Geometrical methods of mathematical physics BERNARD F. SCHUTZ Reader in General Relativity, University College, Cardiff 8 DILIMAN CENTRAL LIBRARY CAMBRIDGE UNIVERSITY PRESS Cambridge London New York New Rochelle Melbourne Sydney Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 32 East-57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia © Cambridge University Press 1980 First published 1980 Reprinted 1982 Printed and bound in Great Britain at The Pitman Press, Bath British Library Cataloguing in Publication Data Schutz, Bernard F Geometrical methods of mathematical physics. 1. Geometry, Differential 2. Mathematical physics I. Title 516'.36'002453 = QC20.7.D52 80-40211 ISBN 0-521-23271-6 ISBN 0-521-29887-3 Pbk CONTENTS Preface ΙΧ Some basic mathematics The space R” and its topology Mappings Real analysis Group theory Linear algebra The algebra of square matrices Bibliography Differentiable manifolds and tensors Definition of a manifold The sphere as a manifold Other examples of manifolds Global considerations Curves Functions on M Vectors and vector fields Basis vectors and basis vector fields Fiber bundles 2.10 Examples of fiber bundles 2.11 A deeper look at fiber bundles 2.12 Vector fields and integral curves 2.13 Exponentiation of the operator d/dA 2.14 Lie brackets and noncoordinate bases 2.15 When is a basis a coordinate basis? 2.16 One-forms 2.17 Examples of one-forms 2.18 The Dirac delta function 2.19 The gradient and the pictorial representation of a one-form 2.20 Basis one-forms and components of one-forms 2.21 Index notation Contents vi Tensors and tensor fields 57 Examples of tensors 58 Components of tensors and the outer product 59 Contraction 59 Basis transformations 60 Tensor operations on components 63 Functions and scalars 64 The metric tensor on a vector space 64 The metric tensor field on a manifold 68 Special relativity 70 Bibliography 71 Lie derivatives and Lie groups 73 Introduction: how a vector field maps a manifold into itself 73 Lie dragging a function 74 Lie dragging a vector field 74 Lie derivatives 76 Lie derivative of a one-form 78 Submanifolds 79 Frobenius’ theorem (vector field version) δΙ Proof of Frobenius’ theorem 83 An example: the generators of S? 85 Invariance 86 Killing vector fields 88 Killing vectors and conserved quantities in particle dynamics 89 Axial symmetry 89 Abstract Lie groups 92 Examples of Lie groups 95 Lie algebras and their groups 101 Realizations and representations 105 Spherical symmetry, spherical harmonics and representations of the rotation group 108 Bibliography 112 Differential forms 113 The algebra and integral calculus of forms 113 Definition of volume — the geometrical role of differential forms 113 Notation and definitions for antisymmetric tensors 115 Differential forms 117 Manipulating differential forms 119 Restriction of forms 120 Fields of forms 120 Contents Vil Handedness and orientability 121 Volumes and integration on oriented manifolds 121 N-vectors, duals, and the symbol εῃ κ 125 Tensor densities 128 Generalized Kronecker deltas 130 Determinants and €;;z 131 Metric volume elements 132 The differential calculus of forms and its applications 134 The exterior derivative 134 Notation for derivatives 135 Familiar examples of exterior differentiation 136 Integrability conditions for partial differential equations 137 Exact forms 138 Proof of the local exactness of closed forms 140 Lie derivatives of forms 142 Lie derivatives and exterior derivatives commute 143 Stokes’ theorem 144 Gauss’ theorem and the definition of divergence 147 A glance at cohomology theory 150 Differential forms and differential equations 152 Frobenius’ theorem (differential forms version) 154 Proof of the equivalence of the two versions of Frobenius’ theorem 157 Conservation laws 158 Vector spherical harmonics 160 Bibliography 161 Applications in physics 163 Thermodynamics 163 Simple systems 163 Maxwell and other mathematical identities 164 Composite thermodynamic systems: Caratheodory’s theorem 165 Hamiltonian mechanics 167 Hamiltonian vector fields 167 Canonical transformations 168 Map between vectors and one-forms provided by @ 169 Poisson bracket 170 Many-particle systems: symplectic forms 170 Linear dynamical systems: the symplectic inner product and conserved quantities 171 Fiber bundle structure of the Hamiltonian equations 174 Electromagnetism 175 Rewriting Maxwell’s equations using differential forms 175 Contents vill Charge and topology 179 The vector potential 180 Plane waves: a simple example 181 Dynamics of a perfect fluid 181 Role of Lie derivatives 181 The comoving time-derivative 182 