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

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136 The initial data problem Figure 8.4: Extended hypersurface Σ ′ 0 depicted in the Kruskal-Szekeres representation of Schwarzschild spacetime. R stands for Schwarzschild radial coordinate and r for the isotropic radial coordinate. R = 0 is the singularity and R = 2m the event horizon. Σ ′ 0 is nothing but a hypersurface t = const, where t is the Schwarzschild time coordinate. In this diagram, these hypersurfaces are straight lines and the Einstein-Rosen bridge S is reduced to a point. 8.2.6 Bowen-York initial data Let us select the same simple free data as above, namely ˜γij = fij, Â ij TT = 0, K = 0, ˜ E = 0 and ˜p i = 0. (8.67) For the hypersurface Σ0, instead of R 3 minus a ball, we choose R 3 minus a point: Σ0 = R 3 \{O}. (8.68) The removed point O is called a puncture [66]. The topology of Σ0 is S2 × R; it differs from the topology considered in Sec. 8.2.5 (R3 minus a ball); actually it is the same topology as that (cf. Fig. 8.3). of the extended manifold Σ ′ 0 Thanks to the choice (8.67), the system to be solved is still (8.39)-(8.40). If we choose the trivial solution X = 0 for Eq. (8.40), we are back to the slice of Schwarzschild spacetime considered in Sec. 8.2.5, except that now Σ0 is the extended manifold previously denoted Σ ′ 0 . Bowen and York [65] have obtained a simple non-trivial solution of Eq. (8.40) (see also Ref. [49]). Given a Cartesian coordinate system (xi ) = (x,y,z) on Σ0 (i.e. a coordinate system such that fij = diag(1,1,1)) with respect to which the coordinates of the puncture O are (0,0,0), this solution writes X i = − 1 7f 4r ij Pj + Pjxjxi r2 − 1 r3ǫij kSjx k , (8.69)
8.2 Conformal transverse-traceless method 137 where r := x2 + y2 + z2 , ǫ ij k is the Levi-Civita alternating tensor associated with the flat metric f and (Pi,Sj) = (P1,P2,P3,S1,S2,S3) are six real numbers, which constitute the six parameters of the Bowen-York solution. Notice that since r = 0 on Σ0, the Bowen-York solution is a regular and smooth solution on the entire Σ0. Example : Choosing Pi = (0,P,0) and Si = (0,0,S), where P and S are two real numbers, leads to the following expression of the Bowen-York solution: ⎧ ⎪⎨ ⎪⎩ Xx = − P xy y + S 4 r3 r3 Xy = − P 7 + 4r y2 r2 − S x r3 Xz = − P xz 4 r3 (8.70) The conformal traceless extrinsic curvature corresponding to the solution (8.69) is deduced from formula (8.9), which in the present case reduces to Âij = (LX) ij ; one gets Â ij = 3 2r3 x i P j + x j P i − f ij − xixj r2 Pkx k + 3 r5 ǫ ik lSkx l x j + ǫ jk lSkx l x i , (8.71) where P i := f ij Pj. The tensor Âij given by Eq. (8.71) is called the Bowen-York extrinsic curvature. Notice that the Pi part of Âij decays asymptotically as O(r −2 ), whereas the Si part decays as O(r −3 ). Remark : Actually the expression of Âij given in the original Bowen-York article [65] contains an additional term with respect to Eq. (8.71), but the role of this extra term is only to ensure that the solution is isometric through an inversion across some sphere. We are not interested by such a property here, so we have dropped this term. Therefore, strictly speaking, we should name expression (8.71) the simplified Bowen-York extrinsic curvature. Example : Choosing Pi = (0,P,0) and Si = (0,0,S) as in the previous example [Eq. (8.70)], we get Â xx = − 3P 2r3y 1 − x2 r2 − 6S Â xy r5 (8.72) xy = 3P 2r3x 1 + y2 r2 + 3S r5 (x2 − y 2 ) (8.73) Â xz = 3P Â 3S 2r5xyz − yz r5 (8.74) yy = 3P 2r3y 1 + y2 r2 + 6S Â xy r5 (8.75) yz = 3P 2r3z 1 + y2 r2 + 3S Â xz r5 (8.76) zz = − 3P 2r3y 1 − z2 r2 . (8.77)
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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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Page 100 and 101: 100 Conformal decomposition discuss
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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 140 and 141: 140 The initial data problem Accord
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Page 146 and 147: 146 The initial data problem 8.4.1
Page 148 and 149: 148 The initial data problem Since
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Page 152 and 153: 152 Choice of foliation and spatial
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Page 180 and 181: 180 Evolution schemes Now the ∇-d
Page 182 and 183: 182 Evolution schemes coordinates (
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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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