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

158 Choice of foliation and spatial coordinates
RC decays from 2m (t = 0) to 3m/2 (t → +∞). Actually, for C = C(t), RC represents
the smallest value of the radial coordinate R in the slice Σt. This maximal slicing of
Schwarzschild spacetime is represented in Fig. 9.4. We notice that, as t → +∞, the
slices Σt accumulate on a limiting hypersurface: the hypersurface R = 3m/2 (let us recall
that for R < 2m, the hypersurfaces R = const are spacelike and are thus eligible for a 3+1
foliation). Actually, it can be seen that the hypersurface R = 3m/2 is the only hypersurface
R = const which is spacelike and maximal [48]. If we compare with Fig. 9.1, we notice
that, contrary to the geodesic slicing, the present foliation never encounters the singularity.
The above example illustrates the singularity-avoidance property of maximal slicing: while
the entire spacetime outside the event horizon is covered by the foliation, the hypersurfaces “pile
up” in the black hole region so that they never reach the singularity. As a consequence, in that
region, the proper time (of Eulerian observers) between two neighbouring hypersurfaces tends
to zero as t increases. According to Eq. (3.15), this implies
N → 0 as t → +∞. (9.21)
This “phenomenon” is called collapse of the lapse. Beyond the Schwarzschild case discussed
above, the collapse of the lapse is a generic feature of maximal slicing of spacetimes describing
black hole formation via gravitational collapse. For instance, it occurs in the analytic solution
obtained by Petrich, Shapiro and Teukolsky [200] for the maximal slicing of the Oppenheimer-
Snyder spacetime (gravitational collapse of a spherically symmetric homogeneous ball of pressureless
matter).
In numerical relativity, maximal slicing has been used in the computation of the (axisymmetric)
head-on collision of two black holes by Smarr, Čadeˇz and Eppley in the seventies [245, 244],
as well as in computations of axisymmetric gravitational collapse by Nakamura and Sato (1981)
[191, 194], Stark and Piran (1985) [249] and Evans (1986) [119]. Actually Stark and Piran used
a mixed type of foliation introduced by Bardeen and Piran [36]: maximal slicing near the origin
(r = 0) and polar slicing far from it. The polar slicing is defined in spherical-type coordinates
(xi ) = (r,θ,ϕ) by
= 0, (9.22)
K θ θ + Kϕ ϕ
instead of K r r + Kθ θ + Kϕ ϕ = 0 for maximal slicing.
Whereas maximal slicing is a nice choice of foliation, with a clear geometrical meaning, a
natural Newtonian limit and a singularity-avoidance feature, it has not been much used in 3D
(no spatial symmetry) numerical relativity. The reason is a technical one: imposing maximal
slicing requires to solve the elliptic equation (9.15) for the lapse and elliptic equations are usually
CPU-time consuming, except if one make uses of fast elliptic solvers [148, 63]. For this reason,
most of the recent computations of binary black hole inspiral and merger have been performed
with the 1+log slicing, to be discussed in Sec. 9.2.4. Nevertheless, it is worth to note that
maximal slicing has been used for the first grazing collisions of binary black holes, as computed
by Brügmann (1999) [68].
To avoid the resolution of an elliptic equation while preserving most of the good properties
of maximal slicing, an approximate maximal slicing has been introduced in 1999 by Shibata
[224]. It consists in transforming Eq. (9.15) into a parabolic equation by adding a term of the