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

74 3+1 equations for matter and electromagnetic field
Let us denote by D the Levi-Civita connection associated with the flat metric f. Obviously
DΦ = DΦ. On the other side, let us express the divergence D · p in terms of the divergence
D · p. From Eq. (5.15), we have γ ij = (1 − 2Φ) −1 f ij ≃ (1 + 2Φ)f ij as well as the relation
√ γ = (1 − 2Φ) 3 f ≃ (1 − 3Φ) √ f between the determinants γ and f of respectively (γij) and
(fij). Therefore
D · p = 1
√ γ
∂
∂xi √ i
γp = 1
√
γ
∂
∂xi √ ij
γγ pj
≃
1
(1 − 3Φ) √ ∂
f ∂xi
(1 − 3Φ) f(1 + 2Φ)f ij
pj ≃ 1 ∂
√
f ∂xi
(1 − Φ) ff ij ≃
pj
1 ∂
√
f ∂xi
ij
ff pj − f ij ∂Φ
pj
∂xi ≃ D · p − p · DΦ. (5.17)
Consequently Eq. (5.16) becomes
∂E
∂t
+ D · p = −p · DΦ. (5.18)
This is the standard energy conservation relation in a Galilean frame with the source term
−p·DΦ. The latter constitutes the density of power provided to the system by the gravitational
field (this will be clear in the perfect fluid case, to be discussed below).
Remark : In the left-hand side of Eq. (5.18), the quantity p plays the role of an energy flux,
whereas it had been defined in Sec. 4.1.2 as a momentum density. It is well known that
both aspects are equivalent (see e.g. Chap. 22 of [155]).
5.2.4 Momentum conservation
Let us now project Eq. (5.2) onto Σt:
Now, from relation (2.79),
Besides
γ ν α ∇µS µ ν − Kpα + γ ν α nµ ∇µpν − Kαµp µ + EDα ln N = 0. (5.19)
DµS µ α = γ ρ µγ µ σγ ν α∇ρS σ ν = γ ρ σγ ν α∇ρS σ ν
= γ ν α(δ ρ σ + n ρ nσ)∇ρS σ ν = γ ν α(∇ρS ρ ν − S σ ν n ρ ∇ρnσ )
=Dσ ln N
= γ ν α∇µS µ ν − S µ αDµ ln N. (5.20)
γ ν α nµ ∇µpν = N −1 γ ν α mµ ∇µpν = N −1 γ ν α (Lmpν − pµ∇νm µ )
= N −1 Lmpα + Kαµp µ , (5.21)