ANALYTICAL AND NUMERICAL APPROACHES TO MATHEMATICAL RELATIVITY

By F. NAVIT

INTRODUCTION SOME ASPECTS OF LIMIT CONSTRUCTIONS THE LORENTZIAN DISTANCE FUNCTION AND CAUSAL DISCONNECTION THE STABILITY OF GEODESIC COMPLETENESS REVISITED THE LORENTZIAN SPLITTING PROBLEM GRAVITATIONAL PLANE WAVES AND THE NONSPACELIKE CUT LOCUS SOME MORE CURRENT ISSUES REFERENCES THE SPACE OF NULL GEODESICS (AND A NEW CAUSAL BOUNDARY) R J LOW SPACE OF NULL GEODESICS STRUCTURES ON THE SPACE OF NULL GEODESICS INSIGHT INTO SPACE-TIME RECOVERING SPACE-TIME A (NEW?) CAUSAL BOUNDARY SOME VARIATIONAL PROBLEMS IN SEMI-RIEMANNIAN GEOMETRY A MASIELLO A REVIEW OF VARIATIONAL METHODS GEODESICS ON RIEMANNIAN MANIFOLDS GEODESICS ON STATIONARY LORENTZIAN MANIFOLDS GEODESICS ON SPLITTING LORENTZIAN MANIFOLDS RESULTS ON MANIFOLDS WITH BOUNDARY OTHER DIRECTIONS ON THE GEOMETRY OF PP-WAVE TYPE SPACETIMES J L FLORES AND M S´ANCHEZ GENERAL PROPERTIES OF THE CLASS OF WAVES DE?NITIONS CURVATURE AND MATTER FINITENESS OF THE WAVE AND DECAY OF H AT IN?NITY CAUSALITY POSITIONS IN THE CAUSAL LADDER CAUSAL CONNECTIVITY TO IN?NITY AND HORIZONS GEODESIC COMPLETENESS GENERIC RESULTS EHLERS–KUNDT QUESTION GEODESIC CONNECTEDNESS AND CONJUGATE POINTS THE LORENTZIAN PROBLEM RELATION WITH A PURELY RIEMANNIAN VARIATIONAL PROBLEM OPTIMAL RESULTS FOR CONNECTEDNESS OF PFWS CONJUGATE POINTS PART II ANALYTICAL METHODS AND DI?ERENTIAL EQUATIONS CONCEPTS OF HYPERBOLICITY AND RELATIVISTIC CONTINUUM MECHANICS R BEIG HYPERBOLIC POLYNOMIALS INITIAL VALUE PROBLEM ELLIPTIC SYSTEMS S DAIN SECOND ORDER ELLIPTIC EQUATIONS ELLIPTIC SYSTEMS DE?NITION OF ELLIPTICITY DE?NITION OF ELLIPTIC BOUNDARY CONDITIONS RESULTS FINAL COMMENTS MATHEMATICAL PROPERTIES OF COSMOLOGICAL MODELS WITH ACCELERATED EXPANSION A D RENDALL PHYSICAL BACKGROUND MATHEMATICAL DEVELOPMENTS MATHEMATICS AND PHYSICS COMPARED SCALAR FIELDS RELATIONS BETWEEN PERFECT FLUIDS AND SCALAR FIELDS TACHYONS AND PHANTOM FIELDS CLOSING REMARKS THE POINCAR´E STRUCTURE AND THE CENTRE-OF-MASS OF ASYMPTOTICALLY FLAT SPACETIMES L B SZABADOS SYMMETRIES AND CONSERVED QUANTITIES IN MINKOWSKI SPACETIME THE KILLING FIELDS OF THE MINKOWSKI SPACETIME QUASI-LOCAL ENERGY-MOMENTUM AND ANGULAR MOMENTUM TOTAL ENERGY-MOMENTUM AND ANGULAR MOMENTUM ASYMPTOTICALLY CARTESIAN COORDINATE SYSTEMS