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

180 Evolution schemes
Now the ∇-divergence of F is related to the D-one by
Besides, we have
DµF µα = γ ρ µ γµ σ γνα ∇ρF σ ν = γρ σ γνα ∇ρF σ ν = γνα (∇ρF ρ ν + nρ nσ∇ρF σ ν )
= γ να (∇ρF ρ ν − F σ νn ρ ∇ρnσ)
= γ να ∇µF µ ν − F αµ Dµ ln N. (10.31)
γ α νn µ ∇µM ν = 1
N γα νm µ ∇µM ν = 1
N γα ν (LmM ν + M µ ∇µm ν )
= 1
N [LmM α + γ α νM µ (∇µN n ν + N∇µn ν )]
= 1
N LmM α − K α µ Mµ , (10.32)
where property (3.32) has been used to write γ α νLmM ν = LmM α .
Thanks to Eqs. (10.31) and (10.32), and to the relation Lm = ∂/∂t −Lβ , Eq. (10.30) yields
an evolution equation for the momentum constraint violation:
∂
− Lβ
∂t
M i = −Dj(NF ij ) + 2NK i j Mj + NKM i + ND i (F − H) + (F − 2H)D i N .
(10.33)
Let us now assume that the dynamical Einstein equation is satisfied, then F = 0 [Eq. (10.12)]
and Eqs. (10.29) and (10.33) reduce to
∂
∂t
− Lβ
∂
− Lβ
∂t
H = −Di(NM i ) + 2NKH − M i DiN (10.34)
M i = −D i (NH) + 2NK i jM j + NKM i + HD i N. (10.35)
If the constraints are satisfied at t = 0, then H|t=0 = 0 and M i |t=0 = 0. The above system gives
then
∂H
∂t
∂M i
∂t
t=0
t=0
= 0 (10.36)
= 0. (10.37)
We conclude that, at least in the case where all the fields are analytical (in order to invoke the
Cauchy-Kovalevskaya theorem),
∀t ≥ 0, H = 0 and M i = 0, (10.38)
i.e. the constraints are preserved by the dynamical evolution equation (4.64). Even if the
hypothesis of analyticity is relaxed, the result still holds because the system (10.34)-(10.35) is
symmetric hyperbolic [136].
Remark : The above result on the preservation of the constraints in a free evolution scheme
holds only if the matter source obeys the energy-momentum conservation law (10.23).