Foundations of Classical Electrodynamics

Progress in Mathematical Physics Volume 33

Editors-in-Chief

Friedrich W. Hehl Yuri N. Obukhov

Anne Boutet de Monvel, Universite' Paris VII Denis Diderot Gerald Kaiser, The Virginia Center for Signals and Waves

Editorial Board D. Bao, University of Houston C . Berenstein, University of Maryland, College Park P. Blanchard, Universitlit Bielefeld A.S. Fokas, Imperial College of Science, Technology and Medicine C. Tracy, University of California, Davis H. van den Berg, Wageningen University

Foundations of Classical Electrodynamics Charge, Flux, and Metric

Birkhauser Boston Base1 Berlin

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Friedrich W. Hehl Institute for Theoretical Physics University of Cologne 50923 Cologne Germany and Department of Physics & Astronomy University of Missouri-Columbia Columbia, MO 6521 1 USA

Yuri N. Obukhov Institute for Theoretical Physics University of Cologne 50923 Cologne Germany and Department of Theoretical Physics Moscow State University 117234 Moscow Russia

Preface

Library of Congress Cataloging-in-Publication Data Hehl, Friedrich W. Foundations of classical electrodynamics : charge, flux, and metric I Friedrich W. Hehl and Yuri N. Obukhov. p. cm. -(Progress in mathematical physics ; v. 33) Includes bibliographical references and index. ISBN 0-8176-4222-6 (alk, paper) - ISBN 3-7643-4222-6 (Basel : alk. paper) 1. Electrodynamics-Mathematics. I. Obukhov, IU. N. (IUrii Nikolaevich) 11. Title. 111. Series.

20030521 87 CIP AMS Subiect Classifications: 78A25,70S20,78A05, 81V10,83C50,83C22

Printed on acid-frcc paper. 02003 BirkhYuser Boston

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All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhluser Boston, c/o Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 0-8176-4222-6 ISBN 3-7643-4222-6

SPIN 10794392

Reformatted from the authors' files by John Spiegelman, Abbington, PA. Printed in the United States of America.

BirkhBuser Boston Basel Berlin A member qf Berrel.7,p,annSpringer Science+Rusiness Media GmbH

I11 this book we display the fundamental structure underlying classical electrodynamics, i.e., the phenomenological theory of electric and magnetic effects. The book can be used as a textbook for an advanced course in theoretical electrodynamics for physics and mathematics students and, perhaps, for some highly motivated electrical engineering students. We expect from our readers that they know elementary electrodynamics in the coiiventional (1 3)-dimensional form including Maxwell's equations. Moreover, they sholild be familiar with linear algebra and elementary analysis, including vector analysis. Some knowledge of differential geometry would help. Our approach rests on the metric-free integral formulation of the conservation laws of electrodynamics in the tradition of F. Kottler (1922), h. Cartan (1923), and D. van Dantzig (1934), and we stress, in particular, the axiomatic point of view. In this manner we are led to an understanding of why the Maxwell equations have their specific form. We hope that our book can be seen in the classical tradition of the book by E. J. Post (1962) on the Formal Strtlcture of Electmmagnetics and of the chapter "Charge and Magnetic Flux" of the encyclopedia article on classical field theories by C. Truesdell and R. A. Toupin (1960), including R. A. Toupin's Bressanone lectures (1965); for the exact references see the end of the introduction on page 11. The manner in which electrodynamics is conventionally presented in physics courses k la R. Feynman (1962), J . D. Jackson (1999), and L. D. Landau & E. M. Lifshitz (1962) is distinctly different, since it is based on a flat spacetime manifold, i.e., on the (rigid) Poincare group, and on H. A. Lorentz's approach (1916) to Maxwell's theory by means of his theory of electrons. We believe that the approach of this book is appropriate and, in our opinion, even superior for

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Preface

a good understandi~igof the structure of electrodynamics ns a classical field theory. In particular, if gravity cannot be neglected, our framework allows for a smooth and trivial transition to thc curved (and contorted) spacetime of general relativistic field tlicolies. This is by no means a minor merit when one has to treat magnetic fields of the order of 10%esla in the neighborhood of a neutron star where spacetime is appreciably curved. Mathematically, intcgrands in the conservation laws are represented by exterior differential forms. Therefore exterior calculus is the appropriate language in which clectrodynaniics shoulcl be spelled out. Accordingly, we exclusively use this formalism (even in our computer algebra programs which we introduce in Scc. A.1.12). In Part A, :tnd later in Part C, we try to motivate and to supply the ncctwary ~nathematicalframework. Readers who are familiar with this formalism may want to skip thesc parts. Thcy could start right away with tlie physics in Part B and tlicn turn to Part D and Part E. In Part B four axioms of classical clectroclynamics are formulated and the consequences derivctl. Tliis general framework lins to be completed by a specific el~ctromagnetzcspacet~merelation as a fifth axiom. This is done in Part D. The Maxwell-Lorentz theory is then recovered under specific conditions. In Part E, we apply clcctrotlynamics to moving continua, inter alia, which requires a sixth axiom on the formulation of clcctroclynamics inside matter. This book grew out of a scientific collaboration with the late Dermott McCrea (University College Dublin). Mainly in Part A and Part C, Dermott's liandwriting call still be seen in numerous places. There are also some contributions to "our" mathematics from Wojtelc KopczyAslci (Warsaw University). At Cologne University in tlic summclr tc.rm of 1991, Martin Zimzbauer started to teach the thcorctical clcctrotlynamics course by using the calculus of exterior differential forms, and lie wrote up successively improvcd notes to his coursc. One of the a~itliors(FWH) also taught this course threc times, partly based on Zirnbaucr's notes. Tliis influcnccd our way of prescnting electrodynamics (and, we believe, also his way). We are very grateful to him for many discussions. There are many collcagucs ant1 friends who helped us in critically reading parts of our book and wlio maclc srlggestions for in~provementor wlio communicated to 11s their own ideas on electrodynamics. We are very grateful to all of them: Carl Brans (Ncw Orleans), Jeff Flowers (Teddington), Dav~dHartley (Aclelaiclc), Cliristia~~ Hcinickc (Cologne), Yakov Itin (Jernsalcm) Martin Janssen (Cologne), Gerry II(aiser (Glen Allen, Virginia), R. M. Kielin (brmerly Houston), Attay II(ovctz (Tcl Aviv), Claus Lammerzalil (Konstanz/Eremen), Bahram Mnslilioon (Columbia, Missouri), Eckeliard Mielke (Mexico City), WeiTou Ni (Hsin-chu), E. Jan Post (Los Angeles), Dirk Piitzfeltl (Cologne), Guillermo Rubilar (Colognc/Conccpci61i), Yasha Shnir (Cologne), Andrzej Dautman (Warsaw), Arkady Tseytlin (Colu~nbus,Ohio), Wolfgang Weller (Leipsig), and otlicrs. We are particularly grateful to the two reviewers of our book, to Jini Nester (Cliung-li) and to an al~onymfor their numerous good suggcstio~lsand for their painstaking work.

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We arc very obliged to Uwe Essmann (Stuttgart) and to Gary Gl~txrnaicr (Santa Cruz, California) for providing beautiful and instructive images. We arc equally grateful to Peter Scherer (Cologne) for his permission to reprint his three comics on computer algebra. The collaboration with the Birkliauser people, with Gerry Kaiser and Ann Kostant, was effective and fruitful. We would like to thank Debra Daugherty (Boston) for improving our English. Pleasc convey critical remarks to our approach or the discovery of mistakes by surface or electronic mail (1iehlQtlip.uni-koeln.de, [email protected]) or by fax +49-221-470-5159. This project has been supportcd by the Alexander von Humboldt Foundation (Bonn), tlie German Academic Exchange Service DAAD, and the Volkswagen Foundation (Hanover). We are very grateful for tlic unbureaucratic help of these institutions.

Friedrich W. Helil Cologne Yuri N. Obukhov Moscow April 2003

Contents

Preface

v

Introduction 1 Five plus one axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 To~~ological approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Elcct.ronii~gnclticspnccttime relation i ~ fifth s axiom . . . . . . . . . . . . 4 Elcctrodynnmics in lilatter and thc sixth axiorrl . . . . . . . . . . . . . 5 List. of axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 A r c ~ n i ~ i d rElect~rodyriamics r: in 3-tlimcnsional Euclidrcx-ln vect.or calculus 5 On t h r litcrnture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

A

References

11

Mathematics: Some Exterior Calculus

17

Why exterior differential forms?

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A.1 Algebra 23 A . l . l A real vcctor space and its dual . . . . . . . . . . . . . . . . . . 23 A . 1.2 Tensors of type [y] . . . . . . . . . . . . . . . . . . . . . . . . . 25 A.1.3 @ Ageneralization of tensors: geometric quantities . . . . . . . . 27 A.1.4 Almost colnplex structure . . . . . . . . . . . . . . . . . . . . . 29 A. 1.5 Ext.erior pforrns . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Contents

A.l.G A.1.7 A.1.8 A.1.9 A.1.10 A.l.11 A.1.12

Exterior multiplication . . . . . . . . . . . . . . . . . . . . . . . 30 Interior m~~ltiplication of a vector wit], a form . . . . . . . . . . 33 @Volumeelcments on a vector space, densities, orientation . . . 34 @Levi-Civitasymbols and generalized Kronecker deltas . . . . . 36 The space M F of two-forms in four dimensions . . . . . . . . . 40 Almost complex structure on Me . . . . . . . . . . . . . . . . . 43 Computer algebra . . . . . . . . . . . . . . . . . . . . . . . . . 45

A.2 E x t e r i o r calculus 57 A.2.1 @Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . 57 A.2.2 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 A.2.3 One-form ficlds, differential p-forms . . . . . . . . . . . . . . . . 62 A.2.4 Pictures of vectors and onc-forms . . . . . . . . . . . . . . . . . 63 A.2.5 @Volume forms and orientability . . . . . . . . . . . . . . . . . 64 A.2.G @Twisted forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 A.2.7 Exterior dcrivativc . . . . . . . . . . . . . . . . . . . . . . . . . G7 A.2.8 Frame and coframe . . . . . . . . . . . . . . . . . . . . . . . . . 70 A.2.9 @Mapsof ~nanifoltls:push-forward and pull-back . . . . . . . . 71 A.2.10 @Licderivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 A.2.11 Excalc, a Rcdrlcc package . . . . . . . . . . . . . . . . . . . . . 78 A.2.12 @Closet1ant1 exact forms, de Rham cohomology groups . . . . 83

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87 A.3 I n t e g r a t i o n o n a manifold A.3.1 Integration of 0-forms and orientability of a manifold . . . . . . 87 A.3.2 Integration of 11-forms . . . . . . . . . . . . . . . . . . . . . . . 88 A.3.3 @Inte g rationof pforms with 0 < p < n . . . . . . . . . . . . . . 89 . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 A.3.4 Stoltcs' tlicorc~i~ A.3.5 @DeRhani's thcorcms . . . . . . . . . . . . . . . . . . . . . . . 9G

B

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References

103

Axioms of Classical Electrodynamics

107

B.l Electric charge conservation 109 B . I . ~ Counting chargcs. Absolute and relative dimension . . . . . . . 109 B. 1.2 spacetit1~cand the first axiom . . . . . . . . . . . . . . . . . . . 114 B.1.3 Elcctrotnagnctic excitation H . . . . . . . . . . . . . . . . . . . 116 B4 Timc-spnce dccom1)osition of the inhomogeneous Maxwell cquat.ion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117 B.2 Lorentz force density 121 ~ . 2 . 1 ~lectromagneticfield strength F . . . . . . . . . . . . . . . . . 121 ~ ~ 2 . Second 2 axiom relatillg ~ecllanicsand electrodynamics . . . . . 123 T'he first t.hrce invariants of the electromagnetic field . . . . . 126 ~ . 2 . 3@

B.3 M a g n e t i c flux conservation 129 B.3.1 Third axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129 B.3.2 Electromagnetic potential . . . . . . . . . . . . . . . . . . . . .132 B.3.3 @AbelianChern-Simons and Kiehn 3-forms . . . . . . . . . . . 134 B.3.4 Measuring the excitation . . . . . . . . . . . . . . . . . . . . . .136 B.4 Basic classical electrodynamics summarized. e x a m p l e B.4.1 Integral version and Maxwell's equations . . . . . . . . . B.4.2 @Len2and anti-Lenz rule . . . . . . . . . . . . . . . . . B.4.3 @Jumpconditions for electromagnetic excitation and field strength . . . . . . . . . . . . . . . . . . . . . . . . B.4.4 Arbitrary local noninertial frame: Maxwell's equations ill components . . . . . . . . . . . . . . . . . . . . . . . 13.4.5 @Electrodynamicsin flatland: 2DEG and QHE . . . . .

143

. . . .143 . . . .146 . . . .150

. . . .151 . . . . 152

163 B.5 Electromagnetic e n e r g y - m o m e n t u m c u r r e n t a n d action B.5.1 Fourth axiom: localization of energy-momentum . . . . . . . . 163 B.5.2 Energy-momentum current, electric/magnetic reciprocity . . . 166 B.5.3 Time-space decomposition of the energy-momentum and the Lenz rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .174 B.5.4 @Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .177 B.5.5 @Couplingof the energy-momentum current t o the coframe . . 180 B.5.6 Maxwell's equations and the energy-momentum current it1 Excalc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 .

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References

187

More Mathematics

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C . 1 Linear connection 195 C.1.1 Covariant differentiation of tensor fields . . . . . . . . . . . . . 195 C.1.2 Linear connection 1-forms . . . . . . . . . . . . . . . . . . . . .197 C.1.3 @Covariant differentiation of a general geometric quantity . . . 199 C.1.4 Parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . .200 C.1.5 @Torsionand curvature . . . . . . . . . . . . . . . . . . . . . .201 (2.1.6 @Cartan'sgeometric interpretation of torsion and curvature . . 205 C.1.7 @Covariantexterior derivative . . . . . . . . . . . . . . . . . . .207 C.1.8 @Theforms o(a), connl (a,b), torsion2(a). curv2(a.b) . . . . . . 208 C.2 M e t r i c 211 C.2.1 Metric vector spaces . . . . . . . . . . . . . . . . . . . . . . . . 212 C.2.2 @Orthonormal. half-null. and null frames. the coframe statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .213 C.2.3 Metric volume 4-form . . . . . . . . . . . . . . . . . . . . . . .216

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E.4.7 Rotating ohscrvcr . . . . . . . . . . . . . . . . . . . . . . . . . .363 E.4.8 Accelerating observer . . . . . . . . . . . . . . . . . . . . . . . . 364 E.4.9 The proper reference frame of the nonincrtial observer ("nonincrtial frame") . . . . . . . . . . . . . . . . . . . . . . . .366 E.4.10 Universality of the Maxwell-Lorcntz spacetirnc relation . . . . SG8

References

371

@Outlook 375 How does gravity affect electrodynamics'? . . . . . . . . . . . . . . . . . 376 Rcissner-Nordstroln solution . . . . . . . . . . . . . . . . . . . . 377 Rotating source: Kerr-Ncwnian solution . . . . . . . . . . . . . . 379 Electrodynamics outside black holes and neutron stars . . . . . . 381 Force-free elcctrodynarnics . . . . . . . . . . . . . . . . . . . . . .383 Re~narkson topology and electrodyl~amics. . . . . . . . . . . . . . . . 385 Superconductivity: Rcnlarks on Ginzb~~rg-Landall theory . . . . . . . . 387 Cl;~~sical (first quantized) Diriic field . . . . . . . . . . . . . . . . . . .388 On thc quantum Hall effect and thc colnpositc ferlnion . . . . . . . . . 390 On quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . . 390 On elcctrowcak unification . . . . . . . . . . . . . . . . . . . . . . . . . 391 References Author Index

397

Subject Index

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Foundations of Classical Electrodynamics Charge. Flux. and Metric

Introduction

Five plus one axioms I11 this book we display the structure underlying classical electrodynamics. For this purpose we formulate six axioms: conservation of electric charge (first axiom), existence of the Lorentz force (sccond axiom), conservation of magnetic flux (tliird axiom), local energy-momentum distribution (fourth axiom), existence of an electromagnetic spacetime relation (fifth axiom), and finally, the splitting of the electric current into material and external pieces (sixth axiom). Tlic axioms expressing the conservation of electric charge and magnetic flux arc formulated as integral laws, whereas the axiom for the Lorentz force is reprcsentcd by a local expression basically defining the electromagnetic field strcngth F = (ElB) as force per unit charge and thereby linking electrodynamics to mechanics; here E is the electric and B the magnetic field strength. Also the energy-momentum distribution is specified as a local law. The fifth axiom, the Maxwell-Lorentz spacetime relation is not as unquestionable as the first four axioms and extensions encompassing dilaton, skewon, and axion fields are cliscussed and nonlocal and nonlinear alternatives mentioned. Wc want t o stress the fundamental nature of the Frst axiom. Electric charge conscrvation is experimentally firmly established. It is valid for single elementary particle processes (like P-dccay, n -,p+ ei7, for instance, with n as neutron, p as proton, e as electron, and i7 as electron antineutrino). In other words, it is a microscopic law valid without any known exception. Accordingly, it is basic to electrodynamics to assume a new type of entity called electric charge, carrying a positive or negative sign, with its own physical

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I I I

II I

II

Introduction

(lilnension, independent of tlic classical fundamental variables mass, length, and time. Furthermore, electric charge is conscrvcd. In an age in which single electrons ailtl (anti)protons arc counted and caught in traps, this law is so deeply ingrained in our tliinking that its explicit formulation as a fundamental law (and not only as a consrqrlcnce of Maxwell's equations) is often forgotten. We show that this first nrzom yields the inhomogeneous Maxwell equation together with a clefinition of the electrolnagnetic excitation H = (X, '23); here 3-t is the excitation ("magnetic field") and 2) the electric excitation ("electric displacemc~~t"). Thc cxcit,ation H is a microscopic field of an analogous quality as thc ficltl strengtli F . There exist operational definitions of the excitations 2) and 3-t (via Maxwcllian clouble plates or a compensating superconducting wire, rcspectivcly). Thc second axiom for the Lorcntz force, as mentioned above, leads to the llotion of the field strength and is thereby exhausted. Thus we need further axioms. The only conservation law that can be naturally formulated in terms of the field strength is the conservation of magnetic flux (lines). This thzrd axiom has the liornogeneous Maxwcll equation - that is, Faraday's induction law and t,hc vanisliing divergence of the magnetic field strength - as a consequence. Moreover, with the help of these first three axioms we arc led, although not completely ~~niqucly, to the elcctlolnngnctic e n ~ r q y - m o m e n t u mcurrent (fourth axiom), which stibsumes the energy and momentum densities of the electromagnct,ic field ant1 their corresponding fluxes, and to the action of the elcctrolnagnct,ic field. In this way, the basic structure of electrodynamics is set up, including the complete set of Maxwell's equations. To make this set of electrodynamic equations wrll determined, we still have to add the fifth axiom. Magnetic nlonopolcs arc alien to the structure of the axiomatics we arc using. In our axiomatic. framrwork, a clear as?ymmetm~is built in between electricity ; ~ l l c l magnetism in tlic scnse of Oersted and Ampbre wherein magnetic effects arc clentcd by moving electric charges. This asymmetry is characteristic for and intrinsic to Maxwell's theory. Therefore the conservation of magnetic flux and llot that of magnetic charge is postulated as the third axiom. The existence of a magnetic charge in violation of our third axiom would have far-reaclling consequences: First of all, the electromagnetic potential A would llot exist. Accordingly, in Hamiltoninn mechanics, we wol~ldhave to give up tllc c o u l ~ l i of ~ ~ag chargcd particle to the clcctrornapnetic field via TI = p - p A. Morcov("~the sccontl axiom on the Lorcntz force could he invalidated since one would IlaVe t,o supplement it with a term carrying the magnetic charge density. By im~)lic:ttion,an extension of the fifth axiom on the energy-momentum current w()tlltl br ncccssary. I11 otIICk words, if ever a magnetzc monopole1 were found, our axiomatics would 11'" its coherence, its compactness, and its plausibility. Or, to formulate / --

'Ollr. ,!'Rull7cnts rcfcr only t o Abclian gauge theory. In non-Abclian gauge theories the situat,ion '"different,. There monopoles seem to be a Inust, a t least if a IIiggs field is present.

Topological approach

3

it more positively: Not long ago, He [22], Abhott et al. [I], and Kalbfleisch et al. [32] determined experimentally new improved limits for the nonexistence of (Abelian or Dirac) magnetic monopoles. This ever increasing accuracy in the exclusion of magnetic monopoles speaks in favor of the axiomatic ap~>roach in Part B.

