Symmetries and curvature structure in general relativity

G. S. Hall9810210515, 9789810210519, 9789812562692

This is a text on classical general relativity from a geometrical viewpoint. Introductory chapters are provided on algebra, topology and manifold theory, together with a chapter on the basic ideas of space-time manifolds and Einstein’s theory. There is a detailed account of algebraic structures and tensor classification in general relativity and also of the relationships between the metric, connection and curvature structures on space-times. The latter includes chapters on holonomy and sectional curvature. An extensive study is presented of symmetries in general relativity, including isometries, homotheties, conformal symmetries and affine, projective and curvature collineations. Several general properties of such symmetries are studied and a preparatory section on transformation groups and on the properties of Lie algebras of vector fields on manifolds is provided.

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Table of contents :
Cover……Page 1
Series Contents……Page 3
Title……Page 4
Date-line ……Page 5
Preface ……Page 6
Contents ……Page 8
1.1 Geometry and Physics ……Page 12
1.2 Preview of Future Chapters ……Page 16
2.2 Groups ……Page 22
2.3 Vector Spaces ……Page 26
2.4 Dual Spaces ……Page 33
2.5 Forms and Inner Products ……Page 34
2.6 Similarity, Jordan Canonical Forms and Segre Types ……Page 39
2.7 Lie Algebras ……Page 49
3.1 Introduction ……Page 52
3.2 Metric Spaces ……Page 53
3.3 Topological Spaces ……Page 56
3.4 Bases ……Page 60
3.5 Subspace Topology ……Page 62
3.6 Quotient Spaces ……Page 63
3.8 Compactness and Paracompactness ……Page 64
3.9 Connected Spaces ……Page 67
3.10 Covering Spaces and the Fundamental Group ……Page 69
3.11 The Rank Theorems ……Page 72
4.1 Introduction ……Page 74
4.2 Calculus on Kn ……Page 75
4.3 Manifolds ……Page 76
4.4 Functions on Manifolds ……Page 79
4.5 The Manifold Topology ……Page 81
4.6 The Tangent Space and Tangent Bundle ……Page 84
4.7 Tensor Spaces and Tensor Bundles ……Page 86
4.8 Vector and Tensor Fields ……Page 88
4.9 Derived Maps and Pullbacks ……Page 92
4.10 Integral Curves of Vector Fields ……Page 94
4.11 Submanifolds ……Page 96
4.13 Distributions ……Page 103
4.14 Curves and Coverings ……Page 108
4.15 Metrics on Manifolds ……Page 110
4.16 Linear Connections and Curvature ……Page 115
4.17 Grassmann and Stiefel Manifolds ……Page 127
5.1 Topological Groups ……Page 130
5.2 Lie Groups ……Page 132
5.3 Lie Subgroups ……Page 133
5.4 Lie Algebras ……Page 135
5.5 One Parameter Subgroups and the Exponential Map ……Page 138
5.7 Lie Transformation Groups ……Page 142
5.8 Orbits and Isotropy Groups ……Page 144
5.9 Complete Vector Fields ……Page 146
5.10 Groups of Transformations ……Page 149
5.11 Local Group Actions ……Page 151
5.12 Lie Algebras of Vector Fields ……Page 153
5.13 The Lie Derivative ……Page 155
6.1 Minkowski Space ……Page 158
6.2 The Lorentz Group ……Page 161
6.3 The Lorentz Group as a Lie Group ……Page 169
6.4 The Connected Lie Subgroups of the Lorentz Group ……Page 174
7.1 Space-Times ……Page 180
7.1.1 Electromagnetic fields ……Page 182
7.1.3 The Vacuum Case ……Page 183
7.2 Bivectors and their Classification ……Page 184
7.3 The Petrov Classification ……Page 195
7.4 Alternative Approaches to the Petrov Classification ……Page 202
7.5 The Classification of Second Order Symmetric Tensors ……Page 213
7.6 The Anti-Self Dual Representation of Second Order Symmetric Tensors ……Page 219
7.7 Examples and Applications ……Page 226
7.8 The Local and Global Nature of Algebraic Classifications ……Page 232
8.2 Holonomy Groups ……Page 238
8.3 The Holonomy Group of a Space-Time ……Page 245
8.4 Vacuum Space-Times ……Page 256
8.5 Examples ……Page 259
9.2 Metric and Connection ……Page 266
9.3 Metric, Connection and Curvature ……Page 270
9.4 Sectional Curvature ……Page 281
9.5 Retrospect ……Page 293
10.1 General Aspects of Symmetries ……Page 296
10.2 Affine Vector Fields ……Page 298
10.3 Subalgebras of the Affine Algebra; Isometries and Homotheties ……Page 302
10.4 Fixed Point Structure ……Page 307
10.5 Orbit Structure ……Page 322
10.6 Space-Times admitting Proper Affine Vector Fields ……Page 334
10.7 Examples and Summary ……Page 344
11.1 Conformal Vector Fields ……Page 352
11.2 Orbit Structure ……Page 356
11.3 Fixed Point Structure ……Page 360
11.4 Conformal Reduction of the Conformal Algebra ……Page 363
11.5 Conformal Vector Fields in Vacuum Space-Times ……Page 369
11.6 Other Examples ……Page 370
11.7 Special Conformal Vector Fields ……Page 374
12.1 Projective Vector Fields ……Page 382
12.2 General Theorems on Projective Vector Fields ……Page 386
12.3 Space-Times Admitting Projective Vector Fields ……Page 392
12.4 Special Projective Vector Fields ……Page 400
12.5 Projective Symmetry and Holonomy ……Page 402
13.2 Curvature Collineations ……Page 408
13.3 Some Techniques for Curvature Collineations ……Page 411
13.4 Further Examples ……Page 419
Bibliography ……Page 424
Index ……Page 432