Equation of motion 183 Conservation of vorticity 184 Cosmology 186 The cosmological principle 186 Lie algebra of maximal symmetry 190 The metric of a spherically symmetric three-space 192 Construction of the six Killing vectors 195 Open, closed, and flat universes 197 Bibliography 199 Connections for Riemannian manifolds and gauge theories 201 Introduction 201 Parallelism on curved surfaces 201 The covariant derivative 203 Components: covariant derivatives of the basis 205 Torsion 207 Geodesics 208 Normal coordinates 210 Riemann tensor 210 Geometric interpretation of the Riemann tensor 212 6.10 Flat spaces 214 6.11 Compatibility of the connection with volume-measure or the metric 215 6.12 Metric connections 216 6.13 The affine connection and the equivalence principle 218 6.14 Connections and gauge theories: the example of electromagnetism 219 6.15 Bibliography 222 Appendix: solutions and hints for selected exercises 224 244 Notation Index 246 PREFACE Why study geometry? This book aims to introduce the beginning or working physicist to a wide range of analytic tools which have their origin in differential geometry and which have recently found increasing use in theoretical physics. It is not uncom- mon today for a physicist’s mathematical education to ignore all but the sim- plest geometrical ideas, despite the fact that young physicists are encouraged to develop mental ‘pictures’ and ‘intuition’ appropriate to physical phenomena. This curious neglect of ‘pictures’ of one’s mathematical tools may be seen as the outcome of a gradual evolution over many centuries. Geometry was certainly extremely important to ancient and medieval natural philosophers; it was in geometrical terms that Ptolemy, Copernicus, Kepler, and Galileo all expressed their thinking. But when Descartes introduced coordinates into Euclidean geometry, he showed that the study of geometry could be regarded as an appli- cation of algrebra. Since then, the importance of the study of geometry in the education of scientists has steadily declined, so that at present a university undergraduate physicist or applied mathematician is not likely to encounter much geometry at all. One reason for this suggests itself immediately: the relatively simple geometry of the three-dimensional Euclidean world that the nineteenth-century physicist believed he lived in can be mastered quickly, while learning the great diversity of analytic techniques that must be used to solve the differential equations of physics makes very heavy demands on the student’s time. Another reason must surely be that these analytic techniques were developed at least partly in response to the profound realization by physicists that the laws of nature could be expressed as differential equations, and this led most mathematical physicists genuinely to neglect geometry until relatively recently. However, two developments in this century have markedly altered the balance between geometry and analysis in the twentieth-century physicist’s outloook. The first is the development of the theory of relativity, according to which the Euclidean three-space of the nineteenth-century physicist is only an approxi- mation to the correct description of the physical world. The second development, which is only beginning to have an impact, is the realization by twentieth-century Preface x mathematicians, led by Cartan, that the relation between geometry and analysis is a two-way street: on the one hand analysis may be the foundation of the study of geometry, but on the other hand the study of geometry leads naturally to the development of certain analytic tools (such as the Lie derivative and the exterior calculus) and certain concepts (such as the manifold, the fiber bundle, and the identification of vectors with derivatives) that have great power in applications of analysis. In the modern view, geometry remains subsidiary to analysis. For example, the basic concept of differential geometry, the differentiable manifold, is defined in terms of real numbers and differentiable functions. But this is no disadvantage: it means that concepts from analysis can be expressed geometri- cally, and this has considerable heuristic power. Because it has developed this intimate connection between geometrical and analytic ideas, modern differential geometry has become more and more import- ant in theoretical physics, where it has led to a greater simplicity in the math- ematics and a more fundamental understanding of the physics. This revolution has affected not only special and general relativity, the two theories whose con- tent is most obviously geometrical, but