Relativistic numerical hydrodynamics

Cover......Page 1
Title......Page 3
Date-line......Page 6
Dedication......Page 7
Contents ......Page 9
Preface ......Page 15
1 Introduction ......Page 19
1.1  Notation and convention ......Page 20
1.2  General relativity ......Page 21
1.2.2 Energy momentum tensor ......Page 23
1.2.3 Covariant differentiation ......Page 24
1.2.4 Bianchi identities ......Page 25
1.3  (3+1) or ADM formalism ......Page 26
1.3.1 Eulerian observer ......Page 27
1.3.3 ADM metric ......Page 29
1.3.4 Fluid observer ......Page 30
1.3.5 Field equations in (3+1) formalism ......Page 31
1.3.6 Constraint equations ......Page 32
1.3.7 Solving the ADM equations ......Page 33
1.3.8 Gauge choices for $\\gamma_{ij}$ ......Page 34
1.3.9 Gauge choices for $\\beta$ ......Page 38
References ......Page 39
2.1  Perfect fluid energy momentum tensor ......Page 41
2.2  Equation of motion ......Page 43
2.2.1 Viscosity and heat flow ......Page 45
2.2.3 Causality and Navier-Stokes ......Page 46
2.4  Difference equations: generalities ......Page 47
2.4.1 Artificial viscosity versus Riemann ......Page 48
2.4.2 Finite difference preliminaries ......Page 50
2.4.3 Relativistic hydrodynamics in one dimension ......Page 52
2.4.4 Operator splitting ......Page 53
2.4.5 Time step calculation ......Page 54
2.4.6 Artificial viscosity ......Page 55
2.4.7 Realistic artificial viscosity in one dimension ......Page 57
2.4.9 Velocity calculation ......Page 59
2.4.10 Pressure work ......Page 60
2.4.11 Grid update ......Page 61
2.4.14 Density advection ......Page 62
2.4.16 Momentum advection ......Page 64
2.5.1 Shocks and jump conditions ......Page 66
2.5.2 Relativistic wall shocks ......Page 67
2.5.3 Accelerating wall shocks ......Page 68
2.5.4 Accelerating shocks in the Eulerian gauge ......Page 69
2.5.5 Stopping wall shocks ......Page 70
2.5.6 Relativistic rarefaction ......Page 71
2.5.7 Newtonian and relativistic shock tube ......Page 73
2.5.8 Newtonian Noh solution ......Page 77
2.5.9 Homologous Newtonian collapse ......Page 78
2.6  Application to heavy ion collisions ......Page 79
2.6.1 Hydrodynamics and heavy ion stopping ......Page 82
2.6.2 Nuclear fluid plus pions ......Page 87
2.6.3 Solving the Navier-Stokes equation with pions ......Page 88
References ......Page 91
3.1  General relativity ......Page 93
3.2.1 State variables ......Page 94
3.2.2 Equations of motion ......Page 96
3.2.4  Grid velocities ......Page 97
3.3.3 Proper volume terms ......Page 98
3.3.5 Grid velocity advection ......Page 99
3.3.7 Metric acceleration ......Page 100
3.4.1 The grid ......Page 101
3.4.2 Advection ......Page 102
3.4.3 Advection in curvilinear coordinates ......Page 103
3.4.4 Pressure terms ......Page 104
3.4.5 Velocity update ......Page 105
3.4.6 Energy PdV velocity terms ......Page 107
3.5 Grid calculation ......Page 108
3.5.3 Shift vector acceleration ......Page 110
3.6 Artificial viscosity ......Page 111
3.8 Time step ......Page 114
References ......Page 116
4.1 Planar cosmology ......Page 117
4.1.1 Hydrodynamics for planar cosmology ......Page 121
4.1.2 Solution of metric equations ......Page 122
4.2.1 Nucleosynthesis ......Page 123
4.2.2 Inflation ......Page 125
4.2.3 Inflaton potential ......Page 127
4.3 Spherical inhomogeneous cosmology ......Page 129
4.3.1 The metric ......Page 131
4.3.3 Boundary conditions in spherical cosmology ......Page 132
References ......Page 133
5.1 Collapse supernovae ......Page 135
5.2.1 The metric ......Page 138
5.2.2 Energy momentum tensor ......Page 139
5.2.3 Evolution equations ......Page 140
5.2.4 Matter equations ......Page 141
5.3.2 Time step ......Page 142
5.3.3 Remap ......Page 143
5.4 Neutrino evolution equation ......Page 144
5.4.1 Flux limited diffusion ......Page 145
5.4.3 Neutrino annihilation beam calculation ......Page 147
5.4.5 Neutrino angular distribution ......Page 148
5.4.6 Operator splitting for the neutrino distribution ......Page 149
5.5 Neutrino-matter interactions ......Page 150
5.5.1 Electron capture ......Page 151
5.5.2 Neutrino-electron elastic scattering ......Page 153
5.5.3 Intermediate density scattering ......Page 154
5.5.4 Neutrino annihilation ......Page 156
5.5.5 Pair production of neutrinos in the core ......Page 157
5.5.6 Neutrino-nucleus interactions ......Page 158
5.6 Equation of state ......Page 159
5.6.1 Nuclear statistical equilibrium (NSE) ......Page 160
5.6.4 Baryons ......Page 161
5.6.5 Baryons below nuclear density not in NSE ......Page 162
5.6.6 Baryons below nuclear density and in NSE ......Page 163
5.6.7 Baryon matter above nuclear density ......Page 166
5.6.8 Numerical implementation of the equation of state ......Page 168
5.7.1 Mixing length theory ......Page 169
5.7.2 Convection phenomenology ......Page 170
5.8 Model of a $20M_\\odot$ supernova explosion ......Page 176
References ......Page 180
6.1.1 Kerr metric ......Page 183
6.1.2 Accretion shocks ......Page 184
6.1.3 Kerr accretion with magnetized gas ......Page 185
6.2.1 Rotating stars ......Page 188
6.2.2 Magnetic rotating stars ......Page 191
6.3 Systems with a dynamic metric ......Page 194
References ......Page 197
7.1 The conformally flat approximation ......Page 199
7.2 Conformally flat model for binary neutron stars ......Page 201
7.2.1 Coordinate system ......Page 202
7.2.2 Hamiltonian constraint ......Page 203
7.2.3 Lapse function ......Page 205
7.2.4 Momentum constraint ......Page 206
7.2.5 Reliability of the conformally flat condition ......Page 207
7.2.6 Other checks on the conformally flat condition ......Page 210
7.3 Relativistic hydrodynamics ......Page 211
7.3.2 Gravitational radiation ......Page 212
7.3.3 Solution of elliptic equations ......Page 216
7.4 Boundary conditions ......Page 217
7.6 Results ......Page 219
7.6.1 Analysis ......Page 222
7.7 Solving the Einstein equation in three dimensions ......Page 225
References ......Page 227
Index ......Page 232
Supercover......Page 235
Back cover......Page 236