Einstein's field equations and their physical implications selected essays in honour of Jürgen Ehlers

Einstein's field equations and their physical implications selected essays in honour of Jürgen Ehlers

Gespeichert in:
Bibliographische Detailangaben
Format: Buch
Sprache:German
Veröffentlicht: Berlin [u.a.] Springer 2000
Schriftenreihe:Lecture notes in physics 540
Schlagworte:
Einstein field equations
Einstein-Feldgleichungen
Feldgleichung
Aufsatzsammlung
Festschrift
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adam_text CONTENTS SELECTED SOLUTIONS OF EINSTEIN*S FIELD EQUATIONS: THEIR ROLE IN GENERAL RELATIVITY AND ASTROPHYSICS JI* R´ * BI* C´ AK ...................................................... 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 SCHWARZSCHILD*KRUSKAL 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 REISSNER*NORDSTR¨ OM SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 REISSNER*NORDSTR¨ OM 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 X CONTENTS 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 ROBINSON*TRAUTMAN 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 EINSTEIN*EULER SYSTEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4.3 THE INITIAL BOUNDARY VALUE PROBLEM . . . . . . . . . . . . . . . . . . . . . . 185 4.4 THE EINSTEIN*DIRAC 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 CONTENTS XI 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 WILL*WISEMAN 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 KA* LUZA*KLEIN 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 XII CONTENTS 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 * -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 NEUGEBAUER*MEINEL 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 EINSTEIN*VLASOV 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 CONTENTS XIII 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 J¨ URGEN EHLERS * BIBLIOGRAPHY ................................. 427
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physical XIII, 433 S.
publishDate 2000
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publisher Springer
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series Lecture notes in physics
series2 Lecture notes in physics
spellingShingle Einstein's field equations and their physical implications selected essays in honour of Jürgen Ehlers
Lecture notes in physics
Einstein field equations
Einstein-Feldgleichungen (DE-588)4013941-4 gnd
Feldgleichung (DE-588)4131471-2 gnd
subject_GND (DE-588)4013941-4
(DE-588)4131471-2
(DE-588)4143413-4
(DE-588)4016928-5
title Einstein's field equations and their physical implications selected essays in honour of Jürgen Ehlers
title_auth Einstein's field equations and their physical implications selected essays in honour of Jürgen Ehlers
title_exact_search Einstein's field equations and their physical implications selected essays in honour of Jürgen Ehlers
title_full Einstein's field equations and their physical implications selected essays in honour of Jürgen Ehlers Bernd G. Schmidt (ed.)
title_fullStr Einstein's field equations and their physical implications selected essays in honour of Jürgen Ehlers Bernd G. Schmidt (ed.)
title_full_unstemmed Einstein's field equations and their physical implications selected essays in honour of Jürgen Ehlers Bernd G. Schmidt (ed.)
title_short Einstein's field equations and their physical implications
title_sort einstein s field equations and their physical implications selected essays in honour of jurgen ehlers
title_sub selected essays in honour of Jürgen Ehlers
topic Einstein field equations
Einstein-Feldgleichungen (DE-588)4013941-4 gnd
Feldgleichung (DE-588)4131471-2 gnd
topic_facet Einstein field equations
Einstein-Feldgleichungen
Feldgleichung
Aufsatzsammlung
Festschrift
url http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008805933&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link (DE-604)BV000003166
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