ISBN: 3540670734
TITLE: Einstein's Field Equations and Their Physical Implications
AUTHOR: Schmidt, Bernd G. (Ed.)
TOC:

Selected Solutions of Einstein's Field Equations: Their Role in General Relativity and Astrophysics
Jir Bick 1
1 Introduction and a Few Excursions 1
1.1 A Word on the Role of Explicit Solutions in Other Parts of Physics and Astrophysics 3
1.2 Einstein's Field Equations 5
1.3 "Just So" Notes on the Simplest Solutions: The Minkowski, de Sitter, and Anti-de Sitter Spacetimes 8
1.4 On the Interpretation and Characterization of Metrics 11
1.5 The Choice of Solutions 15
1.6 The Outline 17
2 The Schwarzschild Solution 19
2.1 Spherically Symmetric Spacetimes 19
2.2 The Schwarzschild Metric and Its Role in the Solar System 20
2.3 Schwarzschild Metric Outside a Collapsing Star 21
2.4 The SchwarzschildKruskal Spacetime 25
2.5 The Schwarzschild Metric as a Case Against Lorentz-Covariant Approaches 28
2.6 The Schwarzschild Metric and Astrophysics 29
3 The ReissnerNordstrm Solution 31
3.1 ReissnerNordstrm Black Holes and the Question of Cosmic Censorship 32
3.2 On Extreme Black Holes, d-Dimensional Black Holes, String Theory and "All That" 39
4 The Kerr Metric 42
4.1 Basic Features 42
4.2 The Physics and Astrophysics Around Rotating Black Holes 47
4.3 Astrophysical Evidence for a Kerr Metric 50
5 Black Hole Uniqueness and Multi-black Hole Solutions 52
6 On Stationary Axisymmetric Fields and Relativistic Disks 55
6.1 Static Weyl Metrics 55
6.2 Relativistic Disks as Sources of the Kerr Metric and Other Stationary Spacetimes 57
6.3 Uniformly Rotating Disks 59
7 Taub-NUT Space 62
7.1 A New Way to the NUT Metric 62
7.2 Taub-NUT Pathologies and Applications 64
8 Plane Waves and Their Collisions 66
8.1 Plane-Fronted Waves 66
8.2 Plane-Fronted Waves: New Developments and Applications 71
8.3 Colliding Plane Waves 72
9 Cylindrical Waves 77
9.1 Cylindrical Waves and the Asymptotic Structure of 3-Dimensional General Relativity 78
9.2 Cylindrical Waves and Quantum Gravity 82
9.3 Cylindrical Waves: a Miscellany 85
10 On the RobinsonTrautman Solutions 86
11 The Boost-Rotation Symmetric Radiative Spacetimes 88
12 The Cosmological Models 93
12.1 Spatially Homogeneous Cosmologies 95
12.2 Inhomogeneous Cosmologies 102
13 Concluding Remarks 105
References 108
The Cauchy Problem for the Einstein Equations
Helmut Friedrich, Alan Rendall 127
1 Introduction 127
2 Basic Observations and Concepts 131
2.1 The Principal Symbol 132
2.2 The Constraints 135
2.3 The Bianchi Identities 137
2.4 The Evolution Equations 137
2.5 Assumptions and Consequences 146
3 PDE Techniques 147
3.1 Symmetric Hyperbolic Systems 147
3.2 Symmetric Hyperbolic Systems on Manifolds 157
3.3 Other Notions of Hyperbolicity 159
4 Reductions 164
4.1 Hyperbolic Systems from the ADM Equations 167
4.2 The EinsteinEuler System 173
4.3 The Initial Boundary Value Problem 185
4.4 The EinsteinDirac System 193
4.5 Remarks on the Structure of the Characteristic Set 200
5 Local Evolution 201
5.1 Local Existence Theorems for the Einstein Equations 201
5.2 Uniqueness 204
5.3 Cauchy Stability 206
5.4 Matter Models 207
5.5 An Example of an Ill-Posed Initial Value Problem 214
5.6 Symmetries 216
6 Outlook 217
References 219
Post-Newtonian Gravitational Radiation
Luc Blanchet 225
1 Introduction 225
1.1 On Approximation Methods in General Relativity 225
1.2 Field Equations and the No-Incoming-Radiation Condition 228
