Numerical Relativity: Solving Einstein's Equations on the Computer

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
What is numerical relativity?......Page 13
About this book......Page 15
Suggestions for using this book......Page 19
Cast of characters......Page 21
Geodesics......Page 25
The Einstein field equations......Page 26
Gravitational radiation......Page 27
1.2 Black holes......Page 29
Schwarzschild black holes......Page 30
Global theorems......Page 33
1.3 Oppenheimer-Volkoff spherical equilibrium stars......Page 35
Polytropes......Page 37
1.4 Oppenheimer-Snyder spherical dust collapse......Page 38
2 The 3+1 decompostion of Einstein\'s equations......Page 43
2.1 Notation and conventions......Page 46
2.2 Maxwell\'s equations in Minkowski spacetime......Page 47
2.3 Foliations of spacetime......Page 49
2.4 The extrinsic curvature......Page 53
2.5 The equations of Gauss, Codazzi and Ricci......Page 56
2.6 The constraint and evolution equations......Page 59
2.7 Choosing basis vectors: the ADM equations......Page 63
3 Constructing initial data......Page 74
3.1.1 Conformal transformation of the spatial metric......Page 76
3.1.2 Elementary black hole solutions......Page 77
3.1.3 Conformal transformation of the extrinsic curvature......Page 84
3.2 Conformal transverse-traceless decomposition......Page 87
A spinning black hole......Page 90
A boosted black hole......Page 93
3.3 Conformal thin-sandwich decomposition......Page 95
3.4 A step further: the ``waveless\'\' approximation......Page 101
3.5 Mass, momentum and angular momentum......Page 103
4 Choosing coordinates: the lapse and shift......Page 118
4.1 Geodesic slicing......Page 120
4.2 Maximal slicing and singularity avoidance......Page 123
4.3 Harmonic coordinates and variations......Page 131
4.4 Quasi-isotropic and radial gauge......Page 134
4.5 Minimal distortion and variations......Page 137
5 Matter sources......Page 143
5.2.1 Perfect gases......Page 144
Wilson scheme......Page 145
High-resolution shock-capturing (HRSC) schemes......Page 149
Smoothed particle hydrodynamics (SPH) schemes......Page 151
Rankine–Hugoniot conditions......Page 154
Tests......Page 156
5.2.2 Imperfect gases......Page 159
Heat and radiation diffusion......Page 160
5.2.3 Radiation hydrodynamics......Page 161
5.2.4 Magnetohydrodynamics......Page 168
Electromagnetic field equations......Page 169
Equations of baryon, energy and momentum conservation......Page 174
Tests......Page 177
5.3 Collisionless matter......Page 183
Particle methods......Page 191
Phase space methods......Page 193
5.4 Scalar fields......Page 195
6.1 Classification of partial differential equations......Page 203
6.2.1 Representation of functions and derivatives......Page 208
6.2.2 Elliptic equations......Page 211
6.2.3 Hyperbolic equations......Page 220
6.2.4 Parabolic equations......Page 229
6.2.5 Mesh refinement......Page 231
6.3.1 Representation of functions and derivatives......Page 233
6.3.2 A simple example......Page 234
6.3.3 Pseudo-spectral methods with Chebychev polynomials......Page 237
6.3.4 Elliptic equations......Page 239
6.3.5 Initial value problems......Page 243
6.3.6 Comparison with finite-difference methods......Page 244
6.4 Code validation and calibration......Page 245
7.1 Concepts......Page 249
7.2 Event horizons......Page 252
7.3 Apparent horizons......Page 255
7.3.1 Spherical symmetry......Page 260
7.3.2 Axisymmetry......Page 261
7.3.3 General case: no symmetry assumptions......Page 266
7.4 Isolated and dynamical horizons......Page 269
8 Spherically symmetric spacetimes......Page 273
Choices! Choices!......Page 274
Familiar gauge choices......Page 276
Maximal slicing......Page 279
8.2 Collisionless clusters: stability and collapse......Page 286
Physical picture......Page 287
Newtonian limit......Page 288
The general relativistic formulation......Page 289
Diagnostics......Page 294
α-Freezing......Page 296
Numerical calculations......Page 299
8.2.2 Phase space method......Page 309
8.3 Fluid stars: collapse......Page 311
8.3.1 Misner-Sharp formalism......Page 314
8.3.2 The Hernandez-Misner equations......Page 317
8.4 Scalar field collapse: critical phenomena......Page 323
Basic equations......Page 324
Numerical results......Page 327
9.1 Linearized waves......Page 331
9.1.1 Perturbation theory and the weak-field, slow-velocity regime......Page 332
9.1.2 Vacuum solutions......Page 339
9.2 Sources......Page 343
9.2.1 The high frequency band......Page 344
Compact binaries......Page 345
Stellar core collapse......Page 346
9.2.2 The low frequency band......Page 348
Supermassive black holes......Page 349
9.2.3 The very low and ultra low frequency bands......Page 350
9.3 Detectors and templates......Page 351
9.3.1 Ground-based gravitational wave interferometers......Page 352
9.3.2 Space-based detectors......Page 354
