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

9.2 Choice of foliation 157
Figure 9.4: Kruskal-Szekeres diagram depicting the maximal slicing of Schwarzschild spacetime defined by the
Reinhart/Estabrook et al. time function t [cf. Eq. (9.18)]. As for Figs. 9.1 and 9.3, R stands for Schwarzschild
radial coordinate (areal radius), so that R = 0 is the singularity and R = 2m is the event horizon, whereas r
stands for the isotropic radial coordinate. At the throat (minimal surface), R = RC where RC is the function of
t defined below Eq. (9.20) (figure adapted from Fig. 1 of Ref. [118]).
[211]. The corresponding time coordinate t is different from Schwarzschild time coordinate
tS, except for t = 0 (initial slice tS = 0). In the coordinates (xα ) = (t,R,θ,ϕ), where R is
Schwarzschild radial coordinate, the metric components obtained by Estabrook et al. [118]
(see also Refs. [50, 48, 210]) take the form
gµνdx µ dx ν = −N 2 dt 2
+ 1 − 2m C(t)2
+
R R4 −1
dR + C(t)
2 N dt + R
R2 2 (dθ 2 + sin 2 θdϕ 2 ),
(9.18)
where
N = N(R,t) =
1 − 2m
R
+ C(t)2
R 4
1 + dC
dt
and C(t) is the function of t defined implicitly by
t = −C
+∞
RC
+∞
R
x 4 dx
[x 4 − 2mx 3 + C(t) 2 ] 3/2
dx
(1 − 2m/x) √ x 4 − 2mx 3 + C 2,
, (9.19)
(9.20)
RC being the unique root of the polynomial PC(x) := x 4 − 2mx 3 + C 2 in the interval
(3m/2, 2m]. C(t) varies from 0 at t = 0 to C∞ := (3 √ 3/4)m 2 as t → +∞. Accordingly,