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

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6 CONTENTS 8 The initial data problem 125 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 8.1.1 The initial data problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 8.1.2 Conformal decomposition of the constraints . . . . . . . . . . . . . . . . . 126 8.2 Conformal transverse-traceless method . . . . . . . . . . . . . . . . . . . . . . . . 127 8.2.1 Longitudinal/transverse decomposition of Âij . . . . . . . . . . . . . . . . 127 8.2.2 Conformal transverse-traceless form of the constraints . . . . . . . . . . . 129 8.2.3 Decoupling on hypersurfaces of constant mean curvature . . . . . . . . . . 130 8.2.4 Lichnerowicz equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 8.2.5 Conformally flat and momentarily static initial data . . . . . . . . . . . . 131 8.2.6 Bowen-York initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8.3 Conformal thin sandwich method . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 8.3.1 The original conformal thin sandwich method . . . . . . . . . . . . . . . . 139 8.3.2 Extended conformal thin sandwich method . . . . . . . . . . . . . . . . . 141 8.3.3 XCTS at work: static black hole example . . . . . . . . . . . . . . . . . . 142 8.3.4 Uniqueness of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 8.3.5 Comparing CTT, CTS and XCTS . . . . . . . . . . . . . . . . . . . . . . 144 8.4 Initial data for binary systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.4.1 Helical symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 8.4.2 Helical symmetry and IWM approximation . . . . . . . . . . . . . . . . . 147 8.4.3 Initial data for orbiting binary black holes . . . . . . . . . . . . . . . . . . 147 8.4.4 Initial data for orbiting binary neutron stars . . . . . . . . . . . . . . . . 149 8.4.5 Initial data for black hole - neutron star binaries . . . . . . . . . . . . . . 150 9 Choice of foliation and spatial coordinates 151 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 9.2 Choice of foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 9.2.1 Geodesic slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 9.2.2 Maximal slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 9.2.3 Harmonic slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 9.2.4 1+log slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 9.3 Evolution of spatial coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 9.3.1 Normal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 9.3.2 Minimal distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 9.3.3 Approximate minimal distortion . . . . . . . . . . . . . . . . . . . . . . . 167 9.3.4 Gamma freezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 9.3.5 Gamma drivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 9.3.6 Other dynamical shift gauges . . . . . . . . . . . . . . . . . . . . . . . . . 172 9.4 Full spatial coordinate-fixing choices . . . . . . . . . . . . . . . . . . . . . . . . . 173 9.4.1 Spatial harmonic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 173 9.4.2 Dirac gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
CONTENTS 7 10 Evolution schemes 175 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 10.2 Constrained schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 10.3 Free evolution schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 10.3.1 Definition and framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 10.3.2 Propagation of the constraints . . . . . . . . . . . . . . . . . . . . . . . . 176 10.3.3 Constraint-violating modes . . . . . . . . . . . . . . . . . . . . . . . . . . 181 10.3.4 Symmetric hyperbolic formulations . . . . . . . . . . . . . . . . . . . . . . 181 10.4 BSSN scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 10.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 10.4.2 Expression of the Ricci tensor of the conformal metric . . . . . . . . . . . 181 10.4.3 Reducing the Ricci tensor to a Laplace operator . . . . . . . . . . . . . . 184 10.4.4 The full scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 10.4.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 A Lie derivative 189 A.1 Lie derivative of a vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 A.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 A.1.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 A.2 Generalization to any tensor field . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 B Conformal Killing operator and conformal vector Laplacian 193 B.1 Conformal Killing operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 B.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 B.1.2 Behavior under conformal transformations . . . . . . . . . . . . . . . . . . 194 B.1.3 Conformal Killing vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 B.2 Conformal vector Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 B.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 B.2.2 Elliptic character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 B.2.3 Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 B.2.4 Solutions to the conformal vector Poisson equation . . . . . . . . . . . . . 198
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Page 4 and 5: 4 CONTENTS 3.3.6 Evolution of the o
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Page 12 and 13: 12 Introduction 3D (no symmetry at
Page 14 and 15: 14 Introduction
Page 16 and 17: 16 Geometry of hypersurfaces in two
Page 18 and 19: 18 Geometry of hypersurfaces 2.2.3
Page 20 and 21: 20 Geometry of hypersurfaces Figure
Page 22 and 23: 22 Geometry of hypersurfaces •
Page 24 and 25: 24 Geometry of hypersurfaces The ei
Page 26 and 27: 26 Geometry of hypersurfaces Figure
Page 28 and 29: 28 Geometry of hypersurfaces The no
Page 30 and 31: 30 Geometry of hypersurfaces Since
Page 32 and 33: 32 Geometry of hypersurfaces 2.4.3
Page 34 and 35: 34 Geometry of hypersurfaces 2.5 Ga
Page 36 and 37: 36 Geometry of hypersurfaces Exampl
Page 38 and 39: 38 Geometry of hypersurfaces
Page 40 and 41: 40 Geometry of foliations Figure 3.
Page 42 and 43: 42 Geometry of foliations 3.3.2 Nor
Page 44 and 45: 44 Geometry of foliations means Eq.
Page 46 and 47: 46 Geometry of foliations Remark :
Page 48 and 49: 48 Geometry of foliations Note that
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56 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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176 Evolution schemes been used by
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178 Evolution schemes Comparing wit
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180 Evolution schemes Now the ∇-d
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182 Evolution schemes coordinates (
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184 Evolution schemes = 1 2 −∆
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186 Evolution schemes corresponds t
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188 Evolution schemes
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190 Lie derivative Figure A.1: Geom
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192 Lie derivative
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194 Conformal Killing operator and
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200 BIBLIOGRAPHY [13] A. Anderson a
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202 BIBLIOGRAPHY [43] T.W. Baumgart
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204 BIBLIOGRAPHY [75] M. Campanelli
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206 BIBLIOGRAPHY [105] G. Darmois :
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208 BIBLIOGRAPHY [135] H. Friedrich
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210 BIBLIOGRAPHY [163] J. Isenberg
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212 BIBLIOGRAPHY [195] S. Nissanke
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214 BIBLIOGRAPHY [226] M. Shibata :
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216 BIBLIOGRAPHY [255] K. Taniguchi
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Index 1+log slicing, 161 3+1 formal
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220 INDEX Ricci identity, 18 Ricci
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