3+1 formalism and bases of numerical relativity - LUTh ...

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Index 1+log slicing, 161 3+1 formalism, 11 3-metric, 21 ADM 4-momentum, 113 ADM mass, 106 ADM momentum, 112 algebraic slicings, 161 approximate maximal slicing, 158 approximate minimal distortion, 168 asymptotically flat, 104 asymptotically maximal gauge, 114 background metric, 85 baryon number 4-current, 78 baryon number current measured by the Eulerian observer, 79 baryon number density measured by the Eulerian observer, 78 Bianchi identity, 178 Bowen-York extrinsic curvature, 137 BSSN scheme, 181 Cauchy surface, 39 CMC, 130 Codazzi relation, 36 Codazzi-Mainardi relation, 36 collapse of the lapse, 158 conformal equivalence class, 84 conformal flatness approximation, 100 conformal isometry, 194 conformal Killing operator, 101, 127, 193 conformal Killing vector, 194 conformal Killing vectors, 127 conformal lapse, 140 conformal metric, 83, 86 conformal thin sandwich, 140 conformal time slicing, 161 conformal transformation, 83 conformal transverse traceless, 129 conformal vector Laplacian, 127, 195 conformally flat, 84 constant mean curvature, 130 constrained scheme, 175 constraint-violating modes, 181 constraints, 64 contracted Codazzi relation, 36 contracted Gauss relation, 35 cotangent space, 16 Cotton tensor, 84 Cotton-York tensor, 84 CTS, 140 CTT, 129 De Donder, 159 Dirac gauge, 174 distortion tensor, 163 dominant energy condition, 111 dynamical equation violation, 177 Einstein tensor, 176 Einstein-Christoffel system, 181 Einstein-Rosen bridge, 135 elliptic, 196 embedding, 19 Eulerian coordinates, 163 Eulerian gauge, 163 Eulerian observer, 42 extended conformal thin sandwich, 141 extrinsic curvature tensor, 24 fiducial observers, 43 first fundamental form of Σ, 21 fluid coordinate velocity, 77
foliation, 40 free evolution scheme, 176 fully constrained scheme, 175 future, 16 Gamma freezing, 168 Gauss relation, 35 Gaussian curvature, 22 Gaussian normal coordinates, 62 geodesic slicing, 62, 152 globally hyperbolic, 39 Hamiltonian constraint, 54 Hamiltonian constraint violation, 176 harmonic, 159 harmonic decomposition generalized, 186 harmonic slicing, 159 helical Killing vector, 146 helical symmetry, 146 hole, 115 hyperbolic Gamma driver, 171 hypersurface, 19 induced metric on Σ, 21 intrinsic curvature, 22 inverse conformal metric, 87 Isenberg-Wilson-Mathews approximation, 100 isolated bodies, 103 IWM, 100 Killing vector, 116 Komar angular momentum, 121 Komar mass, 116 Komar mass of the hole, 118 KST formulation, 181 lapse function, 41 leaf, 40 Levi-Civita connection, 22 Lichnerowicz equation, 98 Lie derivative, 190 Lie dragged, 42 locally nonrotating observers, 43 longitudinal part, 127 INDEX 219 Lorentz factor, 76 matter energy density, 52 matter momentum density, 52 matter stress tensor, 52 maximal slicing, 153 maximal slicings, 100 mean curvature, 24 minimal distortion, 165 minimal surface, 133 momentarily static, 132 momentum constraint, 54 momentum constraint violation, 177 non-expanding horizon, 148 normal coordinates, 163 normal evolution vector, 42 null, 21 orthogonal projector onto Σ, 29 parabolic Gamma driver, 170 partially constrained scheme, 175 past, 16 perfect fluid, 75 Plateau problem, 155 Poincaré transformations, 105 polar slicing, 158 positive Yamabe class, 130 principal curvatures, 24 principal directions, 24 principal part, 62 principal symbol, 195 proper baryon number density, 78 proper internal energy, 80 proper rest-mass energy density, 80 pseudo-minimal distortion, 167 pull-back mapping, 20 puncture, 136 push-forward mapping, 20 quasi-isotropic gauge, 114 quasi-linear, 63 radiation gauge, 166 Ricci equation, 48
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arXiv:gr-qc/0703035v1 6 Mar 2007 3+
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4 CONTENTS 3.3.6 Evolution of the o
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6 CONTENTS 8 The initial data probl
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8 CONTENTS
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10 CONTENTS
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12 Introduction 3D (no symmetry at
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14 Introduction
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16 Geometry of hypersurfaces in two
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18 Geometry of hypersurfaces 2.2.3
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20 Geometry of hypersurfaces Figure
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22 Geometry of hypersurfaces •
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24 Geometry of hypersurfaces The ei
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26 Geometry of hypersurfaces Figure
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28 Geometry of hypersurfaces The no
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30 Geometry of hypersurfaces Since
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32 Geometry of hypersurfaces 2.4.3
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34 Geometry of hypersurfaces 2.5 Ga
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36 Geometry of hypersurfaces Exampl
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38 Geometry of hypersurfaces
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40 Geometry of foliations Figure 3.
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42 Geometry of foliations 3.3.2 Nor
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44 Geometry of foliations means Eq.
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46 Geometry of foliations Remark :
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48 Geometry of foliations Note that
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50 Geometry of foliations
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52 3+1 decomposition of Einstein eq
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72 3+1 equations for matter and ele
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84 Conformal decomposition equivale
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86 Conformal decomposition As an ex
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88 Conformal decomposition 6.2.4 Co
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90 Conformal decomposition 6.3.1 Ge
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92 Conformal decomposition where K
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94 Conformal decomposition to write
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96 Conformal decomposition hence Lm
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98 Conformal decomposition 6.5.2 Ha
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100 Conformal decomposition discuss
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102 Conformal decomposition Remark
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104 Asymptotic flatness and global
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126 The initial data problem Notice
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128 The initial data problem where
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130 The initial data problem 8.2.3
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132 The initial data problem where
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134 The initial data problem Figure
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136 The initial data problem Figure
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138 The initial data problem In par
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140 The initial data problem Accord
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142 The initial data problem Remark
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144 The initial data problem Remark
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146 The initial data problem 8.4.1
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148 The initial data problem Since
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150 The initial data problem • fo
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152 Choice of foliation and spatial
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Page 200 and 201: 200 BIBLIOGRAPHY [13] A. Anderson a
Page 202 and 203: 202 BIBLIOGRAPHY [43] T.W. Baumgart
Page 204 and 205: 204 BIBLIOGRAPHY [75] M. Campanelli
Page 206 and 207: 206 BIBLIOGRAPHY [105] G. Darmois :
Page 208 and 209: 208 BIBLIOGRAPHY [135] H. Friedrich
Page 210 and 211: 210 BIBLIOGRAPHY [163] J. Isenberg
Page 212 and 213: 212 BIBLIOGRAPHY [195] S. Nissanke
Page 214 and 215: 214 BIBLIOGRAPHY [226] M. Shibata :
Page 216 and 217: 216 BIBLIOGRAPHY [255] K. Taniguchi
Page 220: 220 INDEX Ricci identity, 18 Ricci
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