Topological approach Since the notion of metric is a complicated one, which requires measurements with clocks and scales, generally with rigid bodies, wlliclt themselves are systems of great complexity, it seems undesirable to take metric as fun,damental, particularly for ph,enomena which are simpler and actually independent of it. E. Whittaker (1953) The distinctive feature of this type of axiomatic approach is t,hat one only needs minimal assumptions about the structure of the spacetime in which these axioms are formulated. For the first four axioms, a 4-dimensional differentiable manifold is required that allows for a foliatzon into 3-dimensional hyp~.rsurfaccs. Thus no connection and no metric are explicitly introduced in Parts A and B. The Poincard and the Lorentz groups are totally ignored. Nevertheless, we recover Maxwell's equations already in Part B. This shows that elcctrodynan~ics is not as closely related to special relativity theory as is usually supposed. This mininialistic topologzcal type of approach may appear contrived a t first look. We should rccognizc, however, that the metric of spacetime in tlie framcwork of general relativity theory represents the gravitational potential and, similarly, the conncction of spacetime (in the viable Einstrin-Cartan throry of gravity, for example) is intimately linked to gravitational propert,ics of matter. We know that we really live in a curved and, perhaps, contorted spacetime. Consequently our desire should be to formulate the foundations of clcct,rodynamics such that the metric and the connection don't interfere or interfere only in tlie least possible way. Since we know that the gravitational field permeates all the laboratories in which we make experiments with electricity, we should take care that this ever present field doesn't enter the formulation of the first principles of electrodynamics. In other words, a clear separatzon between pure clectrodynamic effects and gravitational effects is desirable and can indeed be achieved by means of the axiomatic approach to be presented in Part B. Eventually, in the spacetime relation (see Part D), the metric does enter. The power of the topological approach is also clearly indicated by its ability to describe the phenomenology (at low frequencies and large distances) of the quantum Hall effect successfully (not, however, its quantization). Insofar as the macroscopic aspects of the quantum Hall effect can be approximately understood in terms of a 2-dimensional electron gas, we can start with (1

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Electrodynamics in 3-tlimensional Euclidean vcctor calculus 2)-dimensional clcctrodynamics, tlic formulation of which is straightforward in our axiomatics. It is then a mcrc finger exercise to show that in this specific casc of 1 2 clinicnsions thcre cxists a lznear constitutivc law that docsn't require a metric. As a conscquencc the action is metric-free too. Tlius the formulation of tlie quantum Hall effect by Incans of a topological (Chern-Simons) Lagrangian is imnlincnt in our way of loolting at clcctrodynamics.

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Electromagnetic spacetime relation as a fifth axiom Let us now turn to that domain wherc tlic metric docs enter tllc 4-dimensional electrodynarnical formalism. Wlicn thc Maxwclliarl structure, including t,hc Lorcntz force and tlic action, is sct up, it docs not rcprcscnt a concrete physical tlicory yct. What is missing is tlic elcctroniagnctic spacctimc relation linking tlic excitation to tlic ficld strength, i.c., V = V ( E , B ) , 3-1 = X(B, E), or written 4-diniensionally, H = H(F). Trying the sinlplcst approach, we assume localzt?/ ancl 11nrnnty bctwt.cn tlie excitation H and the field strcngtli F , tliat is, H = tc(F) witli tlic linear operator tc. Together witli two more "teclinical" assumptions, namely that H = tc(F) is clcctric/magnetic wczprocal and tc symmetric (these propertics will be discussed in detail in Part D), we are able to denve tlie metric of spacetimc from H = u ( F ) up to an arbitrary (conformal) factor. Accordingly, the light cone structure of spacctime is a consequence of a linear electromagnetic spacptirne relation witli the additional properties of reciprocity ancl symmetry. In this sense, tlic light cones arc dcrivcd from electrodynamics. Elcctrotlynalnics docsn't live in a preformed rigitl Minkowslti spacetimc, Rather it ha9 an arbitrary (1 3)-dimensional spacctime manifold as its habitat wliicli, as soon as a linear spacctimc relation witli rcciprocity nncl symrnctry is supplied, is equipped witli local light cones cvcrywherc. With the light concs it is possiblc to definc the Hodge star operator * that maps pforms to (4-11)-forms and, in particular, H to F according to H N * F . Tlius, in the end, that property of spacetimc that tlescribcs its local "constitutivc" structure, namcly the metric, cntcrs the formalisnl of electroclynamics and lnakcs it into a complctc thcory. Onc merit of our approach is tliat it docsn't lnattcr whetllcr it is the rigid, i.e., flat, Minkotuskz lnctric of special relativity or tllc "flexible" Rlemannzan mctric ficlti of general relativity tliat cllangc's from point to point according to Einstein's field equation. In this way, thc traditional discussion of how to translate clectrodynarnics from special to general relativity loses its sense: Thc Maxwell equations remain tlic same, i.c., the exterior dcrivat,ivcs (the " c o m n ~ a .in~ ~coorclinatc language) arc kept and arc not sul~stitutcclby something "covariant" (tlic "scmicolons"), and tlie spacetimc relation H = Xo *Flooks the sanlc (Ao is a suitable factor). However, the Hodge star "feelsv the tlifference in rcfcrring either to a constant or to a spacetimcdependent metric, respectively (see [51, 231).

+

5

Our formalism can accommodate generalizations of classical electrodynamics, including those violating Lorcntz invariance, simply by suitably modifying the fifth tixiom while keeping the first four axioms as indispensable. Then a scalar dllnton field X ( 2 ) , a pseudoscalar axion field a ( x ) , and/or a tensorial traceless skewon field $,J can come up in a most straightforward way. Also tlie Heiscnberg-Eulcr and the Born-Infeld electrodynamics are prime examples of such possiblc modifications. In the latter cases, the spacetime relation becomes cffcctivcly nonlinear, but it still remains a local expression.

Elcctrodynamics in matter and the sixth axiom Evcnt,ually, we have to face thc problem of formrllating clcctrodynamics inside matter. We codify our corresponding approach in the sixth axiom. The total electric cuncnt, cntcring as tlle source in the inlion~ogcneousMaxwell equation, is split into bound charge and free charge. In this way, following Truesdcll & Tolipin [63] (scc also the textbook of Kovetz [34]),we can develop a consistent theory of clcctrotlynamics in matter. For simple cases, wc can amend the axioms by a llnrar constzt~~tive 1au1.Since in our approach (%, V) are n~icroscopicficlds, like (E,R ) , wc bclicve that the conventional theory of elcctrodynanlics inside matter ncctls to be redesigned. I11 ordcr to demonstrate the effcctivencss of our formalism, we apply it t o the clcctrodynamics of moving matter, thereby returning to the post-Maxwellian era of tlic 1880s wlicn a relativistic version of Maxwell's theory had gained momentum. In this contcxt, wc discuss and analysc the cxpcriments of Walkcr & Walkcr ant1 Jnlncs and tliosc of Rontgcn-Eiclicnwalcl aricl Wilson & Wilson.

List of axioms 1. Conscrvation of clectric charge: (B.1.17). 2. Lorentz forcc tlcnsity: (B.2.8). 3. Conscrvation of lnagnctic flux: (B.3.1). 4. Localization of cnergy-mol-ncntum: (B.Ti.7). 5. M:ixwcll-Lorcntz spacct,imc relation: (D.G.13).

6. Splitting of the clectric current in a conscrvcd matter piece and an external piccc: (E.3.1) and (E.3.2).

A rernindcr: Elcctrodynamics in 3-dimensional Euclidcan vector calculus Bcforc wcxstart to clcvclol> elcctrotlynamics in 4-dimensional spacetime in the fralneworlt of the calculus of exterior diffcrential forms, it may be useful to re-

Introduction

6

mind ourselves of electrodynamics in terms of conventional 3-dimensional Euclidean vector calculus. We begin with the laws obeyed by electric charge and current. If 6= (a,, D,, V,) denotes the electric excitation field (historically "electric displacement") and p the electric charge density, then the integral version of the Gauss law, 'flux of V through any closed surface' equals 'net charge inside' reads

with d% as area and dV as volume element. The Oersted-Ampkre law with the magnetic excitation ficld " ;i (X,, X,, 'Hz) (historically "magnetic ficld") and the electric current density j = (j,, j,, j,) is a bit more involved b c c a ~ ~ sofc the presence of the Maxwellian electric excitation current: The 'circulation of 3-1 around any closed contour' equals '$ flux of V tlrrougll surface spanned hy

On the literature

7

Note the minus sign on its right-hand side, which is chosen according t o the Lenz rule (following from energy conservation). Finally, the 'flux of through any closed surface' equals 'zero', that is,

I?

The laws (1.4) and (1.5) are inherently related. Later we formulate the law of magnetic flux conservation and (1.4) and (1.5) just turn out to be consequences of it. Applying the Gauss and the Stokes theorems, the integral form of the Maxwell equations (I.l), (1.2) and (I.4), (1.5) can be transformed into their differential versions:

-4

(

I

contour)' plus 'flux of

4

throagh surface' ( t =time):

-4

Here & is the vectorial line element. Here tllc dot . always denotes the 3dimensional ~netric-dependentscalar product, S denotes a 2-dimensional spatial surface, V a 3-dimensional spatial volume, and BS ant1 BV the respective boundaries. Later we recognize that both, (1.1) and (I.2), can bc derived from the charge conservation law. The homogeneous M,axwcll cquations are formulated in terms of the elcctric field strength l? = (E,, E,, E,) and the magnetic field strength I? = (B,, B1,,B,). They arc defined opcrationally via the expression of the Lorentz force $. An electrically chargecl particle with charge q and velocity ;experiences the force

-.

Here the cross x denotes the 3-dimensional vector product. Then Faradny '.s induction law in its integral version, namely 'circulation of E around any closed contour' equals 'minus $ flux of B through surface spanned by contour ' reads:

(

1

Additionally, we have to specify the spacetime relations V = EO 3 , l? = po 'I?, and if matter is considered, the constitutive laws. This formulation of electrodynamics by means of 3-dimensional Euclidean vector calculus represents only a preliminary version since the 3-dimensional -. the vector products and, in particular, the difmetric enters the scalar and ferential operators div V. and curl V x , with V as the nabla operator. , ~ counting procedures enter, In the Gauss law (1.1) or (I.6)1, for i n ~ t a n c e only namcly counting of elementary charges inside V (taking care of their sign, of course) and counting of flux lines piercing through a closed surface dV. No length or time measurements and thus no metric are involved in such processes, as described in more detail below. Since similar arguments apply also to (1.5) or (1.7)1, respectively, it should be possible t o remove the metric from the Maxwell equations altogether.

-

-

-.

-.

On the literature Basically not too much is new in our book. Probably Part D and Part E are the most original. Most of thc material can be found somewhere in the literature. What we do claim, however, is some originality in the completeness and in the appropriate arrangement of the material, which is fundamental to the structure electrodynamics is based on. Moreover, we try to stress the phen.omena underlying the axioms chosen and the operational interpretation of the quantities introduced. The explicit derivation in Part D of the metric of spacetime from pre-metric electrodynamics by means of linearity, reciprocity, and symmetry, although considered earlier mainly by Toupin [61], Schonbcrg [53], 2 ~ h sr~bscript c 1 refers to the first equation in (1.G).

Introduction

8

and Jadczyk (281, is new and rests on recent results of Fukui, Gross, Rubilar, and the authors [42, 24, 41, 21, 521. Also the generalization encompassing tlie dilaton, the axion, and/or tlie sltcwon field opens a new perspective. In Part E the electroclynamics of moving bodies, including tlie discussions of some classic experiments, colltains much new material. Our marn sources are the works of Post [46, 47, 48, 49, 501, of Truesdell & Toupin [G3], and of Toupin [GI]. Historically, the metric-free approach to elcctroclynarnics, based on integral conservation laws, was pioneered by Kottler [33], h. Cartan [lo], and van Dantzig [64]. The article of Einstein [16] and the books of Mic (391, Weyl (651, and Sornmerfeld [58] should also be consulted on these niatters (see as well the recent textbook of Kovetz [34]).A description of the corresponding historical development, with references to the original papers, can be found in Whittakcr (661and, up to about 1900, in the penetrating account of Darrigol [12]. The driving forces and the results of Maxwell in his research on electrodynamics arc vividly presented in Everitt's [17] concise biography of Maxwell. In our book, we consistently use exterior calculus13including de Rharn's odd (or twisted) differential forms. Telrthooks on electrodynamics using exterior calc~ilusare scarce. In English, we know only of Ingarden & Jamiolkowski [26], in German of Mcetz & Engl (381 and Zirnbauer (671, and in Polish, of Janrewicz [31] (see nlso [30]). Howcvrr, as a discipline of mathematical physics, corresponding presentations can be found in Bamberg & Sternberg [4], in Thirring [GO], and as a short sketch, in Piron [44] (see also [5, 451). Bambcrg & Sternberg arc particularly easy to follow and present clcctrodynamics in a very transparent way. That clcctrodyn~micsin the framework of exterior calculus is also in the scopc of clcctrical engineers can be sccn from Dcschamps [14], Bossavit[8], and Baldolnir & Haliimond [3]. Prcscntations of exterior calculus, partly togcthcr witli applications in physics and electrodynamics, were given amongst many others by Burke [9], ChoquetBruliat ct al. [ll],Edelcn (151, Flanlders [19], Rankel [20], Parrott [43], and ~lebodziriski[57]. For differential geometry we refer to tlie classics of de Rllam [13] ant1 Schouten [54, 551 and to n a u t m a n [G2]. What clsc influenced the writing of our book? The ariomatics of Bopp [7] is different but relatctl to ours. In the more microphysical axiomatic attempt of LRmnierzalll et al. Maxwell's equations (351 (and tlie Dirac equation (21) arc dediiccd from direct experience with electromagnetic (and matter) waves, inter alia. The clcar separation of drffrrrntzal, afine, and metnc structures of spacetime is nowlicrc more 11ron011nced than in Schrodingcr's [56] Space-time stmlcturr. A fiirtlicr presentation of elcctrodynamics in this spirit, somewhat similar 3)-decomposztzon to that of Post, has been given by Stachel (591. Our (1

+

-

%nylis [GI nlso ndvocntes a gcolnctric nppronch, using Clifford algcbrm (see also Janccwicz [20]). In such a. frnmcwork, howcvcr, nt least the way Baylis does it, the metric of 3-dimensional space is i r ~ t r o d ~ ~~.ight c ~ dfrom the beginning. 111this sense, Baylis' Clifford algebra approach is cornplcmrntnry to our metric-free electroclynaniics.

of spacetime is based on the paper by Mielkc & Wallncr [40]. More recently, Hirst (251 has shown, n~ainlybased on experience with neutron scattering on magnetic structures in solitls, that magnetization M is a microscopic quantity. This is in accord witli our axiornatics which yieltls the magnetic excitation 3.1 rrs a microscopic quantity, quite analogously to the field strength B , whereas in conventional texts M is only defined as a niacroscopic average over niicroscopically fll~ctuatingniagnctic fields. Clearly, 'H and also the electric cxcitat,iotl D ,i.e., the electromagnetic cxcit,at,ion H = (3.1,V)altogether, olight to be a microscopic field. Sections and subsections of the book that can be skipped ilt by tllc symbol are ~i~arltcd @.

ii

first reading

References

[I] B. Abbott et al. (DO Collaboration), A search for heavy pointlike Dirac monopoles, Phys. Rev. Lett. 81 (1998) 524-529. [2] J. Audrctsch and C. Lammerzahl, A new constructive axiomatic scheme for the geometnj of space-time In: Semantical Aspects of Space-Time Geornetqy. U. Majer, H.-J. Schmitlt, cds. (BI Wissenschaftsverlag: Mannheim, 1994) pp. 21-39.

[3] D. Baldomir and P. Hammond, Geometry and Electromagnetic Systems (Clarcndon Press: Oxford, 1996). [4]

P.Balnberg and S. Sternberg, A Course in Mathematics for Students of Physics, Vol. 2 (Cambridge University Press: Cambridge, 1990).

[5] A.O. Barut, D.J. Moore and C. Piron, Space-time models from the electromagnetic field, Helv. Phys. Acta 67 (1994) 392-404.

[6] W.E. Baylis, Electrodynamics. A Modern Geometric Approach (Birkhauser: Boston, 1999). [7] I?. Bopp, Prinzipien der Elektrodynamik, 2. Pt~ysik169 (1962) 45-52. [8] A. Bossavit, Differential Geometnj for the Student of Numerical Methods in Electromagnetism, 153 pages, file DGSNME.pdf (1991) (see h t t p : //www. l g e p . supelec . f r/mse/perso/ab/bossavit . html),

[9] W.L. Burlte, Applied Dzflerential Geometry (Cambridge University Press: Cambridge, 1985).

[lo] fi. Cartan, O n Man,ifold.s luith an A f i n e Connection and the Theory of General Relativity, English translation of thc Frcnch original of 1923124 (Bibliopolis: Nagoli, 1986).

1241 F.W. Helil, Yu.N. Ob~~lthov, and G.F. Rubilar, Spacetime metric from linear ele~trod?~n.am.ics II. Ann. Pl~ysik(Lcipzig;) 9 (2000) Spccial issue, SI71-SI-78.

[ll] Y. Choquct-Bruh;Lt, C. DeWitt-Morcttc, ancl M. Dillard-Blcick , Analysis, Manijo1d.s and Pl~?j.sic.s, revised cd. (North-Holland: Amsterdam, 1982).

[25] L.L. Hirst, Th,e m,icroscopic m.agnetizntion: concept and application, Rev. Mod. Pl1.y~.69 (1997) G07-627.

(121 0. Darrigol, ELectrorl~~nam.ics from Ampere to Einstein (Oxford University Press: New York, 2000).

[26] R.. Ingardcn and A. Jamiolkowski, Classical Electrodynamics (Elsevier: Amsterdam, 1985).

[13] G. de Rham, Differentiable Man.ifolds: Forms, Currcnts, Harmonic Forms. Transl. from thc F'rcncli original (Springer: Berlin, 1984).

[27] J.D. Jacltson, Clas.sical Electrodynnm.ics, 3rd cd. (Wiley: Ncw York, 1999).

[14] G.A. Descliamps, Electrom.agn,etics an.d differential fo~m.s,Proc. IEEE 69 (1981) 676-696. [15] D.G.B. Etlelen, Applied Exterior Ca1culzi.s (Wiley: New York, 1985) [16] A. Einstein, Einc neue form.ale Deutun.g rler Maxwellsclt~en.Feldgleicl~ungen dcr Elcktrodyn.am.ik, Sitzungsber. IGnigl. Preuss. Aka.d. Wiss. Bcrlin (1916) p p 184-188; scc also T1j.e collected papers of Albert Ei,n..stein. Vol.6, A..J. Kox et al., eds. (1996) pp. 263-269. [17] C.W.F. Evcritt, James Clerk Maxwell. Physici.st a.ntl Nat1~ra.1Philosoplrer (Charlcs Sribncr's Sons: New York, 1975). [18] R.P. Feynmnn, R.B. Lcighton, ant1 M. Santls, The Feynman Lect~~res on Pl~.~/sics, Vol. 2: Mainly Electromagnetism and Mattcr (Addison-Wesley: Reatling, Mass., 1964). [19] H. Fla~itlcrs,Dzfferentiol Forms with Applications to tlie Physical Sciences. (Academic Press: Ncw York, 1963 and Dovcr: New York, 1989). [20] T. F'rankcl, Tlre Geonletry of Physics: A n In.trod.uction (Cambritlge University Prcss: Calnbridg;e, 1997). (211 A. Gross ancl G.F. Rubilar, O n the derivation of the spacetime metric from linear electrodynnmics, Pl~ys.Lett. A285 (2001) 267-272. [22] Y.D. He, Search for a Dirac m.agnetic nlonopole i n high enwrgy n.u.cleus-nuclet~sco1lisi0n.s~Pllj~s.Rev. Lctt. 79 (1997) 3134-3137. [23] F.W. HcIll and y~1.N.Obukliov, I-Iow does the electromngnctic ficld couplc to grnvit,y, in particular to metric, nonmet,ricity, torsion, ant1 cllrvature? In: Gyros, Clocks, Interferometers . . . : Testing Relativi.stic Gmvit?j i n Space. C. Liimmerzahl et al., eds. Lecture Notes in Physics Vo1.5G2 (Springer: Berlin, 2001) pp. 479-504; see also Los Alanios Eprint Archive gr-qc/0001010.

[28] A.Z. Jadczyk, Electromagnetic permenhilit!/ of the uacu.7~rnand light-conx structure, Bull. Acad. Pol. Sci., Sbr. sci. phys. et mtr. 27 (1979) 91-94. [29] B. Janccwicz, Mvltivector.~and Clzfford Algebra i n Electrodynamzcs (Worlcl Scientific: Singapore, 1989). [30] B. Janccwicz, A variable metric electrodynamics. The Coulomb and BiotSauart laws i n anisotropic media, Ann. Phys. ( N Y ) 245 (1996) 227-274. [31] B. J;xncewicz, Wielko.s'ci skierotuane 111 elektrodynam,ice (in Polish). Directed Quantities in Electrodynamics. (University of Wroclaw Press: Wroclaw, 2000); an English version is u~ldcrpreparation. [32] G.R.. I 0, which means that the bases e, and h, are similarly oriented. Conversely, assuming det (AnP) > 0 for any two bases h, = A,Pep, we find that (A.1.58) holds true for both bases. Clearly, every volume form that is obtained by a "resealing" wl,,., --+ cpl,,,,, = awl..,, with a positive factor a will define the same orientation function (A.1.58): o,(e) = o,,(e). This yields the whole class of equivalent volume forms that we introduced a t the beginning of our discussion. The standard orientation of V for an arbitrary basis e, is determined by the volume form d1 A . . . A d n with cobmis d o . A simple reordering of the vectors (for example, an interchange of the first and the second leg) of a basis may change the orientation.

A.1.9 @Levi-Civitasymbols and generalized Kronecker deltas The Levi-Civita symbols are numerically invariant q~~antities an.d close relatives of the volume form. They can arise by applying the exterior product A or the interior product -I n times, respectively. Levi-Civita symbols are totally antisymmetric tensor densities, and their products can be expressed in terms of th,e generalized Kronecker delta.

