3+1 formalism and bases of numerical relativity - LUTh ...
3+1 formalism and bases of numerical relativity - LUTh ... 3+1 formalism and bases of numerical relativity - LUTh ...
182 Evolution schemes coordinates (x i ): ˜Rij = ∂ ∂x k ˜ Γ k ij − ∂ ∂x j ˜ Γ k ik + ˜ Γ k ij ˜ Γ l kl − ˜ Γ k il ˜ Γ l kj Let us introduce the type 1 2 tensor field ∆ defined by . (10.39) ∆ k ij := ˜ Γ k ij − ¯ Γ k ij , (10.40) where the ¯ Γk ij ’s denote the Christoffel symbols of the flat metric f with respect to the coordinates (xi ). As already noticed in Sec. 9.3.4, the identity (10.40) does define a tensor field, although each set of Christoffel symbols, ˜ Γk ij or ¯ Γk ij , is by no means the set of components of , which is manifestly covariant, is any tensor field. Actually an alternative expression of ∆ k ij ∆ k ij = 1 2 ˜γkl (Di˜γlj + Dj˜γil − Dl˜γij) , (10.41) where Di stands for the covariant derivative associated with the flat metric f. It is not difficult to establish the equivalence of Eqs. (10.40) and (10.41): starting from the latter, we have ∆ k ij = 1 2 ˜γkl hence we recover Eq. (10.40). ∂˜γlj ∂x i − ¯ Γ m il ˜γmj − ¯ Γ m ij ˜γlm + ∂˜γil − ∂˜γij ∂x l + ¯ Γ m li ˜γmj + ¯ Γ m lj ˜γim = ˜ Γ k ij + 1 2 ˜γkl −2 ¯ Γ m ij˜γlm ∂x j − ¯ Γ m ji ˜γml − ¯ Γ m jl ˜γim = Γ˜ k ij − ˜γ kl ˜γlm ¯Γ m ij =δ k m = ˜ Γ k ij − ¯ Γ k ij , (10.42) Remark : While it is a well defined tensor field, ∆ depends upon the background flat metric f, which is not unique on the hypersurface Σt. A useful relation is obtained by contracting Eq. (10.40) on the indices k and j: ∆ k ik = ˜ Γ k ik − ¯ Γ k ik 1 ∂ 1 ∂ = ln ˜γ − ln f, (10.43) 2 ∂xi 2 ∂xi where ˜γ := det ˜γij and f := det fij. Since by construction ˜γ = f [Eq. (6.19)], we get ∆ k ik = 0 . (10.44) Remark : If the coordinates (x i ) are of Cartesian type, then ¯ Γ k ij = 0, ∆k ij = ˜ Γ k ij and Di = ∂/∂x i . This is actually the case considered in the original articles of the BSSN formalism [233, 43]. We follow here the method of Ref. [63] to allow for non Cartesian coordinates, e.g. spherical ones.
Replacing ˜ Γ k ij by ∆k ij + ¯ Γ k ij ˜Rij = ∂ = ∂ 10.4 BSSN scheme 183 [Eq. (10.40)] in the expression (10.39) of the Ricci tensor yields ∂xk (∆kij + ¯ Γ k ∂ ij ) − ∂xj (∆kik + ¯ Γ k ik ) + (∆kij + ¯ Γ k ij )(∆l kl + ¯ Γ l kl ) −(∆ k il + ¯ Γ k il )(∆l kj + ¯ Γ l kj ) ∂xk ∆k ∂ ij + ∂xk ¯ Γ k ∂ ij − ∂xj ∆k ∂ ik − ∂xj ¯ Γ k ik + ∆kij∆l kl + ¯ Γ l kl∆kij + ¯ Γ k ij∆ l kl + ¯ Γ k ij ¯ Γ l kl − ∆kil∆l kj − ¯ Γ l kj∆kil − ¯ Γ k il∆l kj − ¯ Γ k il ¯ Γ l kj Now since the metric f is flat, its Ricci tensor vanishes identically, so that Hence Eq. (10.45) reduces to . (10.45) ∂ ∂xk ¯ Γ k ∂ ij − ∂xj ¯ Γ k ik + ¯ Γ k ij ¯ Γ l kl − ¯ Γ k il ¯ Γ l kj = 0. (10.46) ˜Rij = ∂ ∂xk ∆k ∂ ij − ∂xj ∆kik + ∆kij∆l kl + ¯ Γ l kl∆kij + ¯ Γ k ij∆l kl − ∆kil∆l kj −¯ Γ l kj∆kil − ¯ Γ k il∆l kj . (10.47) Property (10.44) enables us to simplify this expression further: ˜Rij = ∂ ∂x k ∆k ij + ¯ Γ l kl ∆k ij − ¯ Γ l kj ∆k il − ¯ Γ k il ∆l kj − ∆k il ∆l kj = ∂ ∂xk ∆kij + ¯ Γ k kl∆l ij − ¯ Γ l ki∆klj − ¯ Γ l kj∆kil − ∆kil∆l kj . (10.48) We recognize in the first four terms of the right-hand side the covariant derivative Dk∆k ij , hence ˜Rij = Dk∆ k ij − ∆kil∆l kj . (10.49) Remark : Even if ∆k ik would not vanish, we would have obtained an expression of the Ricci tensor with exactly the same structure as Eq. (10.39), with the partial derivatives ∂/∂xi replaced by the covariant derivatives Di and the Christoffel symbols ˜ Γk ij replaced by the tensor components ∆k ij . Indeed Eq. (10.49) can be seen as being nothing but a particular case of the more general formula obtained in Sec. 6.3.1 and relating the Ricci tensors associated with two different metrics, namely Eq. (6.44). Performing in the latter the substitutions γ → ˜γ, ˜γ → f, Rij → ˜ Rij, ˜ Rij → 0 (for f is flat), ˜ Di → Di and Ck ij → ∆k ij [compare Eqs. (6.30) and (10.40)] and using property (10.44), we get immediately Eq. (10.49). If we substitute expression (10.41) for ∆ k ij ˜Rij = 1 2 Dk = 1 Dk 2 ˜γ kl (Di˜γlj + Dj˜γil − Dl˜γij) into Eq. (10.49), we get − ∆ k il ∆l kj Di(˜γ kl ˜γlj ) − ˜γljDi˜γ kl + Dj(˜γ kl ˜γil δ k j δk ) − ˜γilDj˜γ i kl − Dk˜γ kl Dl˜γij − ˜γ kl DkDl˜γij
