《廣義相對論的3+1形式——數值相對論基礎（英文影印版）》是2014年出版的圖書，作者是古爾古隆。
基本介紹
- 書名：廣義相對論的3+1形式——數值相對論基礎（英文影印版）
- 又名：中外物理學精品書系
- 作者：(法)古爾古隆
- ISBN：978-7-301-24831-7
- 頁數：316
- 定價：54
- 出版時間：2014-11-19
- 裝幀：平
主要內容,章節目錄,
主要內容
本書詳細地講解了3+1形式的廣義相對論和數值相對論基礎。本書從研究相對論所必備的數學工具，如微分幾何、超曲面的嵌入等講起，逐步引入了愛因斯坦方程、物質和電磁場方程等的3+1分解。之後，通過更高等的數學工具，如共形變換等，討論了現代相對論的一些重要問題。
章節目錄
1 Introduction
References
2 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
3 Geometry of Hypersurfaces
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 rn
3.4.3 Links Between the r and D Connections
3.5 Gauss-Codazzi Relations.
3.5.1 Gauss Relation
3.5.2 Codazzi Relation
4 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 Tenso
4.4.1 Last Non Trivial Projection of the Spacetime Riemann Tensor .
4.4.2 3+1 Expression of the Spacetime Scalar Curvature .
5 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
6 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
7 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.
8 Asymptotic Flatness and Global Quantities
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.
