广义相对论的3+1形式 : 数值相对论基础

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

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 Decomposition 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 Decomposition of Einstein Equation 5.1 Einstein Equation in 3 1 form 5.1.1 The Einstein Equation 5.1.2 3 1 Decomposition 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 Electromagnetic Field 6.1 Introduction 6.2 Energy and Momentum Conservation 6.2.1 3 1 Decomposition 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 Electromagnetism 6.4.1 Electromagnetic Field 6.4.2 3 1 Maxwell Equations 6.4.3 Electromagnetic Energy, Momentum and Stress... 6.5 3 1 Ideal Magnetohydrodynamics 6.5.1 Basic Settings 6.5.2 Maxwell Equations 6.5.3 Electromagnetic Energy, Momentum and Stress... 6.5.4 MHD-Euler Equation 6.5.5 MHD in Flux-Conservative Form References7 Conformal Decomposition 7.1 Introduction 7.2 Conformal Decomposition 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 Decomposition of the Extrinsic Curvature 7.4.1 Traceless Decomposition 7.4.2 Conformal Decomposition 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 Decomposition 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 Decomposition of the Constraints 9.2 Conformal Transverse-Traceless Method 9.2.1 Longitudinal / Transverse Decomposition 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

《广义相对论的3 1形式》详细地讲解了3 1形式的广义相对论和数值相对论基础。本书从研究相对论所必备的数学工具，如微分几何、超曲面的嵌入等讲起，逐步引入了爱因斯坦方程、物质和电磁场方程等的3 1分解。之后，通过更高等的数学工具，如共形变换等，讨论了现代相对论的一些重要问题。

尽管物理学家提出了一些新理论，但相对论目前依然是唯一成熟的现代引力理论。而对于相对论的研究也远远没有走到尽头，其丰富内涵依然有待发掘。《广义相对论的3 1形式》讲述了相对论的基本理论和数值方法的基础。对于从事或有志于从事相对论研究的研究人员或研究生，本书都是不可错过的杰作。