Contents

Preface page xi

1 The role of gravity 1

2 Differential geometry 10

2.1 Manifolds 11

2.2 Vectors and tensors 15

2.3 Maps of manifolds 22

2.4 Exterior differentiation and the Lie derivative 24

2.5 Covariant differentiation and the curvature tensor 30

2.6 The metric 36

2.7 Hypersurfaces 44

2.8 The volume element and Gauss' theorem 47

2.9 Fibre bundles 50

3 General Relativity 56

3.1 The space-time manifold 56

3.2 The matter fields 59

3.3 Lagrangian formulation 64

3.4 The field equations 71

4 The physical significance of curvature 78

4.1 Timelike curves 78

4.2 Null curves 86

4.3 Energy conditions - 88

4.4 Conjugate points 96

4.5 Variation of arc-length 102

5 Exact solutions 117

5.1 Minkowski space-time 118

5.2 De Sitter and anti-de Sitter space-times 124

5.3 Robertson-Walker spaces 134

5.4 Spatially homogeneous cosmological models 142

5.5 The Schwarzschild and Reissner-NordstrQm solutions page 149

5.6 The Kerr solution 161

5.7 GQdel's universe 168

5.8 Taub-NUT space 170

5.9 Further exact solutions 178

6 Causal structure 180

6.1 Orientability 181

6.2 Causal curves 182

6.3 Achronal boundaries 186

6.4 Causality conditions 189

6.5 Cauchy developments 201

6.6 Global hyperbolicity 206

6.7 The existence of geodesies 213

6.8 The causal boundary of space-time 217

6.9 Asymptotically simple spaces 221

7 The Cauchy problem in General Relativity 226

7.1 The nature of the problem 227

7.2 The reduced Einstein equations 228

7.3 The initial data 231

7.4 Second order hyperbolic equations 233

7.5 Thn oxinfconno iind nniqiinnoan of tlovnlupinoula for the empty space Einstein equations 244

7.6 The maximal development and stability 249

7.7 The Einstein equations with matter 254

8 Space-time singularities 256

8.1 The definition of singularities 256

8.2 Singularity theorems 261

8.3 The description of singularities 276

8.4 The character of the singularities 284

8.5 Imprisoned incompleteness 289

9 Gravitational collapse and black holes 299

9.1 Stellar collapse 299

9.2 Black holes 308

9.3 The final state of black holes 323

10 The initial singularity in the universe page 348

i 0.1 The expansion of the universe 348

10.2 The nature and implications of singularities 359

Appendix A:

Translation of an essay by P. S. Laplace 365 Appendix B:

Spherically symmetric solutions and Birkhoff's theorem 369

References 373

Notation 381

Index

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