This book offers an up-to-date look at black holes from a mathematical perspective containing many results never before published. It can be read as a broadly based introduction to modern techniques in differential geometry.
Author(s): Barrett O'Neill
Publisher: A K Peters/CRC Press
Year: 1992
Language: English
Pages: 399
Preface......Page 10
Introduction......Page 14
Chapter 1 Background......Page 18
1.1 Manifolds......Page 19
1.2 Tensors......Page 25
1.3 Differential Geometry......Page 29
1.4 Extending Manifolds......Page 37
1.5 Lorentz Vector Spaces......Page 42
1.6 Introduction to General Relat......Page 48
1.7 Submanifolds......Page 58
1.8 Cartan Computations......Page 66
1.9 Overview of a Kerr Black Hole......Page 72
Chapter 2 Beginning Kerr Spacetime......Page 74
2.1 The Kerr Metric......Page 75
2.2 Boyer-Lindquist Blocks......Page 78
2.3 Special Submanifolds......Page 84
2.4 Ergosphere and Time Machine......Page 87
2.5 Kerr-star Spacetime......Page 96
2.6 Connection Forms......Page 107
2.7 Kerr Curvature a la Cartan......Page 113
Chapter 3 Maximal Extensions......Page 122
3.1 Star-Kerr Spacetime......Page 123
3.2 Maximal Extreme Kerr Spacetime......Page 128
3.3 Extending Slow Kerr Spacetime......Page 133
3.4 Building the Crossing Spheres......Page 138
3.5 Maximal Slow Kerr Spacetime......Page 148
3.6 Bundle Structure of Kerr Spacetime......Page 157
3.7 Isometries of Boyer-Lindquist Blocks......Page 166
3.8 Isometries of M_e and M_s......Page 172
3.9 Topology of Kerr Spacetime......Page 180
3.10 Kerr Chronology......Page 188
Chapter 4 Kerr Geodesies......Page 194
4.1 First-Integrals......Page 195
4.2 Carter Constant......Page 199
4.3 Equations and Extensions......Page 206
4.4 Crossing Horizons......Page 213
4.5 Control of the ϑ Coordinate......Page 218
4.6 Control of the r Coordinate......Page 224
4.7 r-L Plots......Page 231
4.8 First-Integrals and Orbits......Page 239
4.9 Vortical Timelike Geodesies......Page 253
4.10 Timelike Global Traj ectories......Page 260
4.11 Axial Geodesies......Page 267
4.12 Geodesies in Horizons......Page 272
4.13 Polar Orbits......Page 279
4.14 Equatorial Geodesies......Page 289
4.15 Approaching the Center......Page 305
Chapter 5 Petrov Types......Page 314
5.1 Weyl Tensor......Page 315
5.2 Hodge Star......Page 320
5.3 Commutativity......Page 325
5.4 Petrov Classification......Page 329
5.5 Principal Null Directions......Page 334
5.6 Type D Curvature......Page 339
5.7 The Optical Scalars......Page 344
5.8 Newman-Penrose Formalism......Page 349
5.9 Bianchi Identities and Type D......Page 358
5.10 Goldberg-Sachs Theorem......Page 362
Appendix A Units......Page 368
Appendix B Differential Forms......Page 372
Appendix C Carter Constant......Page 374
Appendix D Exterior Products......Page 378
Index of Notations......Page 382
Bibliography......Page 384
Index......Page 388
