opi for es? Phys.

(1984) 130-138.

Ni,

A non-metri theory of gravity, Dept.

Montana

State

University,

Bozeman.

Referen es

349

Preprint De ember 1973.

The paper is available via http://gravity5.phys.nthu.edu.tw/ .

[19℄ W.-T. Ni, Equivalen e prin iples and ele tromagnetism, Phys. Rev. Lett. 38 (1977) 301-304. [20℄ Yu.N. Obukhov and F.W. Hehl, Spa e-time metri from linear ele trodynami s, Phys. Lett. B458 (1999) 466470; F.W. Hehl, Yu.N. Obukhov, and G.F. Rubilar, Spa etime metri from linear ele trodynami s II, Ann. d. Phys. (Leipzig) 9 (2000) Spe ial issue, SI-71{SI-78; Yu.N. Obukhov, T. Fukui, and G.F. Rubilar, Wave propagation in linear ele trodynami s, Phys. Rev. D62 (2000) 044050 (5 pages). [21℄ Yu.N. Obukhov and S.I. Terty hniy, Va uum Einstein equations in terms of urvature forms, Class. Quantum Grav. 13 (1996) 1623-1640. [22℄ R.D. Pe

ei and H.R. Quinn, CP onservation in the presen e of pseudoparti les, Phys. Rev. Lett. 38 (1977) 14401443. [23℄ A. Peres, Ele tromagnetism, geometry, and the equivalen e prin iple, Ann. Phys. (NY) 19 (1962) 279-286. [24℄ Pham Mau Quan, Indu tions ele tromagnetiques en relativite general et prin ipe de Fermat, Ar h. Rational Me h. Anal. 1 (1957/58) 54-80. [25℄ F.A.E. Pirani and A. S hild, Conformal geometry and the interpretation of the Weyl tensor, in: Perspe tives in Geometry and Relativity. Essays in honor of V. Hlavat y. B. Homann, editor. (Indiana University Press: Bloomington, 1966) pp.291-309. [26℄ C. Piron and D.J. Moore, New aspe ts of eld theory, Turk. J. Phys. 19 (1995) 202-216.

350

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[27℄ E.J. Post, The onstitutive map and some of its rami ations, Ann. Phys. (NY) 71 (1972) 497-518. [28℄ G.F. Rubilar, Pre-metri linear ele trodynami s and the dedu tion of the light one, Thesis (University of Cologne: 2001/02). [29℄ V. de Sabbata and C. Sivaram, Spin and Torsion in Gravitation (World S ienti : Singapore, 1994). [30℄ M. S honberg, Ele tromagnetism and gravitation, Brasileira de Fisi a 1 (1971) 91-122.

Rivista

[31℄ P. Sikivie, ed., Axions '98, in: Pro . of the 5th IFT Workshop on Axions, Gainesville, Florida, USA. Nu l. Phys. B (Pro . Suppl.) 72 (1999) 1-240.  [32℄ W. Slebodzi nski, Exterior Forms and Their Appli ations. Revised translation from the Fren h (PWN{Polish S ienti Publishers: Warszawa, 1970). [33℄ G.E. Stedman, Ring-laser tests of fundamental physi s and geophysi s, Reports on Progress in Physi s 60 (1997) 615688. [34℄ I.E. Tamm, Relativisti rystal opti s in relation with the geometry of bi-quadrati form, Zhurn. Ross. Fiz.-Khim. Ob. 57, n. 3-4 (1925) 209-224 (in Russian). Reprinted in: I.E. Tamm, Colle ted Papers (Nauka: Mos ow, 1975) Vol. 1, pp. 33-61 (in Russian). See also: I.E. Tamm, Ele trodynami s of an anisotropi medium in spe ial relativity theory, ibid, pp. 19-31; A short version in German, together with L.I. Mandelstam, ibid. pp. 62-67. [35℄ R.A. Toupin, Elasti ity and ele tro-magneti s, in:

Non-

Linear Continuum Theories, C.I.M.E. Conferen e, Bressanone, Italy 1965.

Pp.203-342.

C. Truesdell and G. Grioli oordinators.

Referen es

The lassi al eld theo-

[36℄ C. Truesdell and R.A. Toupin,

ries,

in:

Handbu h der Physik,

351

Vol. III/1, S. Fl ugge ed.

(Springer: Berlin, 1960) pp. 226-793. [37℄ H. Urbantke,

Mills eld,

A quasi-metri asso iated with SU (2) YangXIX (1978) 875-

A ta Phys. Austria a Suppl.

876.

Mathemati al Prin iples of Me hani s and Ele tromagnetism, Part B: Ele tromagnetism and Gravitation

[38℄ C. Wang,

(Plenum Press: New York, 1979). [39℄ S. Weinberg,

A new light boson? Phys. Rev. Lett. 40 (1978)

223-226.

Problem of strong P and T invarian e in the presen e of instantons, Phys. Rev. Lett. 40 (1978) 279-282.

[40℄ F. Wil zek,

Part E Ele trodynami s in va uum and in matter

352

353

le birk/partE.tex, with gures [E01folia.eps, E02born.ps, E03walk.eps, E04matt.eps, E05roent.eps, E06wils.eps℄ 2001-06-01

354

E.1

Standard Maxwell{Lorentz theory in va uum

E.1.1

Maxwell-Lorentz equations, impedan e of the va uum

When the Maxwell-Lorentz spa etime relation (D.5.7) is substituted into the Maxwell equations (B.4.8), (B.4.9), we nd the Maxwell-Lorentz equations d ?F = J= ; dF = 0 (E.1.1) of standard ele trodynami s. The numeri al value of the onstant fa tor (D.5.7) is xed by experiment: r "0 e2 1 = = 2 :6544187283 : (E.1.2)  := 0 4f ~ k

Here e is the harge of the ele tron and f = 1=137:036 is the ne stru ture onstant. The inverse 1= is alled the hara teristi impedan e (or wave resistan e) of the va uum. This is a fundamental onstant whi h des ribes the basi ele tromagneti property of spa etime if onsidered as a spe ial type of medium (sometimes alled va uum, or aether, in the old terminology).

MaxLor

lambda

356

E.1.

Standard Maxwell{Lorentz theory in va uum

In this sense, one an understand (D.5.7) as the onstitutive relations for the spa etime itself. The Maxwell-Lorentz spa etime relation (D.5.7) is universal. It is equally valid in Minkowski, Riemannian, and post-Riemannian spa etimes. The ele tri onstant "0 and the magneti onstant 0 (also

alled va uum permittivity and va uum permeability, respe tively) determine the universal onstant of nature

=

p"100

(E.1.3)

em

whi h gives the velo ity of light in va uum. Making use of the homogeneous Maxwell eld equation F = dA and of (E.1.3), we an re ast the inhomogeneous Maxwell equation (E.1.1)1 in the form 1 "0 d ? F = "0 d ? dA = J:

(E.1.4)

linmax

Re alling the de nition of the odierential (C.2.107), dy := ? d ? and of wave operator (d'Alembertian) (C.2.110), we an rewrite (E.1.4) as


d dy A =

1 ? J: "0

(E.1.5)

waveA

If, using the gauge invarian e, one imposes the Lorentz gauge

ondition dyA = 0, a wave equation is found for the ele tromagneti potential 1-form:

A=

1 ? J with dyA = 0: "0

(E.1.6)

In omponents, the left-hand side reads:

A=



rr

e k e k Ai + Ri f i k Ak



dxi :

(E.1.7)

Here Ri ij := Rkij k is the Ri

i tensor, see (C.1.52). The tilde denotes the ovariant dierentiation and the geometri obje ts de ned by the Levi-Civita onne tion (C.2.103).

squareA

E.1.2

A tion

357

Also F obeys a wave equation. We take the Hodge star of (E.1.4) and dierentiate it. We substitute dF = 0 and nd F=

1 ? d J: "0

(E.1.8)

The left hand side of this equation, in terms of omponents, an be determined by substituting (C.2.110) and (C.2.107):  1  ek e f [i k Fj℄k + Rekl ij Fkl dxi ^ dxj : F= r r 2 Ri k Fij 2 (E.1.9)

DeltaF1

A

ordingly, urvature dependent terms surfa e in a natural way both in (E.1.7) and in (E.1.9). E.1.2

A tion

A

ording to (B.5.73), the ex itation an be expressed in terms of the ele tromagneti Lagrangian V by H=

V : F

(E.1.10)

fieldmom1

Be ause of the Maxwell-Lorentz spa etime relation (D.5.7), the ex itation H is linear and homogeneous in F . Therefore the a tion V is homogeneous in F of degree 2. Then by Euler's theorem for homogeneous fun tions, we have F

^ V F

= 2V

or, with (E.1.10) and (D.5.7), V =

1 F 2

^H =

 F 2

^F= ?

1 2

r

"0 F 0

^ ?F :

(E.1.11)

onstitmax

(E.1.12)

onstitv

This is the twisted Lagrangian 4-form of the ele tromagneti eld a la Maxwell-Lorentz.

358

E.1.

Standard Maxwell{Lorentz theory in va uum

σ σ

n

x

hσ Figure E.1.1: The ve tor eld n adapted to the (1+3)-foliation with a metri : n is orthogonal to h . Compare with Fig. B.1.3.

E.1.3

Foliation of a spa etime with a metri . Ee tive permeabilities

In the previous se tions, the standard Maxwell-Lorentz theory is presented in 4-dimensional form. In order to visualize the separate ele tri and magneti pie es, we have to use the (1 + 3) de omposition te hnique. In presen e of the metri , it be omes ne essary to further spe ialize the ve tor eld n whi h is our basi tool in a (1 + 3) de omposition. Before we introdu ed the metri , all possible ve tors n des ribed the same spa etime foliation without really distinguishing `time' and `spa e', see Fig. B.1.3, sin e the very notions of time-like and spa e-like ve tors and subspa es were absent. Now we hoose the three fun tions na in su h a way that

g(n; a ) = g()a + gab nb = 0; where then

gab

:= g(a ; b );

g( )a

g(n; n) = g()()

:= g( ; a ); a b

gab n n

= N2

(E.1.13) g( )( )

>

0;

ortho

:= g( ;  ). If (E.1.14)

N2

E.1.3 Foliation of a spa etime with a metri . Ee tive permeabilities

the ve tor eld n is time-like, and thus we an, indeed, onsider  as a lo al time oordinate. The ondition (E.1.13) guarantees that the folia h of onstant  are orthogonal to n, see Fig. E.1.1. Thus they are really 3-dimensional spa elike hypersurfa es. The metri g(a ; b ) := gab (E.1.15) is evidently a positive de nite Riemannian metri on h . We will denote by  the Hodge star operator de ned in terms of the metri (E.1.15). Applying the general de nitions (C.2.85) to our foliation ompatible oframe (B.1.31), we nd the relations between 4-dimensional and 3-dimensional star operators: p p ? (d ^ ) = ( 1)p 1  ; (E.1.16) (3)

( )

?

359

3met

( )

N

(p)

(p)

= ( 1)p N d ^  :

(E.1.17)

(p)

34hodge1 34hodge2

Here is an arbitrary transversal (i.e., purely spatial) p-form. Note that  = 1 for all forms. Substituting the (1 + 3) de ompositions (B.2.7) and (B.1.36) into (D.5.7), ? H =  ?(? F ); (E.1.18) onst3a H =  (? F ); we nd the three-dimensional form of the Maxwell-Lorentz spa etime relation: D = " " E and B =    H ; (E.1.19) onst3 where we introdu ed the ee tive ele tri and magneti permeabilities

(E.1.20) eff onst " = = ; N g

0

g

g

0

g

see (D.5.7) and (E.1.3). In general, these quantities are fun tions of oordinates sin e N , a

ording to (E.1.14), is determined by

360

E.1.

Standard Maxwell{Lorentz theory in va uum

the geometry of spa etime. Thus the gravitational eld a ts via

its potential, the metri on spa etime and makes it look like a medium with nontrivial polarization properties. In parti ular,

the propagation of light, des ribed by the Maxwell equations, is ae ted by these refra tive properties of urved spa etime. In

at Minkowski spa e N = , and hen e "g = g = 1. E.1.4

Symmetry of the energy-momentum urrent

If the spa etime metri g is given, then there exists a unique torsion-free and metri - ompatible Levi-Civita onne tion e , see (C.2.103), (C.2.134). Consider the onservation law (B.5.43) of the energy-momentum. In a Riemannian spa e, the ovariant e ommutes with the Hodge e + D Lie derivative Le = D ? ? operator, Le = Le . Thus (B.5.44) straightforwardly yields Xb =

 ?

^ Lee F

^ Lee ? F



= 0: (E.1.21) 2 Therefore in general relativity (GR), with the Maxwell-Lorentz spa etime relation, (B.5.43) simply redu es to F

F

e k  = (e F ) ^ J : D

Xalriem

(E.1.22)

The energy-momentum urrent (B.5.8) now reads  F

^ (e

?F )



( e F ) ^ ? F :

(E.1.23) 2 In the absen e of sour es, J = 0, we nd the energy-momentum law k

 =

e k  = 0 : D

(E.1.24)

In the at Minkowski spa etime of SR, we an globally hoose  e = the oordinates in su h a way that e = 0. Thus D d and d k = 0. As we already know from (B.5.14), the urrent (E.1.23) is tra eless # ^ k  = 0. Moreover, we now an use the metri and

maxmomergy

E.1.4

Symmetry of the energy-momentum urrent

361

prove also its symmetry. We multiply (E.1.23) by # = g # and antisymmetrize: 4 # ^  = # ^ F ^ (e ?F ) # ^ (e F ) ^ ?F 



℄

[





?F ) + # 

^ F ^ (e



) ^ ? F: (E.1.25) Be ause of (C.2.133) and (C.2.131), the rst term on the righthand side an be rewritten, # ^ F ^ (e ? F ) = F ^ # ^ ? F (# ^ F ) (E.1.26) = F ^ # ^ (e ? F ); i.e., it is ompensated by the third term. We apply the analogous te hnique to the se ond term. Be ause ?? F = F , we have # ^ ? (? F ^ #  ) ^ ? F = ? (? F ^ #  ) ^ # ^ ? F = ? (# ^ ?F ) ^ ?F ^ # = # ^ (e F ) ^ ?F: (E.1.27) In other words, the se ond term is ompensated by the fourth one and we have # ^  = 0: (E.1.28) Alternatively, we an work with the energy-momentum tensor. We de ompose the 3-form  with respe t to the -basis. This is now possible sin e a metri is available. Be ause of # ^  = Æ   , we nd #

^ (e

k

[

℄

F

momergy1

momergy2

momergy3

momergy4

k




 =: T  or T = ? # ^  ; (E.1.29) EMTensor

ompare this with (B.5.28)-(B.5.29). We have p g: (E.1.30) T = T = Its tra elessness T = 0 has already been established, see (B.5.29), its symmetry T  = 0 (E.1.31) maxsym k

k

k

k

k

k

k

[

℄

362

E.1.

Standard Maxwell{Lorentz theory in va uum

an be either read o from (E.1.25) and (E.1.28) or dire tly from (B.5.40) with Hij  F ij . A manifestly symmetri version of the energy-momentum tensor an be derived from (B.5.28) and (E.1.23): k

T =



 ? ?(e

F ) ^ (e

1 F ) + g (?F 2

^ F)



: (E.1.32)

alt

Thus k T is a tra eless symmetri tensor(-valued 0-form) with 9 independent omponents. Its symmetry is sometimes

alled a bastard symmetry sin e it interrelates two indi es of totally dierent origin as an be seen from (E.1.28). Without using a metri , it annot even be formulated, see (B.5.8). It is a re e tion of the symmetry of k T , that the energy ux density 2-form s (B.5.53) and the momentum density 3-form pa (B.5.54) are losely related: pa =

1 dx ^ s: N2 a

(E.1.33)

In a Minkowski spa e, we have N = . This is the ele tromagneti version of the relativisti formula m = 12 E in a eldtheoreti disguise.

Plan k

E.2

Ele tromagneti spa etime relations beyond lo ality and linearity

E.2.1

Keeping the rst four axioms xed

Lamb shift in the spe trum of an hydrogen atom and the Casimir for e between two un harged ondu ting plates attest The

to the possibility to polarize spa etime ele tromagneti ally. We have ele tromagneti \va uum polarization". These ee ts are to be des ribed in the framework of quantum ele trodynami s. However, sin e the Lamb shift and the Casimir ee t are low energy ee ts with slowly varying ele tromagneti elds involved, they an be des ribed quasi- lassi ally in the framework of lassi al ele trodynami s with an altered spa etime relation. Thus the linear Maxwell-Lorentz relation has to be substituted by the nonlinear Heisenberg-Euler spa etime relation, see Se . E.2.3. It should be understood that in our axiomati set-up, when only the Maxwell-Lorentz relation, the fth axiom, is generalized, the fundamental stru ture of ele trodynami s, namely the rst four axioms, are untou hed. If we turn the argument around: The limits of the Maxwell-Lorentz spa etime relation are visible. The fth axiom is built on shakier grounds than the rst four axioms.

364

E.2.

