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

152 Choice of foliation and spatial coordinates
9.2 Choice of foliation
9.2.1 Geodesic slicing
The simplest choice of foliation one might think about is the geodesic slicing, for it corresponds
to a unit lapse:
N = 1 . (9.1)
Since the 4-acceleration a of the Eulerian observers is nothing but the spatial gradient of ln N
[cf. Eq. (3.18)], the choice (9.1) implies a = 0, i.e. the worldlines of the Eulerian observers are
geodesics, hence the name geodesic slicing. Moreover the choice (9.1) implies that the proper
time along these worldlines coincides with the coordinate time t.
We have already used the geodesic slicing to discuss the basics feature of the 3+1 Einstein
system in Sec. 4.4.2. We have also argued there that, due to the tendency of timelike geodesics
without vorticity (as the worldlines of the Eulerian observers are) to focus and eventually cross,
this type of foliation can become pathological within a finite range of t.
Example : A simple example of geodesic slicing is provided by the use of Painlevé-Gullstrand
coordinates (t,R,θ,ϕ) in Schwarzschild spacetime (see e.g. Ref. [185]). These coordinates
are defined as follows: R is nothing but the standard Schwarzschild radial coordinate 1 ,
whereas the Painlevé-Gullstrand coordinate t is related to the Schwarzschild time coordinate
tS by
R 1
t = tS + 4m +
2m 2 ln
R/2m − 1
. (9.2)
R/2m + 1
The metric components with respect to Painlevé-Gullstrand coordinates are extremely simple,
being given by
gµνdx µ dx ν = −dt 2
2m
+ dR +
R dt
2 + R 2 (dθ 2 + sin 2 θ dϕ 2 ). (9.3)
By comparing with the general line element (4.48), we read on the above expression that
N = 1, β i = ( 2m/R,0,0) and γij = diag(1,R 2 ,R 2 sin 2 θ). Thus the hypersurfaces
t = const are geodesic slices. Notice that the induced metric γ is flat.
Example : Another example of geodesic slicing, still in Schwarzschild spacetime, is provided
by the time development with N = 1 of the initial data constructed in Secs. 8.2.5 and
8.3.3, namely the momentarily static slice tS = 0 of Schwarzschild spacetime, with topology
R × S 2 (Einstein-Rosen bridge). The resulting foliation is depicted in Fig. 9.1. It hits the
singularity at t = πm, reflecting the bad behavior of geodesic slicing.
In numerical relativity, geodesic slicings have been used by Nakamura, Oohara and Kojima
to perform in 1987 the first 3D evolutions of vacuum spacetimes with gravitational waves [193].
However, as discussed in Ref. [233], the evolution was possible only for a pretty limited range of
t, because of the focusing property mentioned above.
1 in this chapter, we systematically use the notation R for Schwarzschild radial coordinate (areal radius), leaving
the notation r for other types of radial coordinates, such that the isotropic one [cf. Eq. (6.24)]