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198 Conformal Killing operator and conformal vector Laplacian B.2.4 Solutions to the conformal vector Poisson equation Let now discuss the existence and uniqueness of solutions to the conformal vector Poisson equation ˜∆L v i = S i , (B.26) where the vector field S is given (the source). Again, we shall distinguish two cases: the closed manifold case and the asymptotically flat one. When Σ is a closed manifold, we notice first that a necessary condition for the solution to exist is that the source must be orthogonal to any vector field in the kernel, in the sense that ∀C ∈ ker ˜L, ˜γijC Σ i S j ˜γ d 3 x = 0. (B.27) This is easily established by replacing S j by ˜ ∆L v i and performing the same integration by part as above to get Σ ˜γijC i S j ˜γ d 3 x = − 1 ˜γij˜γkl( 2 Σ ˜ LC) ik ( ˜ Lv) jl ˜γ d 3 x. (B.28) Since, by definition ( ˜ LC) ik = 0, Eq. (B.27) follows. If condition (B.27) is fulfilled (it may be trivial since the metric ˜γ may not admit any conformal Killing vector at all), it can be shown that Eq. (B.26) admits a solution and that this solution is unique up to the addition of a conformal Killing vector. In the asymptotically flat case, we assume that, in terms of the asymptotically Cartesian coordinates (x i ) introduced in Sec. 7.2 S i = O(r −3 ). (B.29) Moreover, because of the presence of the Ricci tensor in ˜∆L, one must add the decay condition ∂ 2 ˜γij ∂x k ∂x l = O(r−3 ) (B.30) to the asymptotic flatness conditions introduced in Sec. 7.2 [Eqs. (7.1) to (7.4)]. Indeed Eq. (B.30) along with Eqs. (7.1)-(7.2) guarantees that ˜Rij = O(r −3 ). (B.31) Then a general theorem by Cantor (1979) [78] on elliptic operators on asymptotically flat manifolds can be invoked (see Appendix B of Ref. [246] as well as Ref. [91]) to conclude that the solution of Eq. (B.26) with the boundary condition v i = 0 when r → 0 (B.32) exists and is unique. The possibility to add a conformal Killing vector to the solution, as in the compact case, does no longer exist because there is no conformal Killing vector which vanishes at spatial infinity on asymptotically flat Riemannian manifolds. Regarding numerical techniques to solve the conformal vector Poisson equation (B.26), let us mention that a very accurate spectral method has been developed by Grandclément et al. (2001) [148] in the case of the Euclidean space: (Σ, ˜γ) = (R 3 ,f). It is based on the use of Cartesian components of vector fields altogether with spherical coordinates. An alternative technique, using both spherical components and spherical coordinates is presented in Ref. [63].
Bibliography [1] http://www.luth.obspm.fr/IHP06/ [2] http://www.smf.mx/ ∼ dgfm-smf/EscuelaVII/ [3] A.M. Abrahams, G.B. Cook, S.L. Shapiro, and S.A. Teukolsky : Solving Einstein’s equations for rotating spacetimes: Evolution of relativistic star clusters, Phys. Rev. D 49, 5153 (1994). [4] M. Alcubierre : Appearance of coordinate shocks in hyperbolic formalisms of general relativity, Phys. Rev. D 55, 5981 (1997). [5] M. Alcubierre : The status of numerical relativity, in General Relativity And Gravitation: Proceedings of the 17th International Conference, Dublin, 18-23 July 2004, edited by P. Florides, B. Nolan and A. Ottewill, World Scientific (2005). [6] M. Alcubierre, W. Benger, B. Brügmann, G. Lanfermann, L. Nerger, E. Seidel, and R. Takahashi : 3D Grazing Collision of Two Black Holes, Phys. Rev. Lett. 87, 271103 (2001). [7] M. Alcubierre and B. Brügmann : Simple excision of a black hole in 3+1 numerical relativity, Phys. Rev. D 63, 104006 (2001). [8] M. Alcubierre, B. Brügmann, P. Diener, F. S. Guzmán, I. Hawke, S. Hawley, F. Herrmann, M. Koppitz, D. Pollney, E. Seidel, and J. Thornburg : Dynamical evolution of quasicircular binary black hole data, Phys. Rev. D 72, 044004 (2005). [9] M. Alcubierre, B. Brügmann, P. Diener, M. Koppitz, D. Pollney, E. Seidel, and R. Takahashi : Gauge conditions for long-term numerical black hole evolutions without excision, Phys. Rev. D 67, 084023 (2003). [10] M. Alcubierre, B. Brügmann, D. Pollney, E. Seidel, and R. Takahashi : Black hole excision for dynamic black holes, Phys. Rev. D 64, 061501 (2001). [11] M. Alcubierre, A. Corichi, J.A. González, D. Núñez, B. Reimann, and M. Salgado : Generalized harmonic spatial coordinates and hyperbolic shift conditions, Phys. Rev. D 72, 124018 (2005). [12] A. Anderson and J.W. York : Hamiltonian Time Evolution for General Relativity, Phys. Rev. Lett. 81, 1154 (1998).
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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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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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Page 190 and 191: 190 Lie derivative Figure A.1: Geom
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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
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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 :
Page 216 and 217: 216 BIBLIOGRAPHY [255] K. Taniguchi
Page 218 and 219: Index 1+log slicing, 161 3+1 formal
Page 220: 220 INDEX Ricci identity, 18 Ricci
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