other fields where the geometry involved is not always that of physical space but rather of a more abstract space of vari- ables: electromagnetism, thermodynamics, Hamiltonian theory, fluid dynamics, and elementary particle physics. Aims of this book In this book I want to introduce the reader to some of the more important notions of twentieth-century differential geometry, trying always to use that geometrical or ‘pictorial’ way of thinking thati s usually so helpful in developing a physicist’s intuition. The book attempts to teach mathematics, not physics. I have tried to include a wide range of applications of this mathematics to branches of physics which are familiar to most advanced undergraduates. I hope these examples will do more than illustrate the mathematics: the new mathematical formulation of familiar ideas will, if I have been successful, give the reader a deeper understanding of the physics. I will discuss the background I have assumed of the reader in more detail below, but here it may be helpful to give a brief list of some of the ‘familiar’ ideas which are seen in a new light in this book: vectors, tensors, inner products, special relativity, spherical harmonics and the rotation group (and angular- momentum operators), conservation laws, volumes, theory of integration, curl and cross-product, determinants of matrices, partial differential equations and their integrability conditions, Gauss’ and Stokes’ integral theorems of vector calculus, thermodynamics of simple systems, Caratheodory’s theorem (and the second law of thermodynamics), Hamiltonian systems in phase space, Maxwell’s Preface xi equations, fluid dynamics (including the laws governing the conservation of circulation), vector calculus in curvilinear coordinate systems, and the quantum theory of a charged scalar field. Besides these more or less familiar subjects, there are a few others which are not usually taught at undergraduate level but which most readers would certainly have heard of: the theory of Lie groups and symmetry, open and closed cosmologies, Riemannian geometry, and gauge theories of physics. That all of these subjects can be studied by the methods of differential geometry is an indication of the importance differential geometry is likely to have in theoretical physics in the future. I believe it is important for the reader to develop a pictorial way of thinking and a feeling for the ‘naturalness’ of certain geometrical tools in certain situ- ations. To this end I emphasize repeatedly the idea that tensors are geometrical objects, defined independently of any coordinate system. The role played by components and coordinate transformations is submerged into a secondary position: whenever possible I write equations without indices, to emphasize the coordinate-independence of the operations. I have made no attempt to present the material in a strictly rigorous or axiomatic way, and I have had to ignore many aspects of our subject which a mathematician would regard as funda- mental. I do, of course, give proofs of all but a handful of the most important results (references for the exceptions are provided), but I have tried wherever possible to make the main geometrical ideas in the proof stand out clearly from the background of manipulation. 1 want to show the beauty, elegance, and naturalness of the mathematics with the minimum of obscuration. How to use this book The first chapter contains a review of the sort of elementary math- ematics assumed of the reader plus a short introduction to some concepts, par- ticularly in topology, which undergraduates may not be familiar with. The next chapters are the core of the book: they introduce tensors, Lie derivatives, and differential forms. Scattered through these chapters are some applications, but most of the physical applications are left for systematic treatment in chapter 5. The final chapter, on Riemannian geometry, is more advanced and makes con- tact with areas of particle physics and general relativity in which differential geometry is an everyday tool. The material in this book should be suitable for a one-term course, provided the lecturer exercises some selection in the most difficult areas. It should also be possible to teach the most important points as a unit of, say, ten lectures in an advanced course on mathematical methods. I have taught such a unit to graduate students, concentrating mainly on ὃς 2.1—2.3, 2.5-2.8, 2.12—2.14, 2.16, 2.17, 2.19-2.28, 3.1-3.13, 4.1-4.6, 4.8, 4.14-4.18, 4.20-4.23, 4.25, 4.26, 5.1, 5.2, 5 4—5.7, and 5.15—5.18. I hope lecturers will experiment with their own choices