1.3 Method and General Physical Picture 231
2 Multipole Decomposition 233
2.1 The Matching Equation 233
2.2 The Field in Terms of Multipole Moments 236
2.3 Equivalence with the WillWiseman Multipole Expansion 238
3 Source Multipole Moments 240
3.1 Multipole Expansion in Symmetric Trace-Free Form 240
3.2 Linearized Approximation to the Exterior Field 241
3.3 Derivation of the Source Multipole Moments 242
4 Post-Minkowskian Approximation 244
4.1 Multipolar Post-Minkowskian Iteration of the Exterior Field 244
4.2 The "Canonical" Multipole Moments 246
4.3 Retarded Integral of a Multipolar Extended Source 247
5 Radiative Multipole Moments 248
5.1 Definition and General Structure 249
5.2 The Radiative Quadrupole Moment to 3PN Order 250
5.3 Tail Contributions in the Total Energy Flux 251
6 Post-Newtonian Approximation 253
6.1 The Inner Metric to 2.5PN Order 254
6.2 The Mass-Type Source Moment to 2.5PN Order 256
7 Point-Particles 258
7.1 Hadamard Partie Finie Regularization 259
7.2 Multipole Moments of Point-Mass Binaries 261
7.3 Equations of Motion of Compact Binaries 263
7.4 Gravitational Waveforms of Inspiralling Compact Binaries 265
8 Conclusion 267
Duality and Hidden Symmetries in Gravitational Theories
Dieter Maison 273
1 Introduction 273
2 Electromagnetic Duality 277
3 Duality in KaluzaKlein Theories 279
3.1 Dimensional Reduction from D to d Dimensions 280
3.2 Reduction to d = 4 Dimensions 282
3.3 Reduction to d = 3 Dimensions 285
3.4 Reduction to d = 2 Dimensions 290
4 Geroch Group 292
5 Stationary Black Holes 302
5.1 Spherically Symmetric Solutions 306
5.2 Uniqueness Theorems for Static Black Holes 312
5.3 Stationary, Axially Symmetric Black Holes 314
6 Acknowledgments 316
7 Non-linear sigma-Models and Symmetric Spaces 316
7.1 Non-compact Riemannian Symmetric Spaces 316
7.2 Pseudo-Riemannian Symmetric Spaces 319
7.3 Consistent Truncations 319
8 Structure of the Lie Algebra 319
Time-Independent Gravitational Fields
Robert Beig, Bernd Schmidt 325
1 Introduction 325
2 Field Equations 327
2.1 Generalities 327
2.2 Axial Symmetry 333
2.3 Asymptotic Flatness: Lichnerowicz Theorems 334
2.4 Newtonian Limit 339
2.5 Existence Issues and the Newtonian Limit 340
3 Far Fields 341
3.1 Far-Field Expansions 341
3.2 Conformal Treatment of Infinity, Multipole Moments 344
4 Global Rotating Solutions 350
4.1 Lindblom's Theorem 350
4.2 Existence of Stationary Rotating Axi-symmetric Fluid Bodies 353
4.3 The NeugebauerMeinel Disk 357
5 Global Non-rotating Solutions 360
5.1 Elastic Static Bodies 360
5.2 Are Perfect Fluids O(3)-Symmetric? 362
5.3 Spherically Symmetric, Static Perfect Fluid Solutions 365
5.4 Spherically Symmetric, Static EinsteinVlasov Solutions 370
Gravitational Lensing from a Geometric Viewpoint
Volker Perlick 373
1 Introduction 373
2 Some Basic Notions of Spacetime Geometry 375
3 Gravitational Lensing in Arbitrary Spacetimes 378
3.1 Conjugate Points and Cut Points 381
3.2 The Geometry of Light Cones 385
3.3 Citeria for Multiple Imaging 391
3.4 Fermat's Principle 396
3.5 Morse Index Theory for Fermat's Principle 399
4 Gravitational Lensing in Globally Hyperbolic Spacetimes 403
4.1 Criteria for Multiple Imaging in Globally Hyperbolic Spacetimes 405
4.2 Morse Theory in Globally Hyperbolic Spacetimes 408
5 Gravitational Lensing in Asymptotically Simple and Empty Spacetimes 414
References 422
Jrgen Ehlers  Bibliography 427
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