Spacetime: Foundations of general relativity and differential geometry

Lecture Notes in Physics: Monographs 59 ......Page 1
Spacetime: Foundations of General Relativity and Differential Geometry ......Page 3
Copyright ......Page 4
Preface to the first edition ......Page 5
Contents ......Page 11
List of Figures ......Page 15
List of postulates ......Page 17
1.1 Space ......Page 18
1.1.1 Affine space ......Page 19
1.1.2 The fundamental theorem in affine geometry and doubly ruled surfaces ......Page 20
1.1.3 Euclidean geometry ......Page 31
1.2 Absolute space and absolute time ......Page 34
1.2.1 Non-relativistic particles ......Page 37
1.3 Galilei\'s theory of relativity ......Page 39
1.4 Einstein\'s special theory of relativity ......Page 44
1.4.1 Causality in special relativity ......Page 56
1.4.2 Length contraction and time dilatation ......Page 57
1.4.3 Relativistic particles and photons ......Page 60
2. Analysis on manifolds ......Page 64
2.1 Manifolds ......Page 65
2.1.1 Construction of manifolds ......Page 71
2.1.2 Partition of unity ......Page 74
2.2 Vector bundles and the tangent bundle ......Page 78
2.2.1 Construction of the tangent bundle ......Page 80
2.2.2 The derivative of maps between manifolds ......Page 84
2.3 Tensors and tensor fields ......Page 85
2.3.1 Algebraic preliminaries: tensors ......Page 86
2.3.2 Tensor fields ......Page 101
2.4 Vector fields and ordinary differential equations ......Page 104
2.5 Differential forms ......Page 111
2.5.1 The lemma of Poincare ......Page 119
2.5.2 The theorem of Frobenius ......Page 123
2.5.3 Orientable real manifolds ......Page 126
2.5.4 Integration on real manifolds ......Page 129
2.6 Connections and projective structures ......Page 138
2.7.1 The Levi-Civita connection ......Page 149
2.7.2 The Weyl connection ......Page 152
2.8 Curvature ......Page 154
2.8.1 Applications to Weyl structures ......Page 159
2.9 Variation of geodesies ......Page 160
3.1 Light rays: the conformal structure ......Page 168
3.2 Inertial observers: the projective structure ......Page 175
3.3 Compatibility: Weyl structure ......Page 177
3.4 Reduction to the Lorentzian structure ......Page 183
4. Pseudo-Riemannian manifolds ......Page 188
4.1 Existence of Lorentzian and Riemannian manifolds ......Page 192
4.2 The volume form and the Hodge star operator ......Page 193
4.3 Curvature of pseudo-Riemannian manifolds ......Page 201
4.3.1 2-dimensional pseudo-Riemannian manifolds ......Page 208
4.4 Submanifolds ......Page 210
4.4.1 Hyperquadrics ......Page 219
4.4.2 Umbilic and totally geodesic submanifolds ......Page 221
4.4.3 Warped products ......Page 222
4.5 Isometries and Killing vector fields ......Page 226
4.6 Length and energy functionals ......Page 230
4.6.1 Variation of length and energy ......Page 233
4.6.2 Conjugate and focal points ......Page 244
4.6.3 Existence of focal points ......Page 258
5.1 Matter ......Page 272
5.2.1 The perfect fluid ......Page 281
5.2.2 The collisionless gas ......Page 282
5.2.3 The electromagnetic field ......Page 284
5.3 Einstein\'s equation ......Page 285
5.3.1 The Lagrangian formulation of Einstein\'s equation ......Page 288
5.4 The Einstein equation as a system of partial differential equations ......Page 296
6.1 Homogeneity and isotropy ......Page 304
6.2 The initial value problem for infmitesimally isotropic spacetimes ......Page 311
6.3 Geodesies and redshift ......Page 314
6.4 The age of the universe and the big bang ......Page 317
6.5 A simple model for the universe we live in ......Page 321
7. Spherical symmetry ......Page 324
7.1 Pseudo-Riemannian manifolds with spherical symmetry ......Page 325
7.2 The Schwarzschild solution ......Page 332
7.2.1 Experimental tests for the Schwarzschild solution ......Page 339
7.3 Quasi-linear hyperbolic systems of equations in two independent variables ......Page 345
7.4 The initial value problem for spherically symmetric perfect fluid spacetimes with non-interacting electromagnetic fields ......Page 354
7.5 Static perfect fluid stars ......Page 365
8. Causality ......Page 374
8.1 Causality conditions ......Page 375
8.2 Cluster and limit curves ......Page 382
8.3 Achronal submanifolds and Cauchy developments ......Page 391
9. Singularity theorems ......Page 400
9.1 Energy conditions ......Page 401
9.2 Closed trapped surfaces ......Page 406
9.3 The singularity theorem of Hawking and Penrose ......Page 407
9.3.1 Applications of the singularity theorem ......Page 412
9.4.1 The Godel solution ......Page 414
9.4.2 Newman\'s example ......Page 422
9.5 Strength of singularities and cosmic censorship ......Page 426
9.5.1 A simple, 3-dimensional example ......Page 429
References ......Page 442
Index ......Page 446