ISBN: 3-540-66377-0
TITLE: Spacetime
AUTHOR: Kriele, Marcus
TOC:

1. Local theory of space and time 1
1.1 Space 1
1.1.1 Affine space 2
1.1.2 The fundamental theorem in affine geometry and doubly ruled surface 3
1.1.3 Euclidean geometry 10
1.2 Absolute space and absolute time 13
1.2.1 Non-relativistic particles 16
1.3 Galilei's theory of relativity 18
1.4 Einstein's special theory of relativity 23
1.4.1 Causality in special relativity 35
1.4.2 Length contraction and time dilatation 37
1.4.3 Relativistic particles and photons 39
2. Analysis on manifolds 43
2.1 Manifolds 45
2.1.1 Construction of manifolds 50
2.1.2 Partition of unity 53
2.2 Vector bundles and the tangent bundle 57
2.2.1 Construction of the tangent bundle 59
2.2.2 The derivative of maps between manifolds 63
2.3 Tensors and tensor fields 64
2.3.1 Algebraic preliminaries: tensors 64
2.3.2 Tensor fields 79
2.4 Vector fields and ordinary differential equations 83
2.5 Differential forms 90
2.5.1 The lemma of Poincar 98
2.5.2 The theorem of Frobenius 102
2.5.3 Orientable real manifolds 106
2.5.4 Integration on real manifolds 108
2.6 Connections and projective structures 118
2.7 Examples of connections 128
2.7.1 The Levi-Civit a connection 129
2.7.2 The Weyl connection 131
2.8 Curvature 134
2.8.1 Applications to Weyl structures 138
2.9 Variation of geodesics 140
3. Space and time from a global point of view 147
3.1 Light rays: the conformal structure 147
3.2 Inertial observers: the projective structure 154
3.3 Compatibility: Weyl structure 156
3.4 Reduction to the Lorentzian structure 162
4. Pseudo-Riemannian manifolds 167
4.1 Existence of Lorentzian and Riemannian manifolds 171
4.2 The volume form and the Hodge star operator 172
4.3 Curvature of pseudo-Riemannian manifolds 180
4.3.1 2-dimensional pseudo-Riemannian manifolds 187
4.4 Submanifolds 189
4.4.1 Hyperquadrics 198
4.4.2 Umbilic and totally geodesic submanifolds 200
4.4.3 Warped products 201
4.5 Isometries and Killing vector fields 204
4.6 Length and energy functionals 209
4.6.1 Variation of length and energy 211
4.6.2 Conjugate and focal points 222
4.6.3 Existence of focal points 236
5. General relativity 251
5.1 Matter 251
5.2 Some specific matter models 260
5.2.1 The perfect fluid 260
5.2.2 The collisionless gas 261
5.2.3 The electromagnetic field 263
5.3 Einstein's equation 264
5.3.1 The Lagrangian formulation of Einstein's equation 266
5.4 The Einstein equation as a system of partial differential equations 275
6. Robertson-Walker cosmology 283
6.1 Homogeneity and isotropy 283
6.2 The initial value problem for infinitesimally isotropic spacetimes 290
6.3 Geodesies and redshift 293
6.4 The age of the universe and the big bang 296
6.5 A simple model for the universe we live in 300
7. Spherical symmetry 303
7.1 Pseudo-Riemannian manifolds with spherical symmetry 304
7.2 The Schwarzschild solution 311
7.2.1 Experimental tests for the Schwarzschild solution 318
7.3 Quasi-linear hyperbolic systems of equations in two independent variables 324
7.4 The initial value problem for spherically symmetric perfect fluid spacetimes with non-interacting electromagnetic fields 333
7.5 Static perfect fluid stars 344
8. Causality 353
8.1 Causality conditions 354
8.2 Cluster and limit curves 361
8.3 Achronal submanifolds and Cauchy developments 370
9. Singularity theorems 379
9.1 Energy conditions 380
9.2 Closed trapped surfaces 385
9.3 Tlie singularity theorem of Hawking and Penrose 386
9.3.1 Applications of the singularity theorem 391
9.3.2 General problems with Theorem 9.3.1 393
9.4 Singularities and causality violations 393
9.4.1 The Gdel solution 393
9.4.2 Newman's example 401
9.5 Strength of singularities and cosmic censorship 406
9.5.1 A simple, 3-dimensional example 408
References 421
Index 425
END
