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

6.6 Isenberg-Wilson-Mathews approximation to General Relativity 101
so that Eq. (6.118) can be rewritten as
2N Ãij = fkjDiβ k + fikDjβ k − 2
3 Dkβ k fij. (6.124)
Using Ãij = f ik f jl Ãkl, we may rewrite this equation as
where
à ij = 1
2N (Lβ)ij , (6.125)
(Lβ) ij := D i β j + D j β i − 2
3 Dkβ k f ij
(6.126)
is the conformal Killing operator associated with the metric f (cf. Appendix B). Consequently,
the term Dj Ãij which appears in Eq. (6.122) is expressible in terms of β as
Dj Ãij = Dj
1
2N (Lβ)ij
= 1
2N Dj
D i β j + D j β i − 2
3 Dkβ k f ij
− 1
2N2(Lβ)ij DjN
= 1
DjD
2N
j β i + 1
3 DiDjβ j − 2Ãij
DjN , (6.127)
where we have used DjD i β j = D i Djβ j since f is flat. Inserting Eq. (6.127) into Eq. (6.122)
yields
DjD j β i + 1
3 Di Djβ j + 2 Ãij (6NDj ln Ψ − DjN) = 16πNΨ 4 p i . (6.128)
where
The IWM system is formed by Eqs. (6.119), (6.121) and (6.128), which we rewrite as
∆N + 2Di lnΨD i N = N
4π(E + S) + ÃijÃij (6.129)
1
∆Ψ +
8 Ãij Ãij
+ 2πE Ψ 5 = 0 (6.130)
∆β i + 1
3 Di Djβ j + 2 Ãij (6NDj ln Ψ − DjN) = 16πNΨ 4 p i , (6.131)
∆ := DiD i
(6.132)
is the flat-space Laplacian. In the above equations, Ãij is to be understood, not as an independent
variable, but as the function of N and β i defined by Eq. (6.125).
The IWM system (6.129)-(6.131) is a system of three elliptic equations (two scalar equations
and one vector equation) for the three unknowns N, Ψ and β i . The physical 3-metric is fully
determined by Ψ
γij = Ψ 4 fij, (6.133)
so that, once the IWM system is solved, the full spacetime metric g can be reconstructed via
Eq. (4.47).