1 Introduction 1
References 2
2 Basic Differential Geometry 5
2.1 Introduction 6
2.2 Differentiable Manifolds 6
2.2.1 Notion of Manifold 6
2.2.2 Vectors on a Manifold 8
2.2.3 Linear Forms 10
2.2.4 Tensors 12
2.2.5 Fields on a Manifold 13
2.3 Pseudo-Riemannian Manifolds 13
2.3.1 Metric Tensor 13
2.3.2 Signature and Orthonormal Bases 14
2.3.3 Metric Duality 15
2.3.4 Levi-Civita Tensor 17
2.4 Covariant Derivative 17
2.4.1 Affine Connection on a Manifold 17
2.4.2 Levi-Civita Connection 20
2.4.3 Curvature 22
2.4.4 Weyl Tensor 24
2.5 Lie Derivative 25
2.5.1 Lie Derivative of a Vector Field 25
2.5.2 Generalization to Any Tensor Field 27
References 28
3 Geometry of Hypersurfaces 29
3.1 Introduction 29
3.2 Framework and Notations 29
3.3 Hypersurface Embedded in Spacetime 30
3.3.1 Definition 30
3.3.2 Normal Vector 32
3.3.3 Intrinsic Curvature 33
3.3.4 Extrinsic Curvature 34
3.3.5 Examples:Surfaces Embedded in the Euclidean Space R3 36
3.3.6 An Example in Minkowski Spacetime:z The Hyperbolic Space H3 40
3.4 Spacelike Hypersurfaces 43
3.4.1 The Orthogonal Projector 44
3.4.2 Relation Between K and ？n 46
3.4.3 Links Between the ？ and D Connections 47
3.5 Gauss-Codazzi Relations 49
3.5.1 Gauss Relation 50
3.5.2 Codazzi Relation 52
References 54
4 Geometry of Foliations 55
4.1 Introduction 55
4.2 Globally Hyperbolic Spacetimes and Foliations 55
4.2.1 Globally Hyperbolic Spacetimes 55
4.2.2 Definition of a Foliation 56
4.3 Foliation Kinematics 57
4.3.1 Lapse Function 57
4.3.2 Normal Evolution Vector 57
4.3.3 Eulerian Observers 60
4.3.4 Gradients of n and m 63
4.3.5 Evolution of the 3-Metric 64
4.3.6 Evolution of the Orthogonal Projector 66
4.4 Last Part of the 3+1 Decomposition of the Riemann Tensor 67
4.4.1 Last Non Trivial Projection of the Spacetime Riemann Tensor 67
4.4.2 3+1 Expression of the Spacetime Scalar Curvature 69
References 71
5 3+1 Decomposition of Einstein Equation 73
5.1 Einstein Equation in 3+1 form 73
5.1.1 The Einstein Equation 73
5.1.2 3+1 Decomposition of the Stress-Energy Tensor 74
5.1.3 Projection of the Einstein Equation 76
5.2 Coordinares Adapted to the Foliation 78
5.2.1 Definition 78
5.2.2 Shift Vector 79
5.2.3 3+1 Wriring of the Metric Components 82
5.2.4 Choice of Coordinates via the Lapse and the Shift 85
5.3 3+1 Einstein Equation as a PDE System 86
5.3.1 Lie Derivatives Along m as Partial Derivatives 86
5.3.2 3+1 Einstein System 87
5.4 The Cauchy Problem 88
5.4.1 General Relativity as a Three-Dimensional Dynamical System 88
5.4.2 Analysis Within Gaussian Normal Coordinates 89
5.4.3 Constraint Equations 92
5.4.4 Existence and Uniqueness of Solutions to the Cauchy Problem 92
5.5 ADM Hamiltonian Formulation 93
5.5.1 3+1 form of the Hilbert Action 94
5.5.2 Hamiltonian Approach 95
References 98
6 3+1 Equations for Matter and Electromagnetic Field 101
6.1 Introduction 101
6.2 Energy and Momentum Conservation 102
6.2.1 3+1 Decomposition of the 4-Dimensional Equation 102
6.2.2 Energy Conservation 102
6.2.3 Newtonian Limit 104
6.2.4 Momentum Conservation 105
6.3 Perfect Fluid 106
6.3.1 Kinematics 106
6.3.2 Baryon Number Conservation 109
6.3.3 Dynamical Quantities 111
6.3.4 Energy Conservation Law 112
6.3.5 Relativistic Euler Equation 113
6.3.6 Flux-Conservative Form 114
6.3.7 Further Developments 117
6.4 Electromagnetism 117
6.4.1 Electromagnetic Field 117
6.4.2 3+1 Maxwell Equations 119
6.4.3 Electromagnetic Energy,Momentum and Stress 122
6.5 3+1 Ideal Magnetohydrodynamics 123
6.5.1 Basic Settings 123
6.5.2 Maxwell Equations 125
6.5.3 Electromagnetic Energy,Momentum and Stress 127
6.5.4 MHD-Euler Equation 127
6.5.5 MHD in Flux-Conservative Form 129
References 130
7 Conformal Decomposition 133
7.1 Introduction 133
7.2 Conformal Decomposition of the 3-Metric 135
7.2.1 Unit-Determinant Conformal"Metric" 135