9.4 Extracting gravitational waveforms......Page 357
9.4.1 The gauge-invariant Moncrief formalism......Page 358
Even-parity (polar) modes......Page 359
Odd-parity (axial) modes......Page 362
Gravitational wave extraction......Page 363
9.4.2 The Newman-Penrose formalism......Page 366
10.1 Collapse of prolate spheroids to spindle singularities......Page 372
10.2 Head-on collision of two black holes......Page 379
Topology of the event horizon......Page 381
10.3 Disk collapse......Page 384
Close limit approximation......Page 386
10.4 Collapse of rotating toroidal clusters......Page 389
Toroidal black holes and topological censorship......Page 390
11 Recasting the evolution equations......Page 395
11.1 Notions of hyperbolicity......Page 396
11.2 Recasting Maxwell\'s equations......Page 398
11.2.1 Generalized Coulomb gauge......Page 399
11.2.2 First-order hyperbolic formulations......Page 400
11.3 Generalized harmonic coordinates......Page 401
11.4 First-order symmetric hyperbolic formulations......Page 404
11.5 The BSSN formulation......Page 406
Constraint propagation and damping......Page 411
12 Binary black hole initial data......Page 414
12.1 Binary inspiral: overview......Page 415
12.2.1 Solving the momentum constraint......Page 423
12.2.2 Solving the Hamiltonian constraint......Page 425
12.2.3 Identifying circular orbits......Page 427
12.3.1 The notion of quasiequilibium......Page 430
12.3.2 Quasiequilibrium black hole boundary conditions......Page 433
12.3.3 Identifying circular orbits......Page 439
12.4 Quasiequilibrium sequences......Page 441
13 Binary black hole evolution......Page 449
13.1.1 Singularity avoiding coordinates......Page 450
13.1.2 Black hole excision......Page 451
13.1.3 The moving puncture method......Page 452
13.2 Binary black hole inspiral and coalescence......Page 456
13.2.1 Equal-mass binaries......Page 457
13.2.2 Asymmetric binaries, spin and black hole recoil......Page 465
Unequal masses......Page 467
Black hole spin......Page 471
14 Rotating stars......Page 479
14.1.1 Field equations......Page 480
14.1.2 Fluid stars......Page 481
Diagnostics......Page 483
Maximum masses, spins and stability......Page 484
14.1.3 Collisionless clusters......Page 491
Uniformly rotating stars with stiff EOSs......Page 493
Uniformly rotating n = 3 polytropes......Page 496
14.2.2 Bar-mode instability......Page 498
14.2.3 Black hole excision and stellar collapse......Page 501
Uniformly rotating n = 3 polytropes revisited......Page 507
14.2.4 Viscous evolution......Page 511
14.2.5 MHD evolution......Page 515
Hypermassive neutron stars......Page 516
Magnetorotational collapse of n = 3 polytropes......Page 522
Magnetorotational core collapse to neutron stars......Page 524
15.1 Stationary fluid solutions......Page 526
15.1.1 Newtonian equations of stationary equilibrium......Page 528
Corotational binaries......Page 530
Irrotational binaries......Page 531
15.1.2 Relativistic equations of stationary equilibrium......Page 532
15.2 Corotational binaries......Page 534
15.3 Irrotational binaries......Page 543
15.4 Quasiadiabatic inspiral sequences......Page 550
16 Binary neutron star evolution......Page 553
16.1 Peliminary studies......Page 554
16.2 The conformal flatness approximation......Page 555
Validity of the conformal .atness approximation......Page 557
Numerical results......Page 560
16.3 Fully relativistic simulations......Page 565
Equal-mass binaries......Page 566
Binaries with realistic EOSs......Page 572
Magnetized binary neutron star mergers......Page 578
17 Binary black hole--neutron stars......Page 582
17.1.1 The conformal thin-sandwich approach......Page 585
Relativistic equations of stationary equilibrium......Page 586
Quasi-equilibrium black hole boundary conditions......Page 587
Numerical results......Page 588
17.1.2 The conformal transverse-traceless approach......Page 592
17.2.1 The conformal flatness approximation......Page 594
17.2.2 Fully relativistic simulations......Page 598
Corotational neutron stars and nonspinning black holes......Page 599
Irrotational binaries......Page 603
Irrotational neutron stars with spinning black holes......Page 613
18 Epilogue......Page 616
A.1 The Lie derivative......Page 618
A.2 Killing vectors......Page 622
A.3 Tensor densities......Page 623
B Solving the vector Laplacian......Page 627
C The surface element on the apparent horizon......Page 629
D Scalar, vector and tensor spherical harmonics......Page 632
E Post-Newtonian results......Page 636
Post-Newtonian expansions......Page 637
Results for compact binaries......Page 638
Gravitational field equations......Page 649
Matter equations......Page 651
Disks......Page 652
G Rotating equilibria......Page 654
Maximal slicing......Page 657
Stationary 1+log slicing......Page 659
I Binary black hole puncture simulations......Page 662
References......Page 667
Index......Page 704