Ele tromagneti spa etime relations beyond lo ality and linearity

Obviously, besides nonlinearity in the spa etime relation, the nonlo ality an be and has been explored. This is further away from present-day experiments but it may be unavoidable in the end. E.2.2

Mashhoon

One says that a spa etime has dispersion properties when the ele tromagneti elds produ e non-instantaneous polarization and magnetization ee ts. The most general linear spa etime relation is then given, in the omoving system, by means of the Volterra integral Z 1 Hij (;  ) = d Kij kl (;  ) Fkl ( ;  ) : (E.2.1) 2 0

0

0

non-lo al

The oeÆ ients of the kernel Kij kl (;  ) are alled the response fun tions. We expe t the metri to be involved in their set-up. Their form is de ned by the internal properties of spa etime itself. Mashhoon has proposed a physi ally very interesting example of su h a non-lo al ele trodynami s in whi h non-lo ality

omes as a dire t onsequen e of the non-inertial dynami s of observers. In this ase, instead of a de omposition with respe t to dxi ^ dxj , one should use the eld expansions 0

1 1 H # ^ # ; F = F # ^ # (E.2.2) 2 2 with respe t to the oframe of a non-inertial observer # = ei  dxi . The spa etime relation is then repla ed by Z 1 d K Æ (;  ) F Æ ( ;  ) ; (E.2.3) H (;  ) = 2 H

=

0

0

0

and the response kernel in (E.2.3) is now de ned by the a

eleration and rotation of the observer's referen e system. It is a

onstitutive law for the va uum as viewed from a non-inertial frame of referen e.

non-lo alHF

non-lo al1

E.2.3

Heisenberg-Euler

365

Mashhoon imposes an additional assumption that the kernel is of onvolution type, i.e., K Æ (;  ) = K Æ (  ). Then the kernel an be uniquely determined by means of the Volterra te hnique, and often it is possible to use the Lapla e transformation in order to simplify the omputations. Unfortunately, although Mashhoon's kernel is always al ulable in prin iple, in a tual pra ti e one normally annot obtain K expli itly in terms of the observer's a

eleration and rotation. Preserving the main ideas of Mashhoon's approa h, one an abandon the onvolution ondition. Then the general form of the kernel an be worked out expli itly (u is the observer's 4velo ity): 0

 1 K Æ (;  ) =   Æ Æ Æ ( 2 ℄

[

0

0

) u 0





℄ ( 0 ) :

(E.2.4)

NewAnsatz

The in uen e of non-inertiality is manifest in the presen e of the onne tion 1-form. The kernel (E.2.4) oin ides with the original Mashhoon kernel in the ase of onstant a

eleration and rotation, but in general the two kernels are dierent. Perhaps, only the dire t observations would establish the true form of the non-lo al spa etime relation. However, su h a non-lo al ee t has not been on rmed experimentally as yet. 1

E.2.3

Heisenberg-Euler

Quantum ele trodynami al va uum orre tions to the MaxwellLorentz theory an be a

ounted for by an ee tive nonlinear spa etime relation derived by Heisenberg and2 Euler. To the rst order in the ne stru ture onstant  = "e 0 ~ , it is given by f

H=

1 See 2 See

r

" 



0 0

8 1+ 45 B

?

f

2 k

F^

 ?F
?F

2

4

14  + 45 B

f 2

k

Muen h et al. [16℄. Itzykson and Zuber [12℄ or Heyl and Hernquist [10℄, e.g..

?






F ^F F ; (E.2.5)

HE

366

E.2.

Ele tromagneti spa etime relations beyond lo ality and linearity

where the magneti eld strength B = me~  4:4  10 T, with the mass of the ele tron m. Again, post-Riemannian stru tures don't interfere here. This theory is a valid physi al theory. A

ording to (E.2.5), the va uum is treated as a spe i type of a medium whi h polarizability and magnetizability properties are determined by the \ louds" of virtual harges surrounding the real urrents and harges. 2 2

9

k

Born-Infeld

E.2.4

The non-linear Born-Infeld theory represents a lassi al generalization of the Maxwell-Lorentz theory for a

ommodating stable solutions for the des ription of `ele trons'. Its spa etime relation reads ?F + ? " fe2 (F ^ F ) F H= : (E.2.6)  1 2 ? (F ^ ? F ) ? (F ^ F )℄ [ 4 fe fe 3

r

0

0

q

1

2

1

1

2

4

Be ause of the nonlinearity, the eld of a point harge, for example, turns out to be nite at r = 0, in ontrast to the well known 1=r singularity of the Coulomb eld in the MaxwellLorentz ele trodynami s, see Fig. E.2.1. The dimensionful parameter f = E = is de ned by the so- alled maximal eld strength a hieved in the Coulomb on guration of an ele tron: E = e=4" r , where r = re with the lassi al ele tron radius re =  ~=m and a numeri al onstant   1. Expli itly, we 2 2 have E  1:8  10 V=m and f = B = = mf e ~  6:4  10 T. In the quantum string theory, the Born-Infeld spa etime relation arises as an ee tive model with f = 1=2 (where  is the inverse string tension onstant). The spa etime relation (E.2.6) leads to a non-linear equation for the dynami al evolution of the eld strength F . As a onsequen e, the hara teristi surfa e, the light one, depends on the eld strength, and the superposition prin iple for the ele tromagneti eld doesn't hold any longer. 2

e

e

e

0

2

0

0

f

e

20

e

k

11

f

e

3 See

also Gibbons and Rasheed [6℄.

0

0

BI

Plebanski

E.2.5

367

y 1.4 1.2 1 0.8 0.6 0.4 0.2 1

2

3

4

x

Figure E.2.1: Spheri ally symmetri ele tri eld of a point

harge in the Born-Infeld (solid line) and in Maxwell-Lorentz theory (dashed line). On the axes we have dimensionless variables x = r=r0 and y = E =Ee .

Plebanski

E.2.5

Both, Eqs.(E.2.6) and (E.2.5) are spe ial ases of Plebanski's more general non-linear ele trodynami s [17℄. Let the quadrati invariants of the ele tromagneti eld strength be denoted by I1

:=

1 (F 2 ?

^

?

F

1 ~2 ) = (E 2

~2 B

) and

I2

:=

1 (F 2 ?

^

F

~
B ~ ; )=E (E.2.7) Inv

where I1 is an even and I2 is an odd s alar (the Hodge operator is odd). Then Plebanski postulated a non-linear ele trodynami s with the spa etime relation4 H

= U (I1 ; I2 )

?

F

+ V (I1 ; I2 ) F ;

(E.2.8)

where U and V are fun tions of the two invariants. Note that in the Born-Infeld ase U and V depend on both invariants whereas in the Heisenberg-Euler ase we have UHE (I1 ) and VHE (I2 ). Nevertheless, in both ases U is required as well as V . And in both 4 Stri tly, Pleba nski assumed a Lagrangian whi h yields the Maxwell equations together with the stru tural relations F = u(I1 ; I2 ) ? H + v(I1 ; I2 ) H . The latter law, apart from singular ases, is equivalent to (E.2.8).

non-l

368

E.2.

Ele tromagneti spa etime relations beyond lo ality and linearity

ases, see (E.2.6) and (E.2.5),

U

is an even fun tion and

V

and

odd one su h as to preserve parity invarian e. If one hooses V

(I1 ; I2 ) to be an even fun tion, then parity violating terms

would emerge, a ase whi h is not visible in experiment.

E.3 Ele trodynami s in matter,

onstitutive law

E.3.1

Splitting of the urrent: Sixth axiom

In this hapter we will present a onsistent mi ros opi approa h to the ele trodynami s in ontinuous media . Besides the eld strength F , the ex itation H is a mi ros opi eld in its own right, as we have shown in our axiomati dis ussion in Part B. The total urrent density is the sum of the two ontributions originating \from the inside" (bound harge) and \from the outside" (free harge) of the medium: 1

J

=J

mat

+J

ext

(sixth axiom a):

(E.3.1)

A

ordingly, the bound ele tri urrent in matter is denoted by mat and the external urrent by ext. The same notational s heme will also be applied to the ex itation H ; we will have H and H . mat

1 In

ext

a great number of the texts on ele trodynami s the ele tri and magneti proper-

ties of media are des ribed following the

ma ros opi averaging

s heme of Lorentz [14℄.

However, this formalism has a number of serious limitations, see the relevant riti ism of Hirst [11℄, e.g.. An appropriate modern presentation of the mi ros opi approa h to this sub je t has been given in the textbook of Kovetz [13℄.

total

370

E.3.

Ele trodynami s in matter, onstitutive law

Bound harges and bound urrents are inherent hara teristi s of matter determined by the medium itself. They only emerge inside the medium. In ontrast, external harges and external urrents in general appear outside and inside matter. They an be prepared for a spe i purpose by a suitable experimental arrangement. we an, for instan e, prepare a beam of harged parti les, des ribed by J ext , and an s atter them at the medium, or we ould study the rea tion of a medium in response to a pres ribed on guration of harges and urrents, J ext . We assume that the harge bound by matter ful lls the usual

harge onservation law separately:

d J mat = 0

(sixth axiom b):

(E.3.2)

Axiom6

We will all this relation as the 6th axiom whi h spe i es the properties of the lassi al material medium. In view of the rst axiom (B.1.19), this assumption means that there is no physi al ex hange (or onversion) between the bound and the external

harges. The 6th axiom ertainly does not exhaust all possible types of material media, but it is valid for a wide enough lass of ontinua. Mathemati ally, the 6th axiom (E.3.2) has the same form as the 1st axiom. As a onsequen e, we an repeat the arguments of Se . B.1.3 and will nd the orresponding ex itation H mat as a \potential" for the bound urrent:

J mat = d H mat :

(E.3.3)

urexa tM

The (1 + 3)-de omposition, following the pattern of (B.1.36), yields

H mat = ?H mat + H mat = d ^ Hmat + Dmat :

(E.3.4)

de omexiM

The onventional names for these newly introdu ed ex itations are polarization 2-form P and magnetization 1-form M , i.e.,

Dmat 

P;

Hmat  M :

(E.3.5)

PM

E.3.2

Maxwell's equations in matter

371

The minus sign is onventional. Thus, in analogy to the inhomogeneous Maxwell equations (B.1.40)-(B.1.41), we nd d P = mat : (E.3.6) d M + P_ = j mat ;

dP

The identi ations (E.3.5) are only true up to an exa t form. However, if we require Dmat = 0 for E = 0 and Hmat = 0 for B = 0, as we will do in (E.3.14), uniqueness is guaranteed. E.3.2

Maxwell's equations in matter

The Maxwell equations (B.4.8)

(

d H D_ = j dD = 

dH = J or

(E.3.7)

are linear partial dierential equations of rst order. Therefore it is useful to de ne the external ex itation   D := D Dmat = D + P mat IH := H H or H := H Hmat = H M : (E.3.8)

The ex itation IH = (D; H) an be understood as an auxiliary quantity. We dierentiate (E.3.8) and eliminate dH and dH mat by (E.3.7) and (E.3.3), respe tively. Then using (E.3.1), the inhomogeneous Maxwell equation for matter nally reads

dIH = J ext ;

(E.3.9)

evol1a

DHe

MaxMat

or, in (1 + 3)-de omposed form,

dH

d D = ext ; D_ = j ext :

(E.3.10) (E.3.11)

From (E.3.8) and the universal spa etime relation (E.1.19) we obtain

D = "g "0 E + P [E; B ℄ ; H =  1 B M [B; E ℄ : g 0

(E.3.12) (E.3.13)

dDe dHe

372

E.3.

Ele trodynami s in matter, onstitutive law

The polarization P [E; B ℄ is a fun tional of the ele tromagneti eld strengths E and B . In general, it an depend also on the temperature T and possibly on other thermodynami variables spe ifying the material ontinuum under onsideration; 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 elds and sour es. The homogeneous Maxwell equation remains valid in its original form. E.3.3

Linear onstitutive law

\It should be needless to remark that while from the mathemati al standpoint a onstitutive equation is a postulate or a de nition, the rst guide is physi al experien e, perhaps forti ed by experimental data." C. Truesdell and R.A. Toupin (1960)

In the simplest ase of isotropi homogeneous media at rest with nontrivial polarizational/magnetizational properties, we have the linear onstitutive laws

P = "g "0 E  E ;

M=

1  B ; g 0 B

(E.3.14)

sus

with the ele tri and magneti sus eptibilities (E ; B ). If we introdu e the material onstants

" := 1 + E ;

 :=

1

1

B

;

(E.3.15)

epsmu

(E.3.16)

perm

one an rewrite the onstitutive laws (E.3.14) as

D = ""g "0  E

and

B = g 0  H :

In urved spa etime, the quantities (""g ) and (g ), in general, are the fun tions of oordinates, but in at Minkowski spa etime they are usually onstant. However, " 6= , ontrary to the ee tive gravitational permeabilities (E.1.20). In general ase,

E.3.4

Energy-momentum urrents in matter

373

their values are determined by the ele tri and magneti polarizability of a material medium. A medium hara terized by (E.3.16) is alled a simple. For a ondu tive medium, one usually adds one more onstitutive relation, namely, Ohm's law:

j =   E:

(E.3.17)

ohm

Here  is the ondu tivity of the simple medium. An alternative way of writing the onstitutive law (E.3.16) is by using the foliation proje tors expli itly:

IH = "  (?F );

?IH =  ?(? F ):

(E.3.18)

onst4

This form is parti ularly onvenient for the dis ussion of the transition to va uum. Then " =  = 1 and hen e (E.3.18) immediately redu es to the universal spa etime relation (D.5.7). For anisotropi media the onstitutive laws (E.3.16)-(E.3.17) are further generalized by repla ing "; ;  by the linear operators "; ;  a ting on the spa es of transversal 2- and 1-forms. The easiest way to formulate these onstitutive laws expli itly is to use the ve tor omponents of the ele tromagneti elds and ex itations whi h were earlier introdu ed in (D.1.37)-(D.1.38). In this des ription, the linear operators are just 3  3 matri es " = "ab and  = ab . One an write expli itly



Ha Da



=

0

p g N

"ab

N p g

ab 0

!

Eb Bb



:

(E.3.19)

In general, the matri es "ab and ab depend on the spa etime

oordinates. The ontribution of the gravitational eld p(3)is inp

luded in the metri -dependent fa tors N and g = g [In p Minkowski spa etime, N = ; g = 1℄. E.3.4

Energy-momentum urrents in matter

In a medium, the total ele tri urrent J is the sum (E.3.1) of the external or free harge J ext and the material or bound harge

lingen

374

E.3.

Ele trodynami s in matter, onstitutive law

ontributions J mat . Thus, one should arefully distinguish two dierent physi al situations: (i) when we are inspe ting how the ele tri and magneti elds a t on the external or free harges and urrents (whi h are used in a tual observations in media, e.g.) and (ii) when we study the in uen e of the ele tromagneti eld on the material or bound harges and urrents (i.e., on diele tri and magneti bodies). Correspondingly, we have to onsider the Lorentz for e density fext = (e F )^J ext whi h ae ts the external (free) urrent J ext and the Lorentz for e density fmat = (e F ) ^ J mat whi h ae ts the material (bound) harges. We an study these two situations separately be ause, as we assumed in Se . E.3.1, there is no physi al mixing between the free and the bound urrents, in parti ular, they are onserved separately. For the rst ase, by using the inhomogeneous Maxwell equations in matter, (E.3.9) or (E.3.10)-(E.3.11), and repeating the derivations of Se . B.5.3, we obtain fext = (e F ) ^ J ext

= d f  + fX :

(E.3.20)

Here the \free- harge" energy-momentum 3-form of the ele tromagneti eld and the supplementary term are, respe tively, given by f (E.3.21)  := 12 [F ^ (e IH ) IH ^ (e F )℄ ; 1 (F ^ L IH IH ^ L F ) : f X := (E.3.22) e e 2

This energy-momentum des ribes the a tion of the ele tromagneti eld on the free harges (hen e the notation where supers ript \f" stands for \free"). At rst, let us analyze the supplementary term. In (1 + 3)de omposed form, we have F = E ^ d + B , and IH = H ^ d + D. The Lie derivatives of these 2-forms an be easily omputed, Le0 F = E_ ^ d + B;_ (E.3.23) Lea F = (Lea E ) ^ d + Lea B;

fSXe

sigmaM XaM

LieF

E.3.4

Energy-momentum urrents in matter

375

and similarly,

Le IH = H_ ^ d + D_ ; Lea IH = (Lea H) ^ d + Lea D: 0

(E.3.24)

Here Lea := dea + ea d is the purely spatial Lie derivative. Substituting (E.3.23)-(E.3.24) into (E.3.22), we nd h 1 f X0 = d ^ H ^ B_ H_ ^ B 2 i + E ^ D_ E_ ^ D ; (E.3.25) h 1 f Xa = d ^ H ^ Lea B Lea H ^ B 2 i (E.3.26) + E ^ Lea D Lea E ^ D : In Minkowski spa etime for matter with the general linear onstitutive law (E.3.19), we nd   1 1 f ab a b X = Vol "0 ( " ) Ea Eb (  ) B B ; (E.3.27) 2 0  ab where Vol is the 4-form of the spa etime volume. Thus fX vanishes for homogeneous media with onstant ele tri and magneti permeabilities. The stru ture of the free- harge energy-momentum is revealed via the standard (1 + 3)-de omposition: f ^0 = f u d ^ fs ; (E.3.28) f f f a = pa d ^ Sa : (E.3.29) Here, in omplete analogy with (B.5.50)-(B.5.51) and (B.5.52)(B.5.55), we introdu ed the energy density 3-form 1 f u := (E ^ D + B ^ H) ; (E.3.30) 2 the energy ux density (or Poynting) 2-form f s := E ^ H ; (E.3.31)

LieHe

X0mat

Xamat

Xepmu

sig0ext sigaext

enerExt

poyntExt

376

E.3.

