1  Introduction  References2  Basic Differential Geometry  2.1  Introduction  2.2  Differentiable Manifolds  2.2.1  Notion of Manifold  2.2.2  Vectors on a Manifold  2.2.3  Linear Forms  2.2.4  Tensors  2.2.5  Fields on a Manifold  2.3  Pseudo-Riemannian Manifolds  2.3.1   Metric Tensor  2.3.2   Signature and Orthonormal Bases  2.3.3   Metric Duality  2.3.4   Levi-Civita Tensor  2.4  Covariant Derivative  2.4.1  Affine Connection on a Manifold  2.4.2  Levi-Civita Connection  2.4.3  Curvature  2.4.4  Weyl Tensor  2.5  Lie Derivative  2.5.1  Lie Derivative of a Vector Field  2.5.2  Generalization to Any Tensor Field  References3  Geometry of Hypersurfaees  3.1  Introduction  3.2  Framework and Notations  3.3  Hypersurface Embedded in Spacetime  3.3.1  Definition  3.3.2  Normal Vector  3.3.3  Intrinsic Curvature  3.3.4  Extrinsic Curvature  3.3.5  Examples: Surfaces Embedded in the Euclidean Space R3  3.3.6  An Example in Minkowski Spacetime: The Hyperbolic Space H3  3.4  Spacelike Hypersurfaces  3.4.1  The Orthogonal Projector  3.4.2  Relation Between K and Vn  3.4.3  Links Between the ▽ and D Connections  3.5  Gauss-Codazzi Relations  3.5.1  Gauss Relation  3.5.2  Codazzi Relation  References4  Geometry of Foliations  4.1  Introduction  4.2  Globally Hyperbolic Spacetimes and Foliations  4.2.1  Globally Hyperbolic Spacetimes  4.2.2  Definition of a Foliation  4.3  Foliation Kinematics  4.3.1  Lapse Function  4.3.2  Normal Evolution Vector  4.3.3  Eulerian Observers  4.3.4  Gradients of n and m  4.3.5  Evolution of the 3-Metric  4.3.6  Evolution of the Orthogonal Projector  4.4  Last Part of the 3 1 Deposition of the Riemann Tensor.  4.4.1  Last Non Trivial Projection of the Spacetime Riemann Tensor  4.4.2  3 1 Expression of the Spacetime Scalar Curvature.  References5  3 1 Deposition of Einstein Equation  5.1  Einstein Equation in 3 1 form  5.1.1  The Einstein Equation  5.1.2  3 1 Deposition of the Stress-Energy Tensor ..  5.1.3  Projection of the Einstein Equation  5.2  Coordinates Adapted to the Foliation  5.2.1  Definition  5.2.2  Shift Vector  5.2.3  3 1 Writing of the Metric Components  5.2.4  Choice of Coordinates via the Lapse and the Shift  5.3  3 1 Einstein Equation as a PDE System  5.3.1  Lie Derivatives Along m as Partial Derivatives  5.3.2  3 1 Einstein System  5.4  The Cauchy Problem  5.4.1  General Relativity as a Three-Dimensional Dynamical System  5.4.2  Analysis Within Gaussian Normal Coordinates  5.4.3  Constraint Equations  5.4.4  Existence and Uniqueness of Solutions to the Cauchy Problem  5.5  ADM Hamiltonian Formulation  5.5.1   3 1 form of the Hilbert Action  5.5.2   Hamiltonian Approach  References6  3 1 Equations for Matter and Electromagic Field  6.1  Introduction  6.2  Energy and Momentum Conservation  6.2.1  3 1 Deposition of the 4-Dimensional Equation  6.2.2  Energy Conservation  6.2.3  Newtonian Limit  6.2.4  Momentum Conservation  6.3  Perfect Fluid  6.3.1  Kinematics  6.3.2  Baryon Number Conservation  6.3.3  Dynamical Quantities  6.3.4  Energy Conservation Law  6.3.5  Relativistic Euler Equation  6.3.6  Flux-Conservative Form  6.3.7  Further Developments  6.4  Electromagism  6.4.1  Electromagic Field  6.4.2  3 1 Maxwell Equations  6.4.3  Electromagic Energy, Momentum and Stress...  6.5  3 1 Ideal Magohydrodynamics  6.5.1  Basic Settings  6.5.2  Maxwell Equations  6.5.3  Electromagic Energy, Momentum and Stress...  