Introduction to 3+1 numerical relativity

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010283603	QC173.55 A43 2008	Open Access Book	Book	Searching... Unknown
Searching... PSZ KL	33000000017403	QC173.55 A43 2008	Open Access Book	Book	Searching... Unknown

This book is a self-contained introduction to the field of numerical relativity, aimed at graduate students and researchers in physics and astrophysics. Starting from basic general relativity, it introduces all the concepts and tools necessary for the fully relativistic simulation of astrophysical systems with strong and dynamical gravitational fields.

Author Notes

Professor Miguel AlcubierreDepartment of Gravitation and Mathematical PhysicsInstitute of Nuclear ScienceUniversidad Nacional Autonoma de Mexico

1 Brief review of general relativity	p. 1
1.1 Introduction	p. 1
1.2 Notation and conventions	p. 2
1.3 Special relativity	p. 2
1.4 Manifolds and tensors	p. 7
1.5 The metric tensor	p. 10
1.6 Lie derivatives and Killing fields	p. 14
1.7 Coordinate transformations	p. 17
1.8 Covariant derivatives and geodesics	p. 20
1.9 Curvature	p. 25
1.10 Bianchi identities and the Einstein tensor	p. 28
1.11 General relativity	p. 28
1.12 Matter and the stress-energy tensor	p. 32
1.13 The Einstein field equations	p. 36
1.14 Weak fields and gravitational waves	p. 39
1.15 The Schwarzschild solution and black holes	p. 46
1.16 Black holes with charge and angular momentum	p. 53
1.17 Causal structure, singularities and black holes	p. 57
2 The 3+1 formalism	p. 64
2.1 Introduction	p. 64
2.2 3+1 split of spacetime	p. 65
2.3 Extrinsic curvature	p. 68
2.4 The Einstein constraints	p. 71
2.5 The ADM evolution equations	p. 73
2.6 Free versus constrained evolution	p. 77
2.7 Hamiltonian formulation	p. 78
2.8 The BSSNOK formulation	p. 81
2.9 Alternative formalisms	p. 87
2.9.1 The characteristic approach	p. 87
2.9.2 The conformal approach	p. 90
3 Initial data	p. 92
3.1 Introduction	p. 92
3.2 York-Lichnerowicz conformal decomposition	p. 92
3.2.1 Conformal transverse decomposition	p. 94
3.2.2 Physical transverse decomposition	p. 97
3.2.3 Weighted transverse decomposition	p. 99
3.3 Conformal thin-sandwich approach	p. 101
3.4 Multiple black hole initial data	p. 105
3.4.1 Time-symmetric data	p. 105
3.4.2 Bowen-York extrinsic curvature	p. 109
3.4.3 Conformal factor: inversions and punctures	p. 111
3.4.4 Kerr-Schild type data	p. 113
3.5 Binary black holes in quasi-circular orbits	p. 115
3.5.1 Effective potential method	p. 116
3.5.2 The quasi-equilibrium method	p. 117
4 Gauge conditions	p. 121
4.1 Introduction	p. 121
4.2 Slicing conditions	p. 122
4.2.1 Geodesic slicing and focusing	p. 123
4.2.2 Maximal slicing	p. 123
4.2.3 Maximal slices of Schwarzschild	p. 127
4.2.4 Hyperbolic slicing conditions	p. 133
4.2.5 Singularity avoidance for hyperbolic slicings	p. 136
4.3 Shift conditions	p. 140
4.3.1 Elliptic shift conditions	p. 141
4.3.2 Evolution type shift conditions	p. 145
4.3.3 Corotating coordinates	p. 151
5 Hyperbolic reductions of the field equations	p. 155
5.1 Introduction	p. 155
5.2 Well-posedness	p. 156
5.3 The concept of hyperbolicity	p. 158
5.4 Hyperbolicity of the ADM equations	p. 164
5.5 The Bona-Masso and NOR formulations	p. 169
5.6 Hyperbolicity of BSSNOK	p. 175
5.7 The Kidder-Scheel-Teukolsky family	p. 179
5.8 Other hyperbolic formulations	p. 183
5.8.1 Higher derivative formulations	p. 184
5.8.2 The Z4 formulation	p. 185
5.9 Boundary conditions	p. 187
5.9.1 Radiative boundary conditions	p. 188
