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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
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Searching... | 30000010283603 | QC173.55 A43 2008 | Open Access Book | Book | Searching... |
Searching... | 33000000017403 | QC173.55 A43 2008 | Open Access Book | Book | Searching... |
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Summary
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 Notes
Professor Miguel AlcubierreDepartment of Gravitation and Mathematical PhysicsInstitute of Nuclear ScienceUniversidad Nacional Autonoma de Mexico
Table of Contents
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 |