Introduction to 3+1 Numerical Relativity

Contents......Page 12
1.1 Introduction......Page 16
1.3 Special relativity......Page 17
1.4 Manifolds and tensors......Page 22
1.5 The metric tensor......Page 25
1.6 Lie derivatives and Killing fields......Page 29
1.7 Coordinate transformations......Page 32
1.8 Covariant derivatives and geodesics......Page 35
1.9 Curvature......Page 40
1.11 General relativity......Page 43
1.12 Matter and the stress-energy tensor......Page 47
1.13 The Einstein field equations......Page 51
1.14 Weak fields and gravitational waves......Page 54
1.15 The Schwarzschild solution and black holes......Page 61
1.16 Black holes with charge and angular momentum......Page 68
1.17 Causal structure, singularities and black holes......Page 72
2.1 Introduction......Page 79
2.2 3+1 split of spacetime......Page 80
2.3 Extrinsic curvature......Page 83
2.4 The Einstein constraints......Page 86
2.5 The ADM evolution equations......Page 88
2.6 Free versus constrained evolution......Page 92
2.7 Hamiltonian formulation......Page 93
2.8 The BSSNOK formulation......Page 96
2.9.1 The characteristic approach......Page 102
2.9.2 The conformal approach......Page 105
3.2 York–Lichnerowicz conformal decomposition......Page 107
3.2.1 Conformal transverse decomposition......Page 109
3.2.2 Physical transverse decomposition......Page 112
3.2.3 Weighted transverse decomposition......Page 114
3.3 Conformal thin-sandwich approach......Page 116
3.4.1 Time-symmetric data......Page 120
3.4.2 Bowen–York extrinsic curvature......Page 124
3.4.3 Conformal factor: inversions and punctures......Page 126
3.4.4 Kerr–Schild type data......Page 128
3.5 Binary black holes in quasi-circular orbits......Page 130
3.5.1 Effective potential method......Page 131
3.5.2 The quasi-equilibrium method......Page 132
4.1 Introduction......Page 136
4.2 Slicing conditions......Page 137
4.2.2 Maximal slicing......Page 138
4.2.3 Maximal slices of Schwarzschild......Page 142
4.2.4 Hyperbolic slicing conditions......Page 148
4.2.5 Singularity avoidance for hyperbolic slicings......Page 151
4.3 Shift conditions......Page 155
4.3.1 Elliptic shift conditions......Page 156
4.3.2 Evolution type shift conditions......Page 160
4.3.3 Corotating coordinates......Page 166
5.1 Introduction......Page 170
5.2 Well-posedness......Page 171
5.3 The concept of hyperbolicity......Page 173
5.4 Hyperbolicity of the ADM equations......Page 179
5.5 The Bona–Masso and NOR formulations......Page 184
5.6 Hyperbolicity of BSSNOK......Page 190
5.7 The Kidder–Scheel–Teukolsky family......Page 194
5.8 Other hyperbolic formulations......Page 198
5.8.1 Higher derivative formulations......Page 199
5.8.2 The Z4 formulation......Page 200
5.9 Boundary conditions......Page 202
5.9.1 Radiative boundary conditions......Page 203
5.9.2 Maximally dissipative boundary conditions......Page 206
5.9.3 Constraint preserving boundary conditions......Page 209
6.1 Introduction......Page 213
6.2 Isometries and throat adapted coordinates......Page 214
6.3 Static puncture evolution......Page 221
6.4 Singularity avoidance and slice stretching......Page 224
6.5 Black hole excision......Page 229
6.6.1 How to move the punctures......Page 232
6.6.2 Why does evolving the punctures work?......Page 234
6.7 Apparent horizons......Page 236
6.7.1 Apparent horizons in spherical symmetry......Page 238
6.7.2 Apparent horizons in axial symmetry......Page 239
6.7.3 Apparent horizons in three dimensions......Page 241
6.8 Event horizons......Page 245
6.9 Isolated and dynamical horizons......Page 249
7.1 Introduction......Page 253
7.2 Special relativistic hydrodynamics......Page 254
7.3 General relativistic hydrodynamics......Page 260
7.4 3+1 form of the hydrodynamic equations......Page 264
7.5 Equations of state: dust, ideal gases and polytropes......Page 267
7.6.1 Newtonian case......Page 272
7.6.2 Relativistic case......Page 275
7.7 Weak solutions and the Riemann problem......Page 279
7.8.1 Eckart’s irreversible thermodynamics......Page 285
7.8.2 Causal irreversible thermodynamics......Page 288
8.1 Introduction......Page 291
8.2.1 Multipole expansion......Page 292
8.2.2 Even parity perturbations......Page 295
8.2.3 Odd parity perturbations......Page 298
8.2.4 Gravitational radiation in the TT gauge......Page 299
8.3 The Weyl tensor......Page 303
8.4 The tetrad formalism......Page 306
8.5.1 Null tetrads......Page 309
8.5.2 Tetrad transformations......Page 312
8.6 The Weyl scalars......Page 313
8.7 The Petrov classification......Page 314
8.8 Invariants I and J......Page 318
8.9.1 The stress-energy tensor for gravitational waves......Page 319
8.9.2 Radiated energy and momentum......Page 322
8.9.3 Multipole decomposition......Page 328
9.2 Basic concepts of finite differencing......Page 333
9.3 The one-dimensional wave equation......Page 337
9.3.1 Explicit finite difference approximation......Page 338
9.3.2 Implicit approximation......Page 340
9.4 Von Newmann stability analysis......Page 341
9.5 Dissipation and dispersion......Page 344
9.6 Boundary conditions......Page 347
9.7 Numerical methods for first order systems......Page 350
9.8 Method of lines......Page 354
9.9 Artificial dissipation and viscosity......Page 358
9.10.1 Conservative methods......Page 362
9.10.2 Godunov’s method......Page 363
9.10.3 High resolution methods......Page 365
9.11 Convergence testing......Page 368
10.2 Toy 1+1 relativity......Page 372
10.2.1 Gauge shocks......Page 374
10.2.2 Approximate shock avoidance......Page 377
10.2.3 Numerical examples......Page 379
10.3 Spherical symmetry......Page 384
10.3.1 Regularization......Page 385
10.3.2 Hyperbolicity......Page 389
10.3.3 Evolving Schwarzschild......Page 393
10.3.4 Scalar field collapse......Page 398
10.4.1 Evolution equations and regularization......Page 406
10.4.2 Brill waves......Page 410
10.4.3 The “Cartoon” approach......Page 414
A: Total mass and momentum in general relativity......Page 417
B: Spacetime Christoffel symbols in 3+1 language......Page 424
C: BSSNOK with natural conformal rescaling......Page 425
D: Spin-weighted spherical harmonics......Page 428
References......Page 434
B......Page 452
D......Page 453
G......Page 454
K......Page 455
N......Page 456
S......Page 457
T......Page 458
Z......Page 459