This page intentionally left blank General Relativity: An Introduction for Physicists provides a clear mathematical introduction to Einstein’s theory of general relativity. A wide range of applications of the theory are included, with a concentration on its physical consequences. After reviewing the basic concepts, the authors present a clear and intuitive discussion of the mathematical background, including the necessary tools of tensor calculus and differential geometry. These tools are used to develop the topic of special relativity and to discuss electromagnetism in Minkowski spacetime. Gravitation as spacetime curvature is then introduced and the field equations of general relativity are derived. A wide range of applications to physical situations follows, and the conclusion gives a brief discussion of classical field theory and the derivation of general relativity from a variational principle. Written for advanced undergraduate and graduate students, this approachable textbook contains over 300 exercises to illuminate and extend the discussion in the text. Michael Hobson specialised in theoretical physics as an undergraduate at the University of Cambridge and remained at the Cavendish Laboratory to complete a Ph.D. in the physics of star-formation and radiative transfer. As a Research Fellow at Trinity Hall, Cambridge, and later as an Advanced Fellow of the Particle Physics and Astronomy Research Council, he developed an interest in cosmology, in particular in the study of fluctuations in the cosmic microwave background (CMB) radiation. He is currently a Reader in Astrophysics and Cosmology at the Cavendish Laboratory, where he is the principal investigator for the Very Small Array CMB interferometer. He is also joint project scientist for the Arcminute Microkelvin Imager project and an associate of the European Space Agency Planck Surveyor CMB satellite mission. In addition to observational and theo- retical cosmology, his research interests also include Bayesian analysis methods and theoretical optics and he has published over 100 research papers in a wide range of areas. He is a Staff Fellow and Director of Studies in Natural Sciences at Trinity Hall and enjoys an active role in the teaching of undergraduate physics and mathematics. He is a co-author with Ken Riley and Stephen Bence of the well-known undergraduate textbook Mathematical Methods for Physics and Engi- neering (Cambridge, 1998; second edition, 2002; third edition to be published in 2006) and with Ken Riley of the Student’s Solutions Manual accompanying the third edition. George Efstathiou is Professor of Astrophysics and Director of the Institute of Astronomy at the University of Cambridge. After studying physics as an undergraduate at Keble College, Oxford, he gained his Ph.D. in astronomy from Durham University. Following some post-doctoral research at the University of California at Berkeley he returned to work in the UK at the Institute of Astronomy, Cambridge, where he was appointed Assistant Director of Research in 1987. He returned to the Department of Physics at Oxford as Savilian Professor of Astronomy and Head of Astrophysics, before taking on his current posts at the Institute of Astronomy in 1997 and 2004 respectively. He is a Fellow of the Royal Society and the recipient of several awards, including the Maxwell Medal and Prize of the Institute of Physics in 1990 and the Heineman Prize for Astronomy of the American Astronomical Society in 2005. Anthony Lasenby is Professor of Astrophysics and Cosmology at the University of Cambridge and is currently Head of the Astrophysics Group and the Mullard Radio Astronomy Observatory in the Cavendish Laboratory, as well as being a Deputy Head of the Laboratory. He began his astronomical career with a Ph.D. at Jodrell Bank, specializing in the cosmic microwave background, which has remained a major subject of his research. After a brief period at the National Radio Astronomy Observatory in America, he moved from Manchester to Cambridge in 1984 and has been at the Cavendish since then. He is the author or co-author of over 200 papers spanning a wide range of fields and is the co-author of Geometric Algebra for Physicists (Cambridge, 2003) with Chris Doran. General Relativity An Introduction for Physicists M. P. HOBSON, G. P. EFSTATHIOU and A. N. LASENBY cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Informationonthistitle:www.cambridge.org/9780521829519 © M. P. Hobson, G. P. Efstathiou and A. N. Lasenby 2006 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2006 isbn-13 978-0-511-13795-2 eBook (Adobe Reader) isbn-10 0-511-13795-8 eBook (Adobe Reader) isbn-13 978-0-521-82951-9 hardback isbn-10 0-521-82951-8 hardback isbn-13 978-0-521-53639-4 isbn-10 0-521-53639-1 Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. To our families Contents Preface page xv 1 The spacetime of special relativity1 1.1 Inertial frames and the principle of relativity1 1.2 Newtonian geometry of space and time3 1.3 The spacetime geometry of special relativity3 1.4 Lorentz transformations as four-dimensional ‘rotations’5 1.5 The interval and the lightcone6 1.6 Spacetime diagrams8 1.7 Length contraction and time dilation 10 1.8 Invariant hyperbolae 11 1.9 The Minkowski spacetime line element 12 1.10 Particle worldlines and proper time 14 1.11 The Doppler effect 16 1.12 Addition of velocities in special relativity 18 1.13 Acceleration in special relativity 19 1.14 Event horizons in special relativity 21 Appendix 1A: Einstein’s route to special relativity 22 Exercises 24 2 Manifolds and coordinates 26 2.1 The concept of a manifold 26 2.2 Coordinates 27 2.3 Curves and surfaces 27 2.4 Coordinate transformations 28 2.5 Summation convention 30 2.6 Geometry of manifolds 31 2.7 Riemannian geometry 32 2.8 Intrinsic and extrinsic geometry 33 vii viii Contents 2.9 Examples of non-Euclidean geometry 36 2.10 Lengths, areas and volumes 38 2.11 Local Cartesian coordinates 42 2.12 Tangent spaces to manifolds 44 2.13 Pseudo-Riemannian manifolds 45 2.14 Integration over general submanifolds 47 2.15 Topology of manifolds 49 Exercises 50 3 Vector calculus on manifolds 53 3.1 Scalar fields on manifolds 53 3.2 Vector fields on manifolds 54 3.3 Tangent vector to a curve 55 3.4 Basis vectors 56 3.5 Raising and lowering vector indices 59 3.6 Basis vectors and coordinate transformations 60 3.7 Coordinate-independent properties of vectors 61 3.8 Derivatives of basis vectors and the affine connection 62 3.9 Transformation properties of the affine connection 64 3.10 Relationship of the connection and the metric 65 3.11 Local geodesic and Cartesian coordinates 67 3.12 Covariant derivative of a vector 68 3.13 Vector operators in component form 70 3.14 Intrinsic derivative of a vector along a curve 71 3.15 Parallel transport 73 3.16 Null curves, non-null curves and affine parameters 75 3.17 Geodesics 76 3.18 Stationary property of non-null geodesics 77 3.19 Lagrangian procedure for geodesics 78 3.20 Alternative form of the geodesic equations 81 Appendix 3A: Vectors as directional derivatives 81 Appendix 3B: Polar coordinates in a plane 82 Appendix 3C: Calculus of variations 87 Exercises 88 4 Tensor calculus on manifolds 92 4.1 Tensor fields on manifolds 92 4.2 Components of tensors 93 4.3 Symmetries of tensors 94 4.4 The metric tensor 96 4.5 Raising and lowering tensor indices 97 Contents ix 4.6 Mapping tensors into tensors 97 4.7 Elementary operations with tensors 98 4.8 Tensors as geometrical objects 100 4.9 Tensors and coordinate transformations 101 4.10 Tensor equations 102 4.11 The quotient theorem 103 4.12 Covariant derivative of a tensor 104 4.13 Intrinsic derivative of a tensor along a curve 107 Exercises 108 5 Special relativity revisited 111 5.1 Minkowski spacetime in Cartesian coordinates 111 5.2 Lorentz transformations 112 5.3 Cartesian basis vectors 113 5.4 Four-vectors and the lightcone 115 5.5 Four-vectors and Lorentz transformations 116 5.6 Four-velocity 116 5.7 Four-momentum of a massive particle 118 5.8 Four-momentum of a photon 119 5.9 The Doppler effect and relativistic aberration 120 5.10 Relativistic mechanics 122 5.11 Free particles 123 5.12 Relativistic collisions and Compton scattering 123 5.13 Accelerating observers 125 5.14 Minkowski spacetime in arbitrary coordinates 128 Exercises 131 6 Electromagnetism 135 6.1 The electromagnetic force on a moving charge 135 6.2 The 4-current density 136 6.3 The electromagnetic field equations 138 6.4 Electromagnetism in the Lorenz gauge 139 6.5 Electric and magnetic fields in inertial frames 141 6.6 Electromagnetism in arbitrary coordinates 142 6.7 Equation of motion for a charged particle 144 Exercises 145 7 The equivalence principle and spacetime curvature 147 7.1 Newtonian gravity 147 7.2 The equivalence principle 148 7.3 Gravity as spacetime curvature 149 7.4 Local inertial coordinates 151 x Contents 7.5 Observers in a curved spacetime 152 7.6