EINSTEIN FIELD EQUATION SOLUTIONS FOR RELATIVITY & ASTROPHYSICS

By ANTHONY JACKSON

CONTENTS SELECTED SOLUTIONS OF EINSTEIN’S FIELD EQUATIONS: THEIR ROLE IN GENERAL RELATIVITY AND ASTROPHYSICS INTRODUCTION AND A FEW EXCURSIONS A WORD ON THE ROLE OF EXPLICIT SOLUTIONS IN OTHER PARTS OF PHYSICS AND ASTROPHYSICS EINSTEIN’S FIELD EQUATIONS “JUST SO” NOTES ON THE SIMPLEST SOLUTIONS: THE MINKOWSKI, DE SITTER, AND ANTI-DE SITTER SPACETIMES ON THE INTERPRETATION AND CHARACTERIZATION OF METRICS THE CHOICE OF SOLUTIONS THE OUTLINE THE SCHWARZSCHILD SOLUTION SPHERICALLY SYMMETRIC SPACETIMES THE SCHWARZSCHILD METRIC AND ITS ROLE IN THE SOLAR SYSTEM SCHWARZSCHILD METRIC OUTSIDE A COLLAPSING STAR THE SCHWARZSCHILD–KRUSKAL SPACETIME THE SCHWARZSCHILD METRIC AS A CASE AGAINST LORENTZ-COVARIANT APPROACHES THE SCHWARZSCHILD METRIC AND ASTROPHYSICS THE REISSNER–NORDSTR¨OM SOLUTION REISSNER–NORDSTR¨OM BLACK HOLES AND THE QUESTION OF COSMIC CENSORSHIP ON EXTREME BLACK HOLES, D-DIMENSIONAL BLACK HOLES, STRING THEORY AND “ALL THAT” THE KERR METRIC BASIC FEATURES THE PHYSICS AND ASTROPHYSICS AROUND ROTATING BLACK HOLES ASTROPHYSICAL EVIDENCE FOR A KERR METRIC BLACK HOLE UNIQUENESS AND MULTI-BLACK HOLE SOLUTIONS ON STATIONARY AXISYMMETRIC FIELDS AND RELATIVISTIC DISKS STATIC WEYL METRICS RELATIVISTIC DISKS AS SOURCES OF THE KERR METRIC AND OTHER STATIONARY SPACETIMES UNIFORMLY ROTATING DISKS TAUB-NUT SPACE A NEW WAY TO THE NUT METRIC TAUB-NUT PATHOLOGIES AND APPLICATIONS PLANE WAVES AND THEIR COLLISIONS PLANE-FRONTED WAVES PLANE-FRONTED WAVES: NEW DEVELOPMENTS AND APPLICATIONS COLLIDING PLANE WAVES CYLINDRICAL WAVES CYLINDRICAL WAVES AND THE ASYMPTOTIC STRUCTURE OF 3-DIMENSIONAL GENERAL RELATIVITY CYLINDRICAL WAVES AND QUANTUM GRAVITY CYLINDRICAL WAVES: A MISCELLANY ON THE ROBINSON–TRAUTMAN SOLUTIONS THE BOOST-ROTATION SYMMETRIC RADIATIVE SPACETIMES THE COSMOLOGICAL MODELS 12 1 SPATIALLY HOMOGENEOUS COSMOLOGIES 12 2 INHOMOGENEOUS COSMOLOGIES CONCLUDING REMARKS REFERENCES THE CAUCHY PROBLEM FOR THE EINSTEIN EQUATIONS INTRODUCTION BASIC OBSERVATIONS AND CONCEPTS THE PRINCIPAL SYMBOL THE CONSTRAINTS THE BIANCHI IDENTITIES THE EVOLUTION EQUATIONS ASSUMPTIONS AND CONSEQUENCES PDE TECHNIQUES SYMMETRIC HYPERBOLIC SYSTEMS SYMMETRIC HYPERBOLIC SYSTEMS ON MANIFOLDS OTHER NOTIONS OF HYPERBOLICITY REDUCTIONS HYPERBOLIC SYSTEMS FROM THE ADM EQUATIONS THE EINSTEIN–EULER SYSTEM THE INITIAL BOUNDARY VALUE PROBLEM THE EINSTEIN–DIRAC SYSTEM REMARKS ON THE STRUCTURE OF THE CHARACTERISTIC SET LOCAL EVOLUTION LOCAL EXISTENCE THEOREMS FOR THE EINSTEIN EQUATIONS UNIQUENESS CAUCHY STABILITY MATTER MODELS AN EXAMPLE OF AN ILL-POSED INITIAL VALUE PROBLEM SYMMETRIES OUTLOOK POST-NEWTONIAN GRAVITATIONAL RADIATION ON APPROXIMATION METHODS IN