Semi-Riemannian Geometry With Applications to Relativity (eBook)

Front Cover 1
SEMI-RIEMANNIAN GEOMETRY 4
Copyright Page 5
CONTENTS 6
Preface 12
Notation and Terminology 14
CHAPTER 1. MANIFOLD THEORY 16
Smooth Manifolds 16
Smooth Mappings 19
Tangent Vectors 21
Differential Maps 24
Curves 25
Vector Fields 27
One-Forms 29
Submanifolds 30
Immersions and Submersions 34
Topology of Manifolds 36
Some Special Manifolds 39
Integral Curves 42
CHAPTER 2. TENSORS 49
Basic Algebra 49
Tensor Fields 50
Interpretations 51
Tensors at a Point 52
Tensor Components 54
Contraction 55
Covariant Tensors 57
Tensor Derivations 58
Symmetric Bilinear Forms 61
Scalar Products 62
CHAPTER 3. SEMI-RIEMANNIAN MANIFOLDS 69
Isometries 73
The Levi-Civita Connection 74
Parallel Translation 80
Geodesics 82
The Exponential Map 85
Curvature 89
Sectional Curvature 92
Semi-Riemannian Surfaces 95
Type-Changing and Metric Contraction 96
Frame Fields 99
Some Differential Operators 100
Ricci and Scalar Curvature 102
Semi-Riemannian Product Manifolds 104
Local Isometries 105
Levels of Structure 108
CHAPTER 4. SEMI-RIEMANNIAN SUBMANIFOLDS 112
Tangents and Normals 112
The Induced Connection 113
Geodesics in Submanifolds 117
Totally Geodesic Submanifolds 119
Semi-Riemannian Hypersurfaces 121
Hyperquadrics 123
The Codazzi Equation 129
Totally Umbilic Hypersurfaces 131
The Normal Connection 133
A Congruence Theorem 135
Isometric Immersions 136
Two-Parameter Maps 137
CHAPTER 5. RIEMANNIAN AND LORENTZ GEOMETRY 141
The Gauss Lemma 141
Convex Open Sets 144
Arc Length 146
Riemannian Distance 147
Riemannian Completeness 153
Lorentz Causal Character 155
Timecones 158
Local Lorentz Geometry 161
Geodesics in Hyperquadrics 164
Geodesics in Surfaces 165
Completeness and Extendibility 169
CHAPTER 6. SPECIAL RELATIVITY 173
Newtonian Space and Time 173
Newtonian Space–Time 175
Minkowski Spacetime 178
Minkowski Geometry 179
Particles Observed 182
Some Relativistic Effects 186
Lorentz–Fitzgerald Contraction 189
Energy–Momentum 191
Collisions 194
An Accelerating Observer 196
CHAPTER 7. CONSTRUCTIONS 200
Deck Transformations 200
Orbit Manifolds 202
Orientability 204
Semi-Riemannian Coverings 206
Lorentz Time-Orientability 209
Volume Elements 209
Vector Bundles 212
Local Isometries 215
Matched Coverings 218
Warped Products 219
Warped Product Geodesics 222
Curvature of Warped Products 224
Semi-Riemannian Submersions 227
CHAPTER 8. SYMMETRY AND CONSTANT CURVATURE 230
Jacobi Fields 230
Tidal Forces 233
Locally Symmetric Manifolds 234
Isometries of Normal Neighborhoods 236
Symmetric Spaces 239
Simply Connected Space Forms 242
Transvections 246
CHAPTER 9. ISOMETRIES 248
Semiorthogonal Groups 248
Some Isometry Groups 254
Time-Orientability and Space-Orientability 255
Linear Algebra 257
Space Forms 258
Killing Vector Fields 264
The Lie Algebra i(M) 267
I( M ) as Lie Group 269
Homogeneous Spaces 272
CHAPTER 10. CALCULUS OF VARIATIONS 278
First Variation 278
Second Variation 281
The Index Form 283
Conjugate Points 285
Local Minima and Maxima 287
Some Global Consequences 292
The Endmanifold Case 295
Focal Points 296
Applications 301
Variation of E 303
Focal Points along Null Geodesics 305
A Causality Theorem 308
CHAPTER 11. HOMOGENEOUS AND SYMMETRIC SPACES 315
More about Lie Groups 315
Bi-Invariant Metrics 319
Coset Manifolds 321
Reductive Homogeneous Spaces 325
Symmetric Spaces 330
Riemannian Symmetric Spaces 334
Duality 336
Some Complex Geometry 338
CHAPTER 12. GENERAL RELATIVITY COSMOLOGY
Foundations 347
The Einstein Equation 351
Perfect Fluids 352
Robertson–Walker Spacetimes 356
The Robertson–Walker Flow 360
Robertson–Walker Cosmology 362
Friedmann Models 365
Geodesics and Redshift 368
Observer Fields 373
Static Spacetimes 375
CHAPTER 13. SCHWARZSCHILD GEOMETRY 379
Building the Model 379
Geometry of N and B 383
Schwarzschild Observers 386
Schwarzschild Geodesics 387
Free Fall Orbits 389
Perihelion Advance 393
Lightlike Orbits 395
Stellar Collapse 399
The Kruskal Plane 401
Kruskal Spacetime 404
Black Holes 407
Kruskal Geodesics 410
CHAPTER 14. CAUSALITY IN LORENTZ MANIFOLDS 416
Causality Relations 417
Quasi-Limits 419
Causality Conditions 422
Time Separation 424
Achronal Sets 428
Cauchy Hypersurfaces 430
Warped Products 432
Cauchy Developments 434
Spacelike Hypersurfaces 440
Cauchy Horizons 443
Hawking’s Singularity Theorem 446
Penrose’s Singularity Theorem 449
APPENDIX A. FUNDAMENTAL GROUPS AND COVERING MANIFOLDS 456
APPENDIX B. LIE GROUPS 461
Lie Algebras 462
Lie Exponential Map 464
The Classical Groups 465
APPENDIX C. NEWTONIAN GRAVITATION 468
References 471
Index 474