Ele trodynami s in matter, onstitutive law

the momentum density 3-form (E.3.32) pa := B ^ (ea D) ; and the stress (or momentum ux density) 2-form of the ele tromagneti eld 1 Sa := (ea E ) ^ D (ea D) ^ E 2 (E.3.33)  + (ea H) ^ B (ea B ) ^ H : In absen e of free harges and urrents, we have the balan e equations for the ele tromagneti eld energy and momentum d  + X = 0. In the (1 + 3)-de omposed form this reads, analogously to (B.5.62)-(B.5.63): (E.3.34) u_ + d s + ( X )? = 0; (E.3.35) p_ a + d Sa + ( Xa )? = 0: The (Minkowski) energy-momentum (E.3.21) is asso iated with free harges and has no relation to the for es a ting on diele tri and/or magneti bodies and media. It is, however, indispensable for analyzing the wave phenomenae in matter. It seems worthwhile to mention that there is long ontroversy

on erning the la k of symmetry of the Minkowski energymomentum tensor. Let us study this question in Minkowski spa etime for the ase of an isotropi medium at rest in the laboratory frame. Using the de nitions (E.1.29) and (E.3.21), we nd for the \o-diagonal" omponents: f

momExt

f

f

stressExt

f

f

f

f

f

f

^ 0

f

2

f

T0 a

=

H

 ab b E ;

f

Ta 0

= " ab Hb E :

2

(E.3.36)

If the upper index is now lowered with the help of the spa etime metri , Tik gkj , then we nd indeed that Ti k gkj 6= Tj k gki ; (E.3.37) f

f

f

2 Abraham proposed a symmetri energy-momentum tensor whi h turned out to be

obsolete, in ontrast to repeated laims in the literature (see [1℄, e.g.) to the opposite, see below.

k0ext kaext

E.3.4

Energy-momentum urrents in matter

377

be ause f T0b gba = ab Hb E , whereas f Ta 0 g00 = " ab HbE . The extra fa tor is just the square of the refra tive index n2 = " of matter. However it is remarkable that the Minkowski energymomentum tensor is symmetri , provided we use the opti al metri for the lowering of the upper index: f

opt Ti k gkj

= f Tj k gkiopt:

(E.3.38)

maxsymopt

One an easily verify this by means of (E.4.35). The generalized symmetry (E.3.38) holds true also for an arbitrarily moving medium. Then the opti al metri is des ribed by (E.4.33). This fa t highlights the fundamental position whi h the opti al metri o

upies in the Maxwell-Lorentz theory. Thus it is not by

han e that the Minkowski energy-momentum turns out to be most useful for the dis ussion of the opti al phenomenae in material media. Let us now turn our attention ba k to the for es fmat = (e F )^J mat a ting on the bound harges. In omplete analogy with the derivation of (E.3.20)-(E.3.22), we nd fmat

= (e

F)

^ J mat = d b + bX :

(E.3.39)

Here we introdu e a new material energy-momentum 3-form of the ele tromagneti eld and the orresponding supplementary term:   b  := 12 F ^ (e H mat) H mat ^ (e F ) ; (E.3.40) 1 F ^ L H mat H mat ^ L F
: b (E.3.41) X := e e 2

The material energy-momentum des ribes the a tion of the ele tromagneti eld on the bound harges (hen e the notation where supers ript \b" stands for \bound"). In (1+3)-de omposed form, we have H mat = (M ^ d + P ) and, as usual, F = E ^ d + B . Thus the energy-momentum (E.3.40) is ultimately expressed in terms of the polarization P and magnetization M forms (E.3.5). When there are no free harges and urrents, we should use (E.3.40), and not the energy-momentum (E.3.21),

fSXb

sigma-b Xa-b

378

E.3.

Ele trodynami s in matter, onstitutive law

for the omputation of the for es a ting on the diele tri and magneti matter. The (1+3)-de omposition yields a similar stru ture as (E.3.28)(E.3.29): b  = b u d ^ bs ; (E.3.42) sig0mat ^0 b  = bp b d ^ Sa : (E.3.43) sigamat a a Here, in omplete analogy with (B.5.50)-(B.5.51) and (B.5.52)(B.5.55), we introdu ed the bound- harge energy density 3-form bu := 1 (B ^ M E ^ P ) ; (E.3.44) enerMat 2 the bound- harge energy ux density 2-form bs := E ^ M ; (E.3.45) poyntMat the bound- harge momentum density 3-form bp := B ^ (e P ) ; (E.3.46) momMat a a and the bound- harge stress (or momentum ux density) 2-form of the ele tromagneti eld bS := 1 (e P ) ^ E (e E ) ^ P a a 2 a (E.3.47) stressMat  + (ea M ) ^ B (ea B ) ^ M : The (1+3)-de omposed balan e equations for the bound- harge energy-momentum are analogous to (B.5.62)-(B.5.63) and (E.3.34)(E.3.35): bk = bu_ + d bs + (bX ) ; (E.3.48) k0mat ^0 ? 0 bk = bp_ + d bS + (bX ) : (E.3.49) kamat a a a a ? The integral of the 3-form of the for e density (E.3.49) over the 3-dimensional domain mat o

upied by a material body or a medium gives the total 3-for e a ting on the latter: Ka =

Z



mat

bk : a

(E.3.50)

totfor e

E.3.5 Experiment of Walker & Walker

E.3.5

Experiment of Walker

379

& Walker

Let us onsider an expli it example whi h shows how the energymomentum urrent (E.3.40) works. For on reteness, we will analyze the experiment of Walker & Walker who measured the for e a ting on a diele tri dis pla ed in a verti al magneti eld (i.e., between the poles of the ele tromagnet) as shown on Fig. E.3.1. The time-dependent magneti eld was syn hronized with the alternating voltage applied to the inner and outer

ylindri al surfa es of the dis at radius 1 and 2 , respe tively, thus reating the ele tri eld along the radial dire tion. The experiment revealed the torque along the verti al z -axis. We will derive this torque by using the bound- harge energy-momentum

urrent. We have the Minkowski spa etime geometry. In ylindri al oordinates (; '; z ), the torque density along the z -axis evidently is given by produ t  bk' . Hen e the total torque is the integral

Z

Nz =

 bk' :

(E.3.51)

torque

Dis

Assuming the harmoni ally os illating ele tri and magneti elds, we nd for the ex itations inside the dis

  z!n  a2  d' ;

os 

   d D = n sin(!t) dz ^ a1 J1 !n

  

os(!t) dz H =  a1 J0 !n

0 p

p

a2 sin 

 z!n 

(E.3.52)

walD

 d' : (E.3.53)

walH



Here  = "0 =0 and n := " is the refra tive index of the medium. The dis onsists of nonmagneti diele tri material with  = 1. The os illation frequen y is ! , the two onstants a1 ; a2 determine the magnitude of ele tromagneti elds, and

380

E.3.

Ele trodynami s in matter, onstitutive law

z

torque

ρ

ρ

111111 00000 2 11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111

l

E

B Figure E.3.1: Experiment of Walker & Walker J0 ; J1 are Bessel fun tions. The eld strength forms look similar:





 !n   z!n 

a2 a1 J1  d' + d ;

os E = sin(!t) n





B = os(!t) d ^ a1 J0

 !n 

(E.3.54)

 d' +

 z!n  

a2 sin 

dz :

(E.3.55)

One an verify by substitution that the ele tromagneti eld (E.3.52)-(E.3.55) represents an exa t solution of the Maxwell equations plus the onstitutive relations (E.3.16). In the a tual Walker & Walker experiment3 , the dis , made of barium titanate with " = 3340, has l  2 m height and the internal and external radius 1  0:4 m and 2  2:6 m, respe tively. The os illation frequen y is rather low, ! = 60 Hz. Correspondingly, one an verify that everywhere in the dis we have !n

3 See Walker and Walker [22℄

 z!n  10 7  1:

walE

(E.3.56)

walB

E.3.5 Experiment of Walker & Walker Then the eld strengths read approximately   1

a2 2 E = sin(!t) a1 !  d' + d ; 2 n  

a2 !z dz ^ d : B = os(!t) a1 d ^ d'

n

381

(E.3.57)

walEa

(E.3.58)

walBa

Using the onstitutive relations (E.3.14), we nd the polarization 2-form  

a2 1 P = "0 E sin(!t) a1 !  dz ^ d + d' ^ dz : (E.3.59) 2 n

walP

The 1-form of magnetization is vanishing, M = 0, sin e  = 1. For the omputation of the torque around the verti al axis, we need only the azimuthal omponents of the momentum (E.3.46) and of the stress form (E.3.47). Note that e' = 1 ' . A simple

al ulation yields:

p' =

b

d bS' =

a a "0 E sin(!t) os(!t) 1 2 d ^ d' ^ dz; (E.3.60) n

!a a 1 2 "0 E sin2 (!t) d ^ d' ^ dz: (E.3.61) n

Substituting this into (E.3.49) and subsequently omputing the integral (E.3.51), we nd the torque

! 2

os (!t): (E.3.62) n Here l is the height of the dis , whereas 1 and 2 are its inner Nz =

"0 ("

1)  l (22

21 ) a1 a2

walT

and outer radius, as shown on Fig. E.3.1. Formula (E.3.62) was proven experimentally by Walker & Walker.

It is rather urious that this fa t was onsidered as an argument in favour of the

so- alled Abraham energy-momentum tensor4 . A formal oin iden e is taking pla e,

indeed, in the following sense: Re all that our starting point for the deriving the bound harge energy-momentum was the Lorentz for e (E.3.39). Quite generally, for the 3-for e density (E.3.39), we have from (E.3.3)-(E.3.6):

famat

= =

d ^

B ^ ea



d ^ B ^ ea +

4 See [22℄ and [1℄, e.g.

B ^ ea

j mat + E ^ ea P_ dM

+E

^ ea

mat 




dP :

(E.3.63)

for eLP

382

E.3.

Ele trodynami s in matter, onstitutive law

In the Walker & Walker experiment, we have M = 0. By dierentiating (E.3.59), one an prove that dP = 0. Thus the last line in (E.3.63) vanishes. A

ordingly, the for e density redu es to a term  B  P_ whi h resembles the so- alled Abraham for e. However, our derivation was not based on the Abraham energy-momentum and moreover, the argument of the symmetry of the energy-momentum tensor is absolutely irrelevant. As one an see, our bound- harge energy-momentum is manifestly asymmetri sin e energy

ux (E.3.45) is plainly zero whereas the momentum (E.3.46) is nonvanishing.

E.4

Ele trodynami s of moving ontinua

E.4.1

Laboratory and material foliation

The ele tri and magneti parts of urrent, ex itation, and eld strength are only determined with respe t to a ertain foliation of spa etime. In Se . B.1.4 we assumed the existen e of a foliation spe i ed by a formal \time" parameter  and a ve tor eld n. We know of how to express all the physi al and geometri al obje ts in terms of their proje tions into transversal and longitudinal parts by using the oordinate-free (1+3)-de omposition te hnique, see Se . B.1.4. The original spa etime foliation will be alled a laboratory foliation. Moving ma ros opi matter, by means of its own velo ity, de nes another (1 + 3)-splitting of spa etime whi h is dierent from the original foliation dis ussed above. Here we will des ribe this material foliation and its inter-relation with the laboratory foliation. Let us denote a 3-dimensional matter- lled domain by V . Mathemati ally, one starts with a 3-dimensional arithmeti spa e R3 equipped with the oordinates  a , where a = 1; 2; 3, and

onsiders a smooth mapping x(0) : R3 V X4 into spa e-

! 2

384

E.4.

Ele trodynami s of moving ontinua

time whi h de nes a 3-dimensional domain (hypersurfa e) V representing the initial distribution of matter. The oordinates  a (known as the Lagrange oordinates in ontinuum me hani s) serve as labels whi h denote the elements of the material medium. Given the initial on guration V of matter, we parametrize the dynami s of the medium by the oordinate  whi h is de ned as the proper time measured along an element's world line from the original hypersurfa e V . The resulting lo al oordinates (;  a ) are usually alled the normalized omoving oordinates. The motion of matter is thus des ribed by the fun tions xi (;  a ), and we subsequently de ne the (mean) velo ity 4-ve tor eld by u

:=  =

 dxi  d

 a = onst

i :

(E.4.1)

udef

By onstru tion, this ve tor eld is timelike and it is normalized a

ording to g(u; u) = 2 :

(E.4.2)

Evidently, a family of observers omoving with matter is hara terized by the same timelike ongruen e xi (;  a). They are making physi al (in parti ular, ele trodynami al) measurements in their lo al referen e frames drifting with the material motion. By the hypothesis of lo ality it is assumed that the instruments in the omoving frame are not ae ted in an appre iable way by the lo al a

eleration they experien e. They measure the same as if they were in a suitable omoving instantaneous inertial frame. After these preliminaries, we are ready to nd the relation between the two foliations. Te hni ally, the ru ial point is to express the laboratory oframe (d; dxa ) in terms of the oframe adapted to the material foliation. Re all that a

ording to our

onventions formulated in Se . B.1.4, dxa = dxa na d is the transversal proje tion of the spatial oframe. The motion of a medium uniquely determines the (1 + 3)de omposition of spa etime through a material foliation whi h

uu=1

E.4.1

Laboratory and material foliation

385

is obtained by repla ing n; d by u; d . Note that the proper time dierential is d = ui dxi. Thus evidently (E.4.3) u d = 1: We onsider an arbitrary motion of the matter. The velo ity eld u is arbitrary, and one does not assume that the laboratory and moving referen e systems are related by a Lorentz transformation. The te hnique of the (1 + 3)-splitting is similar to that des ribed in Se . B.1.4 for the laboratory foliation. Namely, following the pattern of (B.1.22) and (B.1.23), one de nes the de ompositions with respe t to the material foliation: For any p-form we denote the part longitudinal to the velo ity ve tor u by a := d ^ ; (E.4.4) ` := u ; ` and the part transversal to the velo ity u by f := u (d ^ ) = (1 a) ; u f  0: (E.4.5) 2

Please note that the proje tors are denoted now dierently (a and g ) in order to distinguish them from the orresponding proje tors (? and ) of the laboratory foliation. With the spa etime metri introdu ed on the X by means of the Maxwell{Lorentz spa etime relation, we assume that the laboratory foliation is onsistent with the metri stru ture in the sense outlined in Se . E.1.3. In parti ular, taking into a

ount (E.1.13) and (E.1.14), one nds the line element with respe t to the laboratory foliation oframe: gab dxa dxb : (E.4.6) ds = N d + gab dxa dxb = N d Now it is straightforward to nd the relation between the two foliations. Te hni ally, by using (E.4.4) and (E.4.5), one just needs to (1 + 3)-de ompose the basis 1-forms of the laboratory

oframe (d; dxa) with respe t to the material foliation. Taking into a

ount that, in the lo al oordinates, n =  + na a and

norm1

longiM

transM

4

2

2

2

2

2

(3)

metF

386

E.4.

Ele trodynami s of moving ontinua

Table E.4.1: Two foliations laboratory frame

material frame

n

u





ve tor eld \time" longitudinal transversal time oframe 3D oframe 4D spa etime interval

?

a

d

d

dxa

dxa



ds

2

(3)

=N

f

2

d

ds

2

gab dxa dxb

(3)

gab

2

f

= d 2

2 va vb 1

2




dxa dxb

f f

similarly u = u   + uaa , the result, in a onvenient matrix form, reads: !     d d b : (E.4.7) =

=N vb =( N ) ( )

dxa

v a

dx

Æba

f

dsigmaM

Here we introdu ed for the relative velo ity 3{ve tor the notation v

a

:=

furthermore

N



ua u()

n

a



;

(E.4.8)

:= q 1 : (E.4.9) 1 v Observe that (E.4.7) is not a Lorentz transformation sin e it relates the two frames whi h are both non inertial in general. As usual, the spatial indi es are raised and lowered by the 3{spa e metri of (E.1.15), gab := gab . In parti ular, we expli itly have va = gab vb and v := vava . By means of the

2 2

(3)

(3)

2

relvel

gamM

E.4.2 Ele tromagneti eld in laboratory and material frames

387

normalization (E.4.2), we an express the zeroth (time) omponent of the velo ity as u() = ( =N ). Hen e the expli it form of the matter 4-velo ity reads: u = u()  + ua a = q

1



1 v2

2

N



eb0 + v a ea :

(E.4.10)

Umat

Here (eb0 ; ea ) is the frame dual to the adapted laboratory oframe (d; dxa ), i.e., eb0 = n; ea = a . When the relative 3{velo ity is zero v a = 0, the material and the laboratory foliations oin ide be ause the orresponding foliation 1-forms turn out to be proportional u = ( =N ) n. Substituting (E.4.7) into (E.4.6), we nd for the line element in terms of the new variables ds2 = 2 d 2

bgab dxa dxb ; where bgab = (3) gab f

f

1

2

va vb :

(E.4.11)

metM

Comparing this with (E.4.6), we learly see that the transition from a laboratory referen e frame to a moving material referen e frame hanges the form of the line element from (E.4.6) to (E.4.11). Consequently, this transition orresponds to a linear homogeneous transformation that is anholonomi in general. It is not a Lorentz transformation whi h, by de nition, preserves the form of the metri oeÆ ients. The metri b gab of the material foliation has the inverse gbab = (3) g ab +

2 a b v v:

2

(E.4.12)

For its determinant one nds (det gbab ) = (det gab ) 2 .

E.4.2

Ele tromagneti eld in laboratory and material frames

Let us onsider the ase of a simple medium with homogeneous and isotropi ele tri and magneti properties. The onstitutive

invghat

388

E.4.