In particular, we see that the only nontrivial component is c ' . . . ~= 1. W ith respect to the change of basis, this quantity transforms as the [I;]-valued 0-form density of weight + l :

Recalling the definition of the determinant, we see that the components of the Levi-Civita symbol have the same numerical values with respect to all bases,

... a: -

...a,,

(A.1.64)

They are + I , -1, or 0. Another fundamental antisymmetric object can be obtained from the elementary volume < with the help of the interior product operator. As we have learned from Sec. A.1.7, the interior product of a vector with a pform generates a (p - 1)-form. Thus, starting with the elementary volume n-form and using a vector of basis we find an (n - 1)-form

3See Sokolnikoff (291.

A.1. Algebra

38

I

The trarisforniation law of this ol~jectdefines it as a couector-~inlued(n- 1)-form density of w~igll~t -1: {,I

= CIC~(L,,")-

La," 6,. A

(A.1.66)

39

A.1.9 @Levi-Civitasymbols and generalized Kronecker deltas

Furthermore, let us talce an integer q < p. T h e contraction of (A.1.70) over the (n- q) indices yields the same result (A.1.71) witli p replaced by q. Comparing the two contractions, we then deduce for the generalized Kroncckers:

Applying once more the interior product of the basis t o (A.1.65), one obtains an ( n - 2)-form, anel so 011. Thus, we can construct chain of forms: In particular, we find n! cv1...a,, 6CI1...CIp- ( n - p)! ' T h e last object is a 0-form. Property 4) of the interior product forces all these epsilons to be totally antisy~nmetricin all their indices. Similarly to (A.l.GG), we can verify that for p = 0, . . . , n the object 2,, , is a -valued ( n-p)-form density of tlie weight -1. These forms { i , i,, i, , . . . , i,, ,,, ), alternatively to {d*, 6"' A I ? " ~, . . . ,19"l A . . . A79(Y7~}1 can be used as a basis for arbitrary forms in the exterior a1g~l)riiA*V. In particr~lar,we fintl that (A.1.68) is the [:]-valued 0-form density of weight -1. Tliis quantity is also called the Levi-Civita sy~nholbccause of its c v i d ~ n t sirnilality t o (A.l.61). Analogously t o (A.1.62) we can express (A.1.68) in terms of the generalized Kio~iecltersynlbol (A.1.27):

Thus we find again that the only nontrivial component is il..,,, = $1. Note that dcspit,~the d ~ e psimilarity, wc cannot idclitify the two Levi-Civita symbols in the ahsence of tlic metric; hence the different notation (with and without hat) is nppropriatc. I t is wort.1iwhilc t,o dcrivc a useful idcntity for the product of thc two LcviCivitn symbols: 6 * ~.. n , ,

2"'"'"" ep,, _J . - . J eo, E e ~ ,J, . . . J ep, (29,' A . . . A 29,") - (19"' A . . . A dflVL) (epl , . . . , ep,,)

Let us collect for a vector space of 4 dimensions the decisive formulas for going down the p-form ladder by starting from the 4-form density i and arriving a t the 0-form i,p,a:

i = Zap,a

=

e,

=

ep _r i, =

dP

Elopy6

1 9 ~ ga/3! ,

d Y d6/2! ,

(A.1.74)

fapy = ey J cap = Zapy6 d 6 iolpy6 = e6-]tn/3y Going up the ladder yields: 29''

+

ipy6,L = 6; i o y 6 - 6; iPyIL6; ?paLb- 6; d" igy6= 6; iPY 6; 6; i,s , d a A t p y = 6;ip - 6;iy,

+

19"~ip=

+

iy6,L,

(A.1.75)

6;;.

One can, with respect t o thc ?-system, definc a (prc-mct,ric) duality opcrator 0 t h a t establishes an equivalence bct,wecn pforlns and totally antisynimttric tensor dcnsitics of weight +1 and of type [ n i p ] , In terms of the bases of the corresponding linear spaces, this operator is introduced as

~ p ~ , , , f i= ,,

-

Tlir wholc derivatio~lis bascd just on tlie use of the corresponding definitions. Namely, we use (A.1.68) in t,he first line, (A.1.61) in the second line, (A.1.44) in the third line, anel (A.1.32) in the last line. Tliis identity Iiell~sa lot in calc~llationsof the. different contractions of t,he Levi-Civita symbols. For cxamplc, we easily oI)tain from (A.1.70):

Consequently, given an arbitrary p-form basis as

the map 0 defines a tensor density

expanded with respect t o the 2-

by

For examplc, in n = 4 wc have O i , = e, and 02 = 1. Thus, every 3-form cp = cp" i, is mapped into a vector density Ocp = cp" e,, whereas a 4-form w yields a scalar density Ow.

40

A.1. Algebra

A.l.10 The space M%f two-forms in four dimensions

A.1.10 The space ~

Correspondingly, taking into account that 19' A F^ =: Vol is the elementary 4volumc in V, wc find

Electromagnetic excitation and field strength, are both 2-fomn.s. On the 6-dim.en.sion.alspace of 2-form,s, t11,ere exists a natliral 6-metric, which i s an im,portant property of this space.

i

(A.1.83) (A.1.84)

p a / \ p t ) = 0, ?,/lib 2,r\,Ob /

Lct e, be an arbitrary basis of V with a , P,. . . = 0 , 1 , 2 , 3 . In later applications, the zcrotli leg eo can bc rclatctl t o the time coordinate of spacetinie, but this will not always be tlie case (for the ~lrlllsymlnctric basis (C.2.14), for instance, a11 the re's have thc sanic status witli rcspcct to time). Tlie three rcniaining legs will be clcnotctl by c , with a , 0,.. . = 1 , 2 , 3 . Accordingly, tlic dual basis of V* is reprcsentetl by .19" = (d o , d n ) . I11 tlic linear spacc of 2-forms A 2 V*, every clcliielit can 1>c decomposed according t o p = p,,p d" A d o . Tllc basis 19" A d P consist,^ of six simple 2-forms. Tliis 6-plct can bc altcrnativcly nunil)errtl by a collrctive intlrx. Accordingly, we enumerate the antisymmetric index pairs 01,02,03,23,31,12 by uppercase lcttcrs I, J , . . . from 1 to G:

41

% two-forms f in four dimensions

,A

A

8'

0, = 6:vol, =

=

(A.1.85)

(- 6: 6;: + 6; 6;) Val.

(A.1.86) 1

Every 2-form, being :in elcmcnt of M " , can now be represented as (P = ( P I E by its six coliiponcnts with rcspcct to t,lie basis (A.1.79). A 4-form w, i.e., a form of the liiaximal rank in four dimensions, is expanded with respect t o the wedge protlucts of the Gbnsis as w = i ~GI A~ GJ. .Tlie~coefficients WIJ form R, symmetric G x G matrix since the wedge product between 2-forms is evidently comn~ut,ativc. A 4-for111 has only one component. This simplc observation enables us t o iritrotlucc a natural metric 011 the 6-diniensional spacc Ad%s the syrnmctric bilincar form E(W,(P) := ( W A ( ~ ) ( e o , e l , e z , e s ) ,

w,cp

M6,

(A.1.87)

where e, is a vector basis. Although the metric (A.1.87) apparently depends on the choicc of basis, the linear transformation e,! -t L,! e, induces the pure rescaling E -t det(L,,") E . Using the expansion of the 2-forms with respect to tlic hivcctor bitsis G', tlic bilinear form (A.1.87) turns out t o be (1

whore

E(W,p ) = wI p j E'.',

&IJ =

(G1 A E1)(eo,e l , ez, e3).

(A.1.88)

A rlilrct inspection by using the clefinition (A.1.79) and the identity (A.1.85) shows tllnt the 6-l~lctriccomponents rei~tlexplicitly With tlie GI LS basis (say "cyrillic B" or "Bch"), we can set up a 6-dimensional A 2 V*. Tliis vector spacc will play an important role in our vector spacc A[%= consiticrations in Parts D and E. The extra dcco~npositionwith rcspcct t o P" ant1 ibis convcnicnt for recognizing where tlic clcctric and where the magnetic pieces of tlic field arc locatctl. WP denote the clcmcntary volumc 3-form by i = t9' ~ 6A@. ' Thcn i,, = r , J i is t8hc basis 2-form in t,hc spacr spanned by the 3-cofraliic 79", see (A.1.65). Tliis notat,ion has bcen used in (A.1.79). Moreover, as usrlal, the 1-form basis Citn then be tlcscribccl by in/) = rb J 2,,. Sonir useful algebraic relations can be immedi;~tclydcrivcd: (A.1.80) ('4.1.81) (A.1.82)

Here I3 :=

(

0

: :) 1 0

is tlie 3 x 3 111iitmatrix. Thus we see that the metric

(A.1.87) is ;~lwaysnondegeneratc. Its signature is (+, +,+, -, -, -). Indeed, the cigenvnlucs X of tlic niatrix (A.1.89) are defined by the characteristic equation d c t ( ~ ' . ]- AdrJ) = (A2 = 0. The synimetry group that preserves the 6-metric (A. 1.87) is isomorphic t o 0 ( 3 , 3 ) . By construction, t,he clcmcnts of (A.1.89) numerically coincide witli the components of t,lic Lcvi-Civit,a sylnbol F ' J ~ ' , see (A.1.62):

A.1. Algebra

42

Similarly, the covariant Levi-Civita symbol P,,, sentcd in 6D notation by the matrix

, see

(A.1.69), can be repre-

43

2.1.11 Almost complex structure on M6

&ere the determinant of the transformation matrix (A.1.92) reads det L := det L,,o = [Lo0 - L , O ( L - ~ ) ~ ~ L ~ ~ det ] ~,d.

(A.1.99)

One can write an arbitrary linear transformation L E GL(4, R) as a product One can inlmecliately provc by multiplying the matrices (A.1.89) and (A.1.91) that their product is equal to 6D-unity, in complete agreement with (A.1.71). Thus, the Levi-Civita symbols can be consistently used for raising and lowering indices in M G .

(A.1.100)

L=L1L2L3 of three matrices of the form

@Transformationof the M6-basis What happens in M h h c n the basis in V is changed, i.e., e, --+ eat? As wc know, such a change is descril~edby the linear transformation (A.1.5). Thcn the cobasis transforms in accordance with (A.1.6): dQ = Lo," d") .

In the (1

(A.1.92)

+ 3)-matrix form, this can be written as

+ + +

3ere V n Ub,Roo,A ha , with a , b = 1,2,3, describe 3 3 1 9 = 16 elements )f an arbitrary linear transformation. The matrices {L3) form the group R 8 TL(3, R) which is a subgroup of GL(4,R), whereas the sets of unimodular natrices {L1) and {Lz) evidently form two Abelian subgroups in GL(4, R). In the study of the covariance properties of various objects in M 6 , it is thus sufficient t o consider the three separate cases (A. 1.101)-(A.1.103). Using (A.1.96) and (A.1.97), we find for L = L1 l

Correspondingly, the 2-form basis (A.1.79) transforms into a new bivector b x i s similarly, for L = L2 we have

and for L = L3 Substituting (A.1.92) into (A.1.79), we find that the new and old 2-form bases are related by an induced linear transformation

pah= Roo h b n ,

Qba = (det A ) ( A - ' ) ~ ~ ,

wab= Zab = 0.

(A.1.106)

A.1.11 Almost complex structure on M~

pnb

=

wab=

LoOLha- LoaLbO, Qoa = det L,"L-')~", LcoLdaFbcdl Znb =

The 3 x 3 matrix (L-')ha is inverse to the 3 x 3 subbloclc L,"n invcrsc transformation is easily computed:

(A.1.96) (A.1.97) (A.1.92). The

A n almost complex structure o n the space of 2-forms determines a splitting of the complexification of M6 into two invariant 3-dimensional subspaces. Let us introduce an almost complex structure J on My We recall that every tensor of type [i] represents a linear operator on a vector space. Accordingly, if cp E M G ,it is of type and J(cp) can be defined as a contraction. The result will also be an element of M6. By definition, J(J(cp)) = -I6 cp or

I:[

412

A.1. Algebra

see (A.1.19). the opcrator J can be represented as a G x G matrix. As a tensor of type Since the basis in M G is naturally split into 3 3 parts in (A.1.79), we can write it in terms of the set of four 3 x 3 matrices,

[;I,

+

45

A . l . I 2 Computer algebra

Let us denote the 3-dimensional subspaces of MG(@)that correspond to the eigenvalues +i and -2 by (8)

M

:= {w

E M ~ C I )J(w) = i w ) ,

-

Because of (A.1.107), the 3 x 3 bloclcs A, B , C, D arc constrained by

+

AncBcb C'"CCh

-d;, c ~ ~ A ~ ~ = ~ 0,+ A ~ ~ D ~ ~ Ba,Ccb DoCBc~) = 0, t~ D,,A"" D,"D," = -dn, (A.1.109) =

+ +

The significant tliffcrcncc between these two metrics is that E' assigils a real length to any complex vector, whereas E defines complex vector lengths. We will nssulne that the J operator is defined in MG(@)by the same formula linear operator in MG(@), i.e. as J(w) in Ad? 111other words, J remains a rral for every coml~lex2-form w E A/lG(@)one has J(w) = J(a).The eigenvaluc problem for the operator J(wx) = Xwx is meaningful only in the cornplcxificd space A/lG(Q1) because, in view of the property (A. 1.107), the eigcnvalues are X = fz. Each of these two cigenvalucs has multiplicity three, which follows from the reality of J. Note that the G x G matrix of the J operator has six eigenvcctors, but the numl~crof eigcnvectors wit11 cigenvaluc +z is equal to the number of cigenvcctors with cigenvalue -z beca~lsethey arc complex conjugate to each other. Indccd, if J(w) = zw, tllcn tlie conjugation yields J(LJ)= J ( Z ) = -155.

(a)

-J

) ],

with

2 ) 1-2 [ An allnost complex structure on M G motivates a complex generalization of M G to the complezified linear space Ad G(@).The elements of My(@ are the complex 2-forms w E My(@, i.e. their component,^ WI in a decomposition w = WIG' are conlplex. Alt,ernat,ively,one call consider Ad"(@ as a real 12-dimensional lincar space spanned by the basis (c', i ~ ' ) ,where i is tlie imaginary unit. We denote by M"(@)t,hc complex co~ijugatcspace. The same sylnnlct,ric bilinear form as in (A.1.87) also defines a natural metric in Ad"(@). Note howcvcr, that NOW an orthogonal (complex) basis call always be introtluccd in Ad"(@ so that E'.' = 6I.I in that basis. Incidentally, one can tlefinc nnotllcr scalar product, on a complex spacc MG(C) I)Y

(8)

respectively. Evidently, M = Ad. Therefore, we can restrict our attention only to the self-dual subspace. This will be assumed in our derivatiolls from now on. Accordingly, every form w call be tlecomposed into a self-dual and an antiself-dual piece4,

(*)

1 w = - [w 2

(a)

(3)

+ iJ(w)] .

(a)

It can be checked that J ( w ) = +i w and J ( w ) = -z

. (a) W.

A.1.12 Computer algebra Also in electrodynamics, research usually requires the application of computrrs. Besides numerical methotls and visualization techniques, the manipulation of formulas by means of "computer algchra" systelns is nearly a must. By no means are these nicthods confined to pure algebra. Differentiations ant1 int,egrations, for example, can also he exccutecl with the help of computer algebra tools. "If wc do work on the foundations of clnssical electrodynanlics, wc can dispense wit11 computer algebra," some true fundamentalists will claim. Is this really true? Well later, in Chap. D.2, we analyze the Fresnel equation; we couldn't have done it to the extent we did without using an efficient computer algebra system. Thus, our fundamentalist is well advised to learn some computer algebra. Accordingly, in addition to introducillg some mathemat,ical tools in cxterior calculus, we mention computer algebra systems like Reduce" Maple" and Math4 A discussion of the use of self dual and nnti-self dual 2-forms in general relntivity cnn be found in Kopczyriski and Trnutman [13],e.g. "earn (81 crented this Lisp-based system. For introductions t o Iieduce, see Toussaint 1361, Groain [GI, MncCnllum and Wright [Is],or Winkelmann and I-Iehl [38];in the lnttcr text you can learn how to get hold of n Reduce system for your computer. Reduce as npplied t o general-relntivistic fleld theories is described, for example, by McCren [lG] and by Socorro ct al. [28]. In our presentntion, we pnrtly follow the lectures of Toussaint [36]. ' ~ a p l e , written in C, was created by a group nt the University of Waterloo, Canada. A good introduction is given by Chnr et nl. [3].

A.1. Algchra

46

4*

A.1.12 Computer nlgcbra

brary of McCrcal' and GRG12, in Maple G R T e n ~ o r l I ' ~and , in Mathematica, besides MathTensor, the Cnrtan package14. Computer algcbra systems are almost cxclusively interactive systems nowadays. If onc is inst;~llcdon your computer, you can usually call the system b:r typing its llalne or an abhrcviation thereof, i.c., 'reduce', 'maple', or 'math', and then hit.t,ing the return key, or clicking on the corresponding icon. In the cas? of 'reduce', thc systcm introcluccs itsclf and issues a '1 :'. It waits for your first command. A command is a statcment, usually somc sort of expression, a part cf a formllla or a. formula, followed by a tcrminator15. Thc latter, in Reduce, is i semicolon ; if you want to see the answer of the system, and othcrwisc a dollar sign $. Reduce is case inscnsitivc, i.c., thc lowcrcasc letter a is not clistinguishe~l from the uppercase letter A.

Formulating Reduce input

1

Figure A.1.5: "Hcrc is thc new Rcclucc-updatc

011

a hart1 disk."

emntlcr~~ and specifically explain how to apply thc Rcducc package E : ~ c n l c ~ o fclrms that occur in clcctrotlynal~lics. tllc work in solvilig problems by mcalls of colllpllt,er algcbra, it is our In cxpcricllcc thct it is best to have access to diffcrcnt computer algebra systems. Even tllougll 111 thc coursc of tirrlc good fenturcs of one system "migrated1' to other systems still, for a certain spccificd purpose one systcln may lbc brtter suited tllall alother one - and for different purposes thesc may bc different systems. Thelt' does not cxist as yct, tire optimal system for all purposes. Thcrcfore, it is llot a rare occasion that wc have to feed the results of a calculation by means of d1c systcnl as input into another system. For ~ ~ ~ ~ ~ ~ ~inu clcct,rotlynamics, tztions rcl~tivit~y, and gravitation, we keep the three general-11urpose computer algebra systems: Reduce, Maple, ant1 Mathematica. Otllcr systelils arc available? Our workhorse for corresponding calcu lati on s ill cxtc2rior calculus is the Rcducc package Excalc, but also in tlie Maf/rTcnsor rackage'' of Mnthematica exterior calculrls is implemented. For the m ~ n i ~ u l a i i oofn tensors wc usc tAic followillg packagcs: In Rcducc the li7 W ~ l f r a n r( s c [ N ] ) created the C-based Mathematica software packagc whicll is in very widespreacl use. R S ~ l l r ~[25 ~ f26) ~ ris the creator of that packagr (cf. also [27]). J3xcalc is applied t o Maxwell's theory by P u ~ ~ t i , ' a et m al. [22]. "111 the rcvicv of IIartley [7] possiblc alternative systcnis arc discusscd (scc also IIeinickc ct al. [ l l ] ) . "I'arkrr and Zhristenscn [21] created this package; for a simplc application scc Tsantilis r t nl. 1371.

As a11 illplit statrlncnt to Retlucc, we type in a certain legitimately formcl expression. This means that, with the help of somc opcrators, we compose formulas accortlillg to well-defined rulcs. Most of thc built-in opcrators of Reclucr, like the arithmetic opcrators + (plus), - (minus), * (times), / (divided by), ** (to tlic power of)'%re self-explanatory. They are so-called znfix operators since they arc positiolled zn between their arguments. By means of thcm we can construct combined cxprcssions of the typc (r or x3 sins, which in Redutc read (x+y)**2 ant1 x**3*sin(x), respectively. If thr comlnalltl

+

+

+

is cxccl~t,cd,you will get the expanded for111 r2 2sy y 2 . Therc is a so-callcd switch exp in Rrducc that is usually switched on. You can switch it off by the commalitl I

--

o f f exp; I

See, McCrca's lecturrs [IG]. l21'he GRG s y s t ~ r n creatctl , by Zhytnikov [do], ant1 the GRCrZc system of Tertychniv (35, 34, 201 grew frorn the same root; for an application of GRGEc t o t h e Einstein-Maxwell equations, see [33]. '%ec t h e docrl~nentatiollof Musgrave et al. (191. Maple applications t o t h e EinstcinMaxwcll system arc covered in thc lcctrrres of McLenaghan [17]. '"Solcng (301 is t h c creator of 'Cartan'. l51~1asnothing t o rlo with Arnold Schwarzcncggcr! '%sually one takcs t h c circumflex for exponentiation. I-Iowcvcr, in t h e Excalc package this operator is rcdcfinctl a l ~ dused as tlre wedge syrnbol for exterior multiplication.