CONSERVATION PROPERTIES ASYMPTOTICALLY FLAT SPACETIMES THE BOUNDARY CONDITIONS THE EVOLUTION EQUATIONS THE HAMILTONIAN PHASE SPACE OF VACUUM GR THE PHASE SPACE AND THE GENERAL BEIG–´O MURCHADHA HAMILTONIAN PHYSICAL QUANTITIES FROM THE BEIG–´O MURCHADHA HAMILTONIANS WITH TIME-INDEPENDENT LAPSES AND SHIFTS TRANSFORMATION AND CONSERVATION PROPERTIES THREE DI?CULTIES THE ASYMPTOTIC SPACETIME KILLING VECTORS THE 3 + 1 FORM OF THE LIE BRACKETS AND THE KILLING OPERATORS THE ASYMPTOTIC KILLING VECTORS THE ALGEBRA OF ASYMPTOTIC SYMMETRIES BEIG–´O MURCHADHA HAMILTONIANS WITH ASYMPTOTIC SPACETIME KILLING VECTORS PHYSICAL QUANTITIES FROM THE BEIG–´O MURCHADHA HAMILTONIANS WITH ASYMPTOTIC SPACETIME KILLING VECTORS THE GENERAL DE?NITION OF THE PHYSICAL QUANTITIES TOTAL ENERGY, MOMENTUM, ANGULAR MOMENTUM AND CENTRE-OF-MASS TRANSLATIONS FOR SLOW FALL-O?METRICS SUMMARY PART III NUMERICAL METHODS COMPUTER SIMULATION – A TOOL FOR MATHEMATICAL RELATIVITY – AND VICE VERSA B K BERGER MIXMASTER DYNAMICS AND THE BKL CONJECTURE HOW SPATIALLY HOMOGENEOUS COSMOLOGIES COLLAPSE DO U(1)-SYMMETRIC COSMOLOGIES EXHIBIT LMD? MATHEMATICAL-NUMERICAL SYNERGY IN SPATIALLY INHOMOGENEOUS COSMOLOGIES GOWDY MODELS AS AN EXAMPLE EXPANDING GOWDY SPACE-TIMES GENERAL T 2-SYMMETRIC SPACE-TIMES AS A “LABORATORY” FOR STRONG FIELD GRAVITY CONCLUSIONS ON BOUNDARY CONDITIONS FOR THE EINSTEIN EQUATIONS S FRITTELLI AND R G´OMEZ PRELIMINARIES THE COMPONENTS OF THE PROJECTION GABEB = AS BOUNDARY CONDITIONS STRONGLY HYPERBOLIC FORMULATIONS OF THE EINSTEIN EQUATIONS THE CASE OF THE EINSTEIN–CHRISTO?EL FORMULATION THE PROJECTION GABEB = 0 IN RELATION TO THE PROPAGATION OF THE CONSTRAINTS THE CASE OF THE ADM EQUATIONS CONCLUDING REMARKS RECENT ANALYTICAL AND NUMERICAL TECHNIQUES APPLIED TO THE EINSTEIN EQUATIONS D NEILSEN, L LEHNER, O SARBACH AND M TIGLIO ANALYTICAL AND NUMERICAL TOOLS GUIDELINES FOR A STABLE NUMERICAL IMPLEMENTATION CONSTRAINT-PRESERVING BOUNDARY CONDITIONS DEALING WITH “TOO MANY” FORMULATIONS PARAMETERS VIA CONSTRAINT MONITORING APPLICATIONS BUBBLE SPACE-TIMES BLACK HOLES FINAL WORDS SOME MATHEMATICAL PROBLEMS IN NUMERICAL RELATIVITY M BABIUC, B SZIL´AGYI, J WINICOUR WAVES UNBOUNDED EXPONENTIAL GROWTH MOVING BOUNDARIES GENERAL RELATIVITY: HARMONIC EVOLUTION THE HARMONIC IBVP SOMMERFELD ALTERNATIVES