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E-raamat: Introduction to 3+1 Numerical Relativity

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(Department of Gravitation and Mathematical Physics, Institute of Nuclear Science, Universidad Nacional Autonoma de Mexico)
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Introduction to 3+1 Numerical RelativitySuurem pilt
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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.

Arvustused

An excellent book, and a good starting point for those making an effort to enter the field. * Mathematical Reviews *

1 Brief review of general relativity
1(63)
1.1 Introduction
1(1)
1.2 Notation and conventions
2(1)
1.3 Special relativity
2(5)
1.4 Manifolds and tensors
7(3)
1.5 The metric tensor
10(4)
1.6 Lie derivatives and Killing fields
14(3)
1.7 Coordinate transformations
17(3)
1.8 Covariant derivatives and geodesics
20(5)
1.9 Curvature
25(3)
1.10 Bianchi identities and the Einstein tensor
28(1)
1.11 General relativity
28(4)
1.12 Matter and the stress-energy tensor
32(4)
1.13 The Einstein field equations
36(3)
1.14 Weak fields and gravitational waves
39(7)
1.15 The Schwarzschild solution and black holes
46(7)
1.16 Black holes with charge and angular momentum
53(4)
1.17 Causal structure, singularities and black holes
57(7)
2 The 3+1 formalism
64(28)
2.1 Introduction
64(1)
2.2 3+1 split of spacetime
65(3)
2.3 Extrinsic curvature
68(3)
2.4 The Einstein constraints
71(2)
2.5 The ADM evolution equations
73(4)
2.6 Free versus constrained evolution
77(1)
2.7 Hamiltonian formulation
78(3)
2.8 The BSSNOK formulation
81(6)
2.9 Alternative formalisms
87(5)
2.9.1 The characteristic approach
87(3)
2.9.2 The conformal approach
90(2)
3 Initial data
92(29)
3.1 Introduction
92(1)
3.2 York-Lichnerowicz conformal decomposition
92(9)
3.2.1 Conformal transverse decomposition
94(3)
3.2.2 Physical transverse decomposition
97(2)
3.2.3 Weighted transverse decomposition
99(2)
3.3 Conformal thin-sandwich approach
101(4)
3.4 Multiple black hole initial data
105(10)
3.4.1 Time-symmetric data
105(4)
3.4.2 Bowen-York extrinsic curvature
109(2)
3.4.3 Conformal factor: inversions and punctures
111(2)
3.4.4 Kerr-Schild type data
113(2)
3.5 Binary black holes in quasi-circular orbits
115(6)
3.5.1 Effective potential method
116(1)
3.5.2 The quasi-equilibrium method
117(4)
4 Gauge conditions
121(34)
4.1 Introduction
121(1)
4.2 Slicing conditions
122(18)
4.2.1 Geodesic slicing and focusing
123(1)
4.2.2 Maximal slicing
123(4)
4.2.3 Maximal slices of Schwarzschild
127(6)
4.2.4 Hyperbolic slicing conditions
133(3)
4.2.5 Singularity avoidance for hyperbolic slicings
136(4)
4.3 Shift conditions
140(15)
4.3.1 Elliptic shift conditions
141(4)
4.3.2 Evolution type shift conditions
145(6)
4.3.3 Corotating coordinates
151(4)
5 Hyperbolic reductions of the field equations
155(43)
5.1 Introduction
155(1)
5.2 Well-posedness
156(2)
5.3 The concept of hyperbolicity
158(6)
5.4 Hyperbolicity of the ADM equations
164(5)
5.5 The Bona-Masso and NOR formulations
169(6)
5.6 Hyperbolicity of BSSNOK
175(4)
5.7 The Kidder-Scheel-Teukolsky family
179(4)
5.8 Other hyperbolic formulations
183(4)
5.8.1 Higher derivative formulations
184(1)
5.8.2 The Z4 formulation
185(2)
5.9 Boundary conditions
187(11)
5.9.1 Radiative boundary conditions
188(3)
5.9.2 Maximally dissipative boundary conditions
191(3)
5.9.3 Constraint preserving boundary conditions
194(4)
6 Evolving black hole spacetimes
