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

8.3 Conformal thin sandwich method 139
Substituting expression (8.71) for Âjk as well as expressions (7.60)-(7.62) for (φi) j , we get that
only the Si part contribute to this integral. After some computation, we find
Ji = Si . (8.84)
Hence the parameters Si of the Bowen-York solution are nothing but the three components of
the angular momentum of the hypersurface Σ0.
Remark : The Bowen-York solution with P i = 0 and S i = 0 reduces to the momentarily static
solution found in Sec. 8.2.5, i.e. is a slice t = const of the Schwarzschild spacetime (t
being the Schwarzschild time coordinate). However Bowen-York initial data with P i = 0
and S i = 0 do not constitute a slice of Kerr spacetime. Indeed, it has been shown [138]
that there does not exist any foliation of Kerr spacetime by hypersurfaces which (i) are
axisymmetric, (ii) smoothly reduce in the non-rotating limit to the hypersurfaces of constant
Schwarzschild time and (iii) are conformally flat, i.e. have induced metric ˜γ = f, as the
Bowen-York hypersurfaces have. This means that a Bowen-York solution with S i = 0 does
represent initial data for a rotating black hole, but this black hole is not stationary: it is
“surrounded” by gravitational radiation, as demonstrated by the time development of these
initial data [67, 142].
8.3 Conformal thin sandwich method
8.3.1 The original conformal thin sandwich method
An alternative to the conformal transverse-traceless method for computing initial data has been
introduced by York in 1999 [278]. It is motivated by expression (6.78) for the traceless part of
the extrinsic curvature scaled with α = −4:
Noticing that [cf. Eq. (8.11)]
à ij = 1
2N
and introducing the short-hand notation
we can rewrite Eq. (8.85) as
∂
− Lβ
∂t
The relation between Ãij and Âij is [cf. Eq. (6.102)]
˜γ ij − 2
3 ˜ Dkβ k ˜γ ij
. (8.85)
− Lβ ˜γ ij = ( ˜ Lβ) ij + 2
3 ˜ Dkβ k , (8.86)
˙˜γ ij := ∂
∂t ˜γij , (8.87)
à ij = 1
˙˜γ
2N
ij + ( ˜ Lβ) ij
. (8.88)
 ij = Ψ 6 à ij . (8.89)