5.9.2 Maximally dissipative boundary conditions	p. 191
5.9.3 Constraint preserving boundary conditions	p. 194
6 Evolving black hole spacetimes	p. 198
6.1 Introduction	p. 198
6.2 Isometries and throat adapted coordinates	p. 199
6.3 Static puncture evolution	p. 206
6.4 Singularity avoidance and slice stretching	p. 209
6.5 Black hole excision	p. 214
6.6 Moving punctures	p. 217
6.6.1 How to move the punctures	p. 217
6.6.2 Why does evolving the punctures work?	p. 219
6.7 Apparent horizons	p. 221
6.7.1 Apparent horizons in spherical symmetry	p. 223
6.7.2 Apparent horizons in axial symmetry	p. 224
6.7.3 Apparent horizons in three dimensions	p. 226
6.8 Event horizons	p. 230
6.9 Isolated and dynamical horizons	p. 234
7 Relativistic hydrodynamics	p. 238
7.1 Introduction	p. 238
7.2 Special relativistic hydrodynamics	p. 239
7.3 General relativistic hydrodynamics	p. 245
7.4 3+1 form of the hydrodynamic equations	p. 249
7.5 Equations of state: dust, ideal gases and polytropes	p. 252
7.6 Hyperbolicity and the speed of sound	p. 257
7.6.1 Newtonian case	p. 257
7.6.2 Relativistic case	p. 260
7.7 Weak solutions and the Riemann problem	p. 264
7.8 Imperfect fluids: viscosity and heat conduction	p. 270
7.8.1 Eckart's irreversible thermodynamics	p. 270
7.8.2 Causal irreversible thermodynamics	p. 273
8 Gravitational wave extraction	p. 276
8.1 Introduction	p. 276
8.2 Gauge invariant perturbations of Schwarzschild	p. 277
8.2.1 Multipole expansion	p. 277
8.2.2 Even parity perturbations	p. 280
8.2.3 Odd parity perturbations	p. 283
8.2.4 Gravitational radiation in the TT gauge	p. 284
8.3 The Weyl tensor	p. 288
8.4 The tetrad formalism	p. 291
8.5 The Newman-Penrose formalism	p. 294
8.5.1 Null tetrads	p. 294
8.5.2 Tetrad transformations	p. 297
8.6 The Weyl scalars	p. 298
8.7 The Petrov classification	p. 299
8.8 Invariants I and J	p. 303
8.9 Energy and momentum of gravitational waves	p. 304
8.9.1 The stress-energy tensor for gravitational waves	p. 304
8.9.2 Radiated energy and momentum	p. 307
8.9.3 Multipole decomposition	p. 313
9 Numerical methods	p. 318
9.1 Introduction	p. 318
9.2 Basic concepts of finite differencing	p. 318
9.3 The one-dimensional wave equation	p. 322
9.3.1 Explicit finite difference approximation	p. 323
9.3.2 Implicit approximation	p. 325
9.4 Von Newmann stability analysis	p. 326
9.5 Dissipation and dispersion	p. 329
9.6 Boundary conditions	p. 332
9.7 Numerical methods for first order systems	p. 335
9.8 Method of lines	p. 339
9.9 Artificial dissipation and viscosity	p. 343
9.10 High resolution schemes	p. 347
9.10.1 Conservative methods	p. 347
9.10.2 Godunov's method	p. 348
9.10.3 High resolution methods	p. 350
9.11 Convergence testing	p. 353
10 Examples of numerical spacetimes	p. 357
10.1 Introduction	p. 357
10.2 Toy 1+1 relativity	p. 357
10.2.1 Gauge shocks	p. 359
10.2.2 Approximate shock avoidance	p. 362
10.2.3 Numerical examples	p. 364
10.3 Spherical symmetry	p. 369
10.3.1 Regularization	p. 370
10.3.2 Hyperbolicity	p. 374
10.3.3 Evolving Schwarzschild	p. 378
10.3.4 Scalar field collapse	p. 383
10.4 Axial symmetry	p. 391
10.4.1 Evolution equations and regularization	p. 391
10.4.2 Brill waves	p. 395
10.4.3 The "Cartoon" approach	p. 399
A Total mass and momentum in general relativity	p. 402
B Spacetime Christoffel symbols in 3+1 language	p. 409
C BSSNOK with natural conformal rescaling	p. 410
D Spin-weighted spherical harmonics	p. 413
References	p. 419
Index	p. 437

Available:*

On Order

Summary

Summary

Author Notes

Table of Contents