General relativity and the Einstein equations / / Yvonne Choquet-Bruhat

• Autore: Choquet-Bruhat Yvonne
• Pubblicazione: Oxford ; ; New York, : Oxford University Press, 2009
• Edizione: 1st ed.
• Descrizione fisica: xxiv, 785 p. : ill
• Disciplina: 530.11
• Soggetto topico: General relativity (Physics) - Mathematics; Einstein field equations
• Nota di bibliografia: Includes bibliographical references (p. [771]-779) and index.
• Nota di contenuto: Intro -- CONTENTS -- I: Lorentz geometry -- 1 Introduction -- 2 Manifolds -- 3 Differentiable mappings -- 4 Vectors and tensors -- 4.1 Tangent and cotangent space -- 4.2 Vector fields -- 4.3 Tensors and tensor fields -- 5 Pseudo-Riemannian metrics -- 5.1 General properties -- 5.2 Riemannian and Lorentzian metrics -- 6 Riemannian connection -- 7 Geodesics -- 8 Curvature -- 9 Geodesic deviation -- 10 Maximum of length and conjugate points -- 11 Linearized Ricci and Einstein tensors -- 12 Second derivative of the Ricci tensor -- II: Special Relativity -- 1 Newton's mechanics -- 1.1 The Galileo-Newton spacetime -- 1.2 Newton's dynamics - the Galileo group -- 2 Maxwell's equations -- 3 Minkowski spacetime -- 3.1 Definition -- 3.2 Maxwell's equations on M[sub(4)] -- 4 Poincaré group -- 5 Lorentz group -- 5.1 General formulae -- 5.2 Transformation of electric and magnetic vector fields (case n = 3) -- 5.3 Lorentz contraction and dilatation -- 6 Special Relativity -- 6.1 Proper time -- 6.2 Proper frame and relative velocities -- 7 Dynamics of a pointlike mass -- 7.1 Newtonian law -- 7.2 Relativistic law -- 7.3 Equivalence of mass and energy -- 8 Continuous matter -- 8.1 Case of dust (incoherent matter) -- 8.2 Perfect fluids -- III: General relativity and Einstein's equations -- 1 Introduction -- 2 Newton's gravity law -- 3 General relativity -- 3.1 Physical motivations -- 4 Observations and experiments -- 4.1 Deviation of light rays -- 4.2 Proper time, gravitational time delay -- 5 Einstein's equations -- 5.1 Vacuum case -- 5.2 Equations with sources -- 6 Field sources -- 6.1 Electromagnetic sources -- 6.2 Electromagnetic potential -- 6.3 Yang-Mills fields -- 6.4 Scalar fields -- 6.5 Wave maps -- 6.6 Energy conditions -- 7 Lagrangians -- 7.1 Einstein-Hilbert Lagrangian -- 7.2 Lagrangians and stress energy tensors of sources -- 7.3 Coupled Lagrangian. ; 8 Fluid sources -- 9 Einsteinian spacetimes -- 9.1 Definition -- 9.2 Regularity hypotheses -- 10 Newtonian approximation -- 10.1 Equations for potentials -- 10.2 Equations of motion -- 11 Gravitational waves -- 11.1 Minkowskian approximation -- 11.2 General linear waves -- 12 High-frequency gravitational waves -- 12.1 Phase and polarizations -- 12.2 Radiative coordinates -- 12.3 Energy conservation -- 13 Coupled electromagnetic and gravitational waves -- 13.1 Phase and polarizations -- 13.2 Propagation equations -- IV: Schwarzschild spacetime and black holes -- 1 Introduction -- 2 Spherically symmetric spacetimes -- 3 Schwarzschild metric -- 4 Other coordinates -- 4.1 Isotropic coordinates -- 4.2 Wave coordinates -- 4.3 Painlevé-Gullstrand-like coordinates -- 4.4 Regge-Wheeler coordinates -- 5 Schwarzschild spacetime -- 6 The motion of the planets and perihelion precession -- 6.1 Equations -- 6.2 Results of observations -- 6.3 Escape velocity -- 7 Stability of circular orbits -- 8 Deflection of light rays -- 8.1 Theoretical prediction -- 8.2 Results of observation -- 8.3 Fermat's principle