Ele trodynami s of moving ontinua

law (E.3.18) for su h a medium at rest with respe t to the laboratory frame has to be understood as a result of a laboratory foliation. A moving medium is naturally at rest with respe t to its own material foliation. Consequently, the onstitutive law for su h a simple medium reads IH = "  (?F ); f

 aIH =  a (?F ):

f

(E.4.13)

onst5a

How does the onstitutive law look as seen from the original laboratory frame? For this purpose we will use the results of Se . D.4.4 and the relations between the two foliations established in the previous se tion. To begin with, re all of how the ex itation and the eld strength 2-forms de ompose with respe t to the the laboratory frame

IH =

H ^ d + D;

F = E ^ d + B;

(E.4.14)

HFlab

(E.4.15)

HFmat

and, analogously, with respe t to the material frame: IH =

H0 ^ d + D0;

F = E 0 ^ d + B 0 :

Clearly, we preserve the same symbols IH and F on the lefthand sides of (E.4.14) and (E.4.15) be ause these are just the same physi al obje ts. In ontrast, the right-hand sides are of

ourse dierent, hen e the primes. The onstitutive law (E.4.13), a

ording to the results of the previous se tion, an be rewritten as (E.4.16) D0 = ""0 b E 0; H0 = 1 0 bB 0:

onst5b

Here the Hodge star b orresponds to the metri gbab of the material foliation [please do not mix it up with the Hodge star  de ned by the 3-spa e metri (3) gab of the laboratory foliation℄. Now, (E.4.16) an be presented in the equivalent matrix form



H0a  =  D0a

0

p n g



gbab

p n g

0

gbab

!

Eb0 B 0b



: (E.4.17)

onst5

E.4.2 Ele tromagneti eld in laboratory and material frames

389

The omponents of the onstitutive matri es read expli itly A

0

ab

p

n g ab gb ; Bab = p b gab ; C a b = 0;

n g r "" p 0 ; n := " : with  = 0

=

0

0

(E.4.18)

ABCfm

(E.4.19)

fem

In order to nd the onstitutive law in the laboratory frame, we have to perform some very straightforward matrix algebra manipulations along the lines des ribed in Se . D.4.4. Given is the linear transformation of the oframes (E.4.7). The orresponding transformation of the 2-form basis (A.1.97) turns out to be P

a



b

1

a Æb = N

2

Zab =

^ab v ;



a

Qb a = Æba ;

v vb ; W

ab

=

1



N

ab

(E.4.20)

v :

We use these results in (D.4.27)-(D.4.30). Then, after a lengthy matrix omputation, we obtain from (E.4.18) the onstitutive matri es in the laboratory foliation: ab

A =

Bab =

C ab =

1 1 1 1 1 1

p

(3)

v2

2

v2

2



pN

(3)

n

g

(3)

N

 v2

2

g



(3)

g

 1 n

ab

gab

(3)





v2

2 n

1

Ha  =   C b a Aab Da

v n

2



1

2

+

1

2

a b

v v

Bab C ab



Eb Bb



n



va vb n

v :

The resulting onstitutive law



n +

2

n

 a b





1



n

;

(E.4.21)  1 n

Aem

;

(E.4.22)

Bem

(E.4.23)

Cem

(E.4.24)

CMmoving

390

E.4.

Ele trodynami s of moving ontinua

an be presented in terms of exterior forms as:

H

2 = 0 g

  

B

1 

v2 " 2





+ 2 "0  (v ^ E ) "

D = "0 "g 2

  

E "

v2  2



+ 2 "0 v ^  B "

+ 1

 1 

1

2









v (v ^ B ) " 

;

1

2





v (v ^ E ) " 



:

1





(E.4.25) 1



Hmov



(E.4.26)

Dmov

(E.4.27)

3velo ity

Here we introdu ed the 3-velo ity 1-form v := va dxa :

The 3-velo ity ve tor is de omposed a

ording to v a ea . If we lower the index v a by means of the 3-metri (3) gab , we nd the

ovariant omponents va of the 3-velo ity whi h enter (E.4.27). The dire t inspe tion shows that the onstitutive law (E.4.25)(E.4.26) of above an alternatively be re asted into the pair of equations:

D +  2g v ^ H = ""0 ("g  E + v ^  B ) ; "g B v ^ E = 0 (g  H v ^  D) :

2

(E.4.28)

mov1ex

(E.4.29)

mov2ex

These are the famous Minkowski relations for the elds in a moving medium1 . Originally, the onstitutive relations (E.4.28)-(E.4.29) were derived by Minkowski with the help of the Lorentz transformations for the ase of a at spa etime and a uniformly moving media. We stress, however, that the Lorentz group never entered the s ene in our above derivation. This demonstrates ( ontrary to the traditional view) that the role and the value of the Lorentz invarian e in ele trodynami s should not be overestimated. The onstitutive law (E.4.25)-(E.4.26) or, equivalently, (E.4.28)-(E.4.29), des ribes a moving simple medium on an arbitrary urved ba kground. The in uen e of the spa etime geometry is manifest in "g ; g and in (3) gab whi h enters the Hodge star operator. In at Minkowski spa etime in Cartesian oordinates, these quantities redu e to "g = g = 1; (3) gab = Æab . 1 See the dis ussions of various aspe ts of ele trodynami s of moving media in [2, 18, 19, 4, 13℄.

E.4.3

Opti al metri from the onstitutive law

391

The physi al sour es of the ele tri and magneti ex itations D and H are the free harges and urrents. Re alling the de nitions (E.3.8) and (E.3.5), we an nd the polarization P and the magnetization M whi h have the bound harges and urrents as their sour es. A dire t substitution of (E.4.25) and (E.4.26) into (E.3.8) yields:

( 

2

P = "0 "g E

+ B



v2  E

2

( 

2 M= B 0 g

+ E







1

E

2

1

2

1

B

v2  B

2



2

1

2



v (v ^ E ) + 



v  (v ^  E ) +

1 g

1 g

v^ B

v ^ B



)

"g  (v ^ E )

2

v (v ^ B ) 

"g  (v ^ E )

2

v  (v ^ B )



; (E.4.30)

Pmov



)

: (E.4.31)

Mmov

Here E and B are the ele tri and magneti sus eptibilities (E.3.15). When the matter is at rest, i.e. v = 0, the equations (E.4.30)-(E.4.31) redu e to the rest frame relations (E.3.14). E.4.3

Opti al metri from the onstitutive law

A dire t he k shows that the onstitutive matri es (E.4.21)(E.4.23) satisfy the losure relation (D.2.6), (D.3.17)-(D.3.19). Consequently, a metri of Lorentzian signature is indu ed by the

onstitutive law (E.4.24). The general re onstru tion of a metri from a linear onstitutive law is given by (D.4.9). Starting from (E.4.18), we immediately nd the indu ed metri in the material foliation: 0opt

gij

=

rn

p

g



2 n2

0

0

bgab



:

(E.4.32)

Making use of the relation (E.4.7) between the foliations and of the ovarian e properties proven in Se . D.4.4, we nd the

gijoptfm

392

E.4.

Ele trodynami s of moving ontinua

expli it form of the indu ed metri in the laboratory foliation:



 14 

"



1



ui uj



: (E.4.33) gij 1 det g "

Here gij are the omponents of the metri tensor of spa etime, and ui are the ovariant omponents of the matter 4-velo ity (E.4.10). Note that g ij ui uj = , as usual. The ontravariant indu ed metri reads: 1    ui uj det g 4 ij ij : (E.4.34) g = g (1 ") opt

gij

=

2

gijopt

2

opt

"

2

Su h an indu ed metri gij is usually alled the opti al metri in order to distinguish it from the true spa etime metri gij . It des ribes the \dragging of the aether" (\Mitf uhrung des  Athers" ). The adje tive \opti al" expresses the fa t that all the opti al ee ts in moving matter are determined by the Fresnel equation (D.1.55) whi h redu es to the equation for the light

one determined by the metri (E.4.33) in the present ase. The nontrivial polarization/magnetization properties of matter are manifestly present even when the medium in the laboratory frame is at rest. Let us onsider Minkowski spa etime with gij = diag( ; 1; 1; 1), for example, and a medium at rest in it. Then u = t or, in omponents, ui = (1; 0; 0; 0). We substitute this into (E.4.33) and nd the opti al metri opt

2

2

=

r  n

2 n2

0



: (E.4.35) 0 Æab Evidently, the velo ity of light is repla ed by =n with n as the refra tive index of the diele tri and magneti media. opt

gij

E.4.4

Ele tromagneti eld generated in moving

ontinua

Let us onsider an expli it example whi h demonstrates the power of the generally ovariant onstitutive law. 2 See

Gordon [7℄.

gijoptM

E.4.4 Ele tromagneti eld generated in moving ontinua

S

ext

J(1) =0 ε,µ=1 (1) matter

393

J(2)=0 ε=µ=1

(2) vacuum Figure E.4.1: Two regions divided by a surfa e S .

For simpli ity, we will study ele trodynami s in at Minkowski spa etime in whi h the laboratory referen e frame is determined by the usual time oordinate t and the Cartesian spatial oordinates ~x. Then the metri has the omponents N = ; (3) gab = Æab . Correspondingly, "g = g = 1. The motion of matter will be, as shown in Se . E.4.1, des ribed by the material foliation as spe i ed by the relative velo ity v a . Next, let a surfa e S be the border between the two regions, the rst of whi h [labeled as (1)℄ is lled with matter having nontrivial magneti and ele tri properties with  = 1; " = 1. In the se ond region [labeled as (2)℄, matter is absent, and hen e  = " = 1. We will assume that the matter in the rst region does not

ontain any free (i.e., external) harges and urrents, and that the motion of the medium is stationary. Then all the variables are independent of time. Consider the ase when the se ond (matter-free) region ontains the onstant magneti and/or ele tri elds, i.e., the om6

6

394

E.4.

a B(2)

ponents B(2) E(2)

Ele trodynami s of moving ontinua

and Ea(2) of the eld strengths forms

= B 1 dx2 ^ dx3 + B 2 dx3 ^ dx1 + B 3 dx1 ^ dx2 ; (E.4.36) = E1 dx1 + E2 dx2 + E3 dx3 ;

BEreg2

do not depend on t; ~x. The material law (E.4.24)-(E.4.26) then yields that the omponents H(2) a and Da(2) of the magneti and ele tri ex itations forms

D H

(2) (2)

= "0 1 =

0

^ dx3 + E 2 dx3 ^ dx1 + E 3 dx1 ^ dx2

E 1 dx2

B1 dx1


3

+ B2 dx2 + B3 dx




;

;

(E.4.37)

DHreg2

are also onstant in spa e and time. The spatial indi es are raised and lowered by the spatial metri (3) gab = Æab . These assumptions evidently guarantee that both the homogeneous dF = 0 and inhomogeneous dIH = J Maxwell equations are satis ed for the trivial sour es J = 0 ( = 0 and j = 0) in the se ond region. Let us now verify that the motion of matter generates nontrivial ele tri and magneti elds in the rst region. In order to nd their on gurations, it is ne essary to use the onstitutive law (E.4.24)-(E.4.26) and the boundary onditions at the surfa e S . Re alling the jump onditions on the separating surfa e (B.4.13)-(B.4.14) and (B.4.15)-(B.4.16), we nd, in the absen e of free harges and urrents: A

H

A

E(1)

(1)

S S

= A

H

= A

E(2) ;

(2)

S

;

S

^ D(1) S =  ^ D(2) S; (E.4.38)  ^ B(1) =  ^ B(2) : (E.4.39) S S



Sin e the matter is on ned to the rst region, we on lude that the 3-velo ity ve tor on the boundary surfa e S has only two tangential omponents: v a a = v A A ; A = 1 ; 2: (E.4.40) S Let us assume that the two tangential ve tors are mutually orthogonal and have unit length (whi h is always possible to

HDonS EBonS

E.4.4 Ele tromagneti eld generated in moving ontinua

395

a hieve by the suitable hoi e of the variables  A parametrizing the boundary surfa e). The solution of the Maxwell equations dF = 0 and dIH = 0 in the se ond region is uniquely de ned by the ontinuity

onditions (E.4.38)-(E.4.39). Let us write them down expli itly. Applying A to (E.4.25) and  ^ to (E.4.26), we nd: A

H



2 = 0 g

1

v2 " 2





^



D = " " 2

0 g

"





1

ÆAB + "







1

vA v B B

2



B

AB v B  ( ^  E );   v2  ^ E 2 

(E.4.41)

tauHB

AB vA  (B

(E.4.42)

nuDE



1

+ "0 " 2



1

+ "0 " 2







B ):

Here AB = BA with 1_ 2_ = 1 (and the same for AB ). These equations should be taken on the boundary surfa e S . A simple but rather lengthy al ulation yields the inverse relations: A



B = 0 g 2

 



0  2

^



2 E= "0 "g



1 "



1



"

v2  2

0  2

v2 " 2



1 "



ÆAB



AB v B  ( ^

^





1



"

1



v v B B 2 A

H

D);

(E.4.43)

tauBH

H):

(E.4.44)

nuED

D

AB vA  (B

The 3 equations (E.4.43)-(E.4.44), taken on S , together with the 3 equations (E.4.39) are spe ifying all 6 omponents of the ele tri and magneti eld strength E and B on the boundary S in terms of the onstant values of the eld strengths (E.4.36) in the matter free region. The standard way to nd the stati ele tromagneti elds in region 1 is as follows. The [(1 + 3)-de omposed℄ homogeneous Maxwell equations dE(1) = 0; dB(1) = 0 are solved by

396

E.4.

Ele trodynami s of moving ontinua

A

E(1) = d', B(1) = d . Substituting this, by using the onsti-

tutive law (E.4.25)-(E.4.26), into the inhomogeneous Maxwell equations dH(1) = 0; dD(1) = 0, we obtain the 4 se ond order dierential equations for the 4 independent omponents of the ele tromagneti potential '(x); A(x). The unique solution of the resulting partial dierential system is determined by the boundary onditions (E.4.39) and (E.4.43)-(E.4.44).

E.4.5 The experiments of Rontgen and Wilson & Wilson In general ase, this is a highly nontrivial problem. However, there are two physi ally important spe ial ases for whi h the solution is straightforward. They des ribe the experiments of Rontgen and Wilsons with moving diele tri bodies. In both

ase we will on ne our attention to the hoi e of the boundary as a plane S = fx3 = 0g, so that the tangential ve tors and the normal 1-form are: 1 = 1 ;

2 = 2 ;

 = dx3 :

(E.4.45)

We will assume that the upper half-spa e ( orresponding to the positive x3 ) is lled with the matter moving with the horizontal velo ity v = v1 dx1 + v2 dx2 :

(E.4.46)

Rontgen experiment Let us onsider the ase when the magneti eld is absent in the matter-free region, whereas ele tri eld is dire ted towards the boundary: B(2) = 0;

E(2) = E3 dx3 :

(E.4.47)

= "0 E 3 dx1 ^ dx2 :

(E.4.48)

Then from (E.4.37) we have

H

(2)

= 0;

D

(2)

E.4.5 The experiments of Rontgen and Wilson & Wilson x3

397

x3 (1)

B(1)

v E(2)

x1

B(1)

x2

(2)

v E(2)

Figure E.4.2: Experiment of Rontgen It is straightforward to verify that, for the uniform motion (with

onstant v ), the forms

B(1) = E(1) =

1 1

1 1

v2

2 v2

2

1 "  1 "





v2  2

1 v ^ E3 dx3 ;

2



E3 dx3

(E.4.49)

Broent

(E.4.50)

Eroent

des ribe the solution of the Maxwell equations satisfying the boundary onditions (E.4.39) and (E.4.43)-(E.4.44). Using the

onstitutive law (E.4.25)-(E.4.26), we nd from (E.4.49) and (E.4.50) the orresponding ex itation forms:

H

(1)

= 0;

D

(1)

= "0 E 3 dx1 ^ dx2 :

(E.4.51)

This situation is des ribed on left part of the Figure E.4.2: The magneti eld is generated along the x1 axis by the motion of the matter along the x2 axis. In order to simplify the derivations, here we have studied the

ase of the uniform translational motion of matter. However, in the a tual experiment of Rontgen3 in 1888 he observed this ee t for a rotating diele tri dis , as shown s hemati ally on the right part of Figure E.4.2. One an immediately see that in the 3 See R ontgen [20℄ and the later thorough experimental study of Ei henwald [5℄.

398

E.4.

Ele trodynami s of moving ontinua

x3

x1 (1)

E(1) B(2)

x1

v

E(1)

x2 (2)

v B(2)

Figure E.4.3: Experiment of Wilson and Wilson non-relativisti approximation (negle ting the terms v 2 = 2 ), the formulas (E.4.49)-(E.4.50) des ribe the solution of the Maxwell equations, provided dv = 0. This in ludes, in parti ular, the

ase of the uniform rotation v = ! d with ! = onst. Here,  is the usual polar angle, i.e., d = (x1 dx2 x2 dx1 )=((x1 )2 + (x2 )2 ). The magneti eld generated along the radial dire tion an be dete ted by means of a magneti needle, for example.

