Exact Solutions of Einstein's Field Equations / Edition 2

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"A revised edition of the now classic text, Exact Solutions of Einstein's Field Equations gives a unique survey of the known solutions of Einstein's field equations for vacuum, Einstein-Maxwell, pure radiation and perfect fluid sources. It starts by introducing the foundations of differential geometry and Riemannian geometry and the methods used to characterize, find or construct solutions. The solutions are then considered, ordered by their symmetry group, their algebraic structure (Petrov type) or other invariant properties such as special subspaces or tensor fields and embedding properties." This edition has been expanded and updated to include the new developments in the field since the publication of the first edition. It contains five completely new chapters, covering topics such as generation methods and their application, colliding waves, classification of metrics by invariants and inhomogeneous cosmologies. It is an important source and guide for graduates and researchers in relativity, theoretical physics, astrophysics and mathematics. Parts of the book can also be used for preparing lectures and as an introductory text on some mathematical aspects of general relativity.
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Editorial Reviews

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"This is clearly a most valuable reference book. It comprehensively reviews known local solutions of Einstein's equation and provides a secure base for future research."
Mathematical Reviews
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Product Details

Meet the Author

Hans Stephani gained his diploma, Ph.D. and Habilitation at the Friedrich-Schiller-Universität Jena. He became Professor of Theoretical Physics in 1992, before retiring in 2000. He has been lecturing in theoretical physics since 1964 and has published 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-Universität 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 King's College, Cambridge and went on to complete his M.A. and Ph.D. there. His research covers general relativity and computer algebra, especially tensor manipulators and differential equations. He has published over 100 pages, review articles and books.

Cornelius Hoenselaers gained his Diploma at Technische Universität Karlsruhe, his D.Sc. at Hiroshima Daigaku and his Habilitation at Ludwig-Maximilian Universität München. He is Reader in Relativity Theory at Loughborough University. He has specialized in exact solutions in general relativity and other non-linear partial differential equations, and published a large number of papers, review articles and books.

Eduard Herlt is wissenschaftlicher Mitarbeiter at the Theoretisch Physikalisches Institut der Friedrich-Schiller-Universität Jena. Having studied physics as an undergraduate 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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Table of Contents

List of tables
1 Introduction 1
Pt. I General methods 9
2 Differential geometry without a metric 9
3 Some topics in Riemannian geometry 30
4 The Petrov classification 48
5 Classification of the Ricci tensor and the energy-momentum tensor 57
6 Vector fields 68
7 The Newman-Penrose and related formalisms 75
8 Continuous groups of transformations; isometry and homothety groups 91
9 Invariants and the characterization of geometries 112
10 Generation techniques 129
Pt. II Solutions with groups of motions 157
11 Classification of solutions with isometries or homotheties 157
12 Homogeneous space-times 171
13 Hypersurface-homogeneous space-times 183
14 Spatially-homogeneous perfect fluid cosmologies 210
15 Groups G[subscript 3] on non-null orbits V[subscript 2]. Spherical and plane symmetry 226
16 Spherically-symmetric perfect fluid solutions 247
17 Groups G[subscript 2] and G[subscript 1] on non-null orbits 264
18 Stationary gravitational fields 275
19 Stationary axisymmetric fields: basic concepts and field equations 292
20 Stationary axisymmetric vacuum solutions 304
21 Non-empty stationary axisymmetric solutions 319
22 Groups G[subscript 2]I on spacelike orbits: cylindrical symmetry 341
23 Inhomogeneous perfect fluid solutions with symmetry 358
24 Groups on null orbits. Plane waves 375
25 Collision of plane waves 387
Pt. III Algebraically special solutions 407
26 The various classes of algebraically special solutions. Some algebraically general solutions 407
27 The line element for metrics with [kappa] = [sigma] = 0 = R[subscript 11] = R[subscript 14] = R[subscript 44], [actual symbol not reproducible] 416
28 Robinson-Trautman solutions 422
29 Twisting vacuum solutions 437
30 Twisting Einstein-Maxwell and pure radiation fields 455
31 Non-diverging solutions (Kundt's class) 470
32 Kerr-Schild metrics 485
33 Algebraically special perfect fluid solutions 506
Pt. IV Special methods 518
34 Application of generation techniques to general relativity 518
35 Special vector and tensor fields 553
36 Solutions with special subspaces 571
37 Local isometric embedding of four-dimensional Riemannian manifolds 580
Pt. V Tables 605
38 The interconnections between the main classification schemes 605
References 615
Index 690
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