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

136 The initial data problem
Figure 8.4: Extended hypersurface Σ ′ 0 depicted in the Kruskal-Szekeres representation of Schwarzschild spacetime.
R stands for Schwarzschild radial coordinate and r for the isotropic radial coordinate. R = 0 is the singularity
and R = 2m the event horizon. Σ ′ 0 is nothing but a hypersurface t = const, where t is the Schwarzschild time
coordinate. In this diagram, these hypersurfaces are straight lines and the Einstein-Rosen bridge S is reduced to
a point.
8.2.6 Bowen-York initial data
Let us select the same simple free data as above, namely
˜γij = fij, Â ij
TT = 0, K = 0, ˜ E = 0 and ˜p i = 0. (8.67)
For the hypersurface Σ0, instead of R 3 minus a ball, we choose R 3 minus a point:
Σ0 = R 3 \{O}. (8.68)
The removed point O is called a puncture [66]. The topology of Σ0 is S2 × R; it differs from
the topology considered in Sec. 8.2.5 (R3 minus a ball); actually it is the same topology as that
(cf. Fig. 8.3).
of the extended manifold Σ ′ 0
Thanks to the choice (8.67), the system to be solved is still (8.39)-(8.40). If we choose
the trivial solution X = 0 for Eq. (8.40), we are back to the slice of Schwarzschild spacetime
considered in Sec. 8.2.5, except that now Σ0 is the extended manifold previously denoted Σ ′ 0 .
Bowen and York [65] have obtained a simple non-trivial solution of Eq. (8.40) (see also
Ref. [49]). Given a Cartesian coordinate system (xi ) = (x,y,z) on Σ0 (i.e. a coordinate system
such that fij = diag(1,1,1)) with respect to which the coordinates of the puncture O are (0,0,0),
this solution writes
X i = − 1
7f
4r
ij Pj + Pjxjxi r2
− 1
r3ǫij kSjx k , (8.69)