Table of Contents
Preface xi
Introduction xv
Chapter 1 Background 1
1.1 Manifolds 2
1.2 Tensors 8
1.3 Differential Geometry 12
1.4 Extending Manifolds 20
1.5 Lorentz Vector Spaces 25
1.6 Introduction to General Relativity 31
1.7 Submanifolds 41
1.8 Cartan Computations 49
1.9 Overview of a Kerr Black Hole 55
Chapter 2 Beginning Kerr Spacetime 57
2.1 The Kerr Metric 58
2.2 Boyer-Lindquist Blocks 61
2.3 Special Submanifolds 67
2.4 Ergosphere and Time Machine 70
2.5 Kerr-star Spacetime 79
2.6 Connection Forms 90
2.7 Kerr Curvature à la Cartan 96
Chapter 3 Maximal Extensions 105
3.1 Star-Kerr Spacetime 106
3.2 Maximal Extreme Kerr Spacetime 111
3.3 Extending Slow Kerr Spacetime 116
3.4 Building the Crossing Spheres 121
3.5 Maximal Slow Kerr Spacetime 131
3.6 Bundle Structure of Kerr Spacetime 140
3.7 Isometries of Boyer-Lindquist Blocks 149
3.8 Isometries of Me and Ms 155
3.9 Topology of Kerr Spacetime 163
3.10 Kerr Chronology 171
Chapter 4 Kerr Geodesies 177
4.1 First-Integrals 178
4.2 Carter Constant 182
4.3 Equations and Extensions 189
4.4 Crossing Horizons 196
4.5 Control of the v Coordinate 201
4.6 Control of the r Coordinate 207
4.7 r-L Plots 214
4.8 First-Integrals and Orbits 222
4.9 Vortical Timelike Geodesies 236
4.10 Timelike Global Trajectories 243
4.11 Axial Geodesies 250
4.12 Geodesies in Horizons 255
4.13 Polar Orbits 262
4.14 Equatorial Geodesies 272
4.15 Approaching the Center 288
Chapter 5 Petrov Types 297
5.1 Weyl Tensor 298
5.2 Hodge Star 303
5.3 Commutativity 308
5.4 Petrov Classification 312
5.5 Principal Null Directions 317
5.6 Type D Curvature 322
5.7 The Optical Scalars 327
5.8 Newman-Penrose Formalism 332
5.9 Bianchi Identities and Type D 341
5.10 Goldberg-Sachs Theorem 345
Appendix A Units 351
Appendix B Differential Forms 355
Appendix C Carter Constant 357
Appendix D Exterior Products 361
Index of Notations 365
Bibliography 367
Index 371
