The Geometry of Kerr Black Holes


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.
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The The Geometry of Kerr Black Holes Geometry of Kerr Black Holes

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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.
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Editorial Reviews

An introduction to modern differential geometry and relativity theory that studies the global geometry of Kerr spacetime--the relativistic model of the gravitational field of a rotating central mass. Three of the five chapters deal exclusively with Kerr spacetime (including geodesics and maximal extensions) while one gives general background and the fifth is an exposition of Petrov types and null congruences. Intended to be accessible to students and researchers in fields from abstract Lorentz geometry to observational astronomy. Annotation c. Book News, Inc., Portland, OR (
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Product Details

  • ISBN-13: 9781568810195
  • Publisher: CRC Press
  • Publication date: 6/15/1992
  • Pages: 400

Table of Contents

Ch. 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
Ch. 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 a la Cartan 96
Ch. 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 M[subscript e] and M[subscript s] 155
3.9 Topology of Kerr Spacetime 163
3.10 Kerr Chronology 171
Ch. 4 Kerr Geodesics 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 [actual symbol not reproducible] 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 Geodesics 236
4.10 Timelike Global Trajectories 243
4.11 Axial Geodesics 250
4.12 Geodesics in Horizons 255
4.13 Polar Orbits 262
4.14 Equatorial Geodesics 272
4.15 Approaching the Center 288
Ch. 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
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