General relativity. J. Ehlers honorary volume

Jurgen Ehlers......Page 1
Contents......Page 3
1 Introduction and a Few Excursions......Page 10
1.1 A Word on the Role of Explicit Solutions in Other Parts of Physics and Astrophysics......Page 12
1.2 Einstein’s Field Equations......Page 14
1.3 "Just So" Notes on the Simplest Solutions: The Minkowski, de Sitter, and Anti-de Sitter Spacetimes......Page 17
1.4 On the Interpretation and Characterization of Metrics......Page 20
1.5 The Choice of Solutions......Page 24
1.6 The Outline......Page 26
2.1 Spherically Symmetric Spacetimes......Page 28
2.2 The Schwarzschi d Metric and Its Role in the Solar System......Page 29
2.3 Schwarzschi d Metric Outside a Colapsing Star......Page 30
2.4 The Schwarzschild–Kruska Spacetime......Page 34
2.5 The Schwarzschild Metric as a Case Against Lorentz-Covariant Approaches......Page 37
2.6 The Schwarzschild Metric and Astrophysics......Page 38
3 The Reissner–Nordström Solution......Page 40
3.1 Reissner–Nordström B ack Holes and the Question of Cosmic Censorship......Page 41
3.2 On Extreme B ack Holes, d-Dimensional Black Holes,  String Theora and "All That"......Page 48
4.1 Basic Features......Page 51
4.2 The Physics and Astrophysics Around Rotating Black Holes......Page 56
4.3 Astrophysica Evidence for a Kerr Metric......Page 59
5 Black Hole Uniqueness and Multi-black Solutions......Page 61
6.1 Static Wey Metrics......Page 64
6.2 Relativistic Disks as Sources of the Kerr Metric and Other Stationary Spacetimes......Page 66
6.3 Uniformly Rotating Disks......Page 68
7.1 A New Way to the NUT Metric......Page 71
7.2 Taub-NUT Pathologies and Applications......Page 73
8.1 Plane-Fronted Waves......Page 75
8.2 Plane-Fronted Waves: New Deve Developments and Applications......Page 80
8.3 Colliding Plane Waves......Page 81
9 Cylindrical Waves......Page 86
9.1 Cylindrica Waves and the Asymptotic Structure of 3-Dimensional General Relativity......Page 87
9.2 Cylindrica Waves and Quantum Gravity......Page 91
9.3 Cylindrica Waves: a Miscellany......Page 94
10 On the Robinson–Trautman Solutions......Page 95
11 The Boost–Rotation Symmetric Radiative Spacetimes......Page 97
12 The Cosmological Models......Page 102
12.1 Spatially Homogeneous Cosmologies......Page 104
12.2 Inhomogeneous Cosmologies......Page 111
13 Concluding Remarks......Page 114
Acknowledgements......Page 116
References......Page 117
1 Introduction......Page 136
2 Basic Observations and Concepts......Page 140
2.1 The Principal Symbol......Page 141
2.2 The Constraints......Page 144
2.4 The Evolution Equations......Page 146
2.5 Assumptions and Consequences......Page 155
3.1 Symmetric Hyperbolic Systems......Page 156
3.2 Symmetric Hyperbolic Systems on Manifolds......Page 166
3.3 Other Notions of Hyperbolicity......Page 168
4 Reductions......Page 173
4.1 Hyperbolic Systems from the ADM Equations......Page 176
4.2 The Einstein–Euler System......Page 182
4.3 The Initial Boundary Value Problem......Page 194
4.4 The Einstein–Dirac System......Page 202
4.5 Remarks on the Structure of the Characteristic Set......Page 209
5.1 Local Existence Theorems for the Einstein Equations......Page 210
5.2 Uniqueness......Page 213
5.3 Cauchy Stability......Page 215
5.4 Matter Models......Page 216
5.5 An Example of an Ill-Posed Initial Value Problem......Page 223
5.6 Symmetries......Page 225
6 Outlook......Page 226
References......Page 228
1.1 On Approximation Methods in General Relativity......Page 233
1.2 Field Equations and the No-Incoming-Radiation Condition......Page 236