Wilson and Wilson experiment In the `dual' ase the ele tri eld is absent in the matter-free region whereas a magneti eld is pointing along the boundary:

B(2) = B 1 dx2 ^ dx3 + B 2 dx3 ^ dx1 ;

E(2) = 0: (E.4.52)

Then from (E.4.37) we nd

H

(2)

=

1

0




B1 dx1 + B2 dx2 ;

D

(2)

= 0:

(E.4.53)

The solution of the Maxwell equations satisfying the boundary onditions (E.4.39), (E.4.43)-(E.4.44) is straightforwardly

E.4.5 The experiments of Rontgen and Wilson & Wilson

399

obtained for uniform motion of the matter:

E(1) =

1 1





1

 (v 1 B 2

"

v2

2

v 2 B 1 ) dx3 ;

(E.4.54)

Ewils

v 2 B 1 ) dx3 : (E.4.55)

Bwils

B(1) =  (B 1 dx2 ^ dx3 + B 2 dx3 ^ dx1 ) +

1

1



v2

2

1

"





1

2

v ^ (v 1 B 2

This situation is depi ted on the left part of Figure E.4.3. There, without restri ting generality, we have hosen the velo ity along x2 and the magneti eld B(2) along x1 . Then the generated ele tri eld is dire ted along the x3 axis. The ele tri and magneti ex itations in matter are obtained from the onstitutive law (E.4.25)-(E.4.26) whi h, for (E.4.54) and (E.4.55), yields

H

(1)

=

1

0




B1 dx1 + B2 dx2 ;

D

(1)

= 0:

(E.4.56)

Similarly to the experiment of Rontgen, the experiment of Wilson & Wilson4 was a tually performed for the rotating matter and not for the uniform translational motion des ribed above. The true s heme of the experiment is given on the right side of Figure E.4.3. In fa t, the rotating ylinder is formally obtained from the left gure by identifying x1 = z; x2 = ; x3 =  with the standard ylindri al oordinates (polar angle , radius ). Usually, one should be areful about the use of urvilinear oordinates in whi h the omponents of metri are non onstant. However, the use of exterior al ulus makes all omputations transparent and simple. We leave it as an exer ise to the reader to verify that the Maxwell equations yield the following exa t solution for the ylindri al on guration of the Wilsons experi4 See Wilson and Wilson [23℄.

400

E.4.

Ele trodynami s of moving ontinua

ment: 1 B dz; 0 B(2) = B d ^  d; H(1) = 10 B dz;  1 B(1) =  2 1 (! 2)  1 1 E(1) = 2 1 (! 2) "

H

(2)

D

=

= 0;

(E.4.57)

HD2wilson

E(2) = 0;

(E.4.58)

BE2wilson

D

(E.4.59)

(2)

(1)

= 0;

 (!) B d ^  d; "  2

2

 ! B d:

(E.4.60)

Bwilson

(E.4.61)

Ewilson

The boundary onditions (E.4.39), (E.4.43)-(E.4.44) are satis ed for (E.4.57)-(E.4.61). Note that now  = d and 1 = z ; 2 =  and the velo ity one-form reads

v = !2 d;

(E.4.62)

with onstant angular velo ity ! . The radial ele tri eld (E.4.61) generated in the rotating ylinder an be dete ted by measuring the voltage between the inner and the outer surfa es of the

ylinder. One may wonder what physi al sour e is behind the ele tri and magneti elds that are generated in moving matter. After all, we have assumed that there are no free harges and

urrents inside region 1. However, we have bound harges and

urrents therein des ribed by the polarization and magnetization (E.4.30) and (E.4.31). Substituting (E.4.60)-(E.4.61) into (E.4.30)-(E.4.31), we nd:

P= M=



"0 1 1

(!)2

"0

2

(!)2

2



1  !2 B d ^ dz; (E.4.63) "    (!)2 1  1+ 2

2 B dz: (E.4.64) 1

"

From the de nition of these quantities (E.3.5), we obtain, merely by taking the exterior dierential, the harge and urrent den-

E.4.6 Non-inertial \rotating oordinates"

sities: mat



j

mat

= =



2"0 1

(!)2

2

(!)2

2



"



2"0 1

2



1

2

! B d

^ d ^ dz;



1

! 2  B dz



"

^ d:

401

(E.4.65)

rhomov

(E.4.66)

jmov

It is these harge and urrent densities whi h generate the nontrivial ele tri and magneti elds in the rotating ylinder in the Wilsons experiment. The bound urrent and harge density (E.4.65)-(E.4.66) satisfy the relation j mat

E.4.6



=


mat  v:

(E.4.67)

Non-inertial \rotating oordinates"

How is the Maxwell-Lorentz ele trodynami s seen by a noninertial observer? We need a pro edure of two steps for the installation of su h an observer. In this se tion the rst step is done by introdu ing suitable non-inertial oordinates. We assume the absen e of a gravitational eld. Then spa etime is Minkowskian and a global Cartesian oordinate system t; xa (with a = 1; 2; 3) an be introdu ed whi h spans the inertial (referen e) frame. The spa etime interval reads ds2

Æab dxa dxb :

= 2 dt2

(E.4.68)

metFlat

As usual, the ele tromagneti ex itation and the eld strength are given by IH

=

H ^ dt + D;

F

= E ^ dt + B:

(E.4.69)

HFin

Assuming matter to be at rest in the inertial frame, we have the

onstitutive law

H = 1

 0

B;

D = ""

0



E:

(E.4.70)

matIn1

402

E.4.

Ele trodynami s of moving ontinua

q ""0

Equivalently, we have  = Aab

n ab Æ ;

=

0

Bab

=

n

and the onstitutive matri es C ab

Æab ;

= 0:

(E.4.71)

lawM

The orresponding opti al metri is given by (E.4.35). Now we want to introdu e non-inertial \rotating oordinates" (t ; x a ) by 0

0

t

with the 3  3 matrix Lb a

xa

=t;

= na nb + (Æba

0

= Lb a x b ;

(E.4.72)

0

na nb ) os ' + ^ a b n

sin ' :

(E.4.73)

The matrix de nes a rotation of an angle ' = '(t) around the dire tion spe i ed by the onstant unit ve tor ~n = na , with Æab na nb = 1. The Latin (spatial) indi es are raised and lowered by means of the Eu lidean metri Æab and Æ ab (^a b = Æ ad ^d b , for example). We put \rotating oordinates" in quotes, sin e it is stri tly speaking the natural frame (dt ; dx a ) atta hed to the

oordinates (t ; x a ) that is rotating with respe t to the Cartesian frame. The ele tromagneti two-forms H and F are independent of

oordinates. However, their omponents are dierent in dierent oordinate systems. In exterior al ulus it is easy to nd the

omponents of forms: one only needs to substitute the original natural oframe (dt; dxa ) by the transformed one. The straightforward al ulation, using (E.4.73), yields      1 0 dt dt : (E.4.74) = dx b dxa L a [~ !~ x ℄ Lb a Here the angular 3-velo ity ve tor is de ned by 0

0

rotL

0

0

0

0

0

! ~

:= ! ~n;

!

:=

d' dt

dtdx

(E.4.75)

:

Substituting these dierentials into (E.4.68), we nd the interval in rotating oordinates: 2

ds

h

= (dt ) 1 + (~!
~x = ) 2

2

0

2dt

0

d~ x0

0


[~!  ~x ℄ 0

2

(~!
~!= )(~x 2




d~ x0 d~ x0 :

0


~x ) 0

i

(E.4.76)

metRot

E.4.7

Rotating observer

403

Ele tromagneti ex itation and the eld strength are expanded with respe t to the rotating frame as usual: IH

H0 ^ dt + D0;

=

F

= E 0 ^ dt + B 0 :

(E.4.77)

HFnin1

Note that dt = dt0 . The onstitutive matri es are derived from (E.4.71) with the help of the transformation (D.4.28)-(D.4.30). Given (E.4.74), we nd the matri es (A.1.98)-(A.1.99) as P ab

= La b ;

Qb a

= (L 1 )b a ;

W ab

= 0;

Zab

= (L 1 )a ^b d v d : (E.4.78)

PQrot1

Hereafter we use the abbreviation ~ v

:= [~!  ~x0 ℄:

(E.4.79)

Vrot

We substitute (E.4.78) into (D.4.28)-(D.4.30) and nd the onstitutive matri es in the rotating natural frame as 1

A0ab

=

0 Bab

=



1 n

 n Æ ab ;

Æab



+n

v C 0a b = n Æ a ^ bd

d

p

v2 Æab 2

+

1

2



va vb

;

;

(E.4.80)

Aem1rota

(E.4.81)

Bem1rota

(E.4.82)

Cem1rota

with n = ". These matri es satisfy the algebrai losure relation (D.3.17)-(D.3.19). Thus they de ne the indu ed spa etime metri . The latter is obtained from (D.4.25) by using (E.4.35) and (E.4.74):

0 opt

g ij

E.4.7

r

=

n



2 n2

v2 va

vb Æab



:

(E.4.83)

Rotating observer

However, the behavior of elds with respe t to rotating frame is usually of minor physi al interest to us. The observer rather measures all physi al quantities with respe t to a lo al frame

gijoptrota

404

E.4.

Ele trodynami s of moving ontinua

# whi h is anholonomi in general, i.e. d# = 6 0. The observer

is, in fa t, omoving with that frame and the omponents of ex itation and eld strength should be determined with respe t to # . Consequently, the observer's 4-velo ity ve tor reads eb0 = eib0 i = t : 0

(E.4.84)

;

(E.4.85)

e0rot

The `Lorentz' fa tor 1

=p

1

~v 2 = 2

is determined for the metri (E.4.76) by the normalization ondition g(eb0 ; eb0 ) = eib0 ej b0 gij = 2 . Note that ~v 2 = Æab v a v b . The observer's rotating frame e with (E.4.84) and 0

eba =

x a 0

2 + 2 va  t

(E.4.86)

0

is dual to the orresponding oframe # with b

#0 =

1

~v
d~x ;

2

dt

0

0

#ba = dx a :

(E.4.87)

0

the0rot

Expressed in terms of this oframe, the metri (E.4.76) reads 2

b0 2

2

ds = (# )



2 Æab + 2 va vb



#ba #b : b

(E.4.88)

met orot

Combining (E.4.74) with (E.4.87), we obtain the transformation from the inertial (dt; dxa ) oframe to the non-inertial (#b0 ; #ba ) frame as



dt dxa



=

2  2 vb2 

a

a

L v L Æb + 2 v vb

!

b0

# b #b

!

:

(E.4.89)

dtdx0

With respe t to # , the ele tromagneti ex itation and eld strength read IH =

H ^ #b0 + D ; 0

0

F =E

0

^ #b0 + B : 0

(E.4.90)

HFnin

E.4.8

A

elerating observer

405

Using (E.4.89) in the transformation formulas (D.4.28)-(D.4.30), the onstitutive law in the frame of a rotating observer turns out to be de ned by the onstitutive matri es

 

A =

0 Bab =

0ab

n

 

ab

Æ +

1

2

a b



;

v v

1



2 Æab + 2 va vb + n n

C 0a b = n Æ a ^ bd

(E.4.91)



v2 1 Æab 2 + 2 va vb

d

v :



Aem1rot

;

(E.4.92)

Bem1rot

(E.4.93)

Cem1rot

The matri es (E.4.91)-(E.4.93) satisfy the algebrai losure relation (D.3.17)-(D.3.19). The orresponding opti al metri is obtained from (D.4.25), (E.4.35) and (E.4.89) as g 0 opt ij =

r

n



2 n2

v2

vb



Æab +

va

2

2

va vb



!

(E.4.94)

:

In exterior al ulus, the onstitutive law (E.4.91)-(E.4.93) in the rotating frame reads

H

0

=



1 0

1 

D0 = ""0



0

E0

v2 " 2



0

0

h

B + ""0 v

0 0

(v

0



v0 ^  B0 ; 0

^B ) 0

0

(v

0

^E ) 0

i

;

(E.4.95) (E.4.96)

Here  denotes the Hodge operator with respe t to the or 2 responding 3-spa e metri Æab + 2 va vb [see (E.4.88)℄, and we introdu ed the velo ity 1-form 0

v 0 := Æab v a dx0b : E.4.8

(E.4.97)

A

elerating observer

Let us now analyze the ase of pure a

eleration. It is quite similar to the pure rotation that was onsidered in Se . E.4.6.

gijoptrot

406

E.4.

Ele trodynami s of moving ontinua

More on retely, we will study the motion in a xed spatial dire tion with an a

eleration 3-ve tor parametrized as = a ~n;

~ a

ab

or

= a nb :

(E.4.98)

a

el0

Here a2 := ~a
~a is the magnitude of a

eleration, and the unit ve tor ~n, with ~n
~n = nb nb = 1, gives its dire tion in spa e. Re all that we are in the Minkowski spa etime (E.4.68). The a

elerating oordinates (t ; x a ) an be obtained from the Cartesian ones (t; xa ) by means of the transformation 0

t

=

1

0

Zt

0

sinh  na x a +

d

0

Zt

osh ( );

(E.4.99)

a-0-n

(E.4.100)

a-i-n

(E.4.101)

Kij

0

xa

= Kb a x b + na

d

0

sinh ( ):

Here we denote the 3  3 matrix

Kba = (Æba

na nb ) + na nb

osh :

The s alar fun tion (t ) determines the magnitude of the a

eleration by 0

a(t0 )

=

d dt0

(E.4.102)

:

a

el1

Dierentiating (E.4.99)-(E.4.100), we obtain the transformation from the inertial oframe to the a

elerating one: 

dt dxa



=






1 + ~a
~x2 osh  1 + ~a ~2x na sinh 





0

0

nb

sinh 

Kb a



dt0 dx0b



:

(E.4.103)

dtdx-a-n

Substituting (E.4.103) into (E.4.68), we nd the metri in a

elerating oordinates: ds2

= 2 1 + ~a
~x = 2 0


2

(dt )2 0




d~ x0 d~ x0 :

(E.4.104)

This is one of the possible forms of the well known Rindler spa etime.

metA

E.4.8

A

elerating observer

407

It is straightforward to onstru t the lo al frame of a noninertial observer whi h is omoving with the a

elerating oordinate system. With respe t to the original Cartesian oordinates, it reads: 1 na eba = sinh  t + Ka b xb : (E.4.105) eb0 = u;



frame-a

Here u

= osh  t + na sinh  xa

(E.4.106)

is the observer's 4-velo ity ve tor eld whi h satis es g(u; u) =

2 . Clearly, the ve tors of the basis (E.4.105), in the sense of the Minkowski 4-metri (E.4.68), are mutually orthogonal and normalized: g(eb0 ; e^0 ) = 1;

g(eb0 ; ea^ ) = 0;

g(eba ; e^b ) =

Æab :

(E.4.107)

Be ause of (E.4.103), the oordinate bases are related by 

t xa



=






osh = 1 + ~a ~x2
na sinh = 1 + ~a ~x

2


0




0

nb sinh  b a

K



t0 x0b

 :

(E.4.108)

dtdxinv

Thus, the a

elerated frame (E.4.105), with respe t to the a

el oordinate system, is des ribed by the simple expressions

erating

eb0

=

1




(1 + ~ a ~ x= 2 )

t0 ;

eba

= x a : 0

(E.4.109)

frame-nih

A

ording to the de nition of the ovariant dierentiation, see (C.1.15), one has ru e =  (u)e . This enables us to ompute the proper time derivatives for the frame (E.4.105), (E.4.109): u_ e_bb

:= ru u = := ru ebb =

ab

ebb ;

(E.4.110)

propa

el

ab = 2 u: 1 + ~a ~x= 2

(E.4.111)

dotEa

1 + ~a
~x= 2




408

E.4.

Ele trodynami s of moving ontinua

This means that the frame (E.4.105) is Fermi-Walker transported along the observer's world line. The oframe dual to the frame (E.4.105), (E.4.109) reads, with respe t to the a

elerating oordinate system,
b ba a # = 1 +~ a
~ x= dt ; # = dx : (E.4.112) 0

E.4.9

2

0

0

ofr-a

The proper referen e frame of the noninertial observer (\noninertial frame")

The line elements (E.4.76) and (E.4.104) of spa etime represent the Minkowski spa e in rotating and a

elerating oordinate systems, respe tively. Both are parti ular ases of the interval h


2

i

1 + ~a
~x = + (~!
~x = ) (~!
~!= )(~x
~x ) 2dt d~x
[~!  ~x ℄ d~x
d~x : (E.4.113) Here the 3-ve tors of a

eleration ~a (= ab ) and of angular velo ity ~! (= ! ) an be arbitrary fun tions of time t . The interval (E.4.113) redu es to the diagonal form ds = #b #b #b #b b b b b # # # # in the orthonormal oframe :
b a
~ x=

dt ; (E.4.114) # = 1 +~ a ba a # = dx + [! ~ ~ x ℄ dt : (E.4.115) The orresponding ve tors of the dual frame are
1 [~!  ~x ℄a x a ; (E.4.116) t eb =

(1 + ~ a
~ x = ) (E.4.117) eba = x a : With respe t to the lo al frame hosen, the omponents  of the Levi-Civita onne tion read: (ab = ) bb b b b = = # = (a = ) dt ; (E.4.118) b bb 1 + ~a
~x = ^ab (! = ) bb b

= # = ^ab ! dt : (E.4.119) ba 1 + ~a
~x = 2

ds

= (dt ) 2

0

2

0

0

2

0

2

0

0

0

2

0

0

0

met-ni

0

2

2

2

3

0

3

0

2

0

0

0

1

1

5

0

0

0

0

0

2

0

0

0

ofr0

ofri

HNI0 HNIa

2

0

0

0

0

0

0

5 See

Hehl and Ni [8℄.

