Exact solutions of Einstein's field equations

732 Pages·2003·2.841 MB·English

by  Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt

Exact Solutions of Einstein’s Field Equations
Arevisededitionofthenowclassictext,Exact Solutions of Einstein’s Field Equations
gives a unique survey of the known solutions of Einstein’s field equations for vacuum,
Einstein–Maxwell,pureradiationandperfectfluidsources.Itstartsbyintroducingthe
foundations of differential geometry and Riemannian geometry and the methods used
tocharacterize,findorconstructsolutions.Thesolutionsarethenconsidered,ordered
by their symmetry group, their algebraic structure (Petrov type) or other invariant
properties such as special subspaces or tensor fields and embedding properties.
Thiseditionhasbeenexpandedandupdatedtoincludethenewdevelopmentsinthe
fieldsincethepublicationofthefirstedition.Itcontainsfivecompletelynewchapters,
coveringtopicssuchasgenerationmethodsandtheirapplication,collidingwaves,clas-
sification of metrics by invariants and inhomogeneous cosmologies. It is an important
source and guide for graduates and researchers in relativity, theoretical physics, astro-
physicsandmathematics.Partsofthebookcanalsobeusedforpreparinglecturesand
as an introductory text on some mathematical aspects of general relativity.
hans stephani gained his Diploma, Ph.D. and Habilitation at the Friedrich-
Schiller-Universit¨at Jena. He became Professor of Theoretical Physics in 1992, before
retiring in 2000. He has been lecturing in theoretical physics since 1964 and has pub-
lished numerous papers and articles on relativity and optics. He is also the author of
four books.
dietrich kramer is Professor of Theoretical Physics at the Friedrich-Schiller-
Universit¨at Jena. He graduated from this university, where he also finished his Ph.D.
(1966) and habilitation (1970). His current research concerns classical relativity. The
majority of his publications are devoted to exact solutions in general relativity.
malcolm maccallum is Professor of Applied Mathematics at the School of
Mathematical Sciences, Queen Mary, University of London, where he is also Vice-
Principal for Science and Engineering. He graduated from Kings College, Cambridge
and went on to complete his M.A. and Ph.D. there. His research covers general rela-
tivityandcomputeralgebra,especiallytensormanipulatorsanddifferentialequations.
He has published over 100 papers, review articles and books.
cornelius hoenselaers gained his Diploma at Technische Universit¨at
Karlsruhe,hisD.Sc.atHiroshimaDaigakuandhisHabilitationatLudwig-Maximilian
Universit¨at Mu¨nchen. He is Reader in Relativity Theory at Loughborough University.
He has specialized in exact solutions in general relativity and other non-linear par-
tialdifferentialequations,andpublishedalargenumberofpapers,reviewarticlesand
books.
eduard herlt iswissenschaftlicherMitarbeiterattheTheoretischPhysikalisches
Institut der Friedrich-Schiller-Universit¨at Jena. Having studied physics as an under-
graduate at Jena, he went on to complete his Ph.D. there as well as his Habilitation.
He has had numerous publications including one previous book.

