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

6.6 Isenberg-Wilson-Mathews approximation to General Relativity 99
∂
− Lβ Ãij = −
∂t 2
3 ˜ Dkβ k Ãij + N
+Ψ −4
KÃij − 2˜γ kl ÃikÃjl
− 8π Ψ −4 Sij − 1
3 S˜γij
− ˜ Di ˜ DjN + 2 ˜ Di ln Ψ ˜ DjN + 2 ˜ Dj ln Ψ ˜ DiN
+ 1
˜Dk
3
˜ D k N − 4 ˜ Dk ln Ψ ˜ D k
N ˜γij
+N ˜Rij − 1
3 ˜ R˜γij − 2 ˜ Di ˜ Dj lnΨ + 4 ˜ Di ln Ψ ˜ Dj ln Ψ
+ 2
˜Dk
3
˜ D k ln Ψ − 2 ˜ Dk ln Ψ ˜ D k
ln Ψ ˜γij .
(6.108)
˜Di ˜ D i Ψ − 1
8 ˜
1
RΨ +
8 Ãij Ãij − 1
12 K2
+ 2πE Ψ 5 = 0 (6.109)
˜Dj Ãij + 6 Ãij ˜ Dj ln Ψ − 2
3 ˜ D i K = 8πΨ 4 p i . (6.110)
For the last two equations, which are the constraints, we have the alternative forms (6.103) and
(6.101) in terms of Âij (instead of Ãij ):
˜Di ˜ D i Ψ − 1
8 ˜ RΨ + 1
8 Âij Âij Ψ −7
+ 2πE − 1
12 K2
Ψ 5 = 0 , (6.111)
˜Dj Âij − 2
3 Ψ6 ˜ D i K = 8πΨ 10 p i . (6.112)
Equations (6.105)-(6.110) constitute the conformal 3+1 Einstein system. An alternative
form is constituted by Eqs. (6.105)-(6.108) and (6.111)-(6.112). In terms of the original 3+1
Einstein system (4.63)-(4.66), Eq. (6.105) corresponds to the trace of the kinematical equation
(4.63) and Eq. (6.106) to its traceless part, Eq. (6.107) corresponds to the trace of the dynamical
Einstein equation (4.64) and Eq. (6.108) to its traceless part, Eq. (6.109) or Eq. (6.111) is the
Hamiltonian constraint (4.65), whereas Eq. (6.110) or Eq. (6.112) is the momentum constraint.
If the system (6.105)-(6.110) is solved in terms of ˜γij, Ãij (or Âij), Ψ and K, then the
physical metric γ and the extrinsic curvature K are recovered by
γij = Ψ 4 ˜γij
Kij = Ψ 4
Ãij + 1
3 K˜γij
(6.113)
= Ψ −2 Âij + 1
3 KΨ4 ˜γij. (6.114)
6.6 Isenberg-Wilson-Mathews approximation to General Relativity
In 1978, J. Isenberg [160] was looking for some approximation to general relativity without any
gravitational wave, beyond the Newtonian theory. The simplest of the approximations that
he found amounts to impose that the 3-metric γ is conformally flat. In the framework of the