广义相对论的3+1形式 数值相对论基础 英文、影印版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

广义相对论的3+1形式 数值相对论基础 英文、影印版PDF格式电子书版下载

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1 Introduction1

References2

2 Basic Differential Geometry5

2.1 Introduction6

2.2 Differentiable Manifolds6

2.2.1 Notion of Manifold6

2.2.2 Vectors on a Manifold8

2.2.3 Linear Forms10

2.2.4 Tensors12

2.2.5 Fields on a Manifold13

2.3 Pseudo-Riemannian Manifolds13

2.3.1 Metric Tensor13

2.3.2 Signature and Orthonormal Bases14

2.3.3 Metric Duality15

2.3.4 Levi-Civita Tensor17

2.4 Covariant Derivative17

2.4.1 Affine Connection on a Manifold17

2.4.2 Levi-Civita Connection20

2.4.3 Curvature22

2.4.4 Weyl Tensor24

2.5 Lie Derivative25

2.5.1 Lie Derivative of a Vector Field25

2.5.2 Generalization to Any Tensor Field27

References28

3 Geometry of Hypersurfaces29

3.1 Introduction29

3.2 Framework and Notations29

3.3 Hypersurface Embedded in Spacetime30

3.3.1 Definition30

3.3.2 Normal Vector32

3.3.3 Intrinsic Curvature33

3.3.4 Extrinsic Curvature34

3.3.5 Examples:Surfaces Embedded in the Euclidean Space R336

3.3.6 An Example in Minkowski Spacetime:z The Hyperbolic Space H340

3.4 Spacelike Hypersurfaces43

3.4.1 The Orthogonal Projector44

3.4.2 Relation Between K and ？n46

3.4.3 Links Between the ？ and D Connections47

3.5 Gauss-Codazzi Relations49

3.5.1 Gauss Relation50

3.5.2 Codazzi Relation52

References54

4 Geometry of Foliations55

4.1 Introduction55

4.2 Globally Hyperbolic Spacetimes and Foliations55

4.2.1 Globally Hyperbolic Spacetimes55

4.2.2 Definition of a Foliation56

4.3 Foliation Kinematics57

4.3.1 Lapse Function57

4.3.2 Normal Evolution Vector57

4.3.3 Eulerian Observers60

4.3.4 Gradients of n and m63

4.3.5 Evolution of the 3-Metric64

4.3.6 Evolution of the Orthogonal Projector66

4.4 Last Part of the 3+1 Decomposition of the Riemann Tensor67

4.4.1 Last Non Trivial Projection of the Spacetime Riemann Tensor67

4.4.2 3+1 Expression of the Spacetime Scalar Curvature69

References71

5 3+1 Decomposition of Einstein Equation73

5.1 Einstein Equation in 3+1 form73

5.1.1 The Einstein Equation73

5.1.2 3+1 Decomposition of the Stress-Energy Tensor74

5.1.3 Projection of the Einstein Equation76

5.2 Coordinares Adapted to the Foliation78

5.2.1 Definition78

5.2.2 Shift Vector79

5.2.3 3+1 Wriring of the Metric Components82

5.2.4 Choice of Coordinates via the Lapse and the Shift85

5.3 3+1 Einstein Equation as a PDE System86

5.3.1 Lie Derivatives Along m as Partial Derivatives86

5.3.2 3+1 Einstein System87

5.4 The Cauchy Problem88

5.4.1 General Relativity as a Three-Dimensional Dynamical System88

5.4.2 Analysis Within Gaussian Normal Coordinates89

5.4.3 Constraint Equations92

5.4.4 Existence and Uniqueness of Solutions to the Cauchy Problem92

5.5 ADM Hamiltonian Formulation93

5.5.1 3+1 form of the Hilbert Action94

5.5.2 Hamiltonian Approach95

References98

6 3+1 Equations for Matter and Electromagnetic Field101

6.1 Introduction101

6.2 Energy and Momentum Conservation102

6.2.1 3+1 Decomposition of the 4-Dimensional Equation102

6.2.2 Energy Conservation102

6.2.3 Newtonian Limit104

6.2.4 Momentum Conservation105

6.3 Perfect Fluid106

6.3.1 Kinematics106

6.3.2 Baryon Number Conservation109

6.3.3 Dynamical Quantities111

6.3.4 Energy Conservation Law112

6.3.5 Relativistic Euler Equation113

6.3.6 Flux-Conservative Form114

6.3.7 Further Developments117

6.4 Electromagnetism117

6.4.1 Electromagnetic Field117

6.4.2 3+1 Maxwell Equations119

6.4.3 Electromagnetic Energy,Momentum and Stress122

6.5 3+1 Ideal Magnetohydrodynamics123

6.5.1 Basic Settings123

6.5.2 Maxwell Equations125

6.5.3 Electromagnetic Energy,Momentum and Stress127

6.5.4 MHD-Euler Equation127

6.5.5 MHD in Flux-Conservative Form129

References130

7 Conformal Decomposition133

7.1 Introduction133