198(40)
6.1 Introduction
198(1)
6.2 Isometries and throat adapted coordinates
199(7)
6.3 Static puncture evolution
206(3)
6.4 Singularity avoidance and slice stretching
209(5)
6.5 Black hole excision
214(3)
6.6 Moving punctures
217(4)
6.6.1 How to move the punctures
217(2)
6.6.2 Why does evolving the punctures work?
219(2)
6.7 Apparent horizons
221(9)
6.7.1 Apparent horizons in spherical symmetry
223(1)
6.7.2 Apparent horizons in axial symmetry
224(2)
6.7.3 Apparent horizons in three dimensions
226(4)
6.8 Event horizons
230(4)
6.9 Isolated and dynamical horizons
234(4)
7 Relativistic hydrodynamics
238(38)
7.1 Introduction
238(1)
7.2 Special relativistic hydrodynamics
239(6)
7.3 General relativistic hydrodynamics
245(4)
7.4 3+1 form of the hydrodynamic equations
249(3)
7.5 Equations of state: dust, ideal gases and polytropes
252(5)
7.6 Hyperbolicity and the speed of sound
257(7)
7.6.1 Newtonian case
257(3)
7.6.2 Relativistic case
260(4)
7.7 Weak solutions and the Riemann problem
264(6)
7.8 Imperfect fluids: viscosity and heat conduction
270(6)
7.8.1 Eckart's irreversible thermodynamics
270(3)
7.8.2 Causal irreversible thermodynamics
273(3)
8 Gravitational wave extraction
276(42)
8.1 Introduction
276(1)
8.2 Gauge invariant perturbations of Schwarzschild
277(11)
8.2.1 Multipole expansion
277(3)
8.2.2 Even parity perturbations
280(3)
8.2.3 Odd parity perturbations
283(1)
8.2.4 Gravitational radiation in the TT gauge
284(4)
8.3 The Weyl tensor
288(3)
8.4 The tetrad formalism
291(3)
8.5 The Newman-Penrose formalism
294(4)
8.5.1 Null tetrads
294(3)
8.5.2 Tetrad transformations
297(1)
8.6 The Weyl scalars
298(1)
8.7 The Petrov classification
299(4)
8.8 Invariants I and J
303(1)
8.9 Energy and momentum of gravitational waves
304(14)
8.9.1 The stress-energy tensor for gravitational waves
304(3)
8.9.2 Radiated energy and momentum
307(6)
8.9.3 Multipole decomposition
313(5)
9 Numerical methods
318(39)
9.1 Introduction
318(1)
9.2 Basic concepts of finite differencing
318(4)
9.3 The one-dimensional wave equation
322(4)
9.3.1 Explicit finite difference approximation
323(2)
9.3.2 Implicit approximation
325(1)
9.4 Von Newmann stability analysis
326(3)
9.5 Dissipation and dispersion
329(3)
9.6 Boundary conditions
332(3)
9.7 Numerical methods for first order systems
335(4)
9.8 Method of lines
339(4)
9.9 Artificial dissipation and viscosity
343(4)
9.10 High resolution schemes
347(6)
9.10.1 Conservative methods
347(1)
9.10.2 Godunov's method
348(2)
9.10.3 High resolution methods
350(3)
9.11 Convergence testing
353(4)
10 Examples of numerical spacetimes
357(45)
10.1 Introduction
357(1)
10.2 Tby 1+1 relativity
357(12)
10.2.1 Gauge shocks
359(3)
10.2.2 Approximate shock avoidance
362(2)
10.2.3 Numerical examples
364(5)
10.3 Spherical symmetry
369(22)
10.3.1 Regularization
370(4)
10.3.2 Hyperbolicity
374(4)
10.3.3 Evolving Schwarzschild
378(5)
10.3.4 Scalar field collapse
383(8)
10.4 Axial symmetry
391(11)
10.4.1 Evolution equations and regularization
391(4)
10.4.2 Brill waves
395(4)
10.4.3 The "Cartoon" approach
399(3)
A Total mass and momentum In general relativity 402(7)
B Spacetime Christoffel symbols in 3+1 language 409(1)
C BSSNOK with natural conformal rescaling 410(3)
D Spin-weighted spherical harmonics 413(6)
References 419(18)
Index 437
Miguel Alcubierre, Department of Gravitation and Mathematical Physics, Institute of Nuclear Science, Universidad Nacional Autonoma de Mexico
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