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

94 Conformal decomposition
to write
∂
− Lβ ˜γij = −2N
∂t Ãij − 2
3 ˜ Dkβ k ˜γij . (6.73)
Notice that, as an immediate consequence of Eq. (6.54), Ãij is traceless:
Let us rise the indices of Ãij with the conformal metric, defining
Since ˜γ ij = Ψ 4 γ ij , we get
˜γ ij Ãij = 0 . (6.74)
à ij := ˜γ ik ˜γ jl Ãkl. (6.75)
à ij = Ψ 4 A ij . (6.76)
This corresponds to the scaling factor α = −4 in Eq. (6.58). This choice of scaling has been first
considered by Nakamura in 1994 [192].
We can deduce from Eq. (6.73) an evolution equation for the inverse conformal metric ˜γ ij .
Indeed, raising the indices of Eq. (6.73) with ˜γ, we get
hence
˜γ ik ˜γ jl Lm ˜γkl = −2N Ãij − 2
3 ˜ Dkβ k ˜γ ij
˜γ ik [Lm(˜γ jl ˜γkl ) − ˜γklLm ˜γ jl ] = −2NÃij − 2
3 ˜ Dkβ k ˜γ ij
=δ j
k
− ˜γ ik ˜γkl
=δ i l
2) “Momentum-constraint” scaling: α = −10
Lm ˜γ jl = −2N Ãij − 2
3 ˜ Dkβ k ˜γ ij , (6.77)
∂
− Lβ ˜γ
∂t ij = 2NÃij + 2
3 ˜ Dkβ k ˜γ ij . (6.78)
Whereas the scaling α = −4 was suggested by the evolution equation (6.59) (or equivalently
Eq. (4.63) of the 3+1 Einstein system), another scaling arises when contemplating the momentum
constraint equation (4.66). In this equation appears the divergence of the extrinsic
curvature, that we can write using the twice contravariant version of K and Eq. (6.57):
Now, from Eqs. (6.29), (6.33) and (6.46),
DjK ij = DjA ij + 1
3 Di K. (6.79)
DjA ij = ˜ DjA ij + C i jkAkj + C j
jkAik = ˜ DjA ij
+ 2 δ i j ˜ Dk ln Ψ + δ i k ˜ Dj ln Ψ − ˜ D i ln Ψ ˜γjk
A kj + 6 ˜ Dk ln Ψ A ik
= ˜ DjA ij + 10A ij ˜ Dj lnΨ − 2 ˜ D i ln Ψ ˜γjkA jk . (6.80)