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

Now
6.4 Conformal decomposition of the extrinsic curvature 93
˜γ ij Lm ˜γij = Lm ln det(˜γij). (6.63)
This follows from the general law of variation of the determinant of any invertible matrix A:
δ(ln detA) = tr(A −1 × δA) , (6.64)
where δ denotes any variation (derivative) that fulfills the Leibniz rule, tr stands for the trace
and × for the matrix product. Applying Eq. (6.64) to A = (˜γij) and δ = Lm gives Eq. (6.63).
By construction, det(˜γij) = f [Eq. (6.19)], so that, replacing m by ∂t − β, we get
Lm ln det(˜γij) =
But, as a consequence of Eq. (6.7), ∂f/∂t = 0, so that
∂
− Lβ ln f (6.65)
∂t
Lm ln det(˜γij) = −Lβ ln f = −Lβ ln det(˜γij). (6.66)
Applying again formula (6.64) to A = (˜γij) and δ = Lβ , we get
Hence Eq. (6.63) becomes
Lm ln det(˜γij) = −˜γ ij Lβ ˜γij
= −˜γ ij
β k Dk˜γij
˜
so that, after substitution into Eq. (6.62), we get
=0
i.e. the following evolution equation for the conformal factor:
+˜γkj ˜ Diβ k + ˜γik ˜ Djβ k
= −δ i k ˜ Diβ k − δ j
k ˜ Djβ k
= −2 ˜ Diβ i . (6.67)
˜γ ij Lm ˜γij = −2 ˜ Diβ i , (6.68)
NK + 6Lm ln Ψ = ˜ Diβ i , (6.69)
∂
− Lβ ln Ψ =
∂t 1
˜Diβ
6
i
− NK . (6.70)
Finally, substituting Eq. (6.69) into Eq. (6.61) yields an evolution equation for the conformal
metric:
∂
− Lβ ˜γij = −2NΨ
∂t −4 Aij − 2
3 ˜ Dkβ k ˜γij. (6.71)
This suggests to introduce the quantity
Ãij := Ψ −4 Aij
(6.72)