7.2 Conformal Decomposition of the 3-Metric135

7.2.1 Unit-Determinant Conformal"Metric"135

7.2.2 Background Metric135

7.2.3 Conformal Metric136

7.2.4 Conformal Connection138

7.3 Expression of the Ricci Tensor141

7.3.1 General Formula Relating the Two Ricci Tensors141

7.3.2 Expression in Terms of the Conformal Factor142

7.3.3 Formula for the Scalar Curvature142

7.4 Conformal Decomposition of the Extrinsic Curvature143

7.4.1 Traceless Decomposition143

7.4.2 Conformal Decomposition of the Traceless Part144

7.5 Conformal Form of the 3+1 Einstein System147

7.5.1 Dynamical Part of Einstein Equation147

7.5.2 Hamiltonian Constraint150

7.5.3 Momentum Constraint151

7.5.4 Summary:Conformal 3+1 Einstein System151

7.6 Isenberg-Wilson-Mathews Approximation to General Relativity152

References156

8 Asymptotic Flatness and Global Quantities159

8.1 Introduction159

8.2 Asymptotic Flatness159

8.2.1 Definition160

8.2.2 Asymptotic Coordinate Freedom161

8.3 ADM Mass162

8.3.1 Definition from the Hamiltonian Formulation of GR162

8.3.2 Expression in Terms of the Conformal Decomposition167

8.3.3 Newtonian Limit169

8.3.4 Positive Energy Theorem170

8.3.5 Constancy of the ADM Mass171

8.4 ADM Momentum171

8.4.1 Definition171

8.4.2 ADM 4-Momentum172

8.5 Angular Momentum172

8.5.1 The Supertranslation Ambiguity172

8.5.2 The"Cure"173

8.5.3 ADM Mass in the Quasi-Isotropic Gauge174

8.6 Komar Mass and Angular Momentum176

8.6.1 Komar Mass176

8.6.2 3+1 Expression of the Komar Mass and Link with the ADM Mass179

8.6.3 Komar Angular Momentum182

References185

9 The Initial Data Problem187

9.1 Introduction187

9.1.1 The Initial Data Problem187

9.1.2 Conformal Decomposition of the Constraints188

9.2 Conformal Transverse-Traceless Method189

9.2.1 Longitudinal/Transverse Decomposition of A？189

9.2.2 Conformal Transverse-Traceless Form of the Constraints191

9.2.3 Decoupling on Hypersurfaces of Constant Mean Curvature192

9.2.4 Existence and Uniqueness of Solutions to Lichnerowicz Equation193

9.2.5 Conformally F1at and Momentarily Static Initial Data194

9.2.6 Bowen-York Initial Data200

9.3 Conformal Thin Sandwich Method204

9.3.1 The Original Conformal Thin Sandwich Method204

9.3.2 Extended Conformal Thin Sandwich Method205

9.3.3 XCTS at Work:Static Black Hole Example207

9.3.4 Uniqueness Issue210

9.3.5 Comparing CTT,CTS and XCTS210

9.4 Initial Data for Binary Systems211

9.4.1 Helical Symmetry211

9.4.2 Helical Symmetry and IWM Approximation213

9.4.3 Initial Data for Orbiting Binary Black Holes214

9.4.4 Initial Data for Orbiting Binary Neutron Stars216

9.4.5 Initial Data for Black Hole:Neutron Star Binaries217

References217

10 Choiee of Foliation and Spatial Coordinates223

10.1 Introduction223

10.2 Choice of Foliation224

10.2.1 Geodesic Slicing224

10.2.2 Maximal Slicing225

10.2.3 Harmonic Slicing231

10.2.4 l+log Slicing233

10.3 Evolution of Spatial Coordinates235

10.3.1 Normal Coordinates236

10.3.2 Minimal Distortion236

10.3.3 Approximate Minimal Distortion241

10.3.4 Gamma Freezing242

10.3.5 Gamma Drivers244

10.3.6 Other Dynamical Shift Gauges246

10.4 Full Spatial Coordinate-Fixing Choices247

10.4.1 Spatial Harmonic Coordinates247

10.4.2 Dirac Gauge248

References249

11 Evolution schemes255

11.1 Introduction255

11.2 Constrained Schemes255

11.3 Free Evolution Schemes256

11.3.1 Definition and Framework256

11.3.2 Propagation of the Constraints257

11.3.3 Constraint-Violating Modes261

11.3.4 Symmetric Hyperbolic Formulations262

11.4 BSSN Scheme262

11.4.1 Introduction262

11.4.2 Expression of the Ricci Tensor of the Conformal Metric262

11.4.3 Reducing the Ricci Tensor to a Laplace Operator265

11.4.4 The Full Scheme267

11.4.5 Applications269

References269

Appendix A:Conformal Killing Operator and Conformal Vector Laplacian273

Appendix B:Sage Codes281

Index287