Numerical relativity: solving Einstein's equations on the computer

Preface......Page 9
Suggestions for Using this Book......Page 15
Einstein's Equations In 4-Dimensional Spacetime......Page 17
Black Holes......Page 24
Oppenheimer-Volkoff Spherical Equilibrium Stars......Page 29
Oppenheimer-Snyder Spherical Dust Collapse......Page 32
The 3+1 Decompostion of Einstein's Equations......Page 37
Notation and Conventions......Page 39
Maxwell's Equations in Minkowski Spacetime......Page 40
Foliations of Spacetime......Page 42
The Extrinsic Curvature......Page 46
The Equations of Gauss, Codazzi and Ricci......Page 49
The Constraint and Evolution Equations......Page 51
Choosing Basis Vectors: the ADM Equations......Page 55
Constructing Initial Data......Page 65
Conformal Transformation of the Spatial Metric......Page 66
Elementary Black Hole Solutions......Page 68
Conformal transformation of the extrinsic curvature......Page 74
Conformal Transverse-Traceless Decomposition......Page 77
Conformal Thin-Sandwich Decomposition......Page 84
A Step Further: the ``Waveless'' Approximation......Page 89
Mass, Momentum and Angular Momentum......Page 90
Choosing Coordinates: The Lapse and Shift......Page 103
Geodesic Slicing......Page 105
Maximal Slicing and Singularity Avoidance......Page 107
Harmonic Coordinates and Variations......Page 115
Quasi-isotropic and Radial Gauge......Page 117
Minimal Distortion and Variations......Page 119
Matter Sources......Page 125
Perfect Gases......Page 126
Imperfect Gases......Page 139
Radiation Hydrodynamics......Page 140
Magnetohydrodynamics......Page 146
Collisionless Matter......Page 159
Scalar Fields......Page 170
Classification of Partial Differential Equations......Page 177
Representation of Functions and Derivatives......Page 182
Elliptic Equations......Page 184
Hyperbolic Equations......Page 192
Parabolic Equations......Page 200
Mesh Refinement......Page 201
Representation of Functions and Derivatives......Page 203
A simple example......Page 204
Pseudo-spectral methods with Chebychev polynomials......Page 206
Elliptic equations......Page 209
Initial value problems......Page 211
Comparison with finite difference methods......Page 212
Code validation and calibration......Page 213
Concepts......Page 217
Event Horizons......Page 219
Apparent Horizons......Page 223
Spherical Symmetry......Page 227
Axisymmetry......Page 228
General Case: No Symmetry Assumptions......Page 232
Isolated and Dynamical Horizons......Page 235
Spherically Symmetric Spacetimes......Page 239
Black Holes......Page 241
Collisionless Clusters: Stability and Collapse......Page 250
Particle Method......Page 252
Phase Space Method......Page 268
Fluid Stars: Collapse......Page 273
Misner-Sharp Formalism......Page 276
The Hernandez-Misner Equations......Page 278
Scalar Field Collapse: Critical Phenomena......Page 284
Linearized Waves......Page 291
Perturbation Theory and the Weak-Field, Slow-Velocity Regime......Page 292
Vacuum Solutions......Page 298
Sources......Page 301
The High Frequency Band......Page 302
The Low Frequency Band......Page 306
The Very Low and Ultra Low Frequency Bands......Page 307
Detectors and Templates......Page 308
Ground-based gravitational wave interferometers......Page 309
Space-based detectors......Page 312
The Gauge-Invariant Moncrief Formalism......Page 314
The Newman-Penrose Formalism......Page 322
Collapse of Prolate Spheroids to Spindle Singularities......Page 327
Head-On Collision of Two Black Holes......Page 333
Disk Collapse......Page 338
Collapse of Rotating Toroidal Clusters......Page 342
Recasting the Evolution Equations......Page 349
Notions of Hyperbolicity......Page 350
Recasting Maxwell's Equations......Page 352
Generalized Coulomb gauge......Page 353
Generalized Harmonic Coordinates......Page 354
First-order Symmetric Hyperbolic Formulations......Page 357
The BSSN Formulation......Page 359
Binary Black Hole Initial Data......Page 367
Binary Inspiral: Overview......Page 368
Solving the momentum constraint......Page 375
Solving the Hamiltonian constraint......Page 377
Identifying circular orbits......Page 378
The notion of quasiequilibium......Page 381
Quasiequilibrium black-hole boundary conditions......Page 384
Identifying Circular orbits......Page 389
Quasiequilibrium Sequences......Page 391
Binary Black Hole Evolution......Page 397
Singularity Avoiding Coordinates......Page 398
The Moving Puncture Method......Page 399
Equal-Mass Binaries......Page 404
Asymmetric Binaries, Spin and Black Hole Recoil......Page 411
Rotating Stars......Page 423
Field Equations......Page 424
Fluid Stars......Page 425
Collisionless Clusters......Page 434
Quasi-Radial Stability and Collapse......Page 435
Bar-mode Instability......Page 441
Black Hole Excision and Stellar Collapse......Page 445
Viscous Evolution......Page 452
MHD Evolution......Page 456
Stationary Fluid Solutions......Page 465
Newtonian Equations of Stationary Equilibrium......Page 467
Relativistic Equations of Stationary Equilibrium......Page 470
Corotational Binaries......Page 472
Irrotational Binaries......Page 480
Quasiadiabatic inspiral sequences......Page 486
Binary Neutron Star Evolution......Page 489
Peliminary Studies......Page 490
The Conformal Flatness Approximation......Page 491
Fully Relativistic Simulations......Page 500
Binary Black Hole-Neutron Stars: Initial Data and Evolution......Page 517
Initial Data......Page 519
The conformal thin-sandwich approach......Page 520
The conformal transverse-traceless approach......Page 526
The Conformal Flatness Approximation......Page 528
Fully Relativistic Simulations......Page 532
Epilogue......Page 551
The Lie Derivative......Page 553
Killing Vectors......Page 557
Tensor Densities......Page 559
Solving the Vector Laplacian......Page 563
The Surface Element on the Apparent Horizon......Page 565
Scalar, Vector and Tensor Spherical Harmonics......Page 569
Post-Newtonian Results......Page 573
Collisionless Matter Evolution in Axisymmetry: Basic Equations......Page 585
Rotating Equilibria: Gravitational Field Equations......Page 591
Moving-Puncture Representions of Schwarzschild: Analytical Results......Page 595
Binary Black Hole Puncture Simulations As Test Problems......Page 601