2

0

2

0

GamNI1

GamNI2

E.4.9 The proper referen e frame of the noninertial observer (\noninertial frame")

We an readily he k that d  = 0 and that the exterior produ ts of the onne tion 1-forms are zero. As a result, the Riemannian urvature 2-form of the metri (E.4.113) vanishes, Re = 0. Thus, indeed, we are in a at spa etime as seen by a non-inertial observer moving with a

eleration ~a and angular velo ity ~!. After these geometri al preliminaries, we an address the problem of how a noninertial observer (a

elerating and/or rotating) sees the ele trodynami al ee ts in his proper referen e frame (E.4.116), (E.4.117). In order to apply the results of the previous se tions, let us spe ialize either to the ase of pure rotation or of pure a

eleration. To begin with, we note that the onstitutive relation has its usual form (E.4.70), (E.4.71) in the inertial Cartesian oordinate system (E.4.68). Putting ~a = 0, we nd from (E.4.114), (E.4.115) the proper oframe of a rotating observer: b

#0

= dt ; 0

#ba

= dx a + [~!  ~x ℄a 0

0

(E.4.120)

dt0 :

ofraHN

Combining this with (E.4.74), we nd the transformation of the inertial oframe to the non-inertial (rotating) one,



dt dxa

  =

1 0



0

Lb a

b

#0 b #b

!

(E.4.121)

:

Correspondingly, substituting this into the transformation (D.4.28)(D.4.30), we immediately nd from (E.4.70) the onstitutive law in the rotating observer's frame: (E.4.122) onstNON Hba = 1 Æab B ba; D ba = ""0 Æab Eba: 0

0

0

0

0

The same result holds true for an a

elerating observer. If we put ~! = 0 in (E.4.114), (E.4.115), we arrive at the oframe (E.4.112). Combined with (E.4.103), this yields the transformation of the inertial oframe to the non-inertial (a

elerating) one:



dt dxa

  =

osh 

na sinh 

nb

sinh 

Kb a



b

#0 b #b

!

:

(E.4.123)

409

410

E.4.

Ele trodynami s of moving ontinua

When we use this in (D.4.28)-(D.4.30), the nal onstitutive law again turns out to be (E.4.122). Summing up, despite the fa t that the proper oframe (E.4.114), (E.4.115) is noninertial, the onstitutive relation remains in this

oframe formally the same as in the inertial oordinate system.6 E.4.10

Universality of the Maxwell-Lorentz spa etime relation

The use of foliations and of exterior al ulus for the des ription of the referen e frames enables us to establish the universality of the Maxwell-Lorentz spa etime relation. Let us put  = " = 1 (hen e n = 1) in the formulas above. Physi ally, this orresponds to a transformation from one frame (-foliation) to another frame (-foliation) whi h moves with an arbitrary velo ity u relative to the rst one. Then the relation (E.4.13) redu es to IH =  (?F );

f

f

IH =  a(?F ):

(E.4.124)

a

onst6

On the other hand, from (E.4.21)-(E.4.23), we nd for the onstitutive oeÆ ients:

p

(3) g

Aab =

N

(3) ab

g ;

Bab =

pN g (3)

(3)

gab ; C a b = 0: (E.4.125)

Equivalently, from (E.4.24) we read o that

Ha =

r"

0

0

p N B = g; a

r "0 p D = g= a

1 a E : (E.4.126) 0 N

or, returning to exterior forms,

D = "0 "g  E;

H = 01g  B:

(E.4.127)

6 This shows that it is misleading to asso iate the \Cartesian form" of a onstitutive relation with the inertial frames of referen e. Kovetz [13℄, for example, takes (E.4.122) as a sort of de nition of the inertial frames.

onst6b

E.4.10

Universality of the Maxwell-Lorentz spa etime relation

411

This is nothing but a (1 + 3) de omposition with respe t to the laboratory foliation: IH =  (?F );

IH =  ?(?F ):

?

(E.4.128)

onst6a

Comparing (E.4.124) with (E.4.128), we arrive at the on lusion that (E.4.124) and (E.4.128) are just dierent \proje tions" of the generally valid Maxwell-Lorentz spa etime relation IH =   F:

(E.4.129)

In this form, the Maxwell-Lorentz spa etime relation is valid always and everywhere. Neither the hoi e of oordinates or of a spe i referen e frame (foliation) play any role. Consequently, our fth axiom has a universal physi al meaning.

HastF

412

E.4.

Ele trodynami s of moving ontinua

Referen es

Dete ting Abraham's for e of light by the Fresnel-Fizeau ee t, Eur. Phys. J. D3 (1998) 205-

[1℄ S. Anto i and L. Mihi h, 210.

Current status of ele trodynami s of moving media (in nite media), Sov. Phys.

[2℄ B.M. Bolotovsky and S.N. Stolyarov, Uspekhi

17

(1975) 875-895 [Usp. Fiz. Nauk

114

(1974)

569-608 (in Russian)℄.

[3℄ M. Born and L. Infeld,

Foundations of the new eld theory,

Pro . Roy. So . (London)

A144 (1934) 425-451.

Phenomenologi al ele trodynami s in urvilinear spa e, with appli ation to Rindler spa e, J. Math. Phys. 28

[4℄ I. Brevik,

(1987) 2241-2249.

 die magnetis hen Wirkungen bewegter Uber Korper im elektrostatis hen Felde, Ann. d. Phys. 11 (1903)

[5℄ A. Ei henwald, 1-30; 421-441.

414

Referen es

Magneti duality rotations in nonlinear ele trodynami s, Nu l. Phys. B454

[6℄ G.W. Gibbons and D.A. Rasheed,

(1995) 185-206. Gordon, Zur Li htfortp anzung na h der Relativitatstheorie, Ann. Phys. (Leipzig) 72 (1923) 421-456.

[7℄ W.

[8℄ F.W. Hehl and W.-T. Ni,

Inertial ee ts of a Dira parti le,

D42 (1990) 2045{2048.

Phys. Rev.

Consequen es of Dira 's theory of positrons, Z. Phys. 98 (1936) 714-732 (in German).

[9℄ W. Heisenberg and H. Euler,

[10℄ J.S. Heyl and L. Hernquist,

Birefringen e and di hroism of

the QED va uum, J. Phys. A30 (1997) 6485-6492.

[11℄ L.L. Hirst,

The mi ros opi magnetization: on ept and ap-

pli ation, Rev. Mod. Phys. 69 (1997) 607-627.

[12℄ C. Itzykson and J.-B. Zuber,

Quantum Field Theory (M -

Graw Hill: New York, 1985). [13℄ A.

Ele tromagneti Theory.

Kovetz,

(Oxford

University

Press: Oxford, 2000). [14℄ H.A. Lorentz,

The Theory of Ele trons

and its Appli a-

tions to the Phenomena of Light and Radiant Heat. 2nd ed. (Teubner: Leipzig, 1916). [15℄ B. Mashhoon,

Gravitation,

Nonlo al ele trodynami s, in: Cosmology and

Pro . VII Brasilian S hool of Cosmology and

Gravitation, Rio de Janeiro, August 1993, M. Novello, editor (Editions Frontieres: Gif-sur-Yvette, 1994) pp. 245-295.

A

eleration indu ed nonlo al ele trodynami s in Minkowski spa etime,

[16℄ U. Muen h, F.W. Hehl, and B. Mashhoon,

Phys. Lett. [17℄ J.

A271 (2000) 8-15.

Pleba nski,

Non-Linear Ele trodynami s { a Study

(Nordita: 1968). Our opy is undated and stems from the CINVESTAV Library, Mexi o City ( ourtesy A. Ma  as).

Referen es

415

Applying relativisti ele trodynami s to a rotating material medium, Am. J. Phys. 68 (1998) 114-121;

[18℄ C.T. Ridgely, and

Applying ovariant versus ontravariant ele tromagneti tensors to rotating media, Am. J. Phys. 67

[19℄ C.T. Ridgely,

(1999) 414-421.

Ueber die dur h Bewegung eines im homogenen ele tris hen Felde be ndli hen Diele tri ums hervorgerufene ele trodynamis he Kraft, Ann. d. Phys. 35

[20℄ W.C. R ontgen,

(1888) 264-270. [21℄ J. Van Bladel,

Relativity and Engineering. Springer

Series

in Ele trophysi s Vol.15. (Springer: Berlin, 1984)

Me hani al for es in a diele tri due to ele tromagneti elds, Can. J. Phys. 55 (1977)

[22℄ G.B. Walker and G. Walker, 2121-2127.

Ele tri ee t of rotating a magneti insulator in a magneti eld, Pro . Roy. So .

[23℄ M. Wilson and H.A. Wilson, Lond.

A89 (1913) 99-106.

Part F

Preliminary sket h version of Validity of lassi al ele trodynami s, intera tion with gravity, outlook

416

417

le birk/partO.trun .tex, 2001-06-01

Of ourse, ele trodynami s des ribes but one of the four intera tions in nature. And lassi al ele trodynami s only the nonquantum aspe ts of the ele tromagneti eld. Therefore ele trodynami s relates to the other elds of knowledge in physi s in a multitude of dierent ways. Let us rst explore the lassi al domain.

418

F.1

Classi al physi s (preliminary)

F.1.1

Gravitational eld

The only other intera tion, besides the ele tromagneti one, that

an be des ribed by means of the lassi al eld on ept, is the gravitational intera tion. Einstein's theory of gravity, general relativity (GR), des ribes the gravitational eld su

essfully in the ma rophysi al domain. In GR, spa etime is a 4-dimensional Riemannian manifold with a metri g of Lorentzian signature. The metri is the gravitational potential. The urvature 2-form e , subsuming up to 2nd derivatives of the metri , represents R the gravitational eld strength. Einstein's eld equation, with respe t to an arbitrary oframe # , reads 1

1 e = 8G   :  ^ R (F.1.1) 2

The tilde labels Riemannian obje ts, G is the Newtonian gravitational onstant. The sour e of the right hand side is the symMat

3

1 Compare

Landau-Lifshitz [16℄, e.g.. Einstein's Prin eton le tures [5℄ still give a good

idea of the underlying prin iples and some of the main results of GR. Frankel [6℄ has written a little book on GR underlining its geometri al hara ter and, in parti ular, developing it in terms of exterior al ulus.

Einstein

420

F.1.

metri

Classi al physi s

(preliminary)

energy-momentum urrent of \matter", #[

 ℄ = 0 ; ^ Mat

(F.1.2)

Einsym

embodying all non-gravitational ontributions to energy. Ele trodynami s ts smoothly into this pi ture. The Maxwell equations remain the same, dH

=J;

dF

= 0:

(F.1.3)

Einmax

(F.1.4)

Einstr

However, the Hodge star in the spa etime relation H

=  ?F

\feels" now the dynami al metri g ful lling the Einstein equation. The ele tromagneti eld, in the framework of GR, belongs to the matter side, Mat

mat

 =  

Max

+ 

(F.1.5)

:

Einmatter

In other words, the ele tromagneti eld enters the gravity s ene Max Max via its energy-momentum urrent   . Sin e #[ ^  ℄ = 0, also this is possible in a smooth way. These are the basi s of the gravito-ele tromagneti omplex. Let us illustrate it by an example. Reissner-Nordstr om solution

We onsider the gravitational and the ele tromagneti eld in va uum of a point sour e of mass m and harge Q. The Einstein equation for this ase reads  ^ Re = 8G  [F ^ (e ?F ) ? F ^ (e F )℄ ; 

3





(F.1.6)

for the ele tromagneti equations see (F.1.3) and (F.1.4).

Einstein1

F.1.1 Gravitational eld

421

For solving su h a problem, one will take a spheri ally symmetri oframe and expresses it in spheri al oordinates r; ; : 1 dr ; #^2 = r d ; #^3 = r sin  d : ^ ^ #0 = f dt ; #1 = f

(F.1.7) It ontains the zero-form f = f (r) and is assumed to be orthonormal, i.e., the metri reads 1 dr2 r2 d2 + sin2  d2
: ds2 = o # # = f 2 2 dt2 

oframe1

f2

(F.1.8) In a Minkowski spa etime, i.e., without gravity, we have f = 1. The spheri ally symmetri ele tromagneti eld is des ribed by the Coulomb type ansatz: q

A

=

F

= dA = rq2 #^1 ^ #^0 :

r

dt ;

metri 1

(F.1.9)

Ein oulomb1

(F.1.10)

Ein oulomb2

Here q is a onstant. The homogeneous Maxwell equation dF = 0 as well as the inhomogeneous equation for the va uum ase d ? F = 0 are both ful lled. The energy-momentum urrent an be determined by a substitution of (F.1.10) into the expli it expression (E.1.23) as Max

2  ^ ^ Æ0 ^0 + Æ1 ^1

^

q   =  2r 4

Æ2 ^2

^



Æ3 ^3 :

(F.1.11)

maxenergy1

Clearly, Einstein's equations (F.1.1) are not ful lled: The geometri left-hand side vanishes and does not ounterbalan e the nontrivial right-hand side (F.1.11). The integration onstant q is related to the total harge of the sour e. From (B.1.1) and (B.1.41) we have, by using the Stokes theorem: Q

=

Z



3

=

Z

 3

D;

(F.1.12)

totalQ

422

F.1.

Classi al physi s

(preliminary)

where the 3-dimensional domain 3 ontains the sour e inside. From (F.1.10) and (F.1.4) we nd H

= D =  rq2 #^2 ^ #^3 =  q sin d ^ d:

(F.1.13)

Integration in (F.1.12) is elementary, yielding Q = 4q:

(F.1.14)

Re alling that  = "0=0 and = 1=p"00, we then nd the standard SI form of the Coulomb potential (F.1.9): p

A=

Q 4"0 r dt:

(F.1.15)

Let us now turn to the gravitational ase for an ele tri ally un harged sphere. Then, as is known from GR, we have the S hwarzs hild solution with 2Gm : f2 = 1 (F.1.16)

2 r

Here m is the mass of the sour e. It is an easy exer ise with

omputer algebra to prove that the va uum Einstein equation 1 e = 0 is ful lled for this hoi e of the oframe. 2  ^ R GR is a nonlinear eld theory. Nevertheless, if we now treat the ombined ase with ele tromagneti and gravitational eld, we an sort of superimpose the single solutions be ause of our

oordinate and frame invariant presentation of ele trodynami s. We have now f 6= 1, but still we keep the ansatz for the Coulomb eld (F.1.9). The form of the eld strength (F.1.10) remains the same in terms of the oframe (F.1.8). Also the energy-momentum urrent (F.1.11) does not hange. Hen e we

an write down the Einstein eld equation (F.1.6) with an expli itly known right hand side. For the unknown fun tion f 2 we

an make the ansatz f 2 = 1 2Gm= 2 r + U (r). For U = 0, we re over the S hwarzs hild ase. If we substitute this in the left hand side of (F.1.6), then we nd (also most onveniently by

Ein oulomb3

Eins hwarz

F.1.1 Gravitational eld

423

means of omputer algebra) an ordinary dierential equation of 2nd order for U (r) whi h an be easily solved. The result reads: f2 = 1

2Gm GQ2 +

2 r 4"0 4 r2

(F.1.17)

fun 1

This, together with the ele tri eld (F.1.15), represents the Reissner-Nordstrom solution of GR for a massive harged \parti le". The ele tromagneti eld of the Reissner-Nordstrom solution has the same inno ent appearan e as that of a point harge in

at Minkowski spa etime. It is lear, however, that all relevant geometri obje ts, oframe, metri , onne tion, urvature, `feel' { via the zero-form f { the presen e of the ele tri harge. If the

harge sati es the inequality Q2 4"0

 Gm2 ;

(F.1.18)

then the spa etime metri (F.1.8) has horizons whi h orrespond to the zeros of the fun tion (F.1.17). However, as it is learly seen from (F.1.9), (F.1.10) and espe ially (F.1.11), the ele tromagneti eld is regular everywhere ex ept for the origin. The arising geometry des ribes a harged bla k hole. When the harge is so large that (F.1.18) be omes invalid, a solution is no bla k hole but des ribes a bare singularity. These results an be straightforwardly generalized to gauge theories of gravity with post-Riemannian pie es in the onne tion, see [22, 9℄. Rotating sour e: Kerr-Newman solution

When a sour e is rotating, its ele tromagneti and gravitational elds are no longer spheri ally symmetri . Instead, the ReissnerNordstrom geometry of above is repla ed by the axially symmet-

QlessM

424

ri

F.1.