A.1. Algebra

48

* * * *

* *

*

description if switch is on factorize simple factors divide by the denominator expand all expressions make (common) denominator cancel least common multiples cancel greatest common divisor display as polynomial in f a c t o r display rationals as fraction dominates a l l f a c ,d i v , r a t , r e v p r i display polynom. in opposite order calculate with floats simplify complex expressions don't display zero results display in Reduce input format suppress messages display in Fortran format display in TeX format

Switch allfac div exp mcd lcm gcd rat ratpri pri revpri rounded complex nero nat msg fort tex

The operator neq means not equal. The assignment operator := assigns the value of the expression on its right-hand side (its second argument) t o the identifier on its left-hand side (its first argument). In Reduce, logical (or Boolean) expressions have only theI truth values t (true) or n i l (false). They are only allowed within certain statements (namely in i f , while, r e p e a t , and l e t statements) and in so-called rule lists. A prefix operator stands in front of its argument(s). The arguments are enclosed by parentheses and separated by commas:

examnle

Table A.1.2: Switches for Reduce's reforlnulation rules. Those marked with are turned on by default; the other ones are off.

cos (XI i n t (cos ( X I ,x) f actorial(8)

In ordinary notation, the second statement reads S cos x dx. The following mathematical functions are built-in as prefix operators:

*

Type in again

Now you will filid that Reduce doesn't do anything and gives the expression back as it received it. With on exp; you can go back to the original status. Using the switches is a typical way to influence Reduce's way of how to evaluate an expression. A partial list of switches is presented in a table on the next page. Let us give some more examples of expressions with infix operators: (u+v) * (y-x)/8 (a>b) and (c (greater than), < (less than). Widely used are also the infix operators: neq

>=

18

clear g,x$ a:=sin(g:=(~+7)**6) ; cos (n:=2)*df (x**lO ,x,n) ;

% a and b are declared to be unbound

f;

off gcd,div$ on exp$

are always evaluatcd first. I11 the first case, the value of (x+7)**6 is assigned to g, and then s i n ( ( x + 7 ) **6) is assigned to a. Note that the value of a whole assignmellt statement is always the value of its right-hand side. In the second case, Reduce assigns 2 to n, then computes df (x**lO,x,2), and eventually returns 90*x**8*cos(2) as the value of the whole statement. Note that both of these examples represents bad programming style, which shoulti be avoitfed. One exception to the process of evaluation exists for the assignment opcrator := . Usually, the arguments of an operator are evaluated before the operator is applied to its arguments. In a11 assignment statement, the left side of the assignment operator is not evaluated. Hence 1

clear b, c$ a: =b$ a: =c$ a;

will not assign c to b, but rather c to a. The process of evaluation in an assigllment statement can be studied in the following examples: % if a evaluates to an integer

I

clear h$ g:=l$ a: =(g+h) **3$ a; g:=7$ a;

If you want to display tlie truth valuc of a. B O O ~ Cc~x L ~ )~rIe ~ ~ illse o n ,th(: i f statcmcnt, as ill t l l ~following example: if

53

A.1.12 Computer algebra

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

Rudiments of evaluation A Reduce program is a follow-up of commands. And the evaluation of the commands may be conditioned by switches that we switch on or off (also by a com1naiid). Let us look into t l ~ ccvalr~ationproccss a bit closer. Aftcr a command has been sent to the computer by hitting the return key, the whole command is evaluated. Each expression is evaluatcd from left to right, and the values obtained arc combiricd with the operators specified. Substatements or snbexpressions (.xisting within other expressions, as in

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

1

After the second statement, the variable a hasn't the value (g+h)**3 but rather (i+h) **3. This doesn't change by the fifth statement either where a new valuc is assignctl to g. As onc will recognize, a still has the value of ( l + h ) **3. If we want a to dcpcntl on g, then we must assign (g+h)**3 to a as lollg as g is still unbound: (

clear g,h$ a:=(g+h) **3$ g:=l$ g:=7$ a; I

% all variables are still unbound

% yields: (7+h)**3

54

A.1. Algebra

Now a has the value of (7+h)**3 rather than (g+h)**3. Sometimes it is necessary to remove the assigned value from a variable or an expression. This can be achieved by using the operator clear as in I

clear g,h$ a:=(g+h)**3$ g:=I$ a; clear g$ a;

55

A.1.12 Computer algebra

with denoting the exterior and -I (underline followed by a vertical bar) the interior product sign. Note that before the interior product sign -I (spoken in) there must be a blank; the other blanks are optional. However, before Excalc can understand our intentions, we better declare u to be a (tangential) vector r

tvector u; I

f to be a scalar (i.e., a 0-form), and x, y, z to be 1-forms: 7

pform f=O, x=l, y=l, z=l;

or by overwriting the old value by means of a new ~ssignmentstatement: I

clear b,u,v$ a:=(u+v)**2$ a:=a-v**2$ a; b: =b+l$ b;

A variable that is not declared to be a vector or a form is treated as a constant; thus 0-forms must also be declared. After our declarations, we can input our command I

f*xAy+u - 1 (y-2-x); I

The evaluation of a; results in the value u* (u+2*v) since (u+v) **2 had been assigned to a and a-v**2 (i.e., (u+v) **2-v**2) was reassigned to a. The assignment b: =b+i; will, however, lcad to a difficulty: Since no value was previously assigned to b, the assignment replaces b literally with b+l (wherehs the prcvious a:=a-v**2 statement produces the evaluation a:=(u+v) **2-v**2). The last evaluation b; will lcad to an error or will evcn hang up tho system because b+l is assigncd to b. As soon as b is evaluatccl, Rcducc returns b+l, whereby b still has the value b+l, and so on. Therefore the evaluation process leads to an infinite loop. Hence we should avoid such recursions. Incidentally, if you want to finish a Rcduce session, just type in bye; After these glimpses of Reduce, we will turn to thc real object of our interest.

Loading Excalc We load the Excalc package by I

load-package excalc$

Of course, the system cannot do much with this expression, but it expands the interior product. It also knows, of course, that I

u -If; I

vanishes, that y A z = -x A y, or that x A x = 0. of an expression, we can use

If we want to check the rank

I

exdegree (x-y);

This yields 2 for our example. Quite generally, Excalc can handle scalar-valued exterior forms, vectors and operations between them, as well as nonscalar valued forms (indexed forms). Simple examples of indexed forms are the Kronecker delta 6 i or the connection 1-form I',P of Sec. C.1.2. Their declaration reads

I

pform delta(a,b)=O, gammal(a,b)=l;

The system will tell us that the operator A is redefined since it becamc the new wedge operator. Excalc is designed such that the input to the computer is the same as what would have been written down for a hand calculation. For example, the statement f *xAy + u - I (yAzAx) would be a lcgitimatcly built Excalc expression

The names of the indices are arbitrary. Subsequently, in the program a lower index is marked by a minus sign and an upper index with a plus (or with nothing), i.e., 6: translates into delta(-I , I ) , and so on.

A.1. Algebra

56

A.2 Exterior calculus

Figure A.1.G: "Catastrophic crror," a Reduce crror message. Excalc is a good tool for studying differential equations, for making calcnlntiolls in field t,lieory and general relativity, and for such simple things as calc~~lating the Laplacian of a tensor field for an arbitrarily given frame. Excalc is complotcly cmbccldccl in Rcducc. Thus, a11 fcnturcs and facilities of R c d ~ ~ c c arc available in a calculation. If we tlrclarc the dimension of tlie underlying spacc by spacedim 4 ;

then pform a = 2 , b = 3 ;

a"b;

yicltls 0. Thrsc arc the funtlamcntal commands of Excalc for exterior algebra. As soon as we liavc introduced exterior calculus with frames and coframcs, vector fields and ficltls of forms, not to forgct exterior ant1 Lie differentiation, we will come back to Excalc ant1 brttcr appreciitte its real power.

Having developed the concepts involved in tlie exterior algebra associated witli an 11-dimensional linear vector space V, we now look a t how tliis structr~rccan be "liftetl" ont,o a11 n-dimensional differentiable manifold X,, or, for short, onto X. The procedure for doing tliis is the same as for tlie transition from tensor algebra to t,cllsor calcul~ls.At each point .7: of X t,l~creis an 17,-climcnsional vector spacc X:,, the tangent vector spacc at 2 . Wc identify tlle space X, witli the vector spacc V collsidcrcd in the previous chapt,cr. Tlicn, a t each point 2, tho exterior nlgcbrii of forms is dctcnninctl on V = X,:. Howcvcr, in differential geometry, one is concerned not so mucli witli objects defined at isolatccl points as with fields over tlic lnanifold X or over open sets U C X , A field w of pforms on X is defined l,y assigning a pform t,o each point n: of X and, if this assigllrnclit is pc?rformed in a sinooth manner, wc shall call tlie res~iltiligficltl of pforms a n e3:terioi. differential p-form. For simp1icit.y we sh;\ll take "smootli" to 1ner211 Cwl altliougli in physical applications tlic dcgrcc of diff~rcntiabilit,y may be lcss.

A.2.1 @Differentiable manifolds A topolo!jical space becomes a differen,ti& ~rlnn,.foldrulren 0.71, atlas of coordincl.te clrnrts is introd~~cerl i n it. Coorrlinate trclmsform.ations are smootll i n the intersection,s of the clrnrts. T11.r: a,tlna i.9 o.r.ien.tcr1 lulren irr n.ll in~ter.sectio71.s the ,JncoOian.s of tlre coorrlirl.ate tmn,sfornrc~t~ion,s o,re positive. In ortler to dcscril~cmore rigorously how ficltls arc introducccl on X , we have to recall some basic facts about manifolds. At tllc start, one ~lccds:I topological

-4.2. Exterior calculus

58

~ ~ 2@Differentiable .1 manifolds

59

Figure A.2.1: Nan,-Hausdorff m:~nifold: Take two copies of the line segment {0,1) and identify (paste together) their left halves excluding the points (112) and (112)'. In the resulting manifold, the Hausdorff axiom is violated for the pair of points (112) and (112)'. stm~cture.To be specific, we will normally assume that X is a connected, Hausdorff, and paracompact topological space. A topology on X is introduced by the collection of open sets 'T = {U, c X l n E I) which, by definition, satisfy three conditions: (i) both the empty set Q) and the manifold itself X belong t o that collection, Q),X E 'T, (ii) any un.ion of open sets is again open, i.e., UaEJ U, E T for any subset J E I, and (iii) any intersection of a finite number of open sets is open, i.e., U, E 'T for any finite subset I( E I. A topological space X is con,n,ectedif one cannot represent it by the sum X = X1 U X2 with open Xl,2 and X I X2 = Q).Ust~allyfor a spacetime manifold, one further requires a linwar connectedness, which means that any two points of X can be connected by a continuous path. A topological space X is Hausdorff when for any two points p1 # p2 E X onc can find open sets U1, U2 c X wit,h pl E U1 and 212 E U2 such that Ul U2 = Q). Hausdorff's axiom forbids the "brancl~ecl"manifolds of the sort dcpictcd in Fig. A.2.1. A connected Hausdorff manifold is paracompact when X can be covered by a countable number of open sets, i.c., X = UnEI( U, for a countable subset I< E I. Finally, a manifold X is compact when it can be coveretl by a finite number of open sets selected from itasarbitrary covering. A collection of filnctions {p, : X + R) is called a partition of unity subordinate to a covering {U,) if support of p, C U, and 0 p,(p) 5 1, C , p,(p) = 1 for all points p E X . The partition of unity always exists for paracompact manifolds, and it is a standard tool that helps to derive global constructions from the local ones. A differentiable manifold is a topological spacc X plus a differentiable structure on it. The latter is defined as follows: A coordinate chart on X is a pair (U, $) where U E 'T is an open set and the map $ : U --t Rn is a homeomorphisn~ (i.e., continuous with a continuous inverse map) of U onto an open subset of the arithmetic space of n-tuples Rn. This map aqsigns n labels or coordinates d)(p) = {xl(p),. . . ,xn(p)} to any point p E U c X. Given any two intersecting Up # Q),the map charts, (U,, 4,) and (Uo, $o) with U,

n,,,

n

n


lc wllc11 one consi~lersthe integration tlicory on manifoltls and, in part,icular, on nonorientablc manifolds. I11 Sec. A.1.3, scc Examplcs 4) nntl ti), a twisted form was clcfincd on a vector spacc V as a geornctric quantity. Intuitively, a twistcd forni on the manifolcl X can be dcfinetl as an "orientation-valtiecll' conventional exterior form. Given 2Bott k TI] [I], p. 79. ""l'iui.ristcd tensors were introd~rcedby I l e n n n n W e y l . . . and de R h a m . . . called t h e m tcnsors of odd kind . . . . W e coirld make out a good case tliat the usiinl diflercntiol forms are actualE~/the twisted ones, hut the language is forced o n us by history. Twisted differential for.ms are the natural representations for densities, and sometimes are actually called densities, which u ~ ~ i t lhed a n ideal n a m e were i t n o t already i n iise in tensor analysis. 1 agonized over a notation for twisted tensors, say, a different typeface. I n the end I decided against i t . . . ," Willin~nI,. Burkc [2], p. 183.

"here, as in tlic previous section, J ( f a p ) is the Jacobian of tlie transition fun(ion fao := &% 0 4;'. Example: Consitler the Mobius strip, a nonorientablc 2-dimensional compact nanifold wit,h boundary (sce Fig. A.2.9). I t can be easily rcalized by taking rl ~ectanglc( ( 8 , [) E R2)0< 0 < 27r, -1 < [ < 1) ant1 gluing it together with ore ,wist along vcrtical sides. The simplest atlas for the resulting manifold consisis )f two charts (U1, and (U2,42). Tlie open domains UlY2arc rectangles aril ,hey can be chosen as shown in Fig. A.2.9 with the evidently dcfined loc~l :oordinat,e maps dl = ( x l , x2) and 4 2 = ( y l , y2), where the first coordinaie u n s along tlie rcctanglcs ailcl the second one across them. The intcrsectim ;TI U2 is comprised of two open sets, (Ul U2)left and (Ul U2)rig~lt.Tle ,ransition funct,ions f12 = 41 o qizl are f 1 2 = {x l = yl, x2 = y2) in (UI fl U2)lflt ~ n df12 = {xl = ?,11z2= -y2) in (Ul U2)riKll+,, SO tliat J ( f l z ) = r t l in t,he!e lomains, respectively. Tlie 1-form w = {w(') = dx2,w(2) = dy2) is a twistcd orln on tlie Mobius strip. In general, given a chart (U,, $,,), both a usual a.ntl a twist,ccl ??-form is givm )y its colnponeiits wi,,,,i,,(x),see (A.2.9). With a change of coorclinates, tle :omponcnt,s of a twist,etl form, via (A.2.12), arc transformctl as

n

n

n

n

For a conventional pform, tlic first factor on t,lic right-liantl side, the sign ~f the Jacobian, is abscnt. Normally, in gravity and in fieltl theory one worlts on orientable ninnifolls with an orientcd atlas chosen. Then tlic clifferencc bet,wcen ordinary and twistrtl objects tlisappenrs bccausc of (A.2.2). However, twisted forms are vcry impcrtant 011 ~ionoricnt:tbleli~aliifoltls011 which t,lic ~isualforms ca~inotbc integrat,cI.

A.2.7 Exterior derivative The exterior derivative nrn.ps a p-fornl in.to a ( p ciu.cial propelsty is n.ilpntency, r12 = 0.

+ 1)-form. 'ts

Denotc t,lic set of vector fields on X by Xd. For 0-forms f E Ao(X), tlie ctiffercntial 1-form df is defined by (A.2.7), (A.2.8), i.e., by df = f,, dxZ.We wkh t o extend this map d : Ao(X) A1(X) to a map d : Ap(X) -+ A"+'(J). Ideally this should be performed in a coortlin;~tc-frcc way and we shall glvc such ;I. definition a t tlie cntl of this section. Howrvcr, thc drfinition of cxtcror

-

A.2. Exterior calculus

68

dcrivativc of a pform in terms of a coordinate basis is very transparent. Furt l i e r ~ ~ ~ oitr cis, a simplc matter t o prove that it is, in fact, independent of the local coordinate syste~rithat is used. Starting with the expression (A.2.9) for a w E A Y X ) , wc tlefinc rlw E A"+] ( X ) by

G9

A.2.7 Exterior dcrivative

Thus

Then d ( w ~ 4 )=

d(flt)~dx'~~...~dxJ~l = (f dl!. h rlf) A dz" A . . A dxjq = (df A dzil A . . . A r~n:~l') A (1%dxjl A . . . A dn:j")

+

By (A.2.7), (A.2.8), the riglit-hand side of (A.2.14) is (llp!) wi, ,,,,

,,,,d x j A dxil A . . . A dxip .

(A.2.15)

A

+(-1)l' (f dx"' A . . . A dsi1') A (dh A dxjl A . . . A dzi4)

Hcncc, bccausc of the antisymmetry of the exterior product, we may write

=

dw~d,+(-l)"w~dqi.

(A.2.22)

To prove 4), wc? first of a11 notc that, for a function f E AO(X), Under a coordinate transformation {xi) -+ {n:i'), it is found that sincc partial dcrivat,ivcs comninte. For a pforrn it is sufficient t o consider a monoinial Hence, Then so tliat thc exterior dcrivative, as defined by (A.2.16), is independent of the coordinate systclil clioscn.

-

Proposition 1: T h r exterior tlcrivativc, d : AYX)

,?s

tfcfiricd by (A.2.1G), is a map A"+'(x)

(A.2.19)

with the following properties:

+ A) = dw + dX 2) d(w A 4) = dw A 4 + (-1)"

1) d(w

3) df (11) = d f )

and rcpcntetl application of propt.rty 2) ;ultl (A.2.23) yields tlle drsirctl result d(dw) = 0. By linearity, this m:~ybe extcndctl t o a general pform wliicli is il linear coinbination of tcrlris liltc (A.2.24). Proposztion 2: h v n ~ i a i z tes:pression for tire exterior derivr~tive. For w E A"(X), we can express c(w in a coordinatc-free nlanller as follows:

[linearity], A db

[(anti)Leibniz rule], [partial derivative for f~inctions],

4) d(dw) = 0

[nilpotency].

Here, w, X E Al'(X),4 E A"(X), f E A O ( X ) , uE XA(X). Proof. 1) and 3) are obvious from the definition. Because of 1) and the distributive propcrty of the exterior multiplication, it is sufficient t o prove 2) for w and 4 of the 'monomial' form:

wherc ?LO, u l , . . . ,lip arc arl)it,rnry vcctor ficltls ant1 ii indicates tlint t h r field 11 is omztted as an i~rgunient.It is a straightforwartl m;itter t o verify tliat (A.2.26) is consistent with (A.2.16). Wc shall nlnlte particular use of the casc i11 wliich w is a 1-form nlld (A.2.9) becorncs

A.2. Exterior calculus

70

A.2.8 Frame and coframe A natural frame and natural cofram,e are defined at every local coorrlinate patch by aian.d d z i , respectively. A n arbitranj frame e , and coframe 6" are con.structed by a 1in.ear transformation th.erefrom. T h e object of an,ll.olonom,ity measures how m u c h a cofra,m.e differs from a natural one. A local frame on an n-tlimcnsional differentiable n~anifolclX is a set e,, a = 0 , 1 , . . . , I ? , of n vcctor fields that arc l ? n e a r l ~i n d ~ p e n d ~ an tt each point of an open subset U of X. They thus for111 a basis of tlic tangent (vector) space X, at every point .I' E U . There exist quite ordinary manifolds, the 2-dimensional splicrr for cxamplc, where no continuous frame ficld can be introduced globally, i.r., at racll point of tlle manifold X . Therefore, speaking of frames on X , we will always have in mind local frames. If P , is a framc, then the corresponding coframe is the set 6" of n different 1-forms such that

is valid a t each point of X . In other words 1 9 ~ 1 , a t each point x E X is the dual basis of 1-forms for X:. We note in particular t l ~ a t as , a consequence of (A.2.28), every vcctor ficld u E X,' can be dccomposcd according to

A local coordinate system defines a coortiinate frame aion the open neighborhood U . Tllus an arbitrary framc e , may be expressed on U in terms of 8, in tl1c for111 of

wllcrc e",, arc tliffercntiable functions of the coordinates. For the corresponding coframc 6" wc have

e t a e2p= 6; If

iL

~ofrill~lc do

l i i the ~

p r ~ p ~that r t ~

it is said to be natural or holonornic. In tliis case, in the ncighborliood of each point, thcrc exists a coordinate systcln {.r2) such that

..2.9@Mapsof manifolds: push-forward and pull-back

71

lnder these circumstances, the frame e , is also natural or holonomic with = 6: 8,.The 2-form

--

rith C(p,)" 0, is the object of anholonomity with its 24 independent compoents. It measures how much a given coframe 29" fails to be holonomic. There ; also a version of (A.2.35) in terms of the frame e,. With the help of (A.2.27), ; can be rewritten as

'he object of anholonomity has a nontensorial transformation behavior. Example: On a 2-dimensional manifold with local coordinates {x, y ) , the 1)rms { f i i = x d y , 79% = y dx) are linearly independent. Such a coframe is anolonomic with d ~ = 9 d~ z A d y and d d 2 = - dz A d y , i.e., Ci = - C' = d z A d y .

1.2.9

@Mapsof manifolds: push-forward and pull- back Pull-back cp* and push-forward cp, maps are the companions of every diiffeomorphism cp of the manzfold X . T h e y relate th,e corresponding cotangent and tangent spaces at points x and cp(x). T h e m a p cp* com.mutes with the exterior diflerential.

