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

160 Choice of foliation and spatial coordinates
Thanks to Eq. (9.11), the term in brackets can be replaced by −NK, so that the harmonic
slicing condition becomes
∂
− Lβ N = −KN
∂t 2 . (9.30)
Thus we get an evolution equation for the lapse function. This contrasts with Eq. (9.1) for
geodesic slicing and Eq. (9.15) for maximal slicing.
The harmonic slicing has been introduced by Choquet-Bruhat and Ruggeri (1983) [90] as
a way to put the 3+1 Einstein system in a hyperbolic form. It has been considered more
specifically in the context of numerical relativity by Bona and Masso (1988) [60]. For a review
and more references see Ref. [213].
Remark : The harmonic slicing equation (9.30) was already laid out by Smarr and York in
1978 [246], as a part of the expression of de Donder coordinate condition in terms of 3+1
variables.
Example : In Schwarzschild spacetime, the hypersurfaces of constant standard Schwarzschild
time coordinate t = tS and depicted in Fig. 9.3 constitute some harmonic slicing, in addition
to being maximal (cf. Sec. 9.2.2). Indeed, using Schwarzschild coordinates (t,R,θ,ϕ)
or isotropic coordinates (t,r,θ,ϕ), we have ∂N/∂t = 0 and β = 0. Since K = 0 for these
hypersurfaces, we conclude that the harmonic slicing condition (9.30) is satisfied.
Example : The above slicing does not penetrate under the event horizon. A harmonic slicing
of Schwarzschild spacetime (and more generally Kerr-Newman spacetime) which passes
smoothly through the event horizon has been found by Bona and Massó [60], as well as
Cook and Scheel [96]. It is given by a time coordinate t that is related to Schwarzschild
time tS by
t = tS + 2m ln
2m
1 −
R , (9.31)
where R is Schwarzschild radial coordinate (areal radius). The corresponding expression
of Schwarzschild metric is [96]
gµνdx µ dx ν = −N 2 dt 2 + 1
N2
dR + 4m2
R2 N2 2 dt + R 2 (dθ 2 + sin 2 θdϕ 2 ), (9.32)
where
N =
1 + 2m
1 +
R
4m2
R2 −1/2 . (9.33)
Notice that all metric coefficients are regular at the event horizon (R = 2m). This harmonic
slicing is represented in a Kruskal-Szekeres diagram in Fig. 1 of Ref. [96]. It is
clear from that figure that the hypersurfaces Σt never hit the singularity (contrary to those
of the geodesic slicing shown in Fig. 9.1), but they come arbitrary close to it as t → +∞.
We infer from the above example that the harmonic slicing has some singularity avoidance
feature, but weaker than that of maximal slicing: for the latter, the hypersurfaces Σt never
come close to the singularity as t → +∞ (cf. Fig. 9.4). This has been confirmed by means of
numerical computations by Shibata and Nakamura [233].