Introduction to 3+1 Numerical Relativity

by Alcubierre, Miguel

• ISBN13: 9780199656158
• ISBN10: 0199656150
• Edition: Reprint
• Format: Paperback
• Copyright: 2012-09-07
• Publisher: Oxford University Press

Summary

This book introduces the modern field of 3+1 numerical relativity. The book has been written in a way as to be as self-contained as possible, and only assumes a basic knowledge of special relativity. Starting from a brief introduction to general relativity, it discusses the different concepts and tools necessary for the fully consistent numerical simulation of relativistic astrophysical systems, with strong and dynamical gravitational fields. Among the topics discussed in detail are the following: the initial data problem, hyperbolic reductions of the field equations, gauge conditions, the evolution of black hole space-times, relativistic hydrodynamics, gravitational wave extraction and numerical methods. There is also a final chapter with examples of some simple numerical space-times. The book is aimed at both graduate students and researchers in physics and astrophysics, and at those interested in relativistic astrophysics.

Author Biography

Miguel Alcubierre, Department of Gravitation and Field Theory, Institute of Nuclear Science, Universidad Nacional Autonoma de Mexico

Table of Contents

Brief review of general relativity	p. 1
Introduction	p. 1
Notation and conventions	p. 2
Special relativity	p. 2
Manifolds and tensors	p. 7
The metric tensor	p. 10
Lie derivatives and Killing fields	p. 14
Coordinate transformations	p. 17
Covariant derivatives and geodesies	p. 20
Curvature	p. 25
Bianchi identities and the Einstein tensor	p. 28
General relativity	p. 28
Matter and the stress-energy tensor	p. 32
The Einstein field equations	p. 36
Weak fields and gravitational waves	p. 39
The Schwarzschild solution and black holes	p. 46
Black holes with charge and angular momentum	p. 53
Causal structure, singularities and black holes	p. 57
The 3+1 formalism	p. 64
Introduction	p. 64
3+1 split of spacetime	p. 65
Extrinsic curvature	p. 68
The Einstein constraints	p. 71
The ADM evolution equations	p. 73
Free versus constrained evolution	p. 77
Hamiltonian formulation	p. 78
The BSSNOK formulation	p. 81
Alternative formalisms	p. 87
The characteristic approach	p. 87
The conformal approach	p. 90
Initial data	p. 92
Introduction	p. 92
York-Lichnerowicz conformal decomposition	p. 92
Conformal transverse decomposition	p. 94
Physical transverse decomposition	p. 97
Weighted transverse decomposition	p. 99
Conformal thin-sandwich approach	p. 101
Multiple black hole initial data	p. 105
Time-symmetric data	p. 105
Bowen-York extrinsic curvature	p. 109
Conformal factor: inversions and punctures	p. 111
Kerr-Schild type data	p. 113
Binary black holes in quasi-circular orbits	p. 115
Effective potential method	p. 116
The quasi-equilibrium method	p. 117
Gauge conditions	p. 121
Introduction	p. 121
Slicing conditions	p. 122
Geodesic slicing and focusing	p. 123
Maximal slicing	p. 123
Maximal slices of Schwarzschild	p. 127
Hyperbolic slicing conditions	p. 133
Singularity avoidance for hyperbolic slicings	p. 136
Shift conditions	p. 140
Elliptic shift conditions	p. 141
Evolution type shift conditions	p. 145
Corotating coordinates	p. 151
Hyperbolic reductions of the field equations	p. 155
Introduction	p. 155
Well-posedness	p. 156
The concept of hyperbolicity	p. 158
Hyperbolicity of the ADM equations	p. 164
The Bona-Masso and NOR formulations	p. 169
Hyperbolicity of BSSNOK	p. 175
The Kidder-Scheel-Teukolsky family	p. 179
Other hyperbolic formulations	p. 183
Higher derivative formulations	p. 184
The Z4 formulation	p. 185
Boundary conditions	p. 187
Radiative boundary conditions	p. 188
Maximally dissipative boundary conditions	p. 191
Constraint preserving boundary conditions	p. 194
Evolving black hole spacetimes	p. 198
Introduction	p. 198
Isometries and throat adapted coordinates	p. 199
Static puncture evolution	p. 206
Singularity avoidance and slice stretching	p. 209
Black hole excision	p. 214
Moving punctures	p. 217
How to move the punctures	p. 217
Why does evolving the punctures work?	p. 219
Apparent horizons	p. 221
Apparent horizons in spherical symmetry	p. 223
Apparent horizons in axial symmetry	p. 224
Apparent horizons in three dimensions	p. 226
Event horizons	p. 230
Isolated and dynamical horizons	p. 234
Relativistic hydrodynamics	p. 238
Introduction	p. 238
Special relativistic hydrodynamics	p. 239
General relativistic hydrodynamics	p. 245
3+1 form of the hydrodynamic equations	p. 249
Equations of state: dust, ideal gases and polytropes	p. 252
Hyperbolicity and the speed of sound	p. 257
Newtonian case	p. 257
Relativistic case	p. 260
Weak solutions and the Riemann problem	p. 264
Imperfect fluids: viscosity and heat conduction	p. 270
Eckart's irreversible thermodynamics	p. 270
Causal irreversible thermodynamics	p. 273
Gravitational wave extraction	p. 276
Introduction	p. 276
Gauge invariant perturbations of Schwarzschild	p. 277
Multipole expansion	p. 277
Even parity perturbations	p. 280
Odd parity perturbations	p. 283
Gravitational radiation in the TT gauge	p. 284
The Weyl tensor	p. 288
The tetrad formalism	p. 291
The Newman-Penrose formalism	p. 294
Null tetrads	p. 294
Tetrad transformations	p. 297
The Weyl scalars	p. 298
The Petrov classification	p. 299
Invariants I and J	p. 303
Energy and momentum of gravitational waves	p. 304
The stress-energy tensor for gravitational waves	p. 304
Radiated energy and momentum	p. 307
Multipole decomposition	p. 313
Numerical methods	p. 318
Introduction	p. 318
Basic concepts of finite differencing	p. 318
The one-dimensional wave equation	p. 322
Explicit finite difference approximation	p. 323
Implicit approximation	p. 325
Von Newmann stability analysis	p. 326
Dissipation and dispersion	p. 329
Boundary conditions	p. 332
Numerical methods for first order systems	p. 335
Method of lines	p. 339
Artificial dissipation and viscosity	p. 343
High resolution schemes	p. 347
Conservative methods	p. 347
Godunov's method	p. 348
High resolution methods	p. 350
Convergence testing	p. 353
Examples of numerical spacetimes	p. 357
Introduction	p. 357
Toy 1+1 relativity	p. 357
Gauge shocks	p. 359
Approximate shock avoidance	p. 362
Numerical examples	p. 364
Spherical symmetry	p. 369
Regularization	p. 370
Hyperbolicity	p. 374
Evolving Schwarzschild	p. 378
Scalar field collapse	p. 383
Axial symmetry	p. 391
Evolution equations and regularization	p. 391
Brill waves	p. 395
The "Cartoon" approach	p. 399
Total mass and momentum in general relativity	p. 402
Spacetime Christoffel symbols in 3+1 language	p. 409
BSSNOK with natural conformal rescaling	p. 410
Spin-weighted spherical harmonics	p. 413
References	p. 419
Index	p. 437
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