Foundations of Classical Electrodynamics: Charge, Flux, and Metric

404 Pages·2003·10.824 MB·English

by  Friedrich W. Hehl, Yuri N. Obukhov (auth.)

Progress in Mathematical Physics
Volume 33
Editors-in-Chief
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 ofM aryland, 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
Friedrich W. Hehl
Yuri N. Obukhov
Foundations of
Classical Electrodynamics
Charge, Flux, and Metric
Springer Science+Business Media, LLC
Friedrich W. Hehl Yuri N. Obukhov
Institute for Theoretica1 Physics Institute for Theoretica1 Physics
University of Cologne University of Cologne
50923 Cologne 50923 Cologne
Germany Germany
and and
Department of Physics & Astronomy Department of Theoretica1 Physics
University of Missouri-Columbia Moscow State University
Columbia, MO 65211 117234 Moscow
USA Russia
IJbrary of Congress Cataloging-in-PubHcation 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 978-1-4612-6590-0 ISBN 978-1-4612-0051-2 (eBook)
DOI 10.1007/978-1-4612-0051-2
1. Blectrodynamics-Mathematics. 1. Obukhov, IU. N. (IUrii Nikolaevich) II. Title. III.
Series.
QC631.3.H4S 2003
S37.6-dc21 2003052187
CIP
AMS SubjectC1assitications: 78A2S, 70S20, 78A05, 81VI0, 83C50, 83C22
Printed on acid-free paper.
© 2003 Springer Science+Business Media New York
Origina1ly published by Birkhlluser Boston in 2003
Softcover reprint ofthe hardcover lst edition 2003
Ali rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC),
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
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and Merchandise Marks Act, may accordingly be used free1y by anyone.
ISBN 978-1-4612-6590-0 SPIN 10794392
Reformatted from the authors' files by John Spiegeiman, Abbington, PA.
987 6 5 4 3 2 1
Preface
In this book we display the fundamental structure underlying classical electro
dynamics, 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 conventional (1 + 3)-dimensional form including Maxwell's equations. More
over, they should be familiar with linear algebra and elementary analysis, in
cluding 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), E. 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 equa
tions 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 Structure of Electro
magnetics and of the chapter "Charge and Magnetic Flux" of the encyclopedia
article on classical field theories by C. Truesdell and R. A. Toupin (1960), in
cluding 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
a
courses 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 fiat 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
vi Preface
a good understanding of the structure of electrodynamics as a classical field
theory. In particular, if gravity cannot be neglected, our framework allows for a
smooth and trivial transition to the curved (and contorted) spacetime of general
relativistic field theories. This is by no means a minor merit when one has to
treat magnetic fields of the order of 109 tesla in the neighborhood of a neutron
star where spacetime is appreciably curved.
Mathematically, integrands in the conservation laws are represented by exte
rior differential forms. Therefore exterior calculus is the appropriate language
in which electrodynamics should be spelled out. Accordingly, we exclusively use
this formalism (even in our computer algebra programs which we introduce in
Sec. A.1.12). In Part A, and later in Part C, we try to motivate and to sup
ply the necessary mathematical framework. Readers who are familiar with this
formalism may want to skip these parts. They could start right away with the
physics in Part B and then turn to Part D and Part E.
In Part B four axioms of classical electrodynamics are formulated and the
consequences derived. This general framework has to be completed by a specific
electromagnetic spacetime relation 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 electrodynamics to moving continua, inter alia, which requires a sixth
axiom on the formulation of electrodynamics 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 handwrit
ing can still be seen in numerous places. There are also some contributions to
"our" mathematics from Wojtek Kopczynski (Warsaw University). At Cologne
University in the summer term of 1991, Martin Zirnbauer started to teach the
theoretical electrodynamics course by using the calculus of exterior differential
forms, and he wrote up successively improved notes to his course. One of the
authors (FWH) also taught this course three times, partly based on Zirnbauer's
notes. This influenced our way of presenting electrodynamics (and, we believe,
also his way). We are very grateful to him for many discussions.
There are many colleagues and friends who helped us in critically reading
parts of our book and who made suggestions for improvement or who commu
nicated to us their own ideas on electrodynamics. We are very grateful to all
of them: Carl Brans (New Orleans), Jeff Flowers (Teddington), David Hart
ley (Adelaide), Christian Heinicke (Cologne), Yakov Itin (Jerusalem), Martin
Janssen (Cologne), Gerry Kaiser (Glen Allen, Virginia), R. M. Kiehn (formerly
Houston), Attay Kovetz (Tel Aviv), Claus Lammerzahl (Konstanz/Bremen),
Bahram Mashhoon (Columbia, Missouri), Eckehard Mielke (Mexico City), Wei
Tou Ni (Hsin-chu), E. Jan Post (Los Angeles), Dirk Piitzfeld (Cologne), Guiller
mo Rubilar (Cologne/Concepcion), Yasha Shnir (Cologne), Andrzej Trautman
(Warsaw), Arkady Tseytlin (Columbus, Ohio), Wolfgang Weller (Leipzig), and
others.
We are particularly grateful to the two reviewers of our book, to Jim Nester