and light travel parameter time -- 9 Red shift and time delay -- 10 Spherically symmetric interior solutions -- 10.1 Static solutions. Upper limit on mass -- 10.2 Matching with an exterior solution -- 10.3 Non-static solutions -- 11 The Schwarzschild black hole -- 11.1 The event horizon -- 11.2 The Eddington-Finkelstein extension -- 11.3 Eddington-Finkelstein white hole -- 11.4 Kruskal complete spacetime -- 11.5 Observations -- 12 Spherically symmetric gravitational collapse -- 12.1 Tolman metric -- 12.2 Monotonically decreasing density -- 13 The Reissner-Nordström solution -- 14 Schwarzschild spacetime in dimension n + 1 -- 14.1 Standard coordinates -- 14.2 Wave coordinates -- V: Cosmology -- 1 Introduction -- 2 Cosmological principle. ; 3 Isotropic and homogeneous Riemannian manifolds -- 3.1 Isotropy -- 3.2 Homogeneity -- 4 Robertson-Walker spacetimes -- 4.1 Space metrics -- 4.2 Robertson-Walker spacetime metrics -- 4.3 Robertson-Walker dynamics -- 4.4 Einstein static universe -- 4.5 Cosmological red shift and the Hubble constant -- 4.6 De Sitter spacetime -- 4.7 Anti de Sitter (AdS) spacetime -- 5 Friedmann-Lemaître models. -- 5.1 Equation of state -- 5.2 General properties -- 5.3 Friedmann models -- 5.4 Some other models -- 5.5 Confrontation with observations -- 6 Homogeneous non-isotropic cosmologies -- 7 Bianchi class I universes -- 7.1 Kasner solutions -- 7.2 Models with matter -- 8 Bianchi type IX -- 9 The Kantowski-Sachs models -- 10 Taub and Taub NUT spacetimes -- 10.1 Taub spacetime -- 10.2 Taub NUT spacetime -- 11 Locally homogeneous models -- 11.1 n-dimensional compact manifolds -- 11.2 Compact 3-manifolds -- 12 Recent observations and conjectures -- VI: Local Cauchy problem -- 1 Introduction -- 2 Moving frame formulae -- 2.1 Frame and coframe -- 2.2 Metric -- 2.3 Connection -- 2.4 Curvature -- 3 n + 1 splitting adapted to space slices -- 3.1 Adapted frame and coframe -- 3.2 Structure coefficients -- 3.3 Splitting of the connection. -- 3.4 Extrinsic curvature -- 3.5 Splitting of the Riemann tensor -- 4 Constraints and evolution -- 4.1 Equations. Conservation of constraints -- 5 Hamiltonian and symplectic formulation -- 5.1 Hamilton equations -- 5.2 Symplectic formulation -- 6 Cauchy problem -- 6.1 Definitions -- 6.2 The analytic case -- 7 Wave gauges -- 7.1 Wave coordinates -- 7.2 Generalized wave coordinates -- 7.3 Damped wave coordinates -- 7.4 Globalization in space, ê wave gauges -- 7.5 Local in time existence in a wave gauge -- 8 Local existence for the full Einstein equations -- 8.1 Preservation of the wave gauges -- 8.2 Geometric local existence. ; 8.3 Geometric uniqueness -- 8.4 Causality -- 9 Constraints in a wave gauge -- 10 Einstein equations with field sources -- 10.1 Maxwell constraints -- 10.2 Lorentz gauge -- 10.3 Existence and uniqueness theorems -- 10.4 Wave equation for F -- VII: Constraints -- 1 Introduction -- 2 Linearization and stability -- 2.1 Linearization of the constraints map, adjoint map -- 2.2 Linearization stability -- 3 CF (Conformally Formulated) constraints -- 3.1 Hamiltonian constraint -- 3.2 Momentum constraint -- 3.3 Scaling of the sources -- 3.4 Summary of results -- 3.5 Conformal transformation of the CF constraints -- 3.6 The momentum constraint as an elliptic system -- 4 Case n = 2 -- 5 Solutions on compact manifolds -- 6 Solution of the momentum constraint -- 7 Lichnerowicz equation -- 7.1 The Yamabe classification -- 