If a differcnt,iable map cp : X -+ Y is given, various geometric objects can be transported either from X to Y (pushed forward) or from Y to X (pulled back). A push-forward is denoted by cp, and a pull-back by cp*. Given a tangent vector u at a point x E X, we can define its push-forward cp,u E Y,(,) (which is also called the diflerential) by determining its action on a function f E C(Y) as

However, if is not merely a tangent vector, but a vector field over X, it is in general not possible to define its push-forward to Y. There might be two reasons for that. Firstly, if cp is not injective and cp(x1) = cp(z2) for X I # x2, then the vectors pushed from X,, and X,, are different in general. Secondly, if cp is not surjective, the pushed forward vector field is not, in general, be determined all over Y. It is always possible to define cp,v of a vector field if cp is a diffeomorphism (which can only be considered when d i m X = dimY). Using the rule

A . 2 . Exterior calculus

72

wc can tlefinc the push-forward of an arbitrary contravariant tcnsor a t z E X t o the space of tensors of the same typc a t cp(z) E Y. So cp, bccomes a l~omornorphismof the algebras of contraviiriant tensors a t z E X ant1 cp(x) E Y. In a diagram we can depict the push-forward map p, of tangent vectors 11 a i ~ dtlie pull-back map p* for 1-forms w:

.2.10 @Liederivative

73

If cp is a difleomorphism, or a t least a loct~ldiffeornorphism, we shall use the pull-back cp* of arbitrary tensor fields. For contravariant tcnsors, it can be defined as p* = ( e l * = (p*)-* .

(.!.2.43)

b define it for an arbitrary tensor of type [jj], we liavc to rccluirc only that * is an dgcbra isomorphism. Technically, in local coordinates, this alnounts ) thc invc.itil)ility of the square matrices 6'?j1/6'r2, Wheri p is a (local) tliffcon~orpl~ism, wc can also pull-baclc (or pnsli-fcrwwd) ?omctlic quantit,ics constructed on tlie tangent space. Let [(w, e)] he a gcoletric quantity; llcrc e = ( e l , .. . , e,,) is a frame in the tangcnt space Yy and bclo~igsto the set W in which there is the left action p of GL(n, R). As for vectors, wc clcfinc p*r: = ( p * e l , . .. , p*e,) and I

Let { r Z )be local coordinates in X and {yJ) local coordinates in Y (with thc rnngcs of indices i and J dcfinetl by the dimellsiollality of X and Y , respectively). Thcn thc lnap p is clescribed by a set of smooth functions y3(zZ),and the pushforward map for tcnsors of type ]:[ in components reads

Comparing with (A.2.8) for tlic case when Y = R, it becomes clcar why cp, is also cnllcd a differential map. i c pull-back p'w E A?(X) For a p-form LJ E A;=(P(z)(Y), we can d c t c r ~ n i ~its

I)Y This definition can be straiglitforwardly cxtcntlcd t o a liomomorpllism of tlic algebras of covariant t,ype tcnsors. In local coordinates it reads, analogously to (A.2.39),

I:[

Let w be a11 exterior p-form (i.c., a, pform field) on Y . I11 order t o determine its pull-l~aclcp*w t,o X by (A.2.40), it is sufficient to liavc cp,ul,. . . , p,v, on t,hc right hand sidc of (A.2.40) tiefincd as vectors (i.e., not lieressarily as vcctor fields). Therefore, thc pull-back of cxtcrior forms (and, in general, of covariant tcvsors) is determined for an arbitrary map cp. In cxtcrior calcl~lus,an import,allt property is the corlllnutativity of pull-back and exterior differentiation for any p-form LJ:

i t . , tlic transportcd object has the same components n s the initial objctt with respect t o t,lic transported frame. Certainly, this definition of thc pull-3ack is consistent with that given earlicr for tensors.

A.2.10 @Liederivativc A t~ectorfield generutes a group of diffeornorphisms on z mnnifold. Making use of this group action,, the Lie derivative enables us to com.pare tensors and geometric q.uan,tities at differen.t poin.ts. The main rcsr~ltof the present section will I)(? cquatio~i(A.2.51), thc Lie rlerivntive of a tliffcront,ial for~n.Howovcs, wc slinll first explain tllc conccpt of tho Lit: derivativc of n geiicral gcomctrict~lqr~ant~ity. Note that for Lic tleriva:ivc n,o metric un.d no connection. is required; it can bc tlcfii~cd011 eacl~diffcrcut,iahlc manifold. For cach point 21 E X , a vccf,os field .u witli ~ ( p #) 0 dcternlincs a rlriique curve o,,(t), t 2 0, srlcli that a,(()) = p with IL as tllc tangcnt vector field t o thc curvt:. Tlic family of curvcs dcfincd in this way is called thc congmlen.ce of curvcs generat,ctl by the vcctor field rr. Lct {x i ) be a local coortliliate system with xi,as thc coort1in:itcs of p ant1 decompose 71 according t o 11 = ui(n:', . . . , xTL) (3,.Then the clrrvc o,)(t) is found by solving tlic systcnl of ordinary differential eql~at,iolls

witli init,ial values xl(0) = xi,.Tlic congrllence of curvcs obtained in this way tlcfines (at least locally) a 1-parameter group of diffcomorpliisms cp, on X given by

with thP propcrtics that (a) cp,' = cp-,, (11) cp, o cp, = cp,+,, and (c) cpo is thc idcntity mal). The integral curves of the congruence arc cnlletl tlie frajectones

A.2. Exterior calculus

'igurc A.2.11: Tlie clcfinition of thc Lie derivat,ive L,,v with rcspcct to a vcct,or A: The 1-paramctcr group pt, gcncratcd by the vector field u, is 11scd in ordcr ,o transfer thc vector v(pt(p)) back to the initial point and to compare it with 4~).

Figure A.2.10: Translations ( n ) and circular motion (b) gcnerated on R2. of thc group. F~lrthcrmorc,the equations (A.2.45) are equivalent to

for all p E X alitl all diffcrclltiable functiolls f . Exam.ples in R 2 :

+

1) Thc vector field ?L = 3 / d x generates translations cp,(x, IJ) = (x t,y), -oo < t < 4-oo.Thc tr;~j~ctories arc the lints y = const. See Fig. A.2.10a. 2) The vector ficld IL = (n.a/a?j - y d l a x ) gcneratcs thc circular motion cpt (x, 11) = ( 2 %cos t - y sin t , x sill t y cos t), 0 5 t < 27~.The trajectories are concc~ntriccloscd curvcs around thc origin, see Fig. A.2.10b.

~t is suficirnt to have cxplicit cxprcssions for tllc Lie derivatives of f ~ n c t ~ i o n s , vectors, n~ltl1-forms in order be in a position to do tjhc sallie for a gcnrrnl tensor. Thc two most important c:tsc.s nro ns follows. For vectors 1, E Xd,

+

In g c ~ i ~ r aifl ,we take a coordinate patch U of a differentiable manifold with coordinates { x l ) , tlicll cp, is defined in terms of s q y

where f t ( t ; x1) are diffcrentiablc functions of ( t , x J ) . Property (a) states t,hat r2= x 7 ( t ;y 7 ) = f 2 (- t; yJ). By p r o p ~ r t y(h) we have f 2 (t; f3(s;x k )) = f 2 ( t s;.xJ), while (c) lncans that f "0; r J )= r 2 . For every vnluc of t in a ccrtain interval, tlie diffeomorphism cpt induces corrcsponding pull-backs cp? 011 functions, vcctors, cxterior forms, ancl general tensor firltls of type Accordingly, the Lze denvatzve of a tensor T with respect to a vector ficld IL is defined by

+

[;I.

sec (A.2.6) for a component vcrsion. For p-forlns w E AYX) and p thc ?nuin theorem for t,lic. Lic tlcrivntivc of an cxtcrior form:

1 f ,,w

=

11 J

(dw)

2 0, wc find

+ (I(u J w ) .I

An altcrntitivc coordinnte-free general formllla for this Lie dcrivttt,ive reads: ( w ) ( ,. . )

=L

(

w

(

(, u p-

u

, , [u,vi],. . . , 4 ) (A.2.52)

7=1 Tllc Lie tlerivativc for tlie fullctiolls f E C ( X ) is obtained as a of (A.2.51) for p = 0: Lllf =

1~(f= ) 2LJ

(If.

articular case (A.2.53)

A.2. Exterior calculus

7G

nd we casily find for u = t i ( x ) di,

The last, formula is straightforwardly chcclles. With f domain f =f ( X, y) , h=h ( x ) ; Q(x*f,x); Q(h,y);

0 1

i.e., f +.rD, f ant1 0. T h r partial tlcrivativc symbol can also be 8111 operator with a single arguiiient, :LS in @(z). T~IPIIit rrprescnts the leg 8, of a natural frame. Coming back t,o the cxtcrior derivative, t,lle following cxan~plcis now sclfcxplanatory: pform x=O,y=O,z=O,f=O; f domain f =f ( x , y) ; d f;

A.2. Exterior calculus

80

i.e., clf eva111;~tcst,o (3,j)dn: o l ~ t ix., ,

+ (a,f)dy. Products are normally differentiated +

pf orm x=0, y=p,z=q; d(x*yAz) ;

i.e., ~ ( U JZ ) U J dr. The operator of the Lie derivatives fulfills the rules displayed after (A.2.55). We will check the rule for the rescalcd vector as an example. Already above, the form w has been declared to be a p-form, f to be a scalar, and u to be a vector. Hence we call type in directly

In an ordinary formula, we have d(xyAz) = ( - l ) ~ x y ~ d z + z c l y ~ ~ + d ~ ~ y ~ ~ . This expansion can be suppressed by tlie command noxpnd d ; . Expansion is xpnd d ; is executed. performed again when the coin~na~lcl i.e., The Excalc operator d knows all the rules for the exterior derivative as speciI fied in Proposition 1 in the context of (A.2.19). Let us declare the corresponding ranks of the forms in order to check the first two rules (note that lambda is a I reserved identifier in Reduce ant1 cannot be used):

Then we give the commands I

d(omega+lam) ;

d(omegaaphi) ;

i.c., d(w 4-A) ant1 rl(w A $), a ~ ~find, t l respectively, I

d lam + d omega

(

-

P 1) *omegaad phi + d omega-phi

+

i.e., dw + dX nnd ( - 1 ) " ~A d$ dw A 4. The sccond to last entry in our table is the Lie derivative L . In Excalc, it can be applirrl to an rxtcrior form with respect to a vector or to a vector again with rcspcct to a vector. It is represented by the infix operator I - (vertical bar followed ~y an nnderline). If the Lie derivative is applied to a form, Excalc remembers tl~emain thcorcm of Lie tlerivativcs, namely (A.2.51). Thus, pform z=k; tvector u;

81

,.2.11 Excalc, a Reduce package

ul-z;

I

Lf,, w, and d(u

-I

find omega)*f + u

-I

d omega*f + d f ^u

-I

omega

i.e., d ( u w) ~ f + ( U J clw)f +df A U J W.The rule is verified, but Excalc substituted (A.2.51) immediately. Anyway, we also see that 1- does exactly what we expect from it. In Sec. A.2.8, we introduced the frame e , and the coframe 19" as bases of the tangent and the cotangent space, respectively. In Excalc we use the symbols e(- a) and o(a), respectively. In Excalc a coframe can only be specified protl~drd a metric is given at the same time. This feature of Excalc is not ideal for our purposes. Nevertheless, even if we introduce the metric only in Part C, we have to use it in the Excalc program already here in order to make the programs of Part B executable. As we saw already in Scc. A.1.12, wc can introduce Excalc to the tlimcnsion of a manifold via spacedim 4 ; . This can also be done with the cof rame statementl, since we specify thereby the uildcrlying four 1-forms of the coframc and, if the coframc is orthonormal, the signature of the metric. For a. Minlcowslci spacctimc' with time coordinate t and s~hcricalspatial coordinates r, 0, cp, we state o(t) o(r) theta) = r * = r * sin(theta) o(phi) with signature (1,-1,-1,-1); frame e; coframe

*

d d d d

t, r, theta, phi

With frame e ; , we assigned the identifier e to tlie name of the frame. 111ortlinary mathe~naticallanguage, the coframe statement would read dt

= dt, d r = d r , dO=rdO, d4=~-sin0cl4, g = dt@~9t-dr@dr-de@d0-d1b@d4.

(A.2.72)

,.2.12 @Closed and exact forms, de Rham cohornology groups

A.2. Exterior calculus

82

Of colirss, the frame e ( - a ) ant1 the coframe o ( a ) are inverse to each 0the1., i.e., the conln~nntle ( - a ) -lo(b) ; will yisltl the Kroneckcr drlta (if you switch o n n e r o ; thcn only the c o m p o n ~ i ~which ts nonvanishing values will be printed out). The cofranlc statenlent is very fi~lldamclltalfor Excalc. All quantities will be evalr~ntctlwith resprct to this coframe. This yicltls the anliolonomic (or physical) components of an object. T h s coframe statement of a corrrsponding spherically symmetric Riclnallniall metric with unknown function $(r) reads:

A.2.12 @Closedand exact forms, de Rham cohomology groups Closed f0rm.s are n.ot exact in gen,eral. Two closed f0rm.s belonsgto the same col~omologyclass when they differ by an exact form. Groups of cohomologies are topological invarian,ts. L

1 I

I

load- package e x c a l c $ pform psi=O$ fdomain p s i = p s i ( r ) $ coframe o ( t ) = psi * o(r) = (l/psi) * theta) = r * o(phi) = r * sin(theta) with signature (1,- 1,- 1,- I)$

*

t, r, theta, phi

1 I

r

p = 1 , . . . , n,

(A.2.74)

is also a (real) vector space, ant1 evidently BYX) c Zp(X) (each exact form is closed, since dd r 0). One puts B O(X) = (4. Obviously the rxterior derivative defines an cquivalcncc relation in the space of closed forms: two fornls w, w' E Zl'(X) are said to be cohomologically cquivalent if t,licy differ by an cxact form, i.r., (w - w') E Bp(X). The quotient. space

consists of cohomolog?jclasses of p-forms. Each Hl'(X; IW) is a vector space and, moreover, a n Abclinn group with an evident group action. Tlie spaces H"(X; R) are namcrl as dc Rllam cohomology groups. Unlilte the AYX) which :ire infinite dilnc~~sional functional spaces, the dc Rhnm groups, for compact mnriifolds X , are finite dimensional. Tlie dinlension

d r =

forms a (real) vector s ~ ~ b s p a of c e AP(X). A I)-form w is call~clexact if a (p - 1)form p exists sucli that w = dp. The spacc of all cxact p-forms Bl'(X) := {w E A1'(X)Jw= dp} ,

If a commantl is tcrn~inatcdby a tlollar sign $, the output will be suppressed. Consecluently, if we input the program seglnent into Reduce, only the coframc 49" will be tlisplnycd:

0

7L

let us consider the ext,crior algcbra A*(X) = @ Ap(X) together with the p=o exterior derivative defined in (A.2.19). A pforln w is called closed if dw = 0. The space of all closed p-forms

! d d d d

% d i s p l a y s t h e coframe o ( a ) , i n p u t c a n be checked

displayframe; frame e $

83

----psi

theta

o

is called the p-th Bett~,number of the lnanifold X. Locall?/, an exterior derivative does not yield a difference between closed and xact forms. This fact is usually formulated as the Poincnre' lemma: Locally, in given chart (U,q5) of XI every closed p f o r n ~w with dw = 0 is cxact, i.e., a n - 1)-form cp exists in U C X stlcl~that w = dp. Let us illustrate this by an rxplicit cxamplc. Suppose we have a closet! one-form w. In local coordinates,

= d theta*^

phi o

= d phi*sin(theta)*r

Pcrliaps wc should remind ourselves that $' = 1- 2 m l r rcprescnts tlle Schwarzschild solution of general relativity.

*

12.2. Exterior calculus

84

Then this form is locally exact, w = dq, where the 0-form cp is given explicitly 4) by in the chart (U, q(z) =

I

wi(tx) x i dt.

0

Indeed, let us check directly by differentiation: dq

=

]

.

.

dt [(ajwi(tx))t x' dx2

wi (x) dxi = W.

and exact forms, de Rham cohomology groups

85

With the help of this map onc can prove a fundamental fact: If X and Y are homotopically equivalent manifolds, their de Rham cohomology groups are isomorphic. As a consequence, their Betti numbers are equal, bP(X) = bP(Y). Homotopical equivalence essentially means that the manifolds X and Y can be "continuously deformed into one another". An n-dimensional Euclidean spacc IEn is homotopically equivalent to an n-dimensional disk Dn = { ( x ,. , x n) E l E n l J m 5 I}, for example, and both are homotopically equivalent to a point. Another example: A Euclidean plane lE2 with one point (say, the origin) removed is l~omotopicallyequivalent to a circle S1.More rigorously, manifolds X and Y are homotopically equivalent, if there are two differcntiablc maps f l : X 4 Y and f 2 : Y 4 X such that f 2 o f l : X 4 X and f l o f 2 : Y Y are homotopic to identity maps idx and idy, respectively. Two maps are homotopic if they can be related by a smooth family of maps. Thc alternating sum

+ wi (tx) dxi]

0

=

A.2.12'Closed

(A.2.79)

We used (A.2.77) when moving from the first line to the second one. The explicit construction (A.2.78) is certainly not unique but it is sufficient for demonstrating how the proof works. One can easily generalize (A.2.78) for the casc when w is a pform, p > 1,

is a topological invariant called the Euler characteristic of a manifold X. In two dimensions, every orientable closed (compact without a boundary) manifold is diffeomorphic t o a sphere with a finite number of handles, Mz := S2 " h handles", where 11 = 0 , 1 , 2 , .. . (for h = 1, we find a torus Mf = 'If2from Fig. A.2.3). Euler characteristics of these manifolds is X ( M i )= 2 - 2h. Analogously, for the nonorientable 2-dimensional manifolds N: := S2 "k cross-caps" (Figs. A.2.4, A.2.5 show Nf = P2 and Ni = K 2 , respectively), the Euler characteristic is equal X(N:) = 2 - k.

+

where the vector field u is locally defined by u = x'a,. Its integral lines evidently form a "star-like" structure with the centcr a t the origin of the local coordinatc system. Globally, i.c., on the whole manifold X, however, not every closed form is exact: One usually states that topological obstructions exist. The importance of the de Rham groups is directly related to the fact that they present an example of topological invariants of a smooth manifold. Of course, the Betti numbers then also encode information about, the topology of X . The zeroth number bO(X),for instance, simply counts the connected components of any manifold X. This follows from the fact that 0-forms are just functions of X, and hence, a closed form cp with d q = 0 is a constant on every connected component. Since there are no exact 0-forms, B O ( X ) = 9), the clemcnts of the group H O ( X ; R) are thus N-tuples of constants, with N equal to thc number of connected components. Hence b O (X) = dim H O ( X ; R) = N. Moreover, recall that in Sec. A.2.9 for any differentiable map f : X -+ Y we have described a pull-back map of exterior forms on a manifold Y to the forms on X. Since the pull-back commutes with the extcrior derivative, see (A.2.42), wc immediately find that any such map determines a map between the relevant cohomology groups:

+

Integration on a manifold

[n this chapter we will describc thc integration of exterior forms on a nia.nifold. rhe calculus of differential forms providcs us with a powerful tcchniql~c.This oc:urs because one theorem, known as the Stokes or the Stokes-Poincarb throrem, "eplaccs a ntimbcr of diffcrent theorems known from 3-dimensional vector cal:ulus. Both types of pforms, ordinary and twistcd ones, can be integrated over +dimensional submanifolds, and in botl1 cases onc necds an additional struc;ure, thc oricntation, in order to define them. For ordinary forms onc ~ieedsthe inner and for twisted forms the ovter oricntation. Tllcre are two cxceptio~is:To ntegrate an ordinary 0-form or a twistcd n-form, no oricntation is necessary.

A.3.1 Integration of 0-forms and orientability of a manifold Tlie integral of a 0-form f over a 0-dim.ensiono.1 subrnnn~ifoln! (set of points i n X) is j ~ ~ as sum, t of vn11l.e~o f f at tlrese poin,ts. Let f be a function on X , i.e., f E AO(X),and let Z be a finite collcctio~iof points, Z = ( p l y . .. , p k ) . We can then define the integral of f over Z by

If f is, instead of being an ordinary function, a twisted function, then this definition is not satisfactory. Then thc f (pi)'s change their signs together with

4.3.3 @Integrationof pforrns with 0 < p

tlie change of tlic orieut.ntion of tlic rcfcrcnco frames a t each point p,. If we fix one of the orientations at,, say, tlie point p l , then we call try to propagate this oricntatioil by ~ont~inuity to a11 other points pz, . . . ,pk. If tliis call be don(. unainl~iguously,then we say that tlie manifold is orientable, ant1 we have just chosen an oricntation for X . I11 such a case, the values of the function f , i.e., f (PI), . . . , f (pk), can be t,akcn with respect to any frame wit,h positive orientation, and f o r n ~ ~ l (A.3.1) la defines unambiguously the integral f f of a twisted

a 2 in s11ch a way that the vcrticcs of the 2-simplcx iuc the points Po = (0,0), Pl = (1, 0), ant1 Pz = ( 0 , l ) . The simplex is then callctl starldnrd witli thr canonical clioicc of cooidinutcs. The standard 2-simplex is depicted on Fig. A.3.4. Incidentally, tlic generalizat,ion to lligller-dimcnsiolial sirnpliccs in IW" is straightforward: A standard ysimplex aP = (Po, P I , .. . , P,) is dcfined by the points Po = (0,. . . ,0), Pl = ( 1 , 0 1. . . ,O), . . . , P p = ( 0, . . . ,0 , l ) . Given the parametrization of thc standard 2-simplcx, cf. (A.3.9), 02={(1-t

1

- t 2 ) ~ O + t 1 ~ l + t 2 ~ 2 } O, < t l < l ,

(A.3.22)

its bo~lndaryis described by tlic thrcc 1-simplices (its faccs): o [ ~ )=

{t1Pl+t2P2},

ail

=

{t

at2)

=

{(I - t l ) p0 t1 P I ) ,

For a. 1-for111 w

011

2

P2

+ (1

-

f1

t 2 ) PO),

+

+ t 2 = 1,

0 5 t 2 5 1,

(A.3.23)

o < t1 5 1.