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Exact Solutions of Einstein’s
Field Equations
Second Edition
HANS STEPHANI
Friedrich-Schiller-Universit¨at, Jena
DIETRICH KRAMER
Friedrich-Schiller-Universit¨at, Jena
MALCOLM MACCALLUM
Queen Mary, University of London
CORNELIUS HOENSELAERS
Loughborough University
EDUARD HERLT
Friedrich-Schiller-Universit¨at, Jena
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , United Kingdom
Published in the United States by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521461368
© H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers, E. Herlt 2003
This book 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 2003
ISBN-13 978-0-511-06548-4 eBook (NetLibrary)
ISBN-10 0-511-06548-5 eBook (NetLibrary)
ISBN-13 978-0-521-46136-8 hardback
ISBN-10 0-521-46136-7 hardback
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s for external or third-party internet websites referred to in this book, and does not
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Contents
Preface xix
List of tables xxiii
Notation xxvii
1 Introduction 1
1.1 What are exact solutions, and why study them? 1
1.2 The development of the subject 3
1.3 The contents and arrangement of this book 4
1.4 Using this book as a catalogue 7
Part I: General methods 9
2 Differential geometry without a metric 9
2.1 Introduction 9
2.2 Differentiable manifolds 10
2.3 Tangent vectors 12
2.4 One-forms 13
2.5 Tensors 15
2.6 Exterior products and p-forms 17
2.7 The exterior derivative 18
2.8 The Lie derivative 21
2.9 The covariant derivative 23
2.10 The curvature tensor 25
2.11 Fibre bundles 27
vii
viii Contents
3 Some topics in Riemannian geometry 30
3.1 Introduction 30
3.2 The metric tensor and tetrads 30
3.3 Calculation of curvature from the metric 34
3.4 Bivectors 35
3.5 Decomposition of the curvature tensor 37
3.6 Spinors 40
3.7 Conformal transformations 43
3.8 Discontinuities and junction conditions 45
4 The Petrov classification 48
4.1 The eigenvalue problem 48
4.2 The Petrov types 49
4.3 Principal null directions and determination of the
Petrov types 53
5 Classification of the Ricci tensor and the
energy-momentum tensor 57
5.1 The algebraic types of the Ricci tensor 57
5.2 The energy-momentum tensor 60
5.3 The energy conditions 63
5.4 The Rainich conditions 64
5.5 Perfect fluids 65
6 Vector fields 68
6.1 Vector fields and their invariant classification 68
6.1.1 Timelike unit vector fields 70
6.1.2 Geodesic null vector fields 70
6.2 Vector fields and the curvature tensor 72
6.2.1 Timelike unit vector fields 72
6.2.2 Null vector fields 74
7 The Newman–Penrose and related
formalisms 75
7.1 The spin coefficients and their transformation
laws 75
7.2 The Ricci equations 78
7.3 The Bianchi identities 81
7.4 The GHP calculus 84
7.5 Geodesic null congruences 86
7.6 The Goldberg–Sachs theorem and its generalizations 87
Contents ix
8 Continuous groups of transformations; isometry
and homothety groups 91
8.1 Lie groups and Lie algebras 91
8.2 Enumeration of distinct group structures 95
8.3 Transformation groups 97
8.4 Groups of motions 98
8.5 Spaces of constant curvature 101
8.6 Orbits of isometry groups 104
8.6.1 Simply-transitive groups 105
8.6.2 Multiply-transitive groups 106
8.7 Homothety groups 110
9 Invariants and the characterization of geometries 112
9.1 Scalar invariants and covariants 113
9.2 The Cartan equivalence method for space-times 116
9.3 Calculating the Cartan scalars 120
9.3.1 Determination of the Petrov and Segre types 120
9.3.2 The remaining steps 124
9.4 Extensions and applications of the Cartan method 125
9.5 Limits of families of space-times 126
10 Generation techniques 129
10.1 Introduction 129
10.2 Lie symmetries of Einstein’s equations 129
10.2.1 Point transformations and their generators 129
10.2.2 How to find the Lie point symmetries of a given
differential equation 131
10.2.3 How to use Lie point symmetries: similarity
reduction 132
10.3 Symmetries more general than Lie symmetries 134
10.3.1 Contact and Lie–Ba¨cklund symmetries 134
10.3.2 Generalized and potential symmetries 134
10.4 Prolongation 137
10.4.1 Integral manifolds of differential forms 137
10.4.2 Isovectors, similarity solutions and conservation laws 140
10.4.3 Prolongation structures 141
10.5 Solutions of the linearized equations 145
10.6 B¨acklund transformations 146
10.7 Riemann–Hilbert problems 148
10.8 Harmonic maps 148
10.9 Variational Ba¨cklund transformations 151
10.10 Hirota’s method 152