(Chung-li) and to an anonym for their numerous good suggestions and for their
painstaking work.
Preface vii
We are very obliged to Uwe Essmann (Stuttgart) and to Gary Glatzmaier
(Santa Cruz, California) for providing beautiful and instructive images. We are
equally grateful to Peter Scherer (Cologne) for his permission to reprint his
three comics on computer algebra.
The collaboration with the Birkhiiuser people, with Gerry Kaiser and Ann
Kostant, was effective and fruitful. We would like to thank Debra Daugherty
(Boston) for improving our English.
Please convey critical remarks to our approach or the discovery of mistakes
by surface or electronic mail ([email protected], [email protected]) or
by fax [phone].
This project has been supported by the Alexander von Humboldt Foundation
(Bonn), the German Academic Exchange Service DAAD, and the Volkswagen
Foundation (Hanover). We are very grateful for the unbureaucratic help of these
institutions.
Friedrich W. Hehl
Cologne
Yuri N. Obukhov
Moscow
April 2003
Contents
Preface v
Introduction 1
Five plus one axioms . . . . . . . . . . . . . . . . 1
Topological approach . . . . . . . . . . . . . . . . 3
Electromagnetic spacetime relation as fifth axiom 4
Electrodynamics in matter and the sixth axiom . 5
List of axioms . . . . . . . . . . . . . . . . . . . . 5
A reminder: Electrodynamics in 3-dimensional Euclidean vector calculus 5
On the literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7
References 11
A Mathematics: Some Exterior Calculus 17
Why exterior differential forms? 19
A.1 Algebra 23
A.I.1 A real vector space and its dual .......... . 23
A.1.2 Tensors of type [~] ................. . 25
A.1.3 ® A generalization of tensors: geometric quantities. 27
A.I.4 Almost complex structure 29
A.I.5 Exterior p-forms .................. . 29
x Contents
A.1.6 Exterior multiplication. . . . . . . . . . . . . . . . . . . . . 30
A.l.7 Interior multiplication of a vector with a form . . . . . . . . 33
A.1.8 ®Volume elements on a vector space, densities, orientation. 34
A.1.9 ®Levi-Civita symbols and generalized Kronecker deltas. 36
A.1.l0 The space M6 of two-forms in four dimensions 40
A.l.ll Almost complex structure on M6 43
A.1.l2 Computer algebra ........ 45
A.2 Exterior calculus 57
A.2.l ®Differentiable manifolds ...... 57
A.2.2 Vector fields . . . . . . . . . . . . . . 61
A.2.3 One-form fields, differential p-forms . 62
A.2.4 Pictures of vectors and one-forms. 63
A.2.5 ®Volume forms and orient ability 64
A.2.6 ®Twisted forms . . . 66
A.2.7 Exterior derivative . . . . . . . . 67
A.2.8 Frame and coframe . . . . . . . . 70
A.2.9 ®Maps of manifolds: push-forward and pull-back 71
A.2.l0 ®Lie derivative . . . . . . . . . . . . . . . . . . . 73
A.2.11 Excalc, a Reduce package . . . . . . . . . . . . . 78
A.2.l2 ®Closed and exact forms, de Rham cohomology groups 83
A.3 Integration on a manifold 87
A.3.l Integration of O-forms and orient ability of a manifold. 87
A.3.2 Integration of n-forms . . . . . . . . . . 88
A.3.3 ®Integration of p-forms with 0 < p < n . 89
A.3.4 Stokes' theorem. . . . 93
A.3.5 ®De Rham's theorems 96
References 103
B Axioms of Classical Electrodynamics 107
B.1 Electric charge conservation 109
B.1.l Counting charges. Absolute and relative dimension . 109
B.1.2 Spacetime and the first axiom. . . . . . . . . . . . . 114
B.1.3 Electromagnetic excitation H . . . . . . . . . . . . . 116
B.1.4 Time-space decomposition of the inhomogeneous Maxwell
equation . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 117
B.2 Lorentz force density 121
B.2.l Electromagnetic field strength F . . . . . . . . . . . . . 121
B.2.2 Second axiom relating mechanics and electrodynamics . 123
B.2.3 ®The first three invariants of the electromagnetic field . 126
Contents xi
B.3 Magnetic flux conservation 129
B.3.1 Third axiom. . . . . . . . . . . . . . . . . . . 129
B.3.2 Electromagnetic potential . . . . . . . . . . . 132
B.3.3 ®A belian Chern-Simons and Kiehn 3-forms . 134
B.3.4 Measuring the excitation. . . . . . . . . . . . 136
B.4 Basic classical electrodynamics summarized, example 143
B.4.1 Integral version and Maxwell's equations. . . . . . . . . 143
B.4.2 ®Lenz and anti-Lenz rule . . . . . . . . . . . . . . . . . 146
B.4.3 ®J ump conditions for electromagnetic excitation and
field strength . . . . . . . . . . . . . . . . . . . . . . . . 150
B.4.4 Arbitrary local noninertial frame: Maxwell's equations
in components .................. . 151
B.4.5 ®Electrodynamics in flatland: 2DEG and QHE . . . . . 152
B.5 Electromagnetic energy-momentum current and action 163
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 ®A ction . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 177
B.5.5 ®Coupling of the energy-momentum current to the coframe · 180
B.5.6 Maxwell's equations and the energy-momentum current
in Excalc ......................... . · 184
References 187
C More Mathematics 193
C.l Linear connection 195
C.1.1 Covariant differentiation of tensor fields . . . . . . . . .. . 195
C.1.2 Linear connection I-forms . . . . . . . . . . . . . . . . .. . 197
C.1.3 ®Covariant differentiation of a general geometric quantity . 199
C.1.4 Parallel transport. . . . . . . . . . . . . . . . . . . . . .. . 200
C.1.5 ®Torsion and curvature . . . . . . . . . . . . . . . . . .. . 201
C.1.6 ®Cartan's geometric interpretation of torsion and curvature . 205
C.1.7 ®Covariant exterior derivative. . . . . . . . . . . . . . 207
C.1.8 ®The forms o(a), conn1(a,b), torsion2(a), curv2(a,b) . 208
C.2 Metric 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