广义相对论的3+1形式

内容概要

古尔古隆(E. Gourgoulhon)，法国LUTh教授。

书籍目录

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

References

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 IR3

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 △n

3.4.3 Links Between the △7 and D Connections

3.5 Gauss—Codazzi Relations

3.5.1 Gauss Relation

3.5.2 Codazzi Relation

References

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 Tensor

4.4.1 Last Non Trivial Projection of the Spacetime Riemann Tensor

4.4.23+1 Expression of the Spacetime Scalar Curvature

References

53+1 Decomposition of Einstein Equation

5.1 Einstein Equation in 3+1 form

5.1.1 The Einstein Equation

5.1.23+1 Decomposition of the Stress—Energy Tensor

5.1.3 Projection of the Einsteinon

5.2 Coordinates Adapted to the Foliation

5.2.1 Definition

5.2.2 Shift Vector

5.2.33+1 Writing of the Metric Components

5.2.4 Choice of Coordinates via the Lapse and the Shift

5.33+1 Einstein Equation as a PDE System

5.3.1 Lie Derivatives Along m as Partial Derivatives

5.3.23+1 Einstein System

5.4 The Cauchy Problem

5.4.1 General Relativity as a Three—Dimensional Dynanucal 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

5.5.13+1 form of the Hilbert Action

5.5.2 Hamiltonian Approach

References

63+1 Equations for Matter and Electromagnetic Field

6.1 Introduction

6.2 Energy and Momentum Conservation

6.2.13+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

6.4.23+1 Maxwell Equations

6.4.3 Electromagnetic Energ， Momentum and Stress

6.53+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

References

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

References

8 Asymptotic Flatness and Global Quantities

8.1 Introduction

8.2 Asymptotic Flatness

8.2.1 Definition

8.2.2 Asymptotic totic 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

8.6.23+1 Expression of the Komar Mass and Link with the ADM Mass

8.6.3 Komar Angular Momentum

References

……

9 The Irutial Data Problem

10 Choice of Foliation and Spatial Coordinates

11 Evolution schemes

Appendix A： Conformal Killing Operator and Conformal Vector Laplacian

Appendix B： Sage Codes

Index

作者简介

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。

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