7.2 Non-existence and uniqueness -- 7.3 Existence theorems -- 8 System of constraints -- 8.1 Constant mean curvature & -- #915 -- sources with York-scaled momentum -- 8.2 Solutions with & -- #915 -- & -- #8802 -- constant or J[(0)] & -- #8802 -- 0 -- 9 Solutions on asymptotically Euclidean Manifolds -- 10 Momentum constraint -- 11 Solution of the Lichnerowicz equation -- 11.1 Uniqueness theorem -- 11.2 Generalized Brill-Cantor theorem -- 11.3 Existence theorems -- 12 Solutions of the system of constraints -- 12.1 Decoupled system -- 12.2 Coupled system -- 13 Gluing solutions of the constraint equations -- 13.1 Connected sum gluing -- 13.2 Exterior (Corvino-Schoen) gluing -- VIII: Other hyperbolic-elliptic well-posed systems -- 1 Introduction -- 2 Leray-Ohya non-hyperbolicity of [sup((4))] R[(ij)] = 0 -- 3 Wave equation for K -- 3.1 Hyperbolic system -- 3.2 Hyperbolic-elliptic system -- 4 Fourth-order non-strict and strict hyperbolic systems for g -- 5 First-order hyperbolic systems -- 5.1 FOSH systems -- 6 Bianchi-Einstein equations. ; 6.1 Wave equation for the Riemann tensor -- 6.2 Case n = 3, FOS system -- 6.3 Cauchy-adapted frame -- 6.4 FOSH system for u = (E, H, D, B, g, K, & -- #915 -- ) -- 6.5 Elliptic-hyperbolic system -- 7 Bel-Robinson tensor and energy -- 7.1 The Bel tensor -- 7.2 The Bel-Robinson tensor and energy -- 8 Bel-Robinson energy in a strip -- IX: Relativistic fluids -- 1 Introduction -- 2 Case of dust -- 2.1 Equations -- 2.2 Motion of isolated bodies (Choquet-Bruhat and Friedrichs 2006) -- 3 Charged dust -- 3.1 Equations -- 3.2 Existence and uniqueness theorem in wave and Lorentz gauges -- 3.3 Motion of isolated bodies -- 4 Perfect fluid, Euler equations -- 5 Energy properties -- 6 Particle number conservation -- 7 Thermodynamics -- 7.1 Definitions. Conservation of entropy -- 7.2 Equations of state -- 8 Wave fronts, propagation speeds, shocks -- 8.1 General definitions -- 8.2 Case of perfect fluids -- 8.3 Shocks -- 9 Stationary motion -- 10 Dynamic velocity for barotropic fluids -- 10.1 Fluid index and equations -- 10.2 Vorticity tensor and Helmholtz equations -- 10.3 Irrotational flows -- 11 General perfect fluids -- 12 Hyperbolic Leray system -- 12.1 Hyperbolicity of the Euler equations. -- 12.2 Reduced Einstein-Euler entropy system -- 12.3 Cauchy problem for the Einstein-Euler entropy system -- 12.4 Motion of isolated bodies -- 13 First-order symmetric hyperbolic system -- 14 Equations in a flow adapted frame -- 14.1 n + 1 splitting in a time adapted frame -- 14.2 Bianchi equations (case n = 3) -- 14.3 Vacuum case -- 14.4 Perfect fluid -- 14.5 Conclusion -- 15 Charged fluids -- 15.1 Equations -- 15.2 Fluids with zero conductivity -- 16 Fluids with finite conductivity -- 17 Magnetohydrodynamics -- 17.1 Equations -- 17.2 Wave fronts -- 18 Yang-Mills fluids -- 19 Dissipative fluids -- 19.1 Viscous fluids -- 19.2 The heat equation. ; X: Relativistic kinetic theory.
• Sommario/riassunto: Aimed at researchers in mathematics and physics, this monograph, in which the author overviews the basic ideas in General Relativity, introduces the necessary mathematics and discusses some of the key open questions in the field.
• Titolo autorizzato: General relativity and the Einstein equations
• ISBN: 9780191552267; 0191552267
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910815087203321
• Lo trovi qui: Univ. Federico II
• Serie: Oxford mathematical monographs.