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

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4 CONTENTS 3.3.6 Evolution of the orthogonal projector . . . . . . . . . . . . . . . . . . . . 46 3.4 Last part of the 3+1 decomposition of the Riemann tensor . . . . . . . . . . . . . 47 3.4.1 Last non trivial projection of the spacetime Riemann tensor . . . . . . . . 47 3.4.2 3+1 expression of the spacetime scalar curvature . . . . . . . . . . . . . . 48 4 3+1 decomposition of Einstein equation 51 4.1 Einstein equation in 3+1 form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1.1 The Einstein equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1.2 3+1 decomposition of the stress-energy tensor . . . . . . . . . . . . . . . . 52 4.1.3 Projection of the Einstein equation . . . . . . . . . . . . . . . . . . . . . . 53 4.2 Coordinates adapted to the foliation . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2.1 Definition of the adapted coordinates . . . . . . . . . . . . . . . . . . . . . 54 4.2.2 Shift vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2.3 3+1 writing of the metric components . . . . . . . . . . . . . . . . . . . . 57 4.2.4 Choice of coordinates via the lapse and the shift . . . . . . . . . . . . . . 59 4.3 3+1 Einstein equation as a PDE system . . . . . . . . . . . . . . . . . . . . . . . 59 4.3.1 Lie derivatives along m as partial derivatives . . . . . . . . . . . . . . . . 59 4.3.2 3+1 Einstein system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4 The Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4.1 General relativity as a three-dimensional dynamical system . . . . . . . . 61 4.4.2 Analysis within Gaussian normal coordinates . . . . . . . . . . . . . . . . 61 4.4.3 Constraint equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.4.4 Existence and uniqueness of solutions to the Cauchy problem . . . . . . . 64 4.5 ADM Hamiltonian formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.5.1 3+1 form of the Hilbert action . . . . . . . . . . . . . . . . . . . . . . . . 66 4.5.2 Hamiltonian approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5 3+1 equations for matter and electromagnetic field 71 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2 Energy and momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2.1 3+1 decomposition of the 4-dimensional equation . . . . . . . . . . . . . . 72 5.2.2 Energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2.3 Newtonian limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2.4 Momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.3 Perfect fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.3.1 kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.3.2 Baryon number conservation . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.3.3 Dynamical quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3.4 Energy conservation law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.3.5 Relativistic Euler equation . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.3.6 Further developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.4 Electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.5 3+1 magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
CONTENTS 5 6 Conformal decomposition 83 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.2 Conformal decomposition of the 3-metric . . . . . . . . . . . . . . . . . . . . . . 85 6.2.1 Unit-determinant conformal “metric” . . . . . . . . . . . . . . . . . . . . 85 6.2.2 Background metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.2.3 Conformal metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.2.4 Conformal connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3 Expression of the Ricci tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.3.1 General formula relating the two Ricci tensors . . . . . . . . . . . . . . . 90 6.3.2 Expression in terms of the conformal factor . . . . . . . . . . . . . . . . . 90 6.3.3 Formula for the scalar curvature . . . . . . . . . . . . . . . . . . . . . . . 91 6.4 Conformal decomposition of the extrinsic curvature . . . . . . . . . . . . . . . . . 91 6.4.1 Traceless decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.4.2 Conformal decomposition of the traceless part . . . . . . . . . . . . . . . . 92 6.5 Conformal form of the 3+1 Einstein system . . . . . . . . . . . . . . . . . . . . . 95 6.5.1 Dynamical part of Einstein equation . . . . . . . . . . . . . . . . . . . . . 95 6.5.2 Hamiltonian constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.5.3 Momentum constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.5.4 Summary: conformal 3+1 Einstein system . . . . . . . . . . . . . . . . . . 98 6.6 Isenberg-Wilson-Mathews approximation to General Relativity . . . . . . . . . . 99 7 Asymptotic flatness and global quantities 103 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.2 Asymptotic flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.2.2 Asymptotic coordinate freedom . . . . . . . . . . . . . . . . . . . . . . . . 105 7.3 ADM mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.3.1 Definition from the Hamiltonian formulation of GR . . . . . . . . . . . . . 105 7.3.2 Expression in terms of the conformal decomposition . . . . . . . . . . . . 108 7.3.3 Newtonian limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.3.4 Positive energy theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.3.5 Constancy of the ADM mass . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.4 ADM momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.4.2 ADM 4-momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.5 Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.5.1 The supertranslation ambiguity . . . . . . . . . . . . . . . . . . . . . . . . 113 7.5.2 The “cure” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.5.3 ADM mass in the quasi-isotropic gauge . . . . . . . . . . . . . . . . . . . 115 7.6 Komar mass and angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.6.1 Komar mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.6.2 3+1 expression of the Komar mass and link with the ADM mass . . . . . 119 7.6.3 Komar angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Page 1: arXiv:gr-qc/0703035v1 6 Mar 2007 3+
Page 6 and 7: 6 CONTENTS 8 The initial data probl
Page 8 and 9: 8 CONTENTS
Page 10 and 11: 10 CONTENTS
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
Page 50 and 51: 50 Geometry of foliations
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54 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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