- Page 132 and 133: 132 The initial data problem where
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- Page 180 and 181: 180 Evolution schemes Now the ∇-d
- 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 196 and 197: 196 Conformal Killing operator and
- Page 198 and 199: 198 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
- 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
Replacing ˜ Γ k ij by ∆k ij + ¯ Γ k ij<br />
˜Rij = ∂<br />
= ∂<br />
10.4 BSSN scheme 183<br />
[Eq. (10.40)] in the expression (10.39) <strong>of</strong> the Ricci tensor yields<br />
∂xk (∆kij + ¯ Γ k ∂<br />
ij ) −<br />
∂xj (∆kik + ¯ Γ k ik ) + (∆kij + ¯ Γ k ij )(∆l kl + ¯ Γ l kl )<br />
−(∆ k il + ¯ Γ k il )(∆l kj + ¯ Γ l kj )<br />
∂xk ∆k ∂<br />
ij +<br />
∂xk ¯ Γ k ∂<br />
ij −<br />
∂xj ∆k ∂<br />
ik −<br />
∂xj ¯ Γ k ik + ∆kij∆l kl + ¯ Γ l kl∆kij + ¯ Γ k ij∆ l kl + ¯ Γ k ij ¯ Γ l kl − ∆kil∆l kj − ¯ Γ l kj∆kil − ¯ Γ k il∆l kj − ¯ Γ k il ¯ Γ l kj<br />
Now since the metric f is flat, its Ricci tensor vanishes identically, so that<br />
Hence Eq. (10.45) reduces to<br />
. (10.45)<br />
∂<br />
∂xk ¯ Γ k ∂<br />
ij −<br />
∂xj ¯ Γ k ik + ¯ Γ k ij ¯ Γ l kl − ¯ Γ k il ¯ Γ l kj = 0. (10.46)<br />
˜Rij = ∂<br />
∂xk ∆k ∂<br />
ij −<br />
∂xj ∆kik + ∆kij∆l kl + ¯ Γ l kl∆kij + ¯ Γ k ij∆l kl − ∆kil∆l kj<br />
−¯ Γ l kj∆kil − ¯ Γ k il∆l kj . (10.47)<br />
Property (10.44) enables us to simplify this expression further:<br />
˜Rij = ∂<br />
∂x k ∆k ij + ¯ Γ l kl ∆k ij − ¯ Γ l kj ∆k il − ¯ Γ k il ∆l kj − ∆k il ∆l kj<br />
= ∂<br />
∂xk ∆kij + ¯ Γ k kl∆l ij − ¯ Γ l ki∆klj − ¯ Γ l kj∆kil − ∆kil∆l kj . (10.48)<br />
We recognize in the first four terms <strong>of</strong> the right-h<strong>and</strong> side the covariant derivative Dk∆k ij , hence<br />
˜Rij = Dk∆ k ij − ∆kil∆l kj . (10.49)<br />
Remark : Even if ∆k ik would not vanish, we would have obtained an expression <strong>of</strong> the Ricci<br />
tensor with exactly the same structure as Eq. (10.39), with the partial derivatives ∂/∂xi replaced by the covariant derivatives Di <strong>and</strong> the Christ<strong>of</strong>fel symbols ˜ Γk ij replaced by the<br />
tensor components ∆k ij . Indeed Eq. (10.49) can be seen as being nothing but a particular<br />
case <strong>of</strong> the more general formula obtained in Sec. 6.3.1 <strong>and</strong> relating the Ricci tensors<br />
associated with two different metrics, namely Eq. (6.44). Performing in the latter the<br />
substitutions γ → ˜γ, ˜γ → f, Rij → ˜ Rij, ˜ Rij → 0 (for f is flat), ˜ Di → Di <strong>and</strong> Ck ij →<br />
∆k ij [compare Eqs. (6.30) <strong>and</strong> (10.40)] <strong>and</strong> using property (10.44), we get immediately<br />
Eq. (10.49).<br />
If we substitute expression (10.41) for ∆ k ij<br />
˜Rij = 1<br />
2 Dk<br />
<br />
= 1<br />
<br />
Dk<br />
2<br />
˜γ kl (Di˜γlj + Dj˜γil − Dl˜γij)<br />
<br />
into Eq. (10.49), we get<br />
− ∆ k il ∆l kj<br />
Di(˜γ kl ˜γlj ) − ˜γljDi˜γ kl + Dj(˜γ kl ˜γil<br />
<br />
δ k j<br />
<br />
δk ) − ˜γilDj˜γ<br />
i<br />
kl<br />
<br />
− Dk˜γ kl Dl˜γij − ˜γ kl DkDl˜γij
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