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166 Choice of foliation and spatial coordinates Another way to introduce minimal distortion amounts to minimizing the integral S = QijQ ij√ γ d 3 x (9.59) Σt with respect to the shift vector β, keeping the slicing fixed (i.e. fixing γ, K and N). Indeed, if we replace Q by its expression (9.54), we get S = Σt 4N 2 AijA ij − 4NAij(Lβ) ij + (Lβ)ij(Lβ) ij √ γ d 3 x. (9.60) At fixed values of γ, K and N, δN = 0, δAij = 0 and δ(Lβ) ij = (Lδβ) ij , so that the variation of S with respect to β is δS = −4NAij(Lδβ) ij + 2(Lβ)ij(Lδβ) ij √ 3 γd x = 2 Qij(Lδβ) ij √ γ d 3 x. (9.61) Σt Now, since Q is symmetric and traceless, Qij(Lδβ) ij = Qij(D i δβ j + D j δβ i − 2/3Dkδβ k γ ij ) = Qij(D i δβ j + D j δβ i ) = 2QijD i δβ j . Hence δS = 4 = 4 = 4 Σt Σt ∂Σt QijD i δβ j √ γ d 3 x Σt D i Qijδβ j − D i Qij δβ j √ γ d 3 x Qijδβ j s i √ q d 2 y − 4 Σt D i Qij δβ j √ γ d 3 x (9.62) Assuming that δβ i = 0 at the boundaries of Σt (for instance at spatial infinity), we deduce from the above relation that δS = 0 for any variation of the shift vector if and only if D i Qij = 0. Hence we recover condition (9.52). In stationary spacetimes, an important property of the minimal distortion gauge is to be fulfilled by coordinates adapted to the stationarity (i.e. such that ∂t is a Killing vector): it is immediate from Eq. (9.44) that Q = 0 when ∂t is a symmetry generator, so that condition (9.52) is trivially satisfied. Another nice feature of the minimal distortion gauge is that in the weak field region (radiative zone), it includes the standard TT gauge of linearized gravity [246]. Actually Smarr and York [246] have advocated for maximal slicing combined with minimal distortion as a very good coordinate choice for radiative spacetimes, calling such choice the radiation gauge. Remark : A “new minimal distortion” gauge has been introduced in 2006 by Jantzen and York [164]. It corrects the time derivative of ˜γ in the original minimal distortion condition by the lapse function N [cf. relation (3.15) between the coordinate time t and the Eulerian observer’s proper time τ], i.e. one requires D j 1 N Qij = 0 (9.63)
9.3 Evolution of spatial coordinates 167 instead of (9.52). This amounts to minimizing the integral S ′ = Σt (N −1 Qij)(N −1 Q ij ) √ −g d 3 x (9.64) with respect to the shift vector. Notice the spacetime measure √ −g = N √ γ instead of the spatial measure √ γ in Eq. (9.59). The minimal distortion condition can be expressed in terms of the time derivative of the conformal metric by combining Eqs. (9.48) and (9.52): D j (Ψ 4 ˙˜γ ij) = 0 . (9.65) Let us write this relation in terms of the connection ˜D (associated with the metric ˜γ) instead of the connection D (associated with the metric γ). To this purpose, let us use Eq. (6.81) which relates the D-divergence of a traceless symmetric tensor to its ˜D-divergence: since Qij is traceless and symmetric, we obtain 10 ij Ψ Q . (9.66) DjQ ij = Ψ −10 ˜ Dj Now Q ij = γ ik γ jl Qkl = Ψ −8 ˜γ ik ˜γ jl Qkl = Ψ −4 ˜γ ik ˜γ jl ˙˜γ kl; hence DjQ ij = Ψ −10 ˜ Dj Ψ 6 ˜γ ik ˜γ jl ˙˜γ kl The minimal distortion condition is therefore 9.3.3 Approximate minimal distortion In view of Eq. (9.68), it is natural to consider the simpler condition = Ψ −10 ˜γ ik D˜ l Ψ 6 ˙˜γ kl . (9.67) ˜D j (Ψ 6 ˙˜γ ij) = 0 . (9.68) ˜D j ˙˜γ ij = 0, (9.69) which of course differs from the true minimal distortion (9.68) by a term 6˙˜γ ij ˜ D j ln Ψ. Nakamura (1994) [192, 199] has then introduced the pseudo-minimal distortion condition by replacing (9.69) by D j ˙˜γ ij = 0 , (9.70) where D is the connection associated with the flat metric f. An alternative has been introduced by Shibata (1999) [225] as follows. Starting from Eq. (9.69), let us express ˙˜γ ij in terms of A and β: from Eq. (8.88), we deduce that 2NÃij = ˜γik˜γjl ˙˜γ kl + ( ˜ Lβ) kl ∂ kl = ˜γjl (˜γik˜γ ∂t =δl ) − ˜γ i kl∂˜γik ∂t + ˜γik( ˜ Lβ) kl = −˙˜γ ij + ˜γik˜γjl( ˜ Lβ) kl , (9.71)
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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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Page 116 and 117: 116 Asymptotic flatness and global
Page 126 and 127: 126 The initial data problem Notice
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Page 130 and 131: 130 The initial data problem 8.2.3
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Page 134 and 135: 134 The initial data problem Figure
Page 136 and 137: 136 The initial data problem Figure
Page 138 and 139: 138 The initial data problem In par
Page 140 and 141: 140 The initial data problem Accord
Page 142 and 143: 142 The initial data problem Remark
Page 144 and 145: 144 The initial data problem Remark
Page 146 and 147: 146 The initial data problem 8.4.1
Page 148 and 149: 148 The initial data problem Since
Page 150 and 151: 150 The initial data problem • fo
Page 152 and 153: 152 Choice of foliation and spatial
Page 176 and 177: 176 Evolution schemes been used by
Page 178 and 179: 178 Evolution schemes Comparing wit
Page 180 and 181: 180 Evolution schemes Now the ∇-d
Page 182 and 183: 182 Evolution schemes coordinates (
Page 184 and 185: 184 Evolution schemes = 1 2 −∆
Page 186 and 187: 186 Evolution schemes corresponds t
Page 188 and 189: 188 Evolution schemes
Page 190 and 191: 190 Lie derivative Figure A.1: Geom
Page 192 and 193: 192 Lie derivative
Page 194 and 195: 194 Conformal Killing operator and
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 :
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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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