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

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164 Choice of foliation and spatial coordinates Figure 9.5: Distortion of a spatial domain defined by fixed values of the coordinates (x i ). Q measures the change in shape from Σt to Σt+δt of any spatial domain V which lies at fixed values of the coordinates (x i ) (the evolution of V is then along the vector ∂t, cf. Fig. 9.5). Thanks to the trace removal, Q does not take into account the change of volume, but only the change in shape (shear). From the law (6.64) of variation of a determinant, γ kl∂γkl ∂t ∂ ∂ = ln γ = 12 ∂t ∂t ∂ ln Ψ + ∂t lnf = 12 =0 ∂ ln Ψ, (9.46) ∂t where we have used the relation (6.15) between the determinant γ and the conformal factor Ψ, as well as the property (6.7). Thus we may rewrite Eq. (9.45) as Qij = ∂γij ∂ − 4 ∂t ∂t ln Ψ γij = ∂ ∂t (Ψ4˜γij) − 4Ψ 3∂Ψ ∂t ˜γij = Ψ 4∂˜γij . (9.47) ∂t Hence the relation between the distortion tensor and the time derivative of the conformal metric: Q = Ψ 4 ˙ ˜γ. (9.48) The rough idea would be to choose the coordinates (x i ) in order to minimize Q. Taking into account that it is symmetric and traceless, Q has 5 independent components. Thus it cannot be set identically to zero since we have only 3 degrees of freedom in the choice of the coordinates (x i ). To select which part of Q to set to zero, let us decompose it into a longitudinal part and a TT part, in a manner similar to Eq. (8.9): Qij = (LX)ij + Q TT ij . (9.49) LX denotes the conformal Killing operator associated with the metric γ and acting on some vector field X (cf. Appendix B) 3 : (LX)ij := DiXj + DjXi − 2 3 DkX k γij (9.50) 3 In Sec. 6.6, we have also used the notation L for the conformal Killing operator associated with the flat metric f, but no confusion should arise in the present context.
9.3 Evolution of spatial coordinates 165 and QTT ij is both traceless and transverse (i.e. divergence-free) with respect to the metric γ: DjQTT ij = 0. X is then related to the divergence of Q by Dj (LX)ij = DjQij. It is legitimate to relate the TT part to the dynamics of the gravitational field and to attribute the longitudinal part to the change in γij which arises because of the variation of coordinates from Σt to Σt+δt. This longitudinal part has 3 degrees of freedom (the 3 components of the vector X) and we might set it to zero by some judicious choice of the coordinates (xi ). The minimal distortion coordinates are thus defined by the requirement X = 0 or i.e. Qij = Q TT ij , (9.51) D j Qij = 0 . (9.52) Let us now express Q in terms of the shift vector to turn the above condition into an equation for the evolution of spatial coordinates. By means of Eqs. (4.63) and (9.9), Eq. (9.45) becomes Qij = −2NKijLβ γij + − 1 −2NK + 2Dkβ 3 k γij, (9.53) i.e. (since Lβ γij = Diβj + Djβi) Qij = −2NAij + (Lβ)ij, (9.54) where we let appear the trace-free part A of the extrinsic curvature K [Eq. (6.53)]. If we insert this expression into the minimal distortion requirement (9.52), we get − 2NDjA ij − 2A ij DjN + Dj(Lβ) ij = 0. (9.55) Let then use the momentum constraint (4.66) to express the divergence of A as DjA ij = 8πp i + 2 3 Di K. (9.56) Besides, we recognize in Dj(Lβ) ij the conformal vector Laplacian associated with the metric γ, so that we can write [cf. Eq. (B.11)] Dj(Lβ) ij = DjD j β i + 1 3 Di Djβ j + R i j βj , (9.57) where R is the Ricci tensor associated with γ. Thus we arrive at DjD j β i + 1 3 Di Djβ j + R i j βj = 16πNp i + 4 3 NDi K + 2A ij DjN . (9.58) This is the elliptic equation on the shift vector that one has to solve in order to enforce the minimal distortion. Remark : For a constant mean curvature (CMC) slicing, and in particular for a maximal slicing, the term D i K vanishes and the above equation is slightly simplified. Incidentally, this is the form originally derived by Smarr and York (Eq. (3.27) in Ref. [246]).
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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 114 and 115: 114 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 144 and 145: 144 The initial data problem Remark
Page 146 and 147: 146 The initial data problem 8.4.1
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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
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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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