7.2.2 Background Metric 135
7.2.3 Conformal Metric 136
7.2.4 Conformal Connection 138
7.3 Expression of the Ricci Tensor 141
7.3.1 General Formula Relating the Two Ricci Tensors 141
7.3.2 Expression in Terms of the Conformal Factor 142
7.3.3 Formula for the Scalar Curvature 142
7.4 Conformal Decomposition of the Extrinsic Curvature 143
7.4.1 Traceless Decomposition 143
7.4.2 Conformal Decomposition of the Traceless Part 144
7.5 Conformal Form of the 3+1 Einstein System 147
7.5.1 Dynamical Part of Einstein Equation 147
7.5.2 Hamiltonian Constraint 150
7.5.3 Momentum Constraint 151
7.5.4 Summary:Conformal 3+1 Einstein System 151
7.6 Isenberg-Wilson-Mathews Approximation to General Relativity 152
References 156
8 Asymptotic Flatness and Global Quantities 159
8.1 Introduction 159
8.2 Asymptotic Flatness 159
8.2.1 Definition 160
8.2.2 Asymptotic Coordinate Freedom 161
8.3 ADM Mass 162
8.3.1 Definition from the Hamiltonian Formulation of GR 162
8.3.2 Expression in Terms of the Conformal Decomposition 167
8.3.3 Newtonian Limit 169
8.3.4 Positive Energy Theorem 170
8.3.5 Constancy of the ADM Mass 171
8.4 ADM Momentum 171
8.4.1 Definition 171
8.4.2 ADM 4-Momentum 172
8.5 Angular Momentum 172
8.5.1 The Supertranslation Ambiguity 172
8.5.2 The"Cure" 173
8.5.3 ADM Mass in the Quasi-Isotropic Gauge 174
8.6 Komar Mass and Angular Momentum 176
8.6.1 Komar Mass 176
8.6.2 3+1 Expression of the Komar Mass and Link with the ADM Mass 179
8.6.3 Komar Angular Momentum 182
References 185
9 The Initial Data Problem 187
9.1 Introduction 187
9.1.1 The Initial Data Problem 187
9.1.2 Conformal Decomposition of the Constraints 188
9.2 Conformal Transverse-Traceless Method 189
9.2.1 Longitudinal/Transverse Decomposition of A？ 189
9.2.2 Conformal Transverse-Traceless Form of the Constraints 191
9.2.3 Decoupling on Hypersurfaces of Constant Mean Curvature 192
9.2.4 Existence and Uniqueness of Solutions to Lichnerowicz Equation 193
9.2.5 Conformally F1at and Momentarily Static Initial Data 194
9.2.6 Bowen-York Initial Data 200
9.3 Conformal Thin Sandwich Method 204
9.3.1 The Original Conformal Thin Sandwich Method 204
9.3.2 Extended Conformal Thin Sandwich Method 205
9.3.3 XCTS at Work:Static Black Hole Example 207
9.3.4 Uniqueness Issue 210
9.3.5 Comparing CTT,CTS and XCTS 210
9.4 Initial Data for Binary Systems 211
9.4.1 Helical Symmetry 211
9.4.2 Helical Symmetry and IWM Approximation 213
9.4.3 Initial Data for Orbiting Binary Black Holes 214
9.4.4 Initial Data for Orbiting Binary Neutron Stars 216
9.4.5 Initial Data for Black Hole:Neutron Star Binaries 217
References 217
10 Choiee of Foliation and Spatial Coordinates 223
10.1 Introduction 223
10.2 Choice of Foliation 224
10.2.1 Geodesic Slicing 224
10.2.2 Maximal Slicing 225
10.2.3 Harmonic Slicing 231
10.2.4 l+log Slicing 233
10.3 Evolution of Spatial Coordinates 235
10.3.1 Normal Coordinates 236
10.3.2 Minimal Distortion 236
10.3.3 Approximate Minimal Distortion 241
10.3.4 Gamma Freezing 242
10.3.5 Gamma Drivers 244
10.3.6 Other Dynamical Shift Gauges 246
10.4 Full Spatial Coordinate-Fixing Choices 247
10.4.1 Spatial Harmonic Coordinates 247
10.4.2 Dirac Gauge 248
References 249
11 Evolution schemes 255
11.1 Introduction 255
11.2 Constrained Schemes 255
11.3 Free Evolution Schemes 256
11.3.1 Definition and Framework 256
11.3.2 Propagation of the Constraints 257
11.3.3 Constraint-Violating Modes 261
11.3.4 Symmetric Hyperbolic Formulations 262
11.4 BSSN Scheme 262
11.4.1 Introduction 262
11.4.2 Expression of the Ricci Tensor of the Conformal Metric 262
11.4.3 Reducing the Ricci Tensor to a Laplace Operator 265
11.4.4 The Full Scheme 267
11.4.5 Applications 269
References 269
Appendix A:Conformal Killing Operator and Conformal Vector Laplacian 273
Appendix B:Sage Codes 281
Index 287