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

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150 The initial data problem • for strange quark matter [198, 179]. All these computation are based on a flat conformal metric [choice (8.126)], by solving the helically symmetric XCTS system (8.127)-(8.129), supplemented by an elliptic equation for the velocity potential. Only very recently, configurations based on a non flat conformal metric have been obtained by Uryu, Limousin, Friedman, Gourgoulhon and Shibata [263]. The conformal metric is then deduced from a waveless approximation developed by Shibata, Uryu and Friedman [241] and which goes beyond the IWM approximation. 8.4.5 Initial data for black hole - neutron star binaries Let us mention briefly that initial data for a mixed binary system, i.e. a system composed of a black hole and a neutron star, have been obtained very recently by Grandclément [147] and Taniguchi, Baumgarte, Faber and Shapiro [252, 253]. Codes aiming at computing such systems have also been presented by Ansorg [19] and Tsokaros and Uryu [260].
Chapter 9 Choice of foliation and spatial coordinates Contents 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 9.2 Choice of foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 9.3 Evolution of spatial coordinates . . . . . . . . . . . . . . . . . . . . . . 162 9.4 Full spatial coordinate-fixing choices . . . . . . . . . . . . . . . . . . . 173 9.1 Introduction Having investigated the initial data problem in the preceding chapter, the next logical step is to discuss the evolution problem, i.e. the development (Σt,γ) of initial data (Σ0,γ,K). This constitutes the integration of the Cauchy problem introduced in Sec. 4.4. As discussed in Sec. 4.4.1, a key feature of this problem is the freedom of choice for the lapse function N and the shift vector β, reflecting respectively the choice of foliation (Σt)t∈R and the choice of coordinates (x i ) on each leaf Σt of the foliation. These choices are crucial because they determine the specific form of the 3+1 Einstein system (4.63)-(4.66) that one has actually to deal with. In particular, depending of the choice of (N,β), this system can be made more hyperbolic or more elliptic. Extensive discussions about the various possible choices of foliations and spatial coordinates can be found in the seminal articles by Smarr and York [247, 276] as well as in the review articles by Alcubierre [5], Baumgarte and Shapiro [44], and Lehner [174].
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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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Page 102 and 103: 102 Conformal decomposition Remark
Page 104 and 105: 104 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 146 and 147: 146 The initial data problem 8.4.1
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Page 180 and 181: 180 Evolution schemes Now the ∇-d
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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
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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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