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90 Conformal decomposition 6.3.1 General formula relating the two Ricci tensors The starting point of the calculation is the Ricci identity (2.34) applied to a generic vector field v ∈ T (Σt): (DiDj − DjDi)v k = R k lij vl . (6.39) Contracting this relation on the indices i and k (and relabelling i ↔ j) let appear the Ricci tensor: Rijv j = DjDiv j − DiDjv j . (6.40) Expressing the D-derivatives in term of the ˜D-derivatives via formula (6.29), we get Rijv j = ˜ Dj(Div j ) − C k ji Dkv j + C j jk Div k − ˜ Di(Djv j ) = ˜ Dj( ˜ Div j + C j ik vk ) − C k ji( ˜ Dkv j + C j kl vl ) + C j jk ( ˜ Div k + C k il vl ) − ˜ Di( ˜ Djv j + C j jk vk ) = ˜ Dj ˜ Div j + ˜ DjC j ik vk + C j ik ˜ Djv k − C k ji ˜ Dkv j − C k ji Cj kl vl + C j jk ˜ Div k + C j jk Ck il vl − ˜ Di ˜ Djv j − ˜ DiC j jk vk − C j jk ˜ Div k = ˜ Dj ˜ Div j − ˜ Di ˜ Djv j + ˜ DjC j ik vk − C k jiC j kl vl + C j jk Ck il vl − ˜ DiC j jk vk . (6.41) We can replace the first two terms in the right-hand side via the contracted Ricci identity similar to Eq. (6.40) but regarding the connection ˜D: ˜Dj ˜ Div j − ˜ Di ˜ Djv j = ˜ Rijv j Then, after some relabelling j ↔ k or j ↔ l of dumb indices, Eq. (6.41) becomes (6.42) Rijv j = ˜ Rijv j + ˜ DkC k ij v j − ˜ DiC k jk vj + C l lk Ck ijv j − C k li Cl kj vj . (6.43) This relation being valid for any vector field v, we conclude that where we have used the symmetry of C k ij Rij = ˜ Rij + ˜ DkC k ij − ˜ DiC k kj + Ck ijC l lk − Ck il Cl kj in its two last indices. , (6.44) Remark : Eq. (6.44) is the general formula relating the Ricci tensors of two connections, with the Ck ij ’s being the differences of their Christoffel symbols [Eq. (6.30)]. This formula does not rely on the fact that the metrics γ and ˜γ associated with the two connections are conformally related. 6.3.2 Expression in terms of the conformal factor Let now replace Ck ij in Eq. (6.44) by its expression in terms of the derivatives of Ψ, i.e. Eq. (6.33). First of all, by contracting Eq. (6.33) on the indices j and k, we have = 2 ˜Di ln Ψ + 3 ˜Di ln Ψ − ˜Di ln Ψ , (6.45) i.e. C k ki C k ki = 6 ˜ Di ln Ψ, (6.46)
6.4 Conformal decomposition of the extrinsic curvature 91 whence ˜ DiC k kj = 6 ˜ Di ˜ Dj ln Ψ. Besides, Consequently, Eq. (6.44) becomes ˜DkC k ij = 2 ˜Di ˜ Dj ln Ψ + ˜ Dj ˜ Di ln Ψ − ˜ Dk ˜ D k ln Ψ ˜γij = 4 ˜ Di ˜ Dj ln Ψ − 2 ˜ Dk ˜ D k ln Ψ ˜γij. (6.47) Rij = ˜ Rij + 4 ˜ Di ˜ Dj ln Ψ − 2 ˜ Dk ˜ D k ln Ψ ˜γij − 6 ˜ Di ˜ Dj ln Ψ +2 −4 δ k i ˜ Dj ln Ψ + δ k j ˜ Di ln Ψ − ˜ D k lnΨ ˜γij δ k i ˜ Dl ln Ψ + δ k l ˜ Di ln Ψ − ˜ D k ln Ψ ˜γil Expanding and simplifying, we get × 6 ˜ Dk ln Ψ δ l k ˜ Dj ln Ψ + δ l j ˜ Dk ln Ψ − ˜ D l ln Ψ ˜γkj . Rij = ˜ Rij − 2 ˜ Di ˜ Dj ln Ψ − 2 ˜ Dk ˜ D k ln Ψ ˜γij + 4 ˜ Di ln Ψ ˜ Dj ln Ψ − 4 ˜ Dk ln Ψ ˜ D k ln Ψ ˜γij . (6.48) 6.3.3 Formula for the scalar curvature The relation between the scalar curvatures is obtained by taking the trace of Eq. (6.48) with respect to γ: R = γ ij Rij = Ψ −4 ˜γ ij Rij −4 = Ψ ˜γ ij Rij ˜ − 2 ˜ Di ˜ D i ln Ψ − 2 ˜ Dk ˜ D k lnΨ × 3 + 4 ˜ Di ln Ψ ˜ D i ln Ψ − 4 ˜ Dk ln Ψ ˜ D k ln Ψ × 3 R = Ψ −4 R˜ − 8 ˜Di ˜ D i ln Ψ + ˜ Di ln Ψ ˜ D i ln Ψ , (6.49) where ˜R := ˜γ ij ˜ Rij is the scalar curvature associated with the conformal metric. Noticing that we can rewrite the above formula as (6.50) ˜Di ˜ D i ln Ψ = Ψ −1 ˜ Di ˜ D i Ψ − ˜ Di ln Ψ ˜ D i ln Ψ, (6.51) R = Ψ −4 ˜ R − 8Ψ −5 ˜ Di ˜ D i Ψ . (6.52) 6.4 Conformal decomposition of the extrinsic curvature 6.4.1 Traceless decomposition The first step is to decompose the extrinsic curvature K of the hypersurface Σt into a trace part and a traceless one, the trace being taken with the metric γ, i.e. we define A := K − 1 Kγ, (6.53) 3
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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
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
Page 52 and 53: 52 3+1 decomposition of Einstein eq
Page 72 and 73: 72 3+1 equations for matter and ele
Page 84 and 85: 84 Conformal decomposition equivale
Page 86 and 87: 86 Conformal decomposition As an ex
Page 88 and 89: 88 Conformal decomposition 6.2.4 Co
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Page 98 and 99: 98 Conformal decomposition 6.5.2 Ha
Page 100 and 101: 100 Conformal decomposition discuss
Page 102 and 103: 102 Conformal decomposition Remark
Page 104 and 105: 104 Asymptotic flatness and global
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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
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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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