Tlic right-hand sitlr of Stokcs' tlicorem consists of the three integrals ovcr thc faces (A.3.23). Direct calculation of tlic corresponding line integrals yiclds:

X tlie pull-hack on a 2 c R2 is givcn by

+

(s*w) = f l ( t l l t 2 ) d t 1 f 2 ( t 1 , t 2 ) d t 2 , ' A rigorous proof can bc found in Clioquct-Urul~atct aI. [4].

(A.3.24)

96

A.3. Integration on a manifold

97

A.3.5 @DeItham's theorems

Taking into account that

and recalling (A.3.21), we compare (A.3.26) and (A.3.27) to verify that (A.3.20) holds true for any I-form and any singular 2-simplex on X .

Figure A.3.5: Simplicia1 decomposition (triangulation) of (a) the torus T 2 , (b) the rcal projective plane P 2 , and (c) the Klein bottle R2. is introtluced, in analogy to (A.3.16) and (A.3.21), by defining for every singular pchain c, a singular (p - 1)-chain:

A.3.5 @DeRham's theorems The first theorem of de Rham states that a closed form if and only if all of its periods vanish.

1:s exact

Recall that the de Rham cohomology groups, which wcro defined in Sec. A.2.12 with the help of the exterior derivative

in the algebra of differential forms A*(X), "feel" the topology of the manifold

X. Likewise, singular simplices can also be used to study the topological properties of X . The relevant mathematical structure is represcntecl by the singular homolo.qy groups. They are defined as follows: Like a chain constructed from simplices, see (A.3.15), a singularp-chain on a manifold X is defined a formal sum

with real coefficients ai and singular p-simplices s f . In the space C , ( X ) of all singular p-chains on X, a sum of chains and multiplication by a rcal constant are defined in an obvious way. The bounday map

In complete analogy wit11 the de Rham complex (A*(X),d), a singular psimplex r is callccl a cycle if a z = 0. The set of all p-cycles, Zp(X):={z~CP(X)Idz=0),

p = O , . . . ,n,

is a real vector subspace, ZP(X) C CP(X). A singular pchain b is called a boundarg if a (p b = a c . Tllc space of pboundaries

(A.3.33)

+ 1)-chain c exists such that

B , ( X ) : = { ~ E C , ( X ) I ~ = ~ C ) ,p = 1 ,

... , n ,

(A.3.34)

also forms a. (real) vector space and B , ( X ) C Zp(X), since 813 r 0. One sets B,, ( X ) = Q). Finally, the singular homology groups are defined as the quotient spaces H,(X;R):=Z,(X)/B,(X),

p = O ,...,n.

(A.3.35)

As an instructive example, let us briefly analyze the homological structure of the siinplest compact 2-dimensional manifolds: The sphere S 2 , the torus T 2 (these two are orientable), the real projective plain P 2 , and the Klein bottle R2 (tllese are nonoricntable). The three last n~anifoldsare seen in Fig. A.2.3, Fig. A.2.4, and Fig. A.2.5, respectively. A standard approach to the calculation of lion~ologicsfor a manifold X is to triangulate it, that is, to subdivide X into simplices in such a way that the resulting totality of simplices (called a simplicia1 con~plex)contains, together with each simplex, all of its faces. Every two sinlplices either do not have common points or they intersect over a common

A.3. Integration on a manifold

98

face of lower diincnsion. The triangulation of a sphere obviously reduces just to a collection of four 2-simplices that form the boundary of a 3-sin~plex,that is, tlie surface of a tetrahedron (see Fig. A.3.1). The triangulations of the torus, tlie projective plane, and the Klein bottle arc depicted in Fig. A.3.5.

A.3.5 @DeRham's theorems

dcf,, = (B)- (A) and act2) = (A) - (B),thus proving that z1 = ct,) +ct2) is a l-cycle. Moreover, it is a boundary because of 2z1 = dP2. No other l-cycles exists in P2. Thus we conclude tliat tlic 1st homology group is also trivial. Because of connectedness, the final list reads:

1) S2 lins as the only %cycle the manifold itself, z 2 = S 2. Direct inspection shows that tliere arc no nontrivial l-cycles (they are all boundaries of 2-diniensional chains). Finally, each vertex of the t,etrahedron is trivially a O-cycle, and tliey are all homological to each other because of the connectedness of S 2 . These facts are summarized by displaying the homology groups explicitly:

+

2) T 2 is "cornpo~cd~~ of two 2-simplices, T 2 = Stl) SE), namely Stl) = (A, C, D) ant1 St2)= (A, B, C) with the corresponding identifications (gluing) of sides and points as shown in Fig. A.3.5(a). The direct calculation of the boundaries yields = (A, B ) (B, C ) - (A, C ) and as,",,= (C, D ) - (A, D ) (A, C). Taking tlie identifications into account, we then find dT2 = 0; hence the torus itself is a 2-cycle. There are no other non-trivial 2-cycles. As for the l-cycles, we find two: zfl) = (A, B)lo=n and = ( B ,C)lc=B.A (end points are identified). Geometrically, these cycles are just closed curves, one of which goes along ancl another across the handle. There are no other independent l-cycles (z&) = (C, A)Jc=n, for cxamplc, is l~omologicalto tlie sum of zfl) ancl z(:)). Thus we havc verified t h t the 1st homology group is two-dimensional. For O-cycles the situation is exactly the same as for the sphere. In summary, wc havc for the torus:

+

+

99

+

4) R 2 = Stl) S&, wliere St,) =.(A, C, D ) and St2)= (A, B, C ) witli sides and points glued as shown in Fig. A.3.5~.By an analogous calculation, we find dK2 = 2(B, C ) . There are no nontrivial 2-cycles on the Kleiri bottle. As for the torus, tliere are two independent l-cycles, zfl) = (A, B)IR=A and z(:) = (B, C)Ic=B=n However, tlie second one is a l~ounclary2zt2) = dK2. Hence zll) generates tlie only nontrivial hornology class for t,llc. Klein bottle. T ~ U Sfinally, tlic 11omologygroups are:

+

2 3) P2 = Sfl) S&),where Sfl)= (A,C, D ) and S#) - (A, B, C) with sides and polnts identified a5. shown in Fig. A.3.5b. epeating the calculation for tlie torus, we find dP 2 = 2cf1) t 2cf2),wliere the l-chains are cf,) = (A, B) and cf2)= (B, C ) . Tlius, the projective plane itself is not a 2-cycle. Since there are no other homologically inequivalent 2-cycles, we conclude tliat the 2nd homology group is trivial. Moreover, we immediately verify

Like the dc Rham coliomology groups II"(X; R) (see Scc. A.2.12) tlic singular homology groups Hp(X;R) are topological invarian,ts of a manifoltl. In partic11lar, tliey do not change under a 'smooth deformation' of a manifoltl, i.c.., tliey are hoinotopically invariant. Cohornologies ancl homologics arc tlccply related. In order to demonstrate this (altliough without rigorous proofs), we ncod the central notion of a period. For any closc(1 pform w E P ( X ) and eacli p-c?lclr: z E Z,(X), n period of the form w is t,llc number 1 ' ~ ' (w) ~ :=

[

w.

This real number is not merely a function of w and z; it rather depends on tlie whole coliomology class of the form [w] E Hp(X; R) and on tlie whole llolnology class of the cycle [z] E H,(X;R). Stokes' theorem underlies the proof: For any coliornologically ccluivalcnt p-form w + dv and for any homologically cquivnlcrit pcycle z dc, we find

+

A . 3 . Intcgrntion on a manifold

100

sincc d t = dw = 0. Therefore, in a strict sense, onc has to write a periotl as per[,]([w]).Wc recall thc definition of a form as a linear map from a vcctor space V to thc reals (see Scc. A.1.1). Accordingly, one can treat the pcriod (A.3.40) as a 1-form on the space of cohomologies with V = H"(X; R), i.e., as an elcmcnt of the dual spacc per[,] E H,,(X; R)* ,

I

I

for all [w] E N1'(X; R).

In siniple tcrms, the first theorein tells that DR.([w])= 0 H [w] = 0. A 1-form on a vcctor spacc V is dctcrnlinccl by its components wliicll give the values of of tlic that for111 with respect to a basis of V. Suppose wc have chosen a. basis [ti] p t l i homology group Hr,(X;R), i.e., a complete set of hon~ologicallyinequivalent singular p-cycles ri. (For a compact manifolds this set is finite.) Denotc as Oi E V* = IIp(X;R)" the dual basis to [ri]. Each 1-form on V = II),(X;R) is then an elelllent (ti Oi spccificd Ily a set of real nutnh~rs{ o , ~with } i rrinliing ovcr the wliolc range of t,hc basis t i . Tlic second de Rll,nnr. t11,eoremstatcs that, tlic cle Rhani 111;q) is ~:n,vertible, that is, for every set of rcnl num11c:rs { a i } tlicrc cxist,s a closcd p-form w on X such that i.c.,

[w] = DR-'(a, 0 % ) .

101

n particular, dim Hl'(X; R) = dim H,(X; R). Then one can, for examplc, cal:ulatc the Euler charactcrist~ics(A.2.82) easily. Returning again to the 2-dimen,ional examples, we find: X(S2) = 1 - 0+ 1 = 2, see (A.3.36); X(T2)= 1 - 2 + 1 = 0, scc (A.3.37); X(P2)= 0 - 0 1 = 1, see (A.3.38); and X(JK2)= 0 - 1 1 = 0, see (A.3.39).

+

+

(A.3.41)

Thc linear 111a11DR : HI'(X; R) + Hr,(X;R)*! defincd via the equations (A.3.41) and (A.3.40) as DR([w])([t]):= pcrlz1([w]), is callctl the de R h a m map. A fundamental tllcorem o f de R h a m statcs that this map is an isomorphism. Sometimes, the proof of the cle Rham theorem is subdividetl into tlic two separate propositions known as the first ant1 second de Rham theorems. The first de R h a m tl~eoremrcads: A closet1 f o m zs exact zf and only if all of zts penods vanzsh:

DR([w])= a; O',

i.3.6 @DoItham's theorems

(A.3.43)

In con~binationwith the second theorem, thc first dc Rham tllcorem clcarly guarantees that the dc Rham map is one-t80-onc:Suppose that for a given set {a,) onc can fincl two 1-forms w ancl w' that both satisfy (A.3.43). Then we get DR([w - w']) = 0 ancl (A.3.42) yields [w] = [w'], i.c., w ancl w' differ by an exact, form. Incidentally, our earlier study of the Iiomologica1 st,ructurc of the 2-dimensiona1 manifoltls S2,T 2 , P2,K2 gave explicit constructions of tlic basrs [r,] of the homology groups. Onc can show that for an arbitrary compact lnanifold X both thr cohonlology ttnd homology groups are finite-dimensional vcctor spacrs. Thcn the d r Rliai~lnlap establishes tlic canollical isomorphism

Problem Problem A.1 Show that properties 1)-4) of Proposition 1 in Sec. A.2.7 lead uniquely t o the forlnula (A.2.1G), i.e., they provide also a definition of the exterior derivative.

References

[I] R. Bott and L.W. Tu, Differential Forms i n Algebraic Topology. Corr. 3rd printing (Springer: Berlin, 1995). [2] W.L. Burke, Applied Differential Geometry (Cambridge University Press: Cambridge, 1985). [3] B.W. Char, K.O. Gecldes, G.H. Gonnet, B.L. Leong, M.B. Monagan, and S.M. Watt, First Leaves: A Tutorial Introduction to Maple V (Springer, New York, 1992). [4] Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick, Analysis, Manifolds and Ph?jsics, revised ed. (North-Holland: Amsterdam, 1982). [5] J. Grabmeier, E. Kaltofen, and V. Weispfenning, eds. Computer Algebra Handhook: Foundations, Applications, Systems (Springer: Berlin, 2003). [GI A.G. Grozin, Ussing REDUCE i n High Energy Physics (Cambridge University Press: Cambridge, 1997).

[7] D. I-Iartley, Overview of computer algebra i n relativity. In [lo] pp. 173-191. [8] A.C. Hearn, REDUCE User's Man.ua1, Version 3.6, RAND Publication CP78 (Rev. 7/95). The RAND Corporation, Santa Monica, CA (1995). [9] F.W. Held, J.D. McCrea, E.W. Mielke, and Y. Ne'eman, Metric-ABne Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance, Phys. Rep. 258 (1995) 1171.

104

Part A. Mathematics: Some Exterior Calculus

1101 F.W. Hchl, R.A. Puntigam, and H, Ruder, eds., Relativity and Scientific Com.p~~tin.g: Computer Algebra, Num.erics, Visualization (Springer: Berlin, 1996).

[25] E. Schriifcr, EXCALC: A S?jstem for Doin,g Calculo.tion..s in the Co.lc~~lus of Modern Diflerential Geornetr?j. GMD-SCAI, St. Augustin, Gcrriialiy (1994).

[ l l ] C. Hcinickc and F.W. Hehl, Computer algebra in gravity, 5 pages in 151.

[26] E. Schriifcr, Differen.tin.1Geom.etr?jand Applico,tions, 3 pages in [5].

[12] B. Jancewicz, Wielkos'ci skierowane w elektrodynam,ice (in Polish). Directed Quantities in Electrodynamics. (Univcrsity of Wroclaw Press: Wroclaw, 2000); an English version is under preparation.

[27] E. Schriifcr, F.W. Hchl, nntl J.D. McCrea, Exterior calculus on the corn,puter: Tlre REDUCE-package EXCALC applied to gen.eral relativit?/ and tlre Poincart !lauge thcory. Gen. Rcl. Gr8.v. 1 9 (1987) 197-218.

[13] W. Irovitledby Nelson [34]; see also the references therein.

with

h

a scalar-val~ictlcliargc

which, as a scalar, can bc adtlcd 11p in any coortlinatc system to yield Q. Thus, as alrrady noted, cvcn t.he charge density p carries the same absolute dinielisioll as the net clinrgc Q. In spatial spherical coordinates z" = (r, 0,4), for instance, the coordinates carry different dimcnsions: The .rl or r-coordinate has the dimension of a length, wliercns .r2 = 0 ant1 "r 4 a1c dimensionless. Accordingly, the different coordiliatc coniponcnts pa,,c of p have different dimcnsions that cannot be lneasurecl straiglitforwartlly in an cxpcrimcnt. Thus these components of p are unsuitable for a. gcweral definition of the relative (also called physical) dimension of a quantity.

19"

6 , . , . = 1 , 2 , 3 , atid its d u d frmne r i wit11 r j J 19' = 6;. We mrwk tlie aumbrrs i, 2, by a circumflex in order to bc able to identify thcm as being rclatrd to

*

A

for 6,= 1,2,:3,

P-' ,

A

A

*

,

(13.1.4)

.

for b = 1 , 2 , 3 .

(B.1.5)

Length is here understood :IS n spgmrnt, that is, as part of n straiglit line between two points. Accordingly, lciigtl~is syl~ol~yn~ous with a 1-dimensional 2xtcnsiou i11 ztffinr geometry: it is not, however, the distance in the spnsc of Euclidean geometry. In other words, length as tlinicnsion tlocs not prcs~lpposc :he existence of a metric; it is a, prc-metric conccptg. Now we call dccomposc the cliargc density 3-form with respect to the coframc 19', 1

p=

% PiLCB%

19"

\9?

rcl. dim. := [phi,] = P - ~.

(B.I.G)

The din~cnsionsof all n.nholonomic components /);,je of the charge tlcnsity p arc the samc. We cnll [pibe] the relative dim.ension,of p. Accordingly, t,l~oabsolute dimension of p is charge, i.c., [p] = q, whereas the rc1ativc dilnc1ision of p is c/~.ar~e/(len.~tl~,)', i.c., = q t-'.

112

I3.1. Electric charge conservatiori

In the h~jpotlres~s of local~t?/"t is assumcd that the mcasuring apparatus in the coframta 6" evcn if thc lattcr is accelcratctl, mca-ures thc anholonomic components of a physical quantity, such as the components pAb2,cxactly as in a momentarily comoving inertial frame of rcfercnce. In the special case of the mcasulmcnt of time, Einstein spoke about thc clock hypothcsis. If we assume, as suggested by cxpcricncc, that the electric charge Q has no zntnnszc scrcw-sense, then thc sign of Q docs not tlepend on thc orientation in space. Accordingly, thc charge tlellsity p is reprcsentcd by a twzsted 3-form; fol the definition of twisted quantitics, sec the cnd of Scc. A.1.3. Providcd the coordinates T" arc givcn in Rj, we can dctcrmine dzn and tht. volume 3-form (B.1.2). Thcrc is no nccd to use a lnctric nor a connection, the propertics of a 'bare' differcntial ~nanifoltl(continuum) are sufficient for the definition of ( B . l . l ) . This can also be lccognized as follows: Tlic nct chargc Q in (0.1.1) curl be tlctermined by counting thc chargetl elclnentary particles insidc t3R 3 and adding up their clcmentary electric charges. Nowadays one catches single electrons in traps. Thus the counting of electrons is a11 cxpcriment,ally fcnsihlc procedure, not only a. thought experiment devisetl by a theoretician. This consideration shows that it is not necessary to use a clistancc. concept nor a. length standard in 0 3 for the determination of Q. Only 'counting proccdurcs' arc rcquirctl ant1 a way to delimit an arbitrary finite volume R j of 3-dimensional space by a boundary (303 and to know what is inside aR3 and what outside. Accordingly, p is the prototype of a charge density with nbsolutc tlilncnsioll [p] = q and relative tlinicnsio11 [pabC]= rl P - 3 . It becomes thc convcntional charge densit,y, that is chargc per scnlrd unit volume, if a unit of distance (111 in SI-units) is introduced adclitionally. Then, in SI-units, we have -9 = C 1Tl . Out of the rcgion Rj, crossing its bounding surface i3R,3,thcrc will, in general, flow ;L net elcctric cuncnt, see Fig.B.l.l,

with absolutc dimcnsion qt-I (t = tilnc), whicli must not depend on the orientation of spacc either. Thc int,cgrand 3 , thc twisted electric current density CS-' = ampcre = A, here s tc qt-' 2-form wit,li the same a b s o l ~ ~dimcnsion = sccond, assigns to the area clclncnt 2-for111

4T11c formulation of Mashlloon (301 rcads: " . . . thc hypotliesis of locality-i.e., the p r o sullietl c?quivalenceof an accelerated observer with a ~iiorncntarilyconloving inertial observerunclcrlies the stitndard relativistic f o r ~ n a l i s ~byn relating tlre menqurcrnents of a n accelerntecl observer to tllosc of an inertial observer." T h c two observers are assumed to be otherwise identical. That is, the two obscrvers are copics of one observer: One copy is inertial and the othcr nccelcrated; this is the only difference betwccn them. T h e limitations of the hypothesis of locality are rliscussccl in [31].

3.1.1 Counting charges. Absolute and relative dimension

113

'igure B.1.2: Charge conservation in 4-dimensional spacetime. The 3limensional "cross section" 0 3 sweeps out a 4-.dimensional volume f14 On its Yay from t = t l t o t = t 2 . 1 scalar-valued

charge current

The postulate of electric charge conscrvation requires

xovidcd the area 2-form dxa Adzb is directed in such a way that the outflow is :ounted ~ositivelyin (B.1.7). The time variable t, provisionally i n trodu c ed here, loes not nced to possess scale or a unit. It can be called a "smooth causal time" 11 the sensc of parameterizing a future-directed curve in the spacetime manifold ~ i t tha9. a monotone increasing and sufficiently smooth variable. Substit u ti on of :B.1.1) and (B.1.7) into (B.1.lo) yields an integral form of charge conservation:

By applying the 3-dimensional Stokes theorern, the differential version turns out to be

B. 1. Electric chargc conservation

114

115

3.1.2 Spacetime and the first axiom

Let, us put ( B . l . l l ) into a 4-dimensional form. For this purpose wc integrate (B.1.11) over a certain time interval from tl to t2 (see Fig. B.1.2). Note that this figure depicts the same physical situation as in Fig. B . l . l :

Obviously we are integrating over a 3-dimensional boundary of a compact piece of the 4-dimensional spacetime. If wo int,roclucc in four tliincnsions tlic twisted 3-form J:=-jAdt+p,

with

[J]=q,

(B.1.14)

Figure B.1.3: Spacetime and its (1

+ 3)-foliation.

then the integral can be written as a 4-dimensional boulidary integral,

wlierc Q4 = [ t l lt2] x Q3. The twisted 3-form J of the electric charge-current, tlensity witli absolute dimension q plays the central role as source of the elcctroniagnetic ficltl.

lefinc twisted and ordinary untwisted tensor-valued differential forms. In order o avoid possible violations of causality, we will, as usual, consider oilly non:ompact spacetime manifolds X4. The X4 with the described topological properties is known tjo possess a (1+3)oliation (see Fig. B.1.3); i.e., there exists a set of nonintcrsect,ing 3-di~nensional iypersnrfaces It, that call be parameterized by a monotone increasing (would)e timc) variable a with the clirncnsion of time: [a]= t. Althongh at this stage vc do not introduce any metric on X 4 , it is well known that tlic cxistcncc )f a (1 3)-foliat,ion is closely rclatcd t,o the cxistcncc of pseudo-Rien~:innian tructures. Among the vector fields transverse to the foliation, we choose a vector field z (not to be confiiscd with the dimension n of a manifold discussed in Part A) iormalizcd by tlic condition

+

B.1.2 Spacctime and the first axiom Motivatetl by the integral for111 of charge conservation (B.1.15), we can now turn to an axiolnatic approacli of electlodynamics. First we will forniulatc a set of minimal assumptions tliat we sliall need for clefining an appropriate spacctimc manifold. Let spacetime be given as a 4-dimensional ronnected, Hausdorff, oomentnble, and pnmcompact t1iffcrenti;tblc manifold X 4 . This manifold is "bare," that is, it carries neitlicr a. connection nor a nictric so far. We assume, however, tlic conventional continuity and differelltiability requirements of physics. To recall, a topological space X is Har~stlorffwhcn for any two points $. p2 E X one can fintl ope11 sets pi E U1 c X , pa E U2 c X, such tliat U1 n U2 = Q).An X is connected when any two points can be connected by a continuous curve. Finally, a connected Hausdorff manifold is paracompact whcn X can be covered by a countable number of coordinate cliarts. The (smooth) coorclinatcs in arbitrary cliarts will be called r t ,witli i, 3, k,. . . = 0,1,2,3. The vector basis (frame) of the tangent space will be called e, ailti the 1-form basis (cofrainc) of tlie cotangent space 6" with (anholonomic) illdices a, P, y,. . . = 0,1,2,$. On tlie X4 we call *

.