Classi al physi s

(preliminary)

on guration des ribed by the oframe r
dt a sin2  d
; ^0 # =  r  d r; ^ #1 = p
^2 # =  d ; sin   a dt + r2 + a2
d  ; ^ #3 = p 

(F.1.19)

frameKerr1

where
=
(r);  = (r; ), and a is a onstant. The latter is dire tly related to the angular momentum of the sour e. The ele tromagneti potential 1-form reads ^ A = A^0 #0 ; (F.1.20) AKerr1 with A^0 = A^0(r; ). Substituting the ansatz (F.1.19)-(F.1.20) into Einstein-Maxwell eld equations (F.1.6), (F.1.3) and (F.1.4), one nds: GQ2
= r2 + a2 2Gmr + (F.1.21) sol2a

2 4"0 4 ;  = r2 + a2 os2 ; (F.1.22) sol2b Q r p (F.1.23) sol2 A^0 = 4"
 : 0

A

ordingly, the ele tromagneti eld strength reads: 

= dA = 4"Q2 (a2 os2  r2) #^0 ^ #^1 0 (F.1.24)  2 a2 r sin  os  ^0 ^2 p # ^# : +
Denoting 2 := (r2 + a2 )2 a2
sin2 , we an introdu e the ve tor eld n of the adapted spa etime foliation by (F.1.25) n = n  with n = 2aGmr ; F



2

Phi

F.1.1 Gravitational eld

425

and write the metri of spa etime in the standard form:  dr  d sin   (d + n dt) : ds = N dt
 (F.1.26) Here N =
 = . For large distan es, we nd from (F.1.19), (F.1.21) and (F.1.22) the asymptoti spa etime interval   2 Gm GQ ds = 1 + 4" r + O(r ) dt

r   GQ 4 Gm + O(r ) dt d a sin 

r 2 " r   (F.1.27) GQ 2 Gm 1 + r 4" r + O(r ) dr   
 a d + sin d 1 + O (r ) : r 1+ r When the Lie derivative of the metri vanishes, L g = 0, with respe t to some ve tor eld  , the latter is alled a Killing ve tor of the metri . The Kerr-Newman metri possesses the two Killing ve tors: 2

2

2

2

2

2

2

2



2

2

2

2

2

3

2

0

2

2

0

3

4 2

2

2

0

2

2

4 2

2

2

2

2

3

asymKerr

2

4 2

2

2

3

2

 ij

(t)

()

=  ; and  =  : (F.1.28) Tn GR, the knowledge of the Killing ve tors provides an important information about the gravitating system. In parti ular, for a ompa t sour e the total mass and the total angular momentum an be given in terms of the Killing ve tors by means of the so- alled Komar formulas: Z Z

  (d k ): (F.1.29) M = ( d k ); L= 8G 16G 

t



3

(t)

1

S

()

S

(t)

()

Komar

1

The integrals are taken over the spatial boundary des ribed by the sphere of in nite radius. We used the anoni al map (C.2.3) to de ne the 1-forms e (  ); e(  ) (F.1.30) k =g k =g (t)

KillV

()

KillF

426

F.1.

Classi al physi s

(preliminary)

from the Killing ve tor elds. It is suÆ ient to use the asymptoti formula (F.1.27) in (F.1.29) to prove that for the Kerr-Newman metri we have M

= m;

L

= m a:

(F.1.31)

MLa

This explains the physi al meaning of the parameters m and a in the Kerr-Newman solution. It is easy to see that putting a = 0 brings us ba k to the Reissner-Nordstr om solution.

Ele trodynami s at the outside of bla k holes and neutron stars Neutron stars and bla k holes arise from the gravitational ollaps of the ordinary matter. The gravitational ee ts be ome very strong near su h obje ts, and GR is ne essary for the des ription of orresponding spa etime geometry. Normally, the total ele tri harge of the ollapsing matter is zero, and then we are left, in general, with the Kerr metri obtained from (F.1.19)(F.1.27) by putting Q = 0. Near the surfa e of a neutron star and outside of a bla k hole one an expe t many interesting ele trodynami al ee ts. To des ribe them, we need to solve Maxwell's equations in pres ribed Kerr metri 2 . It is amazingly easy to nd the exa t solution of the Maxwell equations in the Kerr geometry. The ru ial points are: (i) the fa t that Kerr geometry des ribes the va uum (matter-free) spa etime, and (ii) the existen e of the two Killing ve tor elds (F.1.28). It is straightforward to prove that every Killing ve tor e( ) whi h sati es k = 0 and  de nes a harmoni 1-form k = g y d k = 0 in a va uum spa etime. Re alling the Maxwell equation in the form of the wave equation (E.1.5), we immediately nd that the ansatz A

=

B0

2



2a (t)

k

+

()

k



(F.1.32)

2 This is the main idea behind the membrane model, see Straumann [23℄.

Amem1

F.1.1 Gravitational eld

427

yields an exa t solution of the Maxwell equations on the ba kground of the Kerr metri . Here B0 is onstant and the oeÆ ient in the rst term is hosen in a

ordan e with (F.1.29) and (F.1.31) in order to provide a total vanishing harge. Substituting (F.1.30) into (F.1.32), we nd expli itly A

r ^ 2 #0 = aB0 Gm (1 +

os ) p 2




(F.1.33)

+ 2 (r + a ) sin  d: The physi al interpretation is straghtforward: The 1-form potential (F.1.33) is a kind of superposition of the Coulomb-type ele tri pie e [the rst line, f. (F.1.20), (F.1.23)℄ with the asymptoti ally homogeneous magneti pie e [the se ond line℄. With respe t to the oordinate foliation (for whi h n =  ), the ele tromagneti eld strength reads F = dA = E ^ dt + B with B0

2

2

2

Amem2

t

=

2 2 2 2 2 [(1 + 2 os 2)(r a os ) dr + 2(r a ) sin  os  d℄; (F.1.34) B0 GM a B= [(1 + os2)(r2 a2 os2 ) sin2  dr ^ d

2 2 2r sin  os  (2r2 os2 + a2 + a2 os4 ) d ^ d℄ + B0 [r sin2  dr ^ d + (r2 + a2 ) os  sin d ^ d℄: (F.1.35) For large distan es the last line in (F.1.35) dominates yielding asymptoti aly the homogeneous onstant magneti eld F = B0 dx ^ dy + O (1=r 2 ): (F.1.36) Here we performed the usual transformation from spheri al oordinates (r; ; ) to Cartesian (x; y; z) ones. The asymptoti magneti eld is dire ted along the z-axis. It is interesting to note that the ele tri eld vanishes for the non-rotating a = 0 bla k hole. We an draw a dire t parallel to the Wilson and Wilson experiment where the magneti eld indu ed the ele tri eld inside a rotating body. The spa etime of

E

B0 GM a

Bmem

428

F.1.

Classi al physi s

(preliminary)

a rotating Kerr geometry a ts similarly and indu es the ele tri eld around the bla k hole. In the membrane approa h , the physi s outside a rotating bla k hole is des ribed with the help of a model when a horizon is treated as a ondu ting membrane whi h possesses ertain surfa e harge and urrent density, as well as the surfa e resistivity. One an develop, in parti ular, the me hanism of extra ting the (rotational) energy from a bla k hole by means of the external magneti elds. 3

For e-free elds

Near a bla k hole or a neutron star the for e-free elds an naturally emerge in the plasma of ele trons and positrons. In Se . B.2.2, we have de ned su h ele tromagneti elds by the

ondition of vanishing of the Lorentz for e (B.2.12). Using the spa etime relation (F.1.4), we an now develop a more substantial analysis of the situation. The for e-free ondition now reads: (e F ) ^ dF = 0 : (F.1.37) To begin with, let us re all the sour eless solution of above. The magneti eld (F.1.35) has the evident stru ture: B = d ^ d; (F.1.38) with  r + a GM a r  = B sin  (1 + os ) : (F.1.39) 2

 Thus, the magneti eld is manifestly axially symmetri , that is, its Lie derivative with respe t to the ve tor eld  is zero: L B = d( B ) +  dB = dd  0: (F.1.40) Here we used the Maxwell equation dB = 0 whi h are identi ally ful lled for any fun tion whi h does not depend on , = (r; ). 2

2

FFree

2

2

0

3 See,

2

e.g., Straumann [23℄ and the literature therein.

LieB

F.1.1 Gravitational eld

429

Returning to the problem under onsideration (i.e., with nontrivial plasma sour e), we will also demand that the magneti eld be axially symmetri . As more general stru ture of the eld we then expe t B = d ^ d +  dr ^ d:

(F.1.41)

Bfree

Clearly, this 2-form also satis es the axial symmetry ondition (F.1.40) for every . The Maxwell equation dB = 0 is again ful lled provided  = (r; ). Moreover, usually one assumes that  = ( ). Now we need also the ansatz for the ele tri 1-form E . Taking into a

ount the axial symmetry, a natural assumption reads E = v B;

(F.1.42)

v = 

(F.1.43)

Efree1

where the ve tor eld

an be interpreted as the rotational velo ity of the magneti eld lines. The fun tion does not depend on the angular oordinate . Furthermore, substituing (F.1.41) into (F.1.42), we nd E = d :

(F.1.44)

Efree2

The Maxwell equation dE = 0 is ful lled if = ( ). Combining (F.1.41) and (F.1.44), we nd the general ansatz for the ele tromagneti eld strength 2-form: F = d ^ dt + d ^ d +  dr ^ d:

(F.1.45)

This must be inserted into the for e-free ondition (F.1.37) in whi h we will use the natural oordinate frame, e = Æi i . Dire t inspe tion shows that equation (F.1.37) is identi ally ful lled for t and  . Substituting (F.1.45) into (F.1.37) for r and  yields a nontrivial dierential equation: d d(  d ) + d ^  d +   = 0: (F.1.46) d

FBEfree

Pfree

430

F.1.

Classi al physi s

(preliminary)

Here the fun tions  = (r; ), = (r; ), = (r; ) are onstru ted from , its derivative, and from the omponents of the spa etime metri . Equation (F.1.46) is alled Grad-Shafranov equation and, with the given fun tions = ( ) and  = ( ), the solution of (F.1.46) ompletely des ribes the for efree ele tromagneti eld on guration. The orresponding distribution of the harge and urrent density is derived from the Maxwell equation J = dH . F.1.2

Classi al (1st quantized) Dira eld

In lassi al ele trodynami s, the ele tri urrent 3-form J is phenomenologi ally spe i ed, it annot be resolved any further. We know, however, that ele tri harge is arried by the fundamental parti les, namely the leptons and the quarks. For many everyday ee ts, the ele tron and the proton ( onsisting of 3 on ned quarks) are responsible. The 2nd quantized Dira theory governs the behavior of the ele tron. If the energies involved in an experiment, are not too high ( ompared to the mass of the ele tron), then the ele tron an be approximately viewed as a lassi al matter wave, i.e., only a 1st quantized Dira wave fun tion. An ele tron mi ros ope and its resolution may well be des ribed in su h a manner; similarly, an ele tron interferometer for sensing rotation (Sagna type of ee t) needs no more re ned des ription. Let us then assume that the Dira matri es, referred to an orthonormal oframe, are given by

( ) = o :

(F.1.47)

Dira mat

Then we an introdu e Dira -algebra valued 1-forms :=  # . The Dira equation then reads i~ ^ D +  m = 0 ; with

e D = d + i A : (F.1.48) ~

The Dira adjoint is := y. The Dira equation an be derived from a Lagrangian L . The onserved ele tri urrent 0

D

Dira

F.1.2 Classi al (1st quantized) Dira eld

431

turns out to be J :=

ÆLD = ÆA

ie   ;

dJ = 0 :

(F.1.49)

Dira

urrent

The energy-momentum urrent of the Dira eld, a

ording to the Lagrange-Noether pro edure developed in Se .B.5.5, reads ÆLD i~  D =  Æ# 2

 =




D  ;

D = 0 :

(F.1.50)

sigmaDi

Here D := e D. Therefore the self- onsistent Dira -Maxwell system reads d ?F = J ; J=

dF = 0

ie   ;

(F.1.51)

i~ ^ D +  m = 0 : (F.1.52)

DiMax1 DiMax2

With gravity, we have to generalize the spinor ovariant derivative whi h now should read i e D =d+i A+  b  ;

(F.1.53) 4 where, as usual, b := i [ ℄. In order to write the Einstein gravitational eld equation, have to symmetrize the energymomentum urrent, ~

 = 

D

 = #[ ^  ℄ ;

with

(F.1.54)

Disym

where  = (LD =D ) (ib =4) is the anoni al spin urrent 3-form of the Dira eld. Then the omplete Einstein-Dira Maxwell system, together with (E.1.23,F.1.51,F.1.52), reads 1 e = 8G  ^ R 2

3

 Max

D



  +  ;

(F.1.55)

Alternative Dira oupling to gravity via the Einstein-Cartan theory. We allow for a metri ompatible onne tion  =  arrying a torsion pie e. Then the spin urrent of the Dira eld an be de ned a

ording to D

  :=

D

ÆL ~ = # ^ # ^

5 : ; Æ  4 

D

D D +#[ ^  ℄ = 0 :

(F.1.56)

tauDyn

432

F.1.

Classi al physi s

(preliminary)

Then, for the gravitational se tor of the the Einstein-Cartan-Dira -Maxwell system, we nd 1  ^ R = 8G Max + D  ; (F.1.57)    2 

3 1

8G D (F.1.58) 2  ^ T = 3  :

This is a viable alternative to the Einstein-Dira -Maxwell system. It looks very symmetri . Note that the ele tromagneti eld doesn't arry dynami al spin.

If we take either the Einstein-Dira -Maxwell or Einstein-CartanDira -Maxwell system, we an in any ase also study the gravitational properties of the Dira ele tron, see [10℄. Remark on super ondu tivity quasi- lassi ally understood: Ginzburg-Landau theory

Generalizing the lassi al Maxwell-London theory of super ondu tivity, GL a hieved, by introdu ing a omplex `order parameter', a quasi- lassi al des ription of super ondu tivity for T = 0. One onsequen e of this theory is the Abrikosov latti e of magneti ux lines, see Se .B.3.1. The Lagragian... F.1.3

Topology and ele trodynami s

In our book we in fa t did not dis uss genuine topologi al aspe ts of ele trodynami s. However, topology an play a very important and nontrivial role in ele trodynami s and in magnetohydrodynami s4 . A word of aution is in order: One should arefully distinguish the physi al situations in whi h an underlying spa etime (or spa e) has a ompli ated topology from the ase when the ele tromagneti eld on guration is topologi ally nontrivial. Usually the de isive role is played by the pure gauge ontribution to the ele tromagneti potential 1-form. A manifest example is given by the for e-free magneti elds whi h approximately des ribe the twisted ux tubes in the models of solar prominen e (sheets of luminous gas emanating from 4

See the reviews of Moatt and Marsh [19, 17, 18℄.

F.1.3

Topology and ele trodynami s

433

the sun's surfa e) . Re all the equation (B.2.14) whi h determines the for e-free magneti eld. It is identi ally ful lled when 5

dH =  B:

(F.1.59)

dHaB

Indeed, sin e B is a transversal 2-form (i.e., living in 3 spatial dimensions), we have B ^ B  0. Consequently, B ^ ea B = 0. Thus (F.1.59) solves (B.2.14) for any oeÆ ient fun tion  = (x). Note that the ansatz (F.1.59) an be used even in the metri free formulation of ele trodynami s. In Maxwell-Lorentz ele trodynami s, (F.1.59) further redu es to

d  B =  B:

(F.1.60)

dBaB

It is worthwhile to note that this is the eld equation of the so alled \topologi ally massive ele trodynami s" in 3 dimensions provided  is onstant. Spe ializing to the axially symmetri on gurations in Minkowski spa etime, we an easily nd a solution of (F.1.60) for any hoi e of . For example, in ylindri al oordinates (; ; z ), for () = 2=[a(1 +  =a )℄, we obtain 6

2

B=

B  0

2

1 + a2

2



d ^ d

1

a



dz :

(F.1.61)

Btwist

Here a is a onstant parameter whi h determines the twist. This solution des ribes a uniformly twisted ux tube. Although d 6= 0, nevertheless d ^ B = 0 whi h provides the onsisten y of the solution. It is straightforward to read o from (F.1.61) the potential 1-form,



A = 2 log 1 + a Ba 0

2

2



2

d

1

a

dz + d;

where  is an arbitrary gauge fun tion. 5 This is dis ussed in 6 See Deser et al. [4℄.

Marsh [18℄, e.g.



(F.1.62)

Atwist

434

F.1.

Classi al physi s

(preliminary)

There exist various topologi al numbers (or invariants) whi h evaluate the topologi al omplexity of an ele tri and magneti eld on guration. The so- alled magneti heli ity provides an expli it example of su h a number. De ned by the integral Z h := A ^ B; (F.1.63) V

the heli ity measures the \linkage" of the magneti eld lines. One an establish a dire t relation of h to the lassi al Hopf invariant, whi h lassi es the maps of a 3-sphere on a 2-sphere, and an interpret it in terms of the Gauss linking number. It is instru tive to ompare (F.1.63) with (B.3.16) and (B.3.17). Turning again to the twisted ux tube solution above, we see form (F.1.62) and (F.1.61) that in the exterior produ t of A with B only the ontribution from the pure gauge survives: A ^ B = d ^ B = d(B ). As a result, a nonrivial value of the heli ity (F.1.63) an only be obtained by assuming a toroidal topology of spa e. This an be a hieved by gluing two two-dimensional

ross-se tions at some values z1 and z2 of the third oordinate and, moreover, by assigning a nontrivial jump Æ = (z2 ) (z1 ) to the gauge fun tion. There is a lose relation between magneti intera tion energy and heli ity. For the for e-free magneti eld, we have expli itly, Z Z Emag = A ^ j = A ^ B; (F.1.64) where we used the Oersted-Ampere law j = dH and the ansatz (F.1.59). When  is onstant, the energy is proportional to the magneti heli ity. More on the ele trodynami s in multi- onne ted domains an be found in Marsh [17, 18℄. The orresponding ee ts underline the physi al importan e of the ele tromagneti potential 1-form A. Another manifestation of topology in ele trodynami al systems is of a more quantum nature: the Aharonov-Bohm ee t7 . 7 See the theoreti al dis ussion by Aharonov and Bohm [1℄ and the rst experimental ndings by Chambers [3℄, a re ent evaluation has been given by Nambu [20℄.

heli ity

F.1.4 Remark on possible violations of Poin are invarian e

435

Remarks on plasma physi s and magnetohydrodynami s As we saw already above, when we dis ussed solar prominen e, topologi al ee t of mainly magneti on gurations play a role in plasma physi s in general and in magnetohydrodynami s spe ifi ally, see Cap [2℄, Hora [11℄, and also Knoepfel [14℄.

