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 ...
168 Choice of foliation and spatial coordinates where Ãij := ˜γik˜γjl Ãkl = Ψ −4 Aij. Equation (9.69) becomes then or equivalently (cf. Sec. B.2.1), ˜D j ˜γik˜γjl( ˜ Lβ) kl − 2NÃij = 0, (9.72) ˜Dj ˜ D j β i + 1 3 ˜ D i ˜ Djβ j + ˜ R i j βj − 2 Ãij ˜ DjN − 2N ˜ Dj Ãij = 0. (9.73) We can express ˜ Dj Ãij via the momentum constraint (6.110) and get ˜Dj ˜ D j β i + 1 3 ˜ D i Djβ ˜ j + ˜ R i jβ j − 2Ãij DjN ˜ + 4N 3ÃijDj ˜ ln Ψ − 1 3 ˜ D i K − 4πΨ 4 p i = 0. (9.74) At this stage, Eq. (9.74) is nothing but a rewriting of Eq. (9.69) as an elliptic equation for the shift vector. Shibata [225] then proposes to replace in this equation the conformal vector Laplacian relative to ˜γ and acting on β by the conformal vector Laplacian relative to the flat metric f, thereby writing DjD j β i + 1 3 DiDjβ j − 2Ãij DjN ˜ + 4N 3ÃijDj ˜ ln Ψ − 1 3 ˜ D i K − 4πΨ 4 p i = 0. (9.75) The choice of coordinates defined by solving Eq. (9.75) instead of (9.58) is called approximate minimal distortion. The approximate minimal distortion has been used by Shibata and Uryu [239, 240] for their first computations of the merger of binary neutron stars, as well as by Shibata, Baumgarte and Shapiro for computing the collapse of supramassive neutron stars at the mass-shedding limit (Keplerian angular velocity) [229] and for studying the dynamical bar-mode instability in differentially rotating neutron stars [230]. It has also been used by Shibata [227] to devise a 2D (axisymmetric) code to compute the long-term evolution of rotating neutron stars and gravitational collapse. 9.3.4 Gamma freezing The Gamma freezing prescription for the evolution of spatial coordinates is very much related to Nakamura’s pseudo-minimal distortion (9.70): it differs from it only in the replacement of D j by Dj and ˙˜γ ij by ˙˜γ ij := ∂˜γ ij /∂t: Dj ˙˜γ ij = 0 . (9.76) The name Gamma freezing is justified as follows: since ∂/∂t and D commute [as a consequence of (6.7)], Eq. (9.76) is equivalent to ∂ Dj˜γ ∂t ij = 0. (9.77)
9.3 Evolution of spatial coordinates 169 Now, expressing the covariant derivative Dj in terms of the Christoffel symbols ¯ Γ i jk metric f with respect to the coordinates (x i ), we get Dj˜γ ij = ∂˜γij ∂xj + ¯ Γ i jk˜γkj + ¯ Γ j jk ˜γ ik = 1 ∂ 2 ∂xk ln f = ∂˜γij ∂xj + ˜ Γ i jk˜γkj + ¯Γ i jk − ˜ Γ i jk ˜γ kj + 1 ∂ ln ˜γ 2 ∂xk = ˜ Γ j jk = ∂˜γij ∂xj + ˜ Γ i jk˜γkj + ˜ Γ j jk˜γik + ¯Γ i jk − ˜ Γ i jk ˜γ kj = ˜Dj˜γ ij =0 = ˜γ jk ¯ Γ i jk − ˜ Γ i jk ˜γ ik of the , (9.78) where ˜ Γ i jk denote the Christoffel symbols of the metric ˜γ with respect to the coordinates (xi ) and we have used ˜γ = f [Eq. (6.19)] to write the second line. If we introduce the notation then the above relation becomes ˜Γ i := ˜γ jk Γ˜ i jk − ¯ Γ i jk , (9.79) Dj˜γ ij = − ˜ Γ i . (9.80) Remark : If one uses Cartesian-type coordinates, then ¯ Γ i jk = 0 and the ˜ Γ i ’s reduce to the contracted Christoffel symbols introduced by Baumgarte and Shapiro [43] [cf. their Eq. (21)]. In the present case, the ˜ Γ i ’s are the components of a vector field ˜Γ on Σt, as it is clear from relation (9.80), or from expression (9.79) if one remembers that, although the Christoffel symbols are not the components of any tensor field, the differences between two sets of them are. Of course the vector field ˜Γ depends on the choice of the background metric f. By combining Eqs. (9.80) and (9.77), we see that the Gamma freezing condition is equivalent to ∂ ˜ Γ i ∂t = 0 , (9.81) hence