.

A

Physically, the folia h, of collstalit a represent a simple model of a "3-space," vhile the function a serves as a "time" variable. The vector ficltl n is usually ntcrprctetl as a congruence of observer worldlines. In Sec. E.4.1, this rather ormal mathematical construction becomes a full-fledged physical tool whcn the metric is introduced. Now we arc in a position to formulate our first axiom. We require the existence of a twisted charge-c~~~-rent density 3-form J with the absolutc diincnsioll of SI :hargc q, i.e., [ J ] = q = C which, if integrated over an arbitrary closed 3iimensional submanifold C3 c X4, obeys

f

J=O,

c 3

ac3=0

(first axiom) .

(B.1.17)

116

B.1. Electric charge conservation

117

B.1.4 Timc-space decomposition of tllr inhomogoncolls Maxwcllrquation

We recall, a rnanifold is closed if it is conlpact and has no boundary. In particular, tlic 3-dimcnsional boundary C3 = 804 of an arbitrary Cdimensional region 0 4 is a closcd manifold. However, in general, not every closed 3-manifold is a 3-boundary of some spacctime region. This is the first axiom of elcctrotiynamics. It has a firm phenomenological basis.

9.1.4 Time-space decomposition of the

B.1.3 Electromagnetic excitation H

ziven tlie spacetime foliz~tion,wc can decolnposc any exterior form in "timc" tnd "space" pieces. With respect to the fixctl vector ficld n, normalized hy :B.l.lG), we tlefi~lc,for a p-form {I!, thr part longitudinal to tlie vector n by

Since (B.1.17) holds for an arbitrary 3-dimensional boundary C3, we can choose C3 = dCl4. Then, by Stolliisformalism.

first axiom, then the rank of t,lic clcctric current lnust 1)c 17.- 1. The force tlcnsity in mcclianics, in accordance witli its tlcfinition aLli3.x' within the Lilgriillg~ forn~alisn~, slioultl rcnlain a covoctor-va111od11-form. Hcncc wc kccp t,lic? sccontl axiom in it,s origin;tl forln. Accordingly, thc ficltl strc?ngt,h F is again n 2-forni: N

.f

fO=(r(?~F)AJ,

C., 1

.f

F

=

0.

(B.4.34)

((2

This nlay sccnl liltc an academic exercise. However, a t lcmt for 71 = 3, tlicrc cxists an application: Sincc the niidtlle of tlic lSGOs, cxpcrimcntnlists w ~ r ablc c to create [t ,?-d7rnrns1onol rlrctron gas (2DEG) in suitable transisto~sa t sufficiently low tcinl~c.raturcs ant1 to position the 2DEG in a strong cxtc'rn:tl transversal niagnctic field. Undcr such circ~lmstanccs,the electrons can only move in a planc transvcrsc t o B and one spacc tlimension can bc supprcssctl.

B.4.5 @Electrodynamicsin flatland: 2-dimensional electron gas and quantum Hall effect

Electrodynamics in 1

Our formulation of elcctrodynan~icscan be generalized straightforwardly t o arbitrary di~nensionsn. If we assume again the charge conservation law a5 a "o formulate electrodynamics in acceleratcd systems by means of tensor analysis, sce J. Van Blatlel [51]. New experiments in rotating frames (with ring lavers) can be found in Stcclman [47].

J=0,

+ 2 dimensions

In clcct,rotlyna~nicswit,li onc time ant1 t,wo spacc dimensions, wo have from thc first and third axiolns (in arbitrary coordinates),

1

I

and

R.1. I3asic classical electrodynamics summarizcd, examplr

154

respectively, wliere we indicated the rank of thc forms explicitly for better transparency. Here, thc remarkable featurc is tliat field strength F and current J carry thc same rank; this is only possible for n = 3 spacctime dimensions. Moreovcr, the c~irrentJ, the excitation IT, and the field strength F all liavc thc snmc number of indcpcndent components, namely threc. Now wc (1 2)-dccomposc the current and the electromagnetic field:

+

twisted 2-form: twisted 1-form:

(2)

J

=

-j A

d o + p,

H = -'FI d a + V , (1)

(B.4.37) (B.4.38)

Accordingly, in the space of the 2DEG, wc 11:tvc (again in arbitrary coordinates)

We recognize thc rather degcncrate nature of such a system. The magnetic field B , for example, 11213 only one intleprndent colnponcnt B I Z .Such a configuration is visualized in Fig. B.4.2 by using rcctilincar coordinates; they need not bc Cartesian coordinatcs. Chargr conservation (B.4.35) in dccomposrd form and in components rcads,

+

The (1 2)-dcconiposcd Maxwcll cquations look exactly as in (B.4.9) and (13.4.10). Wr also C X P ~ C S S them in componcnt,~:

R.4.5 @Electrodynamics in flatland: 2DEG and QHE

155

Maxwell equations (B.4.44) to (B.4.48) would still be valid since they had been derived for arbitrary (curvilinear) coordinates and arbitrary frames. Before we can apply this formalism to the quantum Hall effect (QHE), we irst remind ourselves of the classical Hall effect (of 1879).

Hall effect (excerpt from the l i t e r a t ~ r e conventional ,~ formalism) rhis subsection on the Hall effect and the next one on QHE are excerpts from ;he literature. Thus equations (B.4.49) to (B.4.56) are written in the conven;ional notation, and they refer to Cartesian coordinates and may contain the metric of 3-dimensional space. However, our subsequent presentation of QHE, which starts with (B.4.57) and which contains all relevant features of a phenomenological description of QHE, is strictly metricfree. We connect the two yz-faces of a (semi)conducting plate of volume I, x 1, x 1, with a battery (see Fig. B.4.3). A current I will flow and in the plate the current density is j,. Transverse to the current, between the contacts P and Q, there exists no voltage. However, if we apply a constant magnetic field B along the z-axis, then the current j is deflected by tlie Lorentz force and the Hall voltage Uw occurs which, according to experiment, turns out to be

BI

i3

(B.4.49) with [AH]= - . UII = R H I = AH - , 1, (I RIj is called the Hall resistance and All the Hall constant. We divide UH by 1,. Because E, = UH/ly, we find

Lct us stress that the classical Hall effect is a volume (or bulk) effect. It is to be dcscribetl in the frameworl< of ordinary (1 3)-dimensional Maxwellian electrodynamics.

+

Quantum Hall effect (excerpt from the literat~re,~ conventional formalism) and

Wc assume an infinitr cxtcnsion of flatland. If tliat cannot be assumed as a valid npproximat,ion, onc has to allow for linc currents a t tlie boundary of flatlancl ("cdgc curr~nts")in ordcr to fulfill t,he Maxwell cquations. In our formulation thc Maxwell cquations don't tlepcnd on the metric. Thus, instead of thc planc, as in Fig. B.4.2, we coultl havc drawn an arbztranj 2din~cnsionnlmanifold, a surfacc of a cylinder or of a sphere, for example; the

A prerequisite for the discovery in 1980 of QHE were the advances in transistor technology. Since the 1960s one was able to assemble Zdimensional electron gas laycrs in certain types of transistors, such as in a metal-oxide-semiconductor (see a schematic view of a Mosfet in Fig. B.4.4). field effcct transistor"Mosfet) Thc electron laycr is only about 50 nanometers thick, whereas its lateral extension may go up to the millimeter region. 4See Landau & Lifshitz (251, pp. 96-98 or Raith [39],p. 502. "ee, for example, von Klitzing [52],Braun [3], Chakraborty and Pietilainen [4], Janssen et al. [20],Yoshioka [54],and references given there. ' A fairly detailed description can be found in Raith [39],pp. 579-582.

156

B.4. Basic classical elcctroclynamics summarized, example

B.4.5 @Electrod y namicsin flatland: 2DEG and QME

157

Figure B.4.3: Hall effcct (schematic): The current density j in tlic collductilig platc is affected by tlic exterlial constant magnctic field l3 (in the figure only sy~nbolizcdby one arrow) such as t o crcatc the Hall voltage UH.

Figure. B.4.5: Sclicmatic view of a quanttun Hall cxpcriment witli a 2dimcnsionnl electron gas (2DEG). Tlie current density gT in tlic 2DEG is cxposed to a strong trarisversc ~iiagncticfield B,,,. Tlic Hall voltage Url can I)c mcasr~rcdin the tralisvrrse direction t,o j , betwccli P ant1 Q. In t l ~ cinset wc clcnoted the longitudinal voltage witli U, and tlie transversal onc witli U,(= UIl).

In the quantum Hall reainic, wc liavc very low temperatrires (between 25 mK ant1 500 mK) and vcry high magtffnetic fields (between 5 T and 15 T ) . Then thc conducting clcctrous of tlie spccirncn, because of a (quantum mechanical) excitation gap, cannot lnovc in the z-direction, thcy are confined t o the zyplanc. Thus an almost idcal %dimensional electron gas (2DEG) is constituted. Tlic Hall coliductance (= llresistance) exhibits very well-defined plateaus a t integral (and, in tlic fractional QHE, a t rational) multiples of tlic funtlamcntal SI cond~~ctance' of e2/11 1/(25 812.807 R), where e is the clemelitary charge and 11 Plallck's constant. Thcrcforc this effcct is instrumental in precision expcrimcnts for measuring, in conjuliction with the Josephson effect, e and 11 vcry accurately. We concentrate here on the integer QHE. nlrliing to Fig. B.4.5, wc colisider thc rectangle in thc ~ y - p l a n ewitli side lcngtlis 1, and l,, rcspcctively. The qualltulli Hall effect is observed in such a

'

Figure 13.4.4: A Mosfct witli a 2-dimclisional clcctron gas (2DEG) laycr between a semiconductor (Si) ancl an insulator (Si02). Adaptcd from Brauli [3].111 1980, with such a transistor, vo11 Ilc,wr citn transvrct its coniponcnts wit11 kCl,. Accordingly, we dcfinc t hr "cliargc" clcnsity

whicli should not be 11iixcd I I with ~ tlic notion of an elrct~xcchargt~.111 fiiit spacctinie, fol example, with t as tilnc coordinate, a suitablc (Killing) vcctol worild 1)c ( = 8, and Q = ["'(C, = k C t would signify tlie enrrgy tlrrisity. Generally, thc integrals J d Q ant1 Q arc wc.11 dcfinc.d.

r3.5. Electromagnetic energy-moment~imcurrent and action

1 GG

B.5.2 Energy-momentum current, electric/magnetic reciprocity

Lct us now comput,c d Q . Wc transvect (B.5.6) with the vector components I n . Then,

167

which amounts to one equation. This property - the vanishing of the "trace" of kC, - is connected with the fact that the electromagnetic field (the "photon") carries no mass and the theory is thus invariant under dilations. Why we call it the trace of the energy-momentum will become clear below (see (B.5.38)).

kC, is electriclmagnetic reciprocal F'urthermore, we can observe another property of kC,. It is remarkable how symmctric H and F enter (B.5.7). This was achieved by our choice of a = 1. The cnergy-momcntum current is electric/magnetic reciprocal; i.e., it remains invariant under the transformation

If wc evalrlatc the last term by subst,itr~ting(B.5.8), we have to remember the rule for the multiplication of a vcctor by a scalar in a Lie derivative (scc Sec. A.2.10): LftLw= f L l L w + d f A ( u - 1 ~ ) .

(B.5.16) with the twisted 0-form (pseudoscalar function) C = ((x) of dimension [C] = [H]/[F]= q 2 / l ~= llresistance. It should be stressed that in spite of kC, being electriclmagnetic reciprocal, Maxwell's equations are not,

WC sr~bstitutef = t", u = em,w = H and reorder: . Aftcr the coframe statement specifying the appropriate value for the coframe, we defincd a suitable 1-form: I

pform wavetocoframel(a)=l$

((3.2.105)

X

The11 the cof'lfferential operator clt can be introduced as an adjoint to thc cxtcrior diffcrc1lt,iald with rcspcct to the scalar product (C.2.105), ( w , d t p ) := (dw, 9 ) .

(C.2.106)

By constrructicnlthe codiffercntial maps p-forms into (p - 1)-forms (contrary to cxteriordifferenti,zlwhich increases the rank of a form by one). Using the properN (C.2.\0)of the Hodge operator, one can verify that in an n-dimensional ~ o r e n t z i "'Pace ~ the codifferential o n p-forms is given explicitly by

The emerging expression we had to treat with switchcs and suitable substitutions, but the quite messy computation of the wave operator was givcn to t,hc machine.

Let a connection an,d a metric be defined independently on the same spacetime manifold. Then the nonmetricity is a m.easure of the incompatibility between metric and connection.

C.2.12 l'ost-Riemannian pieces of the connection @

Lct us consitlcr the general case when on a rnanifoltl X,, tlie metric and connection are defined independently. Such a manifold is denotctl (X,, V, g) ancl called a metric-a@ne spacetime. Sincc the mctric is a tensor field of type its covariant tlifferentiation yields a typc tensor ficltl tliat is called the nonmetricity:

I![

233

The factor l / r z is conventional. Then the nonlnetricity is decomposed into its deviator and its trace according to

['!I,

Thc trace of the curvature, which is called tlie segmental curvature, can be exprcsscd in terms of the Weyl l-form: n R,Y = - dQ. (C.2.119) 2 Thc pliysical importance of thc Wcyl l-form is related to the fact that, during parallel transport, the contribution of the Weyl l-form does not affect the light cone, whereas lengths of non-null vectors are merely scaled with some (pathdependent) factor. A space with gap = 0 is called a Weyl-Cartan, manifold Y,,. In this latter case, tlic position vcctor (C.1.GO) changes according to

for all vcctor fields u, v , ~Nonmetricity . rncaslires the extent to which a conncction V is incomptltible witli the mctric g. Mct,ric conlpatibility (also called mctricity) Q(71,V , w) = 0 implies the conservation of lengths and angles under parallel transport. A manifold that is endowed witli a mctric and a metriccompatible connrction is said to bc a Riemann-Cartan manifold (or a U,). In general, tlie Riemann-Cartan manifold has a nonvanisliing torsion. When the latter is zero, we recover the Riemannian manifold clescribcd in Scc. (3.2.9. Lilte the 2-forms of torsion (C.1.38), (C.1.40) and curvatlire (C.1.45), (C.1.46), we define the nonmetricity l-form

1 r e - R[Pn]rp), (C.2.120) n if it is Cnrtan displaced ovcr a closet1 loop that encircles tlic area clcment AS. T h r first cllrvature t,crm intluces a tliltttion, while the second onc is a purr rotation. A r e = A S (T" -

-R,T

C.2.12 "Post-Riemannian pieces of the connection Since ?~(g,lj)= dgnP (IL), equation (C.2.111) is cquivalcnt to

An arbitrary linear connection can, always be split into the LeviCivita connection plus a post-Riemannian tensorial piece called the distortion. The latter depends on torsion and nonmetricity. Correspondingl?y, nll the geometric objects and operators can be system.ntically decomposecl in-to Riem~n~nian and postRiemannian parts.

wllcre I',p := gp,T',,Y. If t,hc g , ~a" constants, then it follows from (C.2.114) that Q,,p = 2r(,p). Hcncc, in a U,,, where Q,o = 0, we have ~iit~isymmet~ric connection olic-forms

" . . . the qllestion whethcr this [spacctimc]continuum is Euclidean or structured according t,o the Riemannian scheme or still otherwise is a genuine physical question which has to be answered by cxpcricnce rather than being a mere convention to be chosen on the I~asisof c x p ~ d i c n c y . " ~

provided tho g,p arc constants, for example, witli respect to orthonormal coframc fields. We sha11 refer to (C.2.114) as the 0th Cartan struct?~ral relation and shall call the expression obtained as its exterior dcrivntivc,

tlic 0th B~nnchz~dentzfy.The proof of (C.2.116) makes use of the Ricci identity (C.1.65). It is convenient to separate the trace part of the nonmctricity from its traceless piece. Let us define the Weyl I-form (or Weyl cov~ctor)by

I

Thc gcomet,rical properties of an arbitrary metric-affine spacetime arc described by tlie 2-foniis of curvature R e P and torsion T" and by the l-form of nonmctricity Qap. Particular valucs of these funclamental objects specify different gconietrics which may bc rcalizetl on a spacetime manifold. Physically, one can think of a number of "phase transitions" tliat the spacctilne geometry undergoes at different energies (or distance scales). Correspondingly, it is co~ivcnicntt,o study particular realizations of geonietrical structures within the frillncwork of several specific gravitational models. The overview of these models and of tlic relevant geolnetries is given in Fig. C.2.2. 4 A . Einstein: Geo~netrieund Erfahriing [6], our translation.

C.2. Metric

234

-- R=O -T=O )

C.2.12 @Post-Riemannian pieces o f the connection

235

Furthermore, switching off the nonmetricity completely, one recovers the Riemann-Cartan geometry Uq which is the arena of Poincark gauge (PG) gravity in which the spin current of matter, besides its energy-momentum current, is an additional source of the gravitational field. The Riemannian geometry Vq (with Q,p = 0 and Ta = 0) describes, via Einstein's General Relativity (GR), gravitational effects on a macroscopic scale when the energy-momentum current is the only source of gravity. Finally, when the curvature is zero, Rap = 0, one obtains the Weitzenbock space P4 and the teleparallelism theory of gravity (when Q,p = 0) or a generalized teleparallelism theory in the spacetime with The Minkowski spacetime M4 with nontrivial torsion and nonmetricity vanishing Qap = 0, Ta = 0, Rap = 0 underlies Special Relativity (SR) theory. The relations between the different theories and geometries are given in Fig. C.2.2 by means of arrows of different types that specify which object is switched off. In a metric-affine space, curvature, torsion, and nonmetricity satisfy the three Bianchi identities (C.2.116), (C.1.67), and ((2.1.69):

~4f.

DQap

r 2R(ap),

DT" r Rp" D R , ~ = 0,

A dP,

0th Bianchi,

(C.2.121)

1st Bianchi, 2nd Bianchi.

(C.2.122) (C.2.123)

In practical calculations, it is important to know exactly the number of geometrical and physical variables and their algebraic properties (e.g., symmetries, orthogonality relations, etc.). These aspects can be clarified with the help of two types of decompositions. A linear connection can always be decomposed into Riemannian and post-Riemannian parts, Figure C.2.2: MAGic cube: Classification of geo~net~ries and gravit,y thcories in tile threc "dimensions" (R, T, Q).

The most general gravitational model-metric-affine gravity (MAG)-employs tile (L,,g) geometry in which all three main objects, curvaturc, torsion, and nonmetricity are nontrivial. Such a geometry could be rcalized a t extremely snia11 distances (high energies) when the hypermomcntum current of matter fields plays a central role. Other gravitational models and the relevant geometries appear as special ceses when one or several main geometrical objects are complctcly or partly "switched off". The Z4 geometry is characterized by T" = 0 and was used in tlte unified field theory of Eddington and in so-called SKY-gravity (theories of Stephe~lson-Kilmister-Yang). Switching off the traceless nonmetricity, gap = 0, yields the Weyl-Cartan space Y4 (with torsion) or standard Wcyl theory W4 (with Ta = 0).

where the distortion 1-form Nap is expressed in terms of torsion and nonmetricity as follows:

The distortion "measures" a deviation of a particular geometry from the purely Riemannian one. As a by-product of the decomposition ((2.2.124) we verify that a metric-compatible connection without torsion is unique: It is the LeviCivita connection. Nonmetricity and torsion can easily be recovered from the distortion, namely

If we collect the information we have on the splitting of a connection into Riemannian and non-Riemannian pieces, then, in terms of the metric gap, the coframe d", the anholonomity C", the torsion T a , and the nonmetricity Q,p,

C.2. Metric

236

C.2.13 @Excalcagain

wc have the highly symmetric master formula

237

I

coframe o(O)= . . . ; % input 1 frame e; riemannconx chrisl; chrisl (a,b) :=chrisl (b,a) ; pform connl(a,b)=l, distorl(a,b)=l, nonmetl(a,b)=l$ connl(O,O):= . . . ; % input 2 distorl(a,b):=connl(a,b)-chrisl(a,b); nonmetl(a,b):=distorl(a,b)+distorl(b,a);

Thc first linc of this formula refers to the Riemannian part of the connection. Together with the second linc a Ricmann-Cartan geometry is encompassed. Only the third line makcs the connection a really independent quantity. Note that locally the torsion T, can be mimicked by the negative of the anholonomity -C, and the nonmetricity Qep by the exterior derivative dg,p of the metric. In a 4-dimensional metric-affine space, the curvature has 96 components, torsion 24, and nonmetricity 40. In order to make the work with all thcsc variables manageable, onc usually decomposes all geometrical quantities into irrcduciblc pirccs with respect to thc Lorentz group.

and one coulcl continue in this linc and compute torsion :inel curvature according to pf orm torsion2 (a)=2, curv2(a, b) =2;

All other relevant geometrical quantities can be derived therefrom.