1.3 Method and General Physi al Picture......Page 239
2.1 The Matching Equation......Page 241
2.2 The Field in Terms of Multipole Moments......Page 244
2.3 Equivalence with the Will–Wiseman Multipole Expansion......Page 246
3.1 Multipole Expansion in Symmetri Trace-Free Form......Page 248
3.2 Linearized Approximation to the Exterior Field......Page 249
3.3 Derivation of the Source Multipole Moments......Page 250
4.1 Multipolar Post-Minkowskian Iteration of the Exterior Field......Page 252
4.2 The “Canonical” Multipole Moments......Page 254
4.3 Retarded Integral of a Multipolar Extended Source......Page 255
5 Radiative Multipole Moments......Page 256
5.1 Definition and General Structure......Page 257
5.2 The Radiative Quadrupole Moment to 3PN Order......Page 258
5.3 Tail Contributions in the Total Energy Flux......Page 259
6 Post-Newtonian Approximation......Page 261
6.1 The Inner Metri to 2.5PN Order......Page 262
6.2 The Mass-Type Source Moment to 2.5PN Order......Page 264
7 Point-Particles......Page 266
7.1 Hadamard Partie Finie Regularization......Page 267
7.2 Multipole Moments of Point-Mass Binaries......Page 269
7.3 Equations of Motion of Compact Binaries......Page 271
7.4 Gravitational Waveforms of Inspiralling Compact Binaries......Page 273
8 Conclusion......Page 275
References......Page 276
1 Introduction......Page 280
2 Electromagnetic Duality......Page 284
3 Duality in Ka luza–Klein Theories......Page 286
3.1 Dimensionat Reduction from D to d Dimensions......Page 287
3.2 Reduction to d = 4 Dimensions......Page 289
3.3 Reduction to d = 3 Dimensions......Page 292
3.4 Reduction to d = 2 Dimensions......Page 297
4 Geroch Group......Page 299
5 Stationary Black Holes......Page 309
5.1 Spherically Symmetric Solutions......Page 313
5.2 Uniqueness Theorems for Static Black Holes......Page 319
5.3 Stationary, Axially Symmetric Black Holes......Page 321
7.1 Non-compact Riemannian Symmetric Spaces......Page 323
8 Structure of the Lie Algebra......Page 326
References......Page 328
1 Introduction......Page 331
2.1 Generalities......Page 333
2.2 Axial Symmetry......Page 339
2.3 Asymptotic Flatness: Lichnerowicz Theorems......Page 340
2.4 Newtonian Limit......Page 345
2.5 Existence Issues and the Newtonian Limit......Page 346
3.1 Far-Field Expansions......Page 347
3.2 Conformal Treatment of Infinity, Multipole Moments......Page 350
4.1 Lindblom’s Theorem......Page 356
4.2 Existence of Stationary Rotating Axi-symmetric Fluid Bodies......Page 359
4.3 The Neugebauer–Meinel Disk......Page 363
5.1 Elastic Static Bodies......Page 366
5.2 Are Perfect Fluis O(3)-Symmetric?......Page 368
5.3 Spherically Symmetric, Static Perfect Fluid Solutions......Page 371
5.4 Spherically Symmetric, Static Einstein–Vlasov Solutions......Page 376
References......Page 377
1 Introduction......Page 379
2 Some Basic Notions of Spacetime Geometry......Page 381
3 Gravitational Lensing in Arbitrary Spacetimes......Page 384
3.1 Conjugate Points and Cut Points......Page 387
3.2 The Geometry of Light Cones......Page 391
3.3 Citeria for Multiple Imaging......Page 397
3.4 Fermat’s Principle......Page 402
3.5 Morse Index Theory for Fermat’s Principle......Page 405
4 Gravitational Lensing in Globally Hyperbolic Spacetimes......Page 409
4.1 Criteria for Multiple Imaging in Globally Hyperbolic Spacetimes......Page 411
4.2 Morse Theory in Globally Hyperbolic Spacetimes......Page 414
5 Gravitational Lensing in Asymptotically Simple and Empty Spacetimes......Page 420
References......Page 428
Jurgen Ehlers - Bibliography......Page 432