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

184 Evolution schemes
= 1
2
−∆ k il ∆l kj
− Dk˜γlj Di˜γ kl − ˜γljDkDi˜γ kl − Dk˜γil Dj˜γ kl − ˜γilDkDj˜γ kl − Dk˜γ kl Dl˜γij
−˜γ kl DkDl˜γij
− ∆ k il∆l kj . (10.50)
Hence we can write, using DkDi = DiDk (since f is flat) and exchanging some indices k and l,
where
˜Rij = − 1
˜γ
2
kl DkDl˜γij + ˜γikDjDl˜γ kl + ˜γjkDiDl˜γ kl
+ Qij(˜γ,D˜γ) , (10.51)
Qij(˜γ,D˜γ) := − 1
Dk˜γlj Di˜γ
2
kl + Dk˜γil Dj˜γ kl + Dk˜γ kl
Dl˜γij − ∆ k il∆l kj
(10.52)
is a term which does not contain any second derivative of ˜γ and which is quadratic in the first
derivatives.
10.4.3 Reducing the Ricci tensor to a Laplace operator
If we consider the Ricci tensor as a differential operator acting on the conformal metric ˜γ, its
principal part (or principal symbol, cf. Sec. B.2.2) is given by the three terms involving second
derivatives in the right-hand side of Eq. (10.51). We recognize in the first term, ˜γ klDkDl˜γij, a kind of Laplace operator acting on ˜γij. Actually, for a weak gravitational field, i.e. for
˜γ ij = fij + hij with fikfjlhklh ij ≪ 1, we have, at the linear order in h, ˜γ klDkDl˜γij ≃ ∆f˜γij,
where ∆f = fklDkDl is the Laplace operator associated with the metric f. If we combine
Eqs. (6.106) and (6.108), the Laplace operator in ˜ Rij gives rise to a wave operator for ˜γij,
namely 2 ∂
− Lβ −
∂t N2
Ψ4 ˜γkl
DkDl ˜γij = · · · (10.53)
Unfortunately the other two terms that involve second derivatives in Eq. (10.51), namely
˜γikDjDl˜γ kl and ˜γjkDiDl˜γ kl , spoil the elliptic character of the operator acting on ˜γij in ˜ Rij,
so that the combination of Eqs. (6.106) and (6.108) does no longer lead to a wave operator.
To restore the Laplace operator, Shibata and Nakamura [233] have considered the term Dl˜γ kl
which appears in the second and third terms of Eq. (10.51) as a variable independent from ˜γij.
We recognize in this term the opposite of the vector ˜Γ that has been introduced in Sec. 9.3.4
[cf. Eq. (9.80)]:
˜Γ i = −Dj˜γ ij . (10.54)
Equation (10.51) then becomes
˜Rij = 1
−˜γ
2
kl DkDl˜γij + ˜γikDj ˜ Γ k + ˜γjkDi ˜ Γ k
+ Qij(˜γ,D˜γ) . (10.55)