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

9.2 Choice of foliation 159
type ∂N/∂λ in the right-hand side and to compute the “λ-evolution” for some range of the
parameter λ. This amounts to resolve a heat like equation. Generically the solution converges
towards a stationary one, so that ∂N/∂λ → 0 and the original elliptic equation (9.15) is solved.
The approximate maximal slicing has been used by Shibata, Uryu and Taniguchi to compute
the merger of binary neutron stars [226, 239, 240, 237, 238, 236], as well as by Shibata and
Sekiguchi for 2D (axisymmetric) gravitational collapses [227, 228, 221] or 3D ones [234].
9.2.3 Harmonic slicing
Another important category of time slicing is deduced from the standard harmonic or De
Donder condition for the spacetime coordinates (x α ):
gx α = 0, (9.23)
where g := ∇µ∇ µ is the d’Alembertian associated with the metric g and each coordinate
x α is considered as a scalar field on M. Harmonic coordinates have been introduced by De
Donder in 1921 [106] and have played an important role in theoretical developments, notably in
Choquet-Bruhat’s demonstration (1952, [127]) of the well-posedness of the Cauchy problem for
3+1 Einstein equations (cf. Sec. 4.4.4).
The harmonic slicing is defined by requiring that the harmonic condition holds for the
x 0 = t coordinate, but not necessarily for the other coordinates, leaving the freedom to choose
any coordinate (x i ) in each hypersurface Σt:
gt = 0 . (9.24)
Using the standard expression for the d’Alembertian, we get
1 ∂
√
−g ∂x µ
√−gg µν ∂t
∂xν
=δ0
= 0, (9.25)
ν
i.e.
∂
∂x µ
√ µ0
−gg = 0. (9.26)
Thanks to the relation √ −g = N √ γ [Eq. (4.55)], this equation becomes
∂ √ 00
N γg
∂t
+ ∂
∂xi √ i0
N γg = 0. (9.27)
From the expression of g αβ given by Eq. (4.49), g 00 = −1/N 2 and g i0 = β i /N 2 . Thus
Expanding and reordering gives
∂N
∂t
− ∂
∂t
√ γ
− βi∂N − N
∂xi N
+ ∂
∂x i
√ γ
N βi
1 ∂
√γ
√ γ 1
−
∂t
= 0. (9.28)
∂
√
γ ∂xi √ i
γβ
=Diβi
= 0. (9.29)