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

178 Evolution schemes
Comparing with Eq. (10.11), we get
γ ∗ (G − 8πT) = F − 1 4R
+ 8π(S − E) γ. (10.14)
2
Besides, the trace of Eq. (10.11) is
F = trγF = γ ij Fij = γ µν Fµν
= γ µν γ ρ µ γ σ ν 4 Rρσ − 8π
=γ ρν
S + 1
(E − S) × 3
2
= γ ρσ4 Rρσ + 4π(S − 3E) = 4 R + 4 Rρσ n ρ n σ + 4π(S − 3E). (10.15)
Now, from Eq. (10.4), 4 Rρσ n ρ n σ = H − 4 R/2 + 8πE, so that the above relation becomes
F = 4 R + H − 14
R + 8πE + 4π(S − 3E)
2
= H + 1 4R
+ 8π(S − E) .
2
(10.16)
This enables us to write Eq. (10.14) as
γ ∗ (G − 8πT) = F + (H − F)γ. (10.17)
Similarly to the 3+1 decomposition (4.10) of the stress-energy tensor, the 3+1 decomposition
of G − 8πT is
G − 8πT = γ ∗ (G − 8πT) + n ⊗ M + M ⊗ n + H n ⊗ n, (10.18)
γ ∗ (G − 8πT) playing the role of S, M that of p and H that of E. Thanks to Eq. (10.17), we
may write
G − 8πT = F + (H − F)γ + n ⊗ M + M ⊗ n + H n ⊗ n , (10.19)
or, in index notation,
Gαβ − 8πTαβ = Fαβ + (H − F)γαβ + nαMβ + Mαnβ + Hnαnβ. (10.20)
This identity can be viewed as the 3+1 decomposition of Einstein equation (10.2) in terms of
the dynamical equation violation F , the Hamiltonian constraint violation H and the momentum
constraint violation M.
The next step consists in invoking the contracted Bianchi identity:
i.e.
∇ · G = 0 , (10.21)
∇ µ Gαµ = 0 . (10.22)
Let us recall that this identity is purely geometrical and holds independently of Einstein equation.
In addition, we assume that the matter obeys the energy-momentum conservation law (5.1) :
∇ · T = 0 . (10.23)
In view of the Bianchi identity (10.21), Eq. (10.23) is a necessary condition for the Einstein
equation (10.2) to hold.