6.5.4  MHD-Euler Equation  6.5.5  MHD in Flux-Conservative Form  References7  Conformal Deposition  7.1  Introduction  7.2  Conformal Deposition of the 3-Metric  7.2.1  Unit-Determinant Conformal 'Metric'  7.2.2  Background Metric  7.2.3  Conformal Metric  7.2.4  Conformal Connection  7.3  Expression of the Ricci Tensor  7.3.1  General Formula Relating the Two Ricci Tensors  7.3.2  Expression in Terms of the Conformal Factor  7.3.3  Formula for the Scalar Curvature  7.4  Conformal Deposition of the Extrinsic Curvature  7.4.1  Traceless Deposition  7.4.2  Conformal Deposition of the Traceless Part  7.5  Conformal Form of the 3 1 Einstein System  7.5.1  Dynamical Part of Einstein Equation  7.5.2  Hamiltonian Constraint  7.5.3  Momentum Constraint  7.5.4  Summary: Conformal 3 1 Einstein System  7.6  Isenberg-Wilson-Mathews Approximation to General Relativity  References8  Asymptotic Flatness and Global Quantifies  8.1  Introduction  8.2  Asymptotic Flatness  8.2.1  Definition  8.2.2  Asymptotic Coordinate Freedom  8.3  ADM Mass  8.3.1  Definition from the Hamiltonian Formulation of GR  8.3.2  Expression in Terms of the Conformal Deposition  8.3.3  Newtonian Limit  8.3.4  Positive Energy Theorem  8.3.5  Constancy of the ADM Mass  8.4  ADM Momentum  8.4.1  Definition  8.4.2  ADM 4-Momentum  8.5  Angular Momentum  8.5.1  The Supertranslation Ambiguity  8.5.2  The 'Cure'.  8.5.3  ADM Mass in the Quasi-Isotropic Gauge  8.6  Komar Mass and Angular Momentum  8.6.1  Komar Mass  8.6.2  3 1 Expression of the Komar Mass and Link with the ADM Mass  8.6.3  Komar Angular Momentum  References9  The Initial Data Problem  9.1  Introduction  9.1.1  The Initial Data Problem  9.1.2  Conformal Deposition of the Constraints  9.2  Conformal Transverse-Traceless Method  9.2.1  Longitudinal / Transverse Deposition of A ij  9.2.2  Conformal Transverse-Traceless Form of the Constraints  9.2.3  Decoupling on Hypersurfaces of Constant Mean Curvature  9.2.4  Existence and Uniqueness of Solutions to Lichnerowicz Equation  9.2.5  Conformally Flat and Momentarily Static Initial Data  9.2.6  Bowen-York Initial Data  9.3  Conformal Thin Sandwich Method  9.3.1  The Original Conformal Thin Sandwich Method .  9.3.2  Extended Conformal Thin Sandwich Method  9.3.3  XCTS at Work: Static Black Hole Example  9.3.4  Uniqueness Issue  9.3.5  Comparing CTT, CTS and XCTS  9.4  Initial Data for Binary Systems  9.4.1  Helical Symmetry  9.4.2  Helical Symmetry and IWM Approximation  9.4.3  Initial Data for Orbiting Binary Black Holes  9.4.4  Initial Data for Orbiting Binary Neutron Stars  9.4.5  Initial Data for Black Hole: Neutron Star Binaries.  References10  Choice of Foliation and Spatial Coordinates  10.1  Introduction  10.2  Choice of Foliation  10.2.1  Geodesic Slicing  10.2.2  Maximal Slicing  10.2.3  Harmonic Slicing  10.2.4  1 log Slicing  10.3  Evolution of Spatial Coordinates  10.3.1  Normal Coordinates  10.3.2  Minimal Distortion  10.3.3  Approximate Minimal Distortion  10.3.4  Gamma Freezing  10.3.5  Gamma Drivers  10.3.6  Other Dynamical Shift Gauges  10.4  Full Spatial Coordinate-Fixing Choices  10.4.1  Spatial Harmonic Coordinates  10.4.2  Dirac Gauge  References11  Evolution schemes  11.1  Introduction  11.2  Constrained Schemes  11.3  Free Evolution Schemes    11.3.1  Definition and Framework    11.3.2  Propagation of the Constraints    11.3.3  Constraint-Violating Modes    11.3.4  Symmetric Hyperbolic Formulations  11.4  BSSN Scheme    11.4.1  Introduction    11.4.2  Expression of the Ricci Tensor of the Conformal Metric    11.4.3  Reducing the Ricci Tensor to a Laplace Operator    11.4.4  The Full Scheme    11.4.5  Applications  ReferencesAppendix A: Conformal Killing Operator and Conformal Vector LaplacianAppendix B: Sage CodesIndex

作者介紹

古爾古隆(E. Gourgoulhon)，法國LUTh教授。

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廣義相對論的3+1形式——數值相對論基礎(英文影印版) 下載 mobi epub pdf txt 電子書格式