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General Relativity: An Introduction for Physicists provides a clear mathematical
introductiontoEinstein’stheoryofgeneralrelativity.Awiderangeofapplications
of the theory are included, with a concentration on its physical consequences.
After reviewing the basic concepts, the authors present a clear and intuitive
discussionofthemathematicalbackground,includingthenecessarytoolsoftensor
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
FellowatTrinityHall,Cambridge,andlaterasanAdvancedFellowoftheParticle
PhysicsandAstronomyResearchCouncil,hedevelopedaninterestincosmology,
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-knownundergraduatetextbookMathematicalMethodsforPhysicsandEngi-
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
CaliforniaatBerkeleyhereturnedtoworkintheUKattheInstituteofAstronomy,
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
InstituteofAstronomyin1997and2004respectively.HeisaFellowoftheRoyal
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.
AnthonyLasenbyisProfessorofAstrophysicsandCosmologyattheUniversity
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
remainedamajorsubjectofhisresearch.AfterabriefperiodattheNationalRadio
AstronomyObservatoryinAmerica,hemovedfromManchestertoCambridgein
1984 and has been at the Cavendish since then. He is the author or co-author of
over200papersspanningawiderangeoffieldsandistheco-authorofGeometric
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
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Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
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©M.P.Hobson,G.P.EfstathiouandA.N.Lasenby2006
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without the written permission of Cambridge University Press.
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To our families
Contents
Preface page xv
1 The spacetime of special relativity 1
1.1 Inertial frames and the principle of relativity 1
1.2 Newtonian geometry of space and time 3
1.3 The spacetime geometry of special relativity 3
1.4 Lorentz transformations as four-dimensional ‘rotations’ 5
1.5 The interval and the lightcone 6
1.6 Spacetime diagrams 8
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
Appendix1A:Einstein’sroutetospecialrelativity 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
Appendix3A:Vectorsasdirectionalderivatives 81
Appendix3B:Polarcoordinatesinaplane 82
Appendix3C:Calculusofvariations 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
