Introduction to 3+1 numerical relativity / Miguel Alcubierre.

Format:

Book

Author/Creator:

Alcubierre, Miguel.

Series:

International series of monographs on physics ; 140.

International series of monographs on physics ; 140

Language:

English

Subjects (All):

Relativity (Physics).

Space and time--Mathematics.

Space and time.

Physical Description:

xiv, 444 pages : illustrations ; 25 cm.

Place of Publication:

Oxford ; New York : Oxford University Press, 2008.

Contents:

1 Brief review of general relativity 1

1.2 Notation and conventions 2

1.3 Special relativity 2

1.4 Manifolds and tensors 7

1.5 The metric tensor 10

1.6 Lie derivatives and Killing fields 14

1.7 Coordinate transformations 17

1.8 Covariant derivatives and geodesics 20

1.9 Curvature 25

1.10 Bianchi identities and the Einstein tensor 28

1.11 General relativity 28

1.12 Matter and the stress-energy tensor 32

1.13 The Einstein field equations 36

1.14 Weak fields and gravitational waves 39

1.15 The Schwarzschild solution and black holes 46

1.16 Black holes with charge and angular momentum 53

1.17 Causal structure, singularities and black holes 57

2 The 3+1 formalism 64

2.2 3+1 split of spacetime 65

2.3 Extrinsic curvature 68

2.4 The Einstein constraints 71

2.5 The ADM evolution equations 73

2.6 Free versus constrained evolution 77

2.7 Hamiltonian formulation 78

2.8 The BSSNOK formulation 81

2.9 Alternative formalisms 87

2.9.1 The characteristic approach 87

2.9.2 The conformal approach 90

3 Initial data 92

3.2 York-Lichnerowicz conformal decomposition 92

3.2.1 Conformal transverse decomposition 94

3.2.2 Physical transverse decomposition 97

3.2.3 Weighted transverse decomposition 99

3.3 Conformal thin-sandwich approach 101

3.4 Multiple black hole initial data 105

3.4.1 Time-symmetric data 105

3.4.2 Bowen-York extrinsic curvature 109

3.4.3 Conformal factor: inversions and punctures 111

3.4.4 Kerr-Schild type data 113

3.5 Binary black holes in quasi-circular orbits 115

3.5.1 Effective potential method 116

3.5.2 The quasi-equilibrium method 117

4 Gauge conditions 121

4.2 Slicing conditions 122

4.2.1 Geodesic slicing and focusing 123

4.2.2 Maximal slicing 123

4.2.3 Maximal slices of Schwarzschild 127

4.2.4 Hyperbolic slicing conditions 133

4.2.5 Singularity avoidance for hyperbolic slicings 136

4.3 Shift conditions 140

4.3.1 Elliptic shift conditions 141

4.3.2 Evolution type shift conditions 145

4.3.3 Corotating coordinates 151

5 Hyperbolic reductions of the field equations 155

5.2 Well-posedness 156

5.3 The concept of hyperbolicity 158

5.4 Hyperbolicity of the ADM equations 164

5.5 The Bona-Masso and NOR formulations 169

5.6 Hyperbolicity of BSSNOK 175

5.7 The Kidder-Scheel-Teukolsky family 179

5.8 Other hyperbolic formulations 183

5.8.1 Higher derivative formulations 184

5.8.2 The Z4 formulation 185

5.9 Boundary conditions 187

5.9.1 Radiative boundary conditions 188

5.9.2 Maximally dissipative boundary conditions 191

5.9.3 Constraint preserving boundary conditions 194

6 Evolving black hole spacetimes 198

6.2 Isometries and throat adapted coordinates 199

6.3 Static puncture evolution 206

6.4 Singularity avoidance and slice stretching 209

6.5 Black hole excision 214

6.6 Moving punctures 217

6.6.1 How to move the punctures 217

6.6.2 Why does evolving the punctures work? 219

6.7 Apparent horizons 221

6.7.1 Apparent horizons in spherical symmetry 223

6.7.2 Apparent horizons in axial symmetry 224

6.7.3 Apparent horizons in three dimensions 226

6.8 Event horizons 230

6.9 Isolated and dynamical horizons 234

7 Relativistic hydrodynamics 238

7.2 Special relativistic hydrodynamics 239

7.3 General relativistic hydrodynamics 245

7.4 3+1 form of the hydrodynamic equations 249

7.5 Equations of state: dust, ideal gases and polytropes 252

7.6 Hyperbolicity and the speed of sound 257

7.6.1 Newtonian case 257

7.6.2 Relativistic case 260

7.7 Weak solutions and the Riemann problem 264

7.8 Imperfect fluids: viscosity and heat conduction 270

7.8.1 Eckart's irreversible thermodynamics 270

7.8.2 Causal irreversible thermodynamics 273

8 Gravitational wave extraction 276

8.2 Gauge invariant perturbations of Schwarzschild 277

8.2.1 Multipole expansion 277

8.2.2 Even parity perturbations 280

8.2.3 Odd parity perturbations 283

8.2.4 Gravitational radiation in the TT gauge 284

8.3 The Weyl tensor 288

8.4 The tetrad formalism 291

8.5 The Newman-Penrose formalism 294

8.5.1 Null tetrads 294

8.5.2 Tetrad transformations 297

8.6 The Weyl scalars 298

8.7 The Petrov classification 299

8.8 Invariants I and J 303

8.9 Energy and momentum of gravitational waves 304

8.9.1 The stress-energy tensor for gravitational waves 304

8.9.2 Radiated energy and momentum 307

8.9.3 Multipole decomposition 313

9 Numerical methods 318

9.2 Basic concepts of finite differencing 318

9.3 The one-dimensional wave equation 322

9.3.1 Explicit finite difference approximation 323

9.3.2 Implicit approximation 325

9.4 Von Newmann stability analysis 326

9.5 Dissipation and dispersion 329

9.6 Boundary conditions 332

9.7 Numerical methods for first order systems 335

9.8 Method of lines 339

9.9 Artificial dissipation and viscosity 343

9.10 High resolution schemes 347

9.10.1 Conservative methods 347

9.10.2 Godunov's method 348

9.10.3 High resolution methods 350

9.11 Convergence testing 353

10 Examples of numerical spacetimes 357

10.2 Toy 1+1 relativity 357

10.2.1 Gauge shocks 359

10.2.2 Approximate shock avoidance 362

10.2.3 Numerical examples 364

10.3 Spherical symmetry 369

10.3.1 Regularization 370

10.3.2 Hyperbolicity 374

10.3.3 Evolving Schwarzschild 378

10.3.4 Scalar field collapse 383

10.4 Axial symmetry 391

10.4.1 Evolution equations and regularization 391

10.4.2 Brill waves 395

10.4.3 The "Cartoon" approach 399

A Total mass and momentum in general relativity 402

B Spacetime Christoffel symbols in 3+1 language 409

C BSSNOK with natural conformal rescaling 410

D Spin-weighted spherical harmonics 413.

Notes:

Includes bibliographical references and index.

ISBN:

9780199205677

0199205671

OCLC:

191929824