F.1.4

Remark on possible violations of Poin are invarian e

The rst four axioms are ompletely free of the metri on ept. The Poin are group doesn't play a role at all. The onformal group is brought in by the fth axiom. In fa t, we start from a linear ansatz and end up, via some `te hni al assumptions', at the Maxwell-Lorentz spa etime relation. Can we modify the fth axiom in a suitable way? Opti al properties of the osmos near the big bang. Is there opti al a tivity, birefringen e et . of the va uum? If yes, then the fth axiom has to go willy nilly. See Kostele ky [15℄ and referen es given there.

436

F.1.

Classi al physi s

(preliminary)

F.2

Quantum physi s (preliminary)

F.2.1

QED

Our approa h is essentially lassi al. This is a legitimate assumption within the well known and rather wide limits of the idealized representation of the ele tri harges, urrents by means of parti les (and ontinuous media) and ele tromagneti radiation by means of lassi al waves of ele tri and magneti elds. However it is experimentally well established that an ele tron under ertain onditions may display the wave properties, whereas the ele tromagneti radiation may sometimes be treated in terms of parti les, i.e. photons. Quantum ele trodynami s (QED) takes into a

ount these fa ts. The mathemati al framework of QED is the s heme of the se ond quantization in whi h the ele tromagneti (Maxwell) and ele tron (Dira spinor) elds are repla ed by the eld operators a ting in the Hilbert spa e of quantum states. Physi al pro esses are then des ribed in terms of reation, annihilation, and propagation of quanta of ele tromagneti and spinor elds. The photon is massless and has spin 1, whereas ele tron is massive and has spin 1/2.

438

F.2.

Quantum physi s

(preliminary)

In QED, the ele tromagneti 1-form potential A plays a fundamental role, representing the \generalized oordinate" with the ex itation H being its anoni ally onjugated \momentum". Also the gauge symmetry A ! A + d moves to the enter of the theory: One an onsistently build the ele trodynami s on the basis of the lo al gauge invarian e prin iple. The underlying symmetry group is the Abelian U (1) and A naturally arizes as the U (1)-gauge eld potential whi h is geometri ally interpreted as the onne tion in the prin ipal U (1)-bundle of the spa etime. The gauge freedom is responsible for the masslessness of a photon, and thus ultimately for the long range hara ter of ele tromagneti intera tion. QED orre tly des ribes many quantum phenomena involving ele trons and photons. It is very well experimentally veri ed, in fa t this is the most pre isely tested physi al theory. The most famous and pre ise experiments are the proof of the predi ted value of the anomalous magneti moment of the ele tron and the observation of the shift of energy levels in atoms. The su

ess of QED is to a great extent related to the small2 ness of the oupling onstant f = 4"e 0 ~ of the theory whi h makes it possible to ee tively use the perturbation approa h. At the same time, QED is not free of de ien ies. The most serious is the problem of divergen es and the need of the regularization and renormalization methods.

F.2.2

Ele tro-weak uni ation

Re ently, the onsiderable progress has been a hieved in the

onstru tion of uni ed theories of the physi al intera tions. In parti ular, QED is found to be naturally uni ed with the weak intera tion. A typi al example of a quantum pro ess governed by the weak intera tion is the de ay of a neutron into the proton, ele tron and antineutrino. The modern understanding of the weak for es is based on the gauge approa h. In the Weinberg-Salam model, the fundamental symmetry group SU (2)  U (1) gives rise to the

F.2.2 Ele tro-weak uni ation

439

four gauge elds as the mediators of the ele tro-weak intera tion: harged W + and W ve tor bosons, a neutral Z 0 intermediate ve tor boson, and the photon . The spin 1 parti les W  ; Z 0 be ome massive via the me hanism of the spontaneous symmetry breaking of the gauge group, whereas orresponds to the unbroken exa t subgroup and remains massless. In the so alled standard model, the group SU (2)  U (1) is enlarged to SU (3)  SU (2)  U (1) and the resulting theory represents the uni ation of ele tromagneti , weak and strong for es. The olor group SU (3) brings to life the 8 additional gauge eld 1-forms Aa , a = 1; : : : ; 8 whi h are alled the gluon potentials. The standard model is responsible for the des ription of intera tion of quarks (fermioni onstituents of the barions) and leptons (ele tron, muon, tau, and neutrinos) by means of the gluons Aa and the intermediate gauge bosons W  ; Z 0 and

. Of the most ambitious development of the uni ation program it is worthwhile to mention the (super)string model whi h aims ultimately in onstru ting the onsitent quantum gravity theory. In the string model the point parti les are repla ed by the extended fundamental obje ts. The main advantage of su h an approa h is the possibility of elimination of the quantum divergen es.1 What an we learn from our approa h for the quantum eld theory? Perhaps that the Abelian axion is a very natural andidate for a parti le, in ontrast to the Dira monopole. Also our approa h ould give hints how a possible violation of Poin are invarian e ould be brought about. Will the topologi al results, see above, turn out to be important in higer-dimensional theories? 1 There exist a lot of texts on the string theory. As a good introdu tion, one may

onsult [21℄.

440

F.2.

F.2.3

Quantum physi s

(preliminary)

Quantum Chern-Simons and the QHE

In Se . B.4.4 we have presented a lassi al des ription of the quantum Hall ee t. Su h an approa h is however only qualitatively orre t. The true understanding of the QHE is possible when the quantum aspe ts of the phenomenon are arefully studied. We have no tools in our book to pursue this goal. The best thing we an do is to address the interested reader to the orresponding literature . The QHE is des ribed within the framework of the ee tive topologi al quantum eld theory with a Chern-Simons a tion. The explanation of the quantization of the Hall ondu tan e  is the ultimate goal a hieved in these studies. In identally, if one des ribes the quantum Hall ee t for low lying Landau levels, then the on ept of a omposite fermion is very helpful: it onsists of one ele tron and an even number of uxoids is atta hed to it. Isn't that a very lear adiitional indi ation of what the fundamental quantities are in ele trodynami s? Namely, ele tri harge (see rst axiom) and magneti

ux (see third axiom). 2

H

3

2 See 3 See

Fr ohli h and Pedrini [7℄, e.g.. Jain [12, 13℄ and, in this general ontext also Nambu [20℄.

Referen es

Signi an e of ele tromagneti potentials in the quantum theory, Phys. Rev. 115 (1959)

[1℄ Y. Aharonov and D. Bohm,

485-491.

[2℄ F. Cap,

Lehrbu h der Plasmaphysik und Magnetohydrody-

namik (Springer: Wien, 1994).

Shift of an ele tron interferen e pattern by en losed magneti ux, Phys. Rev. Lett. 5 (1960) 3-5.

[3℄ R. G. Chambers,

Three-dimensional 48 (1982) 975978; S. Deser, R. Ja kiw, and S. Templeton, Topologi ally massive gauge theories, Ann. Phys. 140 (1982) 372-411 [Erratum, Ann. Phys. 185 (1988) 406℄.

[4℄ S. Deser, R. Ja kiw, and S. Templeton,

massive gauge theories,

[5℄ A. Einstein,

Phys. Rev. Lett.

The Meaning of Relativity, 5th ed. (Prin eton

University Press: Prin eton 1955).

[6℄ T. Frankel,

Gravitational Curvature.

An Introdu tion to

Einstein's Theory (Freeman: San Fran is o, 1979).

442

Referen es

[7℄ J. Fr ohli h and B. Pedrini,

anomaly,

in:

New appli ations of the hiral

\Mathemati al Physi s 2000",

Ed. T. Kib-

ble (Imperial College Press: London, 2000), 39 pp.; hepth/0002195. [8℄ F.W.

Hehl,

J.

Lemke,

and

on fermions and gravity, Physi s,

in:

E.W.

Mielke,

Two le tures

Geometry and Theoreti al

Pro . of the Bad Honnef S hool 12{16 Feb. 1990,

J. Debrus and A.C. Hirshfeld, eds. (Springer: Heidelberg, 1991) pp. 56{140.

Metri -aÆne gauge theory of gravity II. exa t solutions, Int. J. Mod. Phys. D8 (1999)

[9℄ F.W. Hehl and A. Ma ias,

399-416. [10℄ F.W. Hehl, A. Ma  as, E.W. Mielke, and Yu.N. Obukhov,

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Index

rsted-Ampere law, 151, 175 ABS, 67 a

eleration, 406 ACOS, 67 ACOSD, 67 ACOSH, 67 ACOT, 67 ACOTD, 67 ACOTH, 67 ACSC, 67 ACSCD, 67 ACSCH, 67 a tion prin iple, 216 almost omplex stru ture, 40, 60, 264 AND, 71 anholonomity obje t, 94, 290 anti-self-dual form, 62, 265 ASEC, 67 ASECD, 67

ASECH, 67 ASIN, 67 ASIND, 67 ASINH, 67 ATAN, 67 ATAND, 67 ATANH, 67 atlas, 79 oriented, 79, 87 autoparallel, 241 Axion, 299 basis, 34  -forms, 273 ^-forms, 53

o-, 34

ove tor, 34 half-null, 257 null Finkelstein, 259 Newman-Penrose, 258

Index

orthonormal, 255 spa e of 2-forms, 56 transformation, 58 transformation law, 35 ve tor, 34 Betti numbers, 109 Bian hi identity rst, 249, 284 se ond, 249, 284 zeroth, 280, 284 Boolean expressions, 71 boundary, 118, 125 map, 124 Cartan's displa ement, 247, 281 stru ture equation rst, 249 se ond, 249 zeroth, 280 CBRT, 67

hain, 118 singular, 124

harge bound, 370 external, 370 free, 373 material, 373 Christoel symbols, 277 CLEAR, 73

losure relation, 262

obasis, 34

odierential, 278

oframe, 93 foliation ompatible, 150 COFRAME (Ex al ), 107

ohomology, 109

445

ondu tivity, 373

onne tion, 234 1-forms, 236 Riemannian, 277 transposed, 243

onstitutive law anisotropi media, 373 inertial frame, 401 laboratory frame, 388 Minkowski, 390 moving medium, 389 simple medium, 373

oordinate hart, 78

oordinates, 78 a

elerating, 406 Cartesian, 28

omoving, 384 Lagrange, 384 noninertial, 408 rotating, 402 COS, 67 COSD, 67 COSH, 67 COT, 67 COTD, 67 COTH, 67

ovariant dierential, 237 dierentiation, 234 of geometri quantity, 239 exterior derivative, 248 Lie derivative, 250

ove tor, 34 CSC, 67 CSCD, 67

446

Index

CSCH, 67

urrent bound, 370 external, 370

urvature, 244 2-form, 244 geometri al meaning, 244 tensor, 245

y le, 125 d'Alembertian, 279 de Rham

ohomology groups, 111

omplex, 125 map, 128 theorem, 128 rst, 128 se ond, 128 density, 31 dieomorphism, 95 1-parameter group, 98, 219 dierentiable manifold, 27, 77, 78 non-orientable, 80, 89 orientable, 79, 87 map, 95 stru ture, 78 dierential, 83 form, 83 tensor-valued, 40 twisted, 32, 88 map, 95 twisted form transformation law, 89 DILOG, 67 distortion 1-form, 284 duality operator, 262, 299

losure ondition, 312 Einstein's theory, 27 ele tri

harge

onservation, 143 density, 140

onstant, 356

urrent, 143 3-form, 145 density, 143 dimension, 145 ee tive permittivity, 359 ex itation 2-form, 151 dimension, 171 on a boundary, 173 eld strength 1-form, 156 sus eptibility, 372 ele tromagneti energy ux density 2-form, 212, 375, 378 energy-momentum, 199 (1+3)-de omposition, 212

anoni al urrent, 219

onservation law, 220

onservation law, 360 free- harge, 374 kinemati 3-form, 201 material, 377 symmetry, 362 ex itation, 147 2-form, 147 dimension, 147 external, 371 in matter, 370 eld strength 2-form, 155

Index

invariants, 367 momentum density 3-form, 213, 376, 378 potential 1-form, 166 stress 2-form, 213, 376, 378 ele tromagneti eld, 176 energy density 3-form, 212, 375, 378 energy-momentum Abraham, 376, 381 Minkowski, 376 ERF, 67 Euler hara teristi , 111, 129 Euler-Lagrange equation, 221 evaluation, 71 EXP, 67 experiment of Rontgen, 396 of Walker & Walker, 379 of Wilson & Wilson, 398 EXPINT, 67 exterior derivative properties, 91 dierential form, 83 dierentiation, 90 form, 41

losed, 108 exa t, 108 longitudinal, 148, 385 transversal, 148, 385 produ t, 43 FACTORIAL, 67 Faraday's indu tion law, 161 frame, 93

447

foliation ompatible, 150 holonomi , 94 inertial, 384 normal, 238 FRAME (Ex al ), 107 Fresnel equation, 304 gauge transformation, 31, 216 gauge eld momentum, 217 Gauss law, 151, 170 general relativity (GR), 27, 253, 284 geometri quantity, 39 Hall quantum ee t, 191 resistan e, 193 Heisenberg-Euler ele trodynami s, 366 homology, 124 homotopi equivalen e, 111 HYPOT, 67 ideal 2-dimensional ele tron gas, 192 ideal ondu tor, 171 ideal super ondu tor, 174 in x operator, 65 integer expressions, 69 integral, 114 of exterior n-form, 115 of exterior p-form, 119 of twisted n-form, 116 interior produ t, 46 Ja obian determinant, 30, 87 jump onditions, 394

448

Index

Klein bottle, 80 homology groups, 127 Krone ker symbol, 51 Lagrangian 4-form, 216 Maxwell, 357 Lapla e-Beltrami operator, 279 Levi-Civita

onne tion, 277 permutation symbol, 51, 261 tensor densities, 276 Lie derivative, 99

ovariant, 250 of exterior forms, 101 of geometri quantity, 102 of s alar density, 103 LN, 67 lo ality hypothesis, 384 LOG, 67 LOG10, 67 LOGB, 67 Lorentz for e

ove tor-valued 4-form, 154 on bound harge, 374, 381 on free harge, 374 Mobius strip, 80, 89, 121 magneti

onstant, 356 ee tive permeability, 359 ex itation 1-form, 151 dimension, 175 on a boundary, 174 eld strength 2-form, 156

ux, 161

sus eptibility, 372 magnetization, 370, 372, 391, 400 manifold, 77 metri -aÆne, 282 Riemann-Cartan, 280 Riemannian, 275 spa etime, 145 matter urrent, 217 Maxwell's equations, 178, 356 homogeneous, 166 in anholonomi oordinates, 185 in matter, 371 inhomogeneous, 147 Maxwell's theory axioms, 177 Maxwell-Lorentz relation, 337 Maxwellian double plates, 170 Meissner ee t, 174 metri , 255 Minkowski, 253, 256, 338, 401 on spa e of 2-forms, 57, 263 opti al, 377, 392 Riemannian, 275 S honberg-Urbantke, 270 tensor, 255 eld, 276 three-spa e, 386 ve tor spa e, 255 Noether urrent, 218 nonmetri ity, 279 1-form, 280 NOT, 71

Index

observer a

elerating, 406

omoving, 384 noninertial, 409 rotating, 404 one-form, 33 image, 84 open set, 78 OR, 71 orientation, 40, 49, 87 inner, 113, 120 of a ve tor spa e, 49 outer, 113, 120 standard, 50 parallel transport, 241 period, 127, 147, 152 permeability of va uum, 356 permittivity of va uum, 356 Poin are lemma, 109 polarization, 370, 372, 391, 400 position ve tor eld, 246 pre x operator, 66 proje tive plane, 80 homology groups, 126 pull-ba k map, 96 Ri

i identity, 249 s alar density, 48 s alar expressions, 69 S honberg-Urbantke formulas, 270

449

SEC, 67 SECD, 67 SECH, 67 segmental urvature, 281 self-dual form, 62, 265 simplex, 116 fa es, 117 singular, 119 standard, 122 SIN, 67 SIND, 67 singular homology group, 124 SINH, 67 spa e

onne ted, 78, 145

at, 245 Hausdor, 78, 145 of two-forms, 55

omplexi ed, 61 para ompa t, 78, 145 spa etime, 145 foliation, 146 laboratory, 383 material, 383 Rindler, 406 spa etime relation, 218, 294 linear lo al, 345 nonlo al, 364 Maxwell-Lorentz universality, 410 nonlinear lo al, 365, 366 spe ial relativity (SR), 253, 284 sphere homology groups, 126

450

Index

SQRT, 67 Stokes' theorem, 29, 122

ombinatorial, 122 submanifold, 120 surfa e harge, 172 surfa e urrent, 174 TAN, 67 TAND, 67 TANH, 67 tensor, 37 de omposable, 37 density, 48 transformation law, 37 twisted, 40 topologi al invariant, 110, 127 stru ture, 78 topology, 78 torsion, 242 2-form, 243 geometri al meaning, 242 tensor, 243 torus, 80 homology groups, 126 triangulation, 125 triplet of 2-forms

ompleteness, 267 self-dual, 266 unbound variables, 70 variables unbound, 70 variational derivative, 217 ve tor, 33

ontravariant, 36

ovariant, 36

eld, 83 integral urve, 98 of foliation, 148 image, 84 length, 255 null, 256 spa e, 33 aÆne, 247 dual, 34 oriented, 49 spa e-like, 256 tangent, 82 time-like, 256 transversal, 120 velo ity angular, 402 mean material, 384 of light, 356 one-form, 390 relative, 386 volume, 47 element, 48 elementary, 51 form, 87 twisted 4-form, 261 wave operator, 279 wave operator, 356 wedge, 43 Weyl

ove tor, 281 one-form, 281