GENERAL RELATIVITY FIELD EQUATIONS AND THE NO-INCOMING-RADIATION CONDITION METHOD AND GENERAL PHYSICAL PICTURE MULTIPOLE DECOMPOSITION THE MATCHING EQUATION THE FIELD IN TERMS OF MULTIPOLE MOMENTS EQUIVALENCE WITH THE WILL–WISEMAN MULTIPOLE EXPANSION SOURCE MULTIPOLE MOMENTS MULTIPOLE EXPANSION IN SYMMETRIC TRACE-FREE FORM LINEARIZED APPROXIMATION TO THE EXTERIOR FIELD DERIVATION OF THE SOURCE MULTIPOLE MOMENTS POST-MINKOWSKIAN APPROXIMATION MULTIPOLAR POST-MINKOWSKIAN ITERATION OF THE EXTERIOR FIELD THE “CANONICAL” MULTIPOLE MOMENTS RETARDED INTEGRAL OF A MULTIPOLAR EXTENDED SOURCE RADIATIVE MULTIPOLE MOMENTS DE?NITION AND GENERAL STRUCTURE THE RADIATIVE QUADRUPOLE MOMENT TO 3PN ORDER TAIL CONTRIBUTIONS IN THE TOTAL ENERGY FLUX POST-NEWTONIAN APPROXIMATION THE INNER METRIC TO 2 5PN ORDER THE MASS-TYPE SOURCE MOMENT TO 2 5PN ORDER POINT-PARTICLES HADAMARD PARTIE FINIE REGULARIZATION MULTIPOLE MOMENTS OF POINT-MASS BINARIES EQUATIONS OF MOTION OF COMPACT BINARIES GRAVITATIONAL WAVEFORMS OF INSPIRALLING COMPACT BINARIES CONCLUSION DUALITY AND HIDDEN SYMMETRIES IN GRAVITATIONAL THEORIES ELECTROMAGNETIC DUALITY DUALITY IN KAELUZA–KLEIN THEORIES DIMENSIONAL REDUCTION FROM D TO D DIMENSIONS REDUCTION TO D = 4 DIMENSIONS REDUCTION TO D = 3 DIMENSIONS REDUCTION TO D = 2 DIMENSIONS GEROCH GROUP STATIONARY BLACK HOLES SPHERICALLY SYMMETRIC SOLUTIONS UNIQUENESS THEOREMS FOR STATIC BLACK HOLES STATIONARY, AXIALLY SYMMETRIC BLACK HOLES ACKNOWLEDGMENTS NON-LINEAR ?-MODELS AND SYMMETRIC SPACES NON-COMPACT RIEMANNIAN SYMMETRIC SPACES PSEUDO-RIEMANNIAN SYMMETRIC SPACES CONSISTENT TRUNCATIONS STRUCTURE OF THE LIE ALGEBRA TIME-INDEPENDENT GRAVITATIONAL FIELDS FIELD EQUATIONS GENERALITIES AXIAL SYMMETRY ASYMPTOTIC FLATNESS: LICHNEROWICZ THEOREMS NEWTONIAN LIMIT EXISTENCE ISSUES AND THE NEWTONIAN LIMIT FAR FIELDS FAR-FIELD EXPANSIONS CONFORMAL TREATMENT OF IN?NITY, MULTIPOLE MOMENTS GLOBAL ROTATING SOLUTIONS LINDBLOM’S THEOREM EXISTENCE OF STATIONARY ROTATING AXI-SYMMETRIC FLUID BODIES THE NEUGEBAUER–MEINEL DISK GLOBAL NON-ROTATING SOLUTIONS ELASTIC STATIC BODIES ARE PERFECT FLUIDS O(3)-SYMMETRIC? SPHERICALLY SYMMETRIC, STATIC PERFECT FLUID SOLUTIONS SPHERICALLY SYMMETRIC, STATIC EINSTEIN–VLASOV SOLUTIONS GRAVITATIONAL LENSING FROM A GEOMETRIC VIEWPOINT SOME BASIC NOTIONS OF SPACETIME GEOMETRY GRAVITATIONAL LENSING IN ARBITRARY SPACETIMES CONJUGATE POINTS AND CUT POINTS THE GEOMETRY OF LIGHT CONES CITERIA FOR MULTIPLE IMAGING FERMAT’S PRINCIPLE MORSE INDEX THEORY FOR FERMAT’S PRINCIPLE GRAVITATIONAL LENSING IN GLOBALLY HYPERBOLIC SPACETIMES CRITERIA FOR MULTIPLE IMAGING IN GLOBALLY HYPERBOLIC SPACETIMES MORSE THEORY IN GLOBALLY HYPERBOLIC SPACETIMES GRAVITATIONAL LENSING IN ASYMPTOTICALLY SIMPLE AND EMPTY SPACETIMES