the name Gamma freezing: for such a choice, the vector ˜Γ does not evolve, in the sense that L∂t ˜Γ = 0. The Gamma freezing prescription has been introduced by Alcubierre and Brügmann in 2001 [7], in the form of Eq. (9.81). Let us now derive the equation that the shift vector must obey in order to enforce the Gamma freezing condition. If we express the Lie derivative in the evolution equation (6.106) for ˜γ ij in terms of the covariant derivative D [cf. Eq. (A.6)], we get ˙˜γ ij = 2N Ãij + β k Dk˜γ ij − ˜γ kj Dkβ i − ˜γ ik Dkβ j + 2 3 Dkβ k ˜γ ij . (9.82)
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- Page 200 and 201: 200 BIBLIOGRAPHY [13] A. Anderson a
- Page 202 and 203: 202 BIBLIOGRAPHY [43] T.W. Baumgart
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- 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
168 Choice <strong>of</strong> foliation <strong>and</strong> spatial coordinates<br />
where Ãij := ˜γik˜γjl Ãkl = Ψ −4 Aij. Equation (9.69) becomes then<br />
or equivalently (cf. Sec. B.2.1),<br />
<br />
˜D<br />
j<br />
˜γik˜γjl( ˜ Lβ) kl − 2NÃij <br />
= 0, (9.72)<br />
˜Dj ˜ D j β i + 1<br />
3 ˜ D i ˜ Djβ j + ˜ R i j βj − 2 Ãij ˜ DjN − 2N ˜ Dj Ãij = 0. (9.73)<br />
We can express ˜ Dj Ãij via the momentum constraint (6.110) <strong>and</strong> get<br />
˜Dj ˜ D j β i + 1<br />
3 ˜ D i Djβ ˜ j + ˜ R i jβ j − 2Ãij <br />
DjN ˜ + 4N 3ÃijDj ˜ ln Ψ − 1<br />
3 ˜ D i K − 4πΨ 4 p i<br />
<br />
= 0. (9.74)<br />
At this stage, Eq. (9.74) is nothing but a rewriting <strong>of</strong> Eq. (9.69) as an elliptic equation for<br />
the shift vector. Shibata [225] then proposes to replace in this equation the conformal vector<br />
Laplacian relative to ˜γ <strong>and</strong> acting on β by the conformal vector Laplacian relative to the flat<br />
metric f, thereby writing<br />
DjD j β i + 1<br />
3 DiDjβ j − 2Ãij <br />
DjN ˜ + 4N 3ÃijDj ˜ ln Ψ − 1<br />
3 ˜ D i K − 4πΨ 4 p i<br />
<br />
= 0. (9.75)<br />
The choice <strong>of</strong> coordinates defined by solving Eq. (9.75) instead <strong>of</strong> (9.58) is called approximate<br />
minimal distortion.<br />
The approximate minimal distortion has been used by Shibata <strong>and</strong> Uryu [239, 240] for their<br />
first computations <strong>of</strong> the merger <strong>of</strong> binary neutron stars, as well as by Shibata, Baumgarte<br />
<strong>and</strong> Shapiro for computing the collapse <strong>of</strong> supramassive neutron stars at the mass-shedding<br />
limit (Keplerian angular velocity) [229] <strong>and</strong> for studying the dynamical bar-mode instability<br />
in differentially rotating neutron stars [230]. It has also been used by Shibata [227] to devise<br />
a 2D (axisymmetric) code to compute the long-term evolution <strong>of</strong> rotating neutron stars <strong>and</strong><br />
gravitational collapse.<br />
9.3.4 Gamma freezing<br />
The Gamma freezing prescription for the evolution <strong>of</strong> spatial coordinates is very much related<br />
to Nakamura’s pseudo-minimal distortion (9.70): it differs from it only in the replacement <strong>of</strong> D j<br />
by Dj <strong>and</strong> ˙˜γ ij by ˙˜γ ij := ∂˜γ ij /∂t:<br />
Dj ˙˜γ ij = 0 . (9.76)<br />
The name Gamma freezing is justified as follows: since ∂/∂t <strong>and</strong> D commute [as a consequence<br />
<strong>of</strong> (6.7)], Eq. (9.76) is equivalent to<br />
∂ <br />
Dj˜γ<br />
∂t<br />
ij = 0. (9.77)
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