Special relativity The foundation ofm acroscopic physics W. G. DIXON Cambridge University Press ¢ Digitized by the Internet Archive in 2022 with funding from Kahle/Austin Foundation https://archive.org/details/specialrelativitO000dixo SPECIAL RELATIVITY THE FOUNDATION OF MACROSCOPIC PHYSICS Wawel AEA TTVITY LHEBOUNDATION OF ., MACROSCOPIC PHYSICS W.G. DIXON Fellow of Churchill College, Cambridge CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON - NEW YORK +: MELBOURNE Published by the Syndics of the Cambridge University Press The Pitt Building, Trumpington Street, Cambridge CB2 1RP Bentley House, 200 Euston Road, London NW1 2DB 32. East 57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia © Cambridge University Press 1978 First published 1978 Printed in Great Britain at the University Press, Cambridge Library of Congress Cataloguing in Publication Data Dixon, William Graham, 1940— Special relativity Includes bibliographical references and index 1. Relativity (Physics) I. Title QC173.65.D58 530.1’1 77-83991 ISBN 0 521 21871 3 Contents Preface page ix 1 The physics of space and time 1 Introduction 2 Frames of reference 3 Newtonian conceptions Eoe we 4 Foundations of the special theory of relativity 11 5 The transformation between inertial frames 15 6 The uniformity of space and time 21 7 The invariant measure of interval 25 8 Something for nothing? 30 9 The Principle of Equivalence 32 10 Gravitation as curvature 35 2 Affine spaces in mathematics and physics 1 Introduction 42 2 Basic properties of affine spaces 44 2a Algebraic properties 44 2b Geometric properties 51 3 Fundamental tensors 55 4 Specializations and limits 62 4a Cartesian. tensors 62 4b The Newtonian limit c > co 64 5 Symmetries and alternating tensors 68 5a The bracket notations 68 5b Oriented affine spaces 72 6 Integration and orientation for surfaces in £,, 76 6a Definitions 76 6b Stokes’ Theorem 80 7 Spacetime geometry in special relativity 86 8 Spacetime geometry in Newtonian physics 92 8a Intrinsic theory 92 8b The Newtonian situation as a limiting case 96 vi CONTENTS 3 Foundations of dynamics 1 The free motion of particle systems page 99 1a Conservation laws 99 1b Centroid lines 104 Mass and energy in Newtonian physics 108 2a Three-dimensional formulation 108 2b Four-dimensional formulation 111 Mass and energy in the relativistic theory 113 3a The relativistic unification of mass and energy 113 3b The relativistic distinction between mass and energy 115 3c The non-relativistic limit 118 The free motion of a continuum 121 4a Consequences of momentum conservation 121 4b Consequences of angular momentum conservation 124 4c The symmetric energy-momentum tensor 126 Spacetime decomposition of 7'%4 128 Three-dimensional interpretation of 7'*/ 135 for) Energy in Newtonian continuum mechanics 140 7a The Newtonian stress—energy tensor 140 7b Relation to the relativistic description 143 4 Relativistic simple fluids 1 The role of thermodynamics in continuum mechanics 146 2 The entropy law 148 3 The nature of equilibrium 151 + Thermostatics of a relativistic simple fluid 152 4a Detailed characterization of a fluid in equilibrium 152 4b The local state in equilibrium 155 4c The global state in equilibrium 159 4d Hyperbolic motion 161 Description of near-equilibrium states 163 Phenomenological laws — first approximation 166 6a The entropy source strength near local equilibrium 166 6b The Lie derivative — an aid to interpretation 168 6c General form of the phenomenological laws 170 6d Implications of the entropy law 172 6e Linearization 174