C.2.13 @Excalcagain

Problems

r,P

Excalc has ;I cornmoclity: If a coframe o(a) and a metric g are prescribed, it cdculates thc Riemannian picce of the connection on demand. One just has to issue the command riemannconx chrisi; Then chrisl (one could take any ot,hcr namc) is, without further dcclaration, a 1-form with the index structure chrisi(a,b). Since we use Schouten's conventions in this book, we have to redefine Schriifcr's ricmannconx according to chrisl ( a , -b) : =chrisl (-b, a) ; If you distrust Excalc, you could also compute your own Riemannian connection according to (C.2.127), i.c.,

Problem C.1. I I

Problem C.2. Prove ((3.1.52) in thc infinitesimal case, approximating thc curve parallelogram.

r

pf orm anhol2(a)=2,christl(a,b)=l$

I anhol2(a) : = d o(a)$ christl(-a,-b) := (1/2)*d g(-a,+) +(1/2)*((e(-a)-1 ( d g(-b,-c)))-(e(-b)-l (d g(-a,-c))))^o(c) +(1/2) * ( e(-a)- 1 anhol2(-b) - e(-b)- 1 anhol%(-a)) -(1/2)*( e(-a)-1 (e(-b)-lanhol2(-c)))^o(c)$

Rut wc can asslirc you that Excalc does its job correctly. In any casc, with chrisi(a,b) or with christl(a,b) you can equally well compute the clistortion 1-form anel the nonmetricity 1-form in terms of coframc o(a), metric g, and conncction conni(a, b). The calculation would run a? follows:

Check the geometrical interpretation of torsion given in Fig. C.1.1 by clircct, calcrllation using thc definition of the parallel transport.

(7

by a small

Problem C.3. Prove the following relations involving the transposed connection:

i

-

,-.

1. T" = -To, that is, T" = -D29";

I

4. The covariant Lie derivative of an arbitrary pform 9 = @,,.,, . . . A dmr1lp!:

79"l

A

1. Find a transformnt,ion matrix from a11 orthonormal basis t o tlie half-null franic in which the mct,ric has the form ((2.2.11). 2. Fintl a transfornlatioll matrix from an orthonormal lxc~ist o a. NewnlnriPenrosc null framc in which the metric has the form ((3.2.13). 3. Find a linear transformation e, = Lap f p that brings tlie null symmetric Coll-XiIornles basis f p back t o an orthonormal frame e,. Solution,:

JZ

0

-JZ

0

1

1 1 4

1

Checlnl nr~llsymmetric frnmc - - -f,_N,ote that the 3-vcctors OA, O B , etc. t o tllc planes B C D , C D A , etc. arc pcrpcntlic~ilr~r 4. Show that in 4-dimcnsional space (a) 79" A qp = 6; q , (1)) dl' A qp, = (5; ~1p- 6; q, , (c) 79" A rlprs = 6; r/pr 4- 6; qsp 6; 71-,6 , ('1) 7 9 " 7)/j'-y(i11 = 6;; 7 1 - ~6; r/pyl~ ~ ~ + 6; 11/36/~- 'J; 'ITS/' .

+

Hint:19"

A

71

= 0 because it is x 5-form in n 4-dilncnsional spttcc.

1. Prove that for 9 , $ E AI'V*, *J,A1//= * $ A $ , 1. Prove that Chri~t~offcl symbols dcfinc n covnriant diffcrcntiation by clieclr recover tlie rigid (global) Poincard group unlrss the mctric is h4inkowski;ui. In our whole axiomatics in Part B alid Part D we did not use tlic Poinc;ti6 group nor its G-paramrtcr Lorrntz subgroup at rill. In tlir conventional app~oacli to clcctrotlynamics, tlic Poincnr6 group is an cssclitial ingrctliclit for cooltil~g 1111 the formalism of clcctrotlynamics. I11 thr general covariant approacli, witli electric charge and magnetic flux conservation as its basis, which wc followc~d, the mctric is tlist,illctl from a linear spacctime relation witli reciprocity ant1 symmetry as adtlitives. The mctric, and thus the gravitational potcwtial, is a derivccl conccpt. The mctric gets its meaning from clcctrodynamics; it is not i) fundamental field.

304

D.6. Fifth axiom: Maxwell-Lorentz spacetime relation

D.6.3 @Extensions.Dissolving Lorentz invariance? We formulated our deductions of the Maxwell-Lorentz spacetime relation already in such a way that the possible generalizations of the Maxwell-Lorentz framework are apparent: Have a dilaton, have an axion, have a skewon, either one of them, some of them, or all of them. But all of these possible generalizations to dilaton electrodynamics, axion(-dilaton) electrodynamics, and so on take place in the framework of the linear spacetime relation (D.G.1) cum (D.6.2). This linearity gave birth to the axion, the skcwon, and an unstructured prin/- (1) xk / %. ~The subsequent assumption of reciprocity and cipal part ( l ) X ~ 3 , symmetry created the metric g of spacetime. Had we given up linearity, that is, had we allowed nonlinearity at a fundamental level in the spacetime relation (D.G.l), thcn thc metric and their siblings axion and skewon would bc irretrievably lost. In fact, we arc not aware of any attempt to construct a nonlinear electrodynamic model a t this fundamental level, and we see no way of doing so. However, if we take the mctric for granted, i.e., fundamental linearity, reciprocity, and symmetry, then, with the help of the metric, one can construct nonlocal and nonlinear nlodels that, nevertheless, are based on fundamental linparity b la (D.6.2). Presupposing a metric of spacetime implies a linear spacetime relation willy-nilly. To fix this linear relation later by some decorative nonlinear and nonlocal terms doesn't change the underlying rationale. For this reason, we discuss these nonlocal ancl nonlinear models, which respect the linear spacetime relation in the sense described, in Chap. E.2. Let us come back to the possible extensions of the Maxwell-Lorentz electrodynamics by means of introducing dilaton, axion, and/or skewon fields. Therc are two distinct cla.~ses:Those that respect the light cone (one could call them the harmless extensions) and those that don't. To the former belong axion, dilaton, ant1 axion-dilaton electrodynamics, to the latter skewon electrodynamics (that is, with $ # 0) with possible admixtures of axion and/or dilaton fields. As soon as one mixes a skewon field into the spacetime relation (see (D.6.5))) one cannot recover the light cone any longer and the Lorentz group is dissolved. Several such attempts arc discussed in the l i t e r a t ~ r e If. ~one wants t o look for possible new physics violating Lorentz invariance, then the assumption of a skcwon appears to be tlle most natural possibility. As a first attempt one could try to simplify (D.6.5) by brute force to

D.6.3 @Extensions.Dissolving Lorentz invariance?

305

Moreover, recently, there was some observational evidence from astronomy that Sommerfeld's fine structure constant may depend on (cosmic) time:

(e = elementary charge, h = reduced Planck constant). Even though these observations did not find independent support, the posed question may be interesting for future. Provided that we want to uphold the pre-metric Maxwellian structure (B.4.1), then, in the light of (B.4.2), only EOC could be time dependent: The speed of light would not be necessarily constant 6 We saw that EOC codifies a response of spacetime to the propagation of electromagnetic disturbances. Dirnensionwise, we have [ E ~ c=] l/resistance, that is, it represents an impedance of spacetime ("vacuum"). It is conceivable that such a "vacuum" impedance picks up some time dependence or, more generally, becomes a field dependig on time and space. Such an approach would dismantle local Lorentz invariance and thereby general relativity too. s

and then investigate the influence of $ on light propag a t'lon. "ee,

for instance, KostelcckJ; [25]

"his

is the main result of

a recent article of Peres [40]. Here we partly follow his arguments.

References

[ I ] A.O. Barut and R. Rgczka, Tlzeory of Group Representations and Applications ( P W N - Polish Scientific Publishers: Warsaw, 1977). [2] J.D. Bekenstein, Fine-stm~ctureconstant ~~ariability, equivalence principle and cosmology, Eprint Archive gr-qc/0208081, 18 pagcs (August 2002). [3] C.H. Brans, Complex 2-forms representation of the Einstein, equations: The Petrov Type 111 solutions, J. Math. P11,y.s. 12 (1971) 1616-1619. (41 C.H. Brans and R.H. Dicke, Mach's prin.ciple and a relativistic t h e o ~ ~ofj gravitation, Pti-ys. Rev. 124 (1961) 925-935. [5] R. Capovilla, T . Jacobson, and J . Dell, General relativity witl~outthe metric, Phys. Rev. Lett. 63 (1989) 2325-2328. [6] L. Cooper and G.E. Stedman, Axion detection by ring lasers, Phys. Lett. B357 (1995) 464-468. [7] R.H. Dicke, The theoretical significance of experimental relativity, (Gordon and Breach: New Y o r k , 1964). [8] B. Fauser, Projective relativit?~:Present status and outlook, Gcn. Relat. Grnv. 33 (2001) 875-887. (91 G.B. Field and S.M. Carroll, Cosmological magnetic fields from primordial helicity, P11.y.s. Rev. D62 (2000) 103008, 5 pages.

308

Part D. The Maxwell-Lorentz Spacetime Relation

[lo] J.L. Flowers and B.W. Petley, Progress in our knowledge of the fundamental constanh in physics, R,ep. Progr. Phys. 64 (2001) 1191-1246. [ll] F.R. Gantmachcr, Matritenrechnung. Tcil I: Allgemeine Theorie (VEB Deutschcr Vcrlag der Wisscnschaften: Berlin, 1958). [12] A. Gross and G.F. Rubilar, On the derivation of the spacetime metric from linear electrodynamics, Pl~ys.Lett. A285 (2001) 267-272. [13] J. Hadanlard, L e ~ o n ssur la propagation des ondes et les e'quations de 1'lz?jdrod~jnaml:q~~e (Hermann: Paris, 1903). [14] G. Harnett, Metrics and dual operators, J. Math. Pliys. 32 (1991) 84-91. [15] G. Harnett, TIze hivector Clifford algebra and the geometry of Hodge dual operators, J. Pl;ys. A25 (1992) 5649-5662. [16] M. Haugan ancl C. Liimmerzahl, On the experimental foundations of the Maxwell equations, Ann. Physik (Leipzig) 9 (2000) Special Issue, SI-119SI-124. [I71 F.W. Hchl, Yu.N. Obukliov, G.F. Rubilar, On a possible new type of a T odd skeuron field linked to electromagnetism. In: Developments in Mathematical and Experimental Ph.ysics, A. Macias, F. Uribe, and E. Diaz, eds. Voluinc A: Cosmology ancl Gravitation (Kluwer Academic/Plenum Publishers: New York, 2002) pp.241-256. [18] G. 't I-Iooft, A clriral alternative to the vierbein field in general relativity, Nl~cl.Pliys. B357 (1991) 211-221. [I91 K. Huang, Quarks, Leptons Ed Gauge Fields, 2nd ed. (World Scientific: Singapore, 1992). [20] C.J. Isham, Abdl~sSalam, and J . Strathdee, Broken chiral and conformal sym.metry in an effective-Lagrangian formalism, Phys.Rev. D2 (1970) 685690. [21] A.Z. Jadczyk, Electromagnetic permeability of the vacuum and light-cone structure, B1111. Acad. Pol. Sci., Sdr. sci. pli,ys, et astr. 27 (1979) 91-94. [22] P. Jordan, Schwerkraft und Weltall, 2nd ed. (Vieweg: Braunschweig, 1955). [23] R.M. Kielin, G.P. Kichn, and J.B. Roberds, Parity and time-reversal symmetry breaking, singular solutions, and Fresnel surfaces, Phys. Rev. A43 (1991) 5665-5671. [24] E.W. Iersurfacc V. The resulting local coordinates (7,lishcd in the previous scction. To begin with, recall of how tlie excitation ant1 the field strength 2-forms dccompose wit,ll respect to the the laborutoql frame,

ant1 a~~alogorisly, with rcspcct to the matem:al frame: Here (e6,e,) is tlie frame dual to the adapted laboratory coframe ( d a , b a ) , i.c., ec = n,ea = 8,. When the rclativc 3-vclocity is zero v" = 0, the material and the laboratory folintioiis coincide bccause the corresponding foliation I-forms turn out to be proportional to each otlicr v = (c/N) n. Substituting (E.4.7) into (E.4.G), we find for the line element in terms of the new variables

Clcarly, we preserve the same sylnbols If and d 011 tllc left-liantl sidcs of (E.4.14) ant1 (E.4.15) because tlicsc arc just the samc physical objects. In contrast, tlic right-hand sidcs arc of course tliffcrent, hence we usc prirncs. The constit,utivc law (E.4.13), ;iccortling to the rcs~iltsof the prcvior~ssection, can hc rewritten as

cab

Comparing this with (E.4.6), wc recognize that the transition from a laboratory frarne to a moving material frame cliangcs the form of the line element from (E.4.G) t,o (E.4.11). Consequently, this transitioli corresponds to a linear hoinogcrleous transformation tliat is aiiholonornic in general. It is not a Lorcntz transformation which, by definition, preserves the forin of the metric coefficients. The metric of the material foliation has the inverse

For its determinant one finds (dctFab)= (det gab) y-2.

Rcrc the Hotlgc star ;corrcsponds to tlic metric of the matenal foliation. (Plcasc do not mix it up with tlic Hotlgc star 5 clcfincd by the 3-space nictric ( ' ' ) g , , ~ lof tllc l i \ l ) ~ r ~ tfoliation.) ~~.y Now, (E.4.1G) can be prcsentctl in t l l ~ cquivalcnt matrix form

The coniponcnt,s of the constitutivc matrices read explicitly

1C.4.2 Elrctrornagnctic firltl in laboratory and material frames

In order to find the constitutive law in the laboratory frame, we have to perform some very straightforward mallipulations in matrix algebra along the lines described in Sec. D.5.4. Given is the linear transformation of the coframes (E.4.7). The corresponcling trnnsformation of t,hc 2-form basis (A.1.95) turns out to be

We use these results in (D.5.27)-(D.5.30). Then, after a lengthy matrix computation, we obtain from (E.4.18) the constitutive matrices in the laboratory foliation:

The resulting constitutive law

359

3-velocity that enter (E.4.27). Direct inspection shows that the constitutive law (E.4.25), (E.4.26) of above can alternatively be recast into the pair of equations:

These are the famous Minkowski relations for the electromagnetic field in a moving meclium.' Originally, the constitutive relations (E.4.28), (E.4.29) were derived by Minkowski with the help of the Lorentz transformations for the case of a flat spacetime and a uniformly moving medium. We stress, however, that the Lorentz group never entered the scene in our derivation above. This dcmonstratcs (contrary to the traditional view) that the role and the value of the Lorentz invariance in electrodynamics should not be overestimated. The constitutive law (E.4.25), (E.4.26) or, equivalently, (E.4.28), (E.4.29) describes a moving simple medium on an arbitranj curved background. The influence of the spacetime geometry is manifest in E ~pg, and in which enters the Hodge star operator. In flat Minkowski spacetime in Cartesian coordinates, these quantities reduce to Eg = pg = 1, ( 3 ) g , b = dab. The physical sources of the electric and magnetic excitations 9 and d are free charges and currents, Recalling the definitions (E.3.8) and (E.3.5), we can find the polarization P and the magnetization M that have bound charges and currents as their sources. A direct substitution of (E.4.25) and (E.4.26) into (E.3.8) yields:

can be presented in terms of exterior forms as:

Here we introdr~cedthe 3-velocity 1-form

The 3-velocity vector is deconlposed according to vn e,. If we lower the index v" by means of thc 3-metric (")CJ,~,, we find the covariant components v, of the

Here XE alld X B are the electric and magnetic susceptibilities (E.3.57). When the matter is a t rest, i.e., v = 0, the equations (E.4.30), (E.4.31) reduce t o the rest frame relations (E.3.56). 'See the discussions of various aspects of the electrodyna~nicsof moving media in [3, 6, 20, 30, 31, 351.

354

E.4. Elcctrodynarnics of moving continua.

E.4.4 Electromagnetic field generated in moving continua

E.4.3 Optical metric from the constitutivc law A direct check shows tliat the constitutive matrices (E.4.21)-(E.4.23) satisfy the closure relation (D.3.4), (D.4.13), (D.4.14). Consequently, a metric of Lorcntzian signature is induced by the constitutivc law (E.4.24). The geilcral reconstructioll of a metric from a linear constitutive law is given by (D.5.9). Starting from (E.4.18))we immecliatcly find the induced metric in the material foliation:

Making use of relation (E.4.7) between the foliations and the covariance properties proven in Sec. D.5.4, we find tlie explicit form of the iilduced metric in the laboratory foliation:

Here g,, are the components of the metric tensor of spacetime and uz are tllr covariant components of the 4-velocity of matter (E.4.10). Note that g" uzL, u, = c2, m ~islial.The contravariallt inducecl metric reads:

(2) vacuum

Figure E.4.1: Two regions divided by a surface S.

E.4.4 Electromagnetic field generated in moving continua Such an indr~ccdmctric' ; :JC is usually calletl the optical rnctmc in ordcr to distinguish it from the true spacetimc metric g T 3 .It describes the "drnggiilg of ) . adjective "optical" expresses tlic. the nether" ("Mitfiihrung tles ~ t l i e r s " ~The fact tliat a11 the optical effects in moving matter are determined by the Fresiirl equation (D.2.44), which reduces to the equation for the light cone detcrminctl by the mctric (E.4.33) in the present case. The nontrivial polarization/magnetimtion properties of matter arc manifcst,ly present cvcil when t,lle medium in the laboratory frame is at rest. Let 11s coilsider Minkowski spacctiinc with g,) = diag(c2,-1, -1, -I), for example, ant1 a lnediuin a t rest in it. Then u = dt or, in components, 11' = (1,0,0,O). WP substitute this into (E.4.33) ailcl find the optical metric

E~idently,the vclocit,y of light c is replacctl by c l n , with n as the refractive index of the dielectric and inagnetic media.

Let us collsiclcr an explicit example that demonstrates the power of the generally covariant const,itutivc law. For simplicity, we study electrodynamics in flat M~nl I~ountlaryas a plane S = {x"

01, so t1i;lt thc

tangential vcctors ant1 thc normal 1-foim ore 71 =

al,

72 = 82;

u = d~3 .

We assume tliat the uppcr half-space (corresponding to "x matter moving witli t,hc horizontal velocity v = vl d r l

+ v2 d.r2.

(3.4.45) 0) is filled witli

(E.4.46)

E.4. Elcct,rodynamics of moving continua

E.4.5 Thc experiments of Rontgen and Wilson & Wilson

t x3

Figure E.4.2: Experiment of Rontgen.

Figure E.4.3: Experiment of Wilson & Wilson.

Let us consider the case when the magnetic field is absent in the matter-free region, whereas the electric field is clirectcd towards the boundary:

the nonrelativistic approximation (neglecting terms with v2/c2), the formulas (E.4.49), (E.4.50) describe a solution of the Maxwell equations provided dv = 0. This includes, in particular, the case of the slow uniform rotation of a small disk. The magnetic field generated along the radial direction can be detected by means of a magnetic needle, for example.

Then, from (E.4.37), we have

Wilson and Wilson experiment

Rontgen experiment

In the "dual" case, the electric field is absent in the matter-free region whereas a magnetic field is pointing along the boundary: It is straightforward to verify that, for uniform motion (with constant v), the for~ns

B(2) = B' dz2 A dz3

+ 'B

dz3 A dxl,

E(2)=

0.

(E.4.52)

Then, from (E.4.37), we find

A solution of the Mysubsequently integration tlic eincrging second-order ordinary cliffercntial cquat,ion (ODE). GR is a nonlinear field theory. Nevcrtl~cless,if wc now trvat thc con~hir~c~l case with c1e~tro1nag1~eti~ nnd gravitational fielcls, wc can sort of superimposc~ tlie single solutions bccitrlsc of our coordinntr and frame invariant prcscntatio~~ of electrotlynamics. Wc now havc f # 1, but we still lic cqllivalcnce, 85 llypotl~csisof locality, 112, 348 idcal 2-tlilncnsional clectron gas, 157 colitluct,or, 137 suprrcontluctor, 140 IF, 52 int,cgr;rl, 87

Subject Index

of cxterior n-form, 89 of exterior p-form, 92 of twisted exterior n-form, 89 interior prodrlct, 33 (Excalc), 55 invariants of the elcctromagnetic field, 127 reciprocity transformation, 167 Jacobian determinant, 21, 65 jump conditions, 356 I<err-Newman solution, 379 Kiclin 3-form, 134, 135, 162 I