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

40 Geometry of foliations
Figure 3.1: Foliation of the spacetime M by a family of spacelike hypersurfaces (Σt)t∈R.
The topology of a globally hyperbolic spacetime M is necessarily Σ × R (where Σ is the
Cauchy surface entering in the definition of global hyperbolicity).
Remark : The original definition of a globally hyperbolic spacetime is actually more technical
that the one given above, but the latter has been shown to be equivalent to the original one
(see e.g. Ref. [88] and references therein).
3.2.2 Definition of a foliation
Any globally hyperbolic spacetime (M,g) can be foliated by a family of spacelike hypersurfaces
(Σt) t∈R . By foliation or slicing, it is meant that there exists a smooth scalar field ˆt on M,
which is regular (in the sense that its gradient never vanishes), such that each hypersurface is a
level surface of this scalar field:
∀t ∈ R, Σt := p ∈ M, ˆt(p) = t . (3.1)
Since ˆt is regular, the hypersurfaces Σt are non-intersecting:
Σt ∩ Σt ′ = ∅ for t = t′ . (3.2)
In the following, we do no longer distinguish between t and ˆt, i.e. we skip the hat in the name
of the scalar field. Each hypersurface Σt is called a leaf or a slice of the foliation. We assume
that all Σt’s are spacelike and that the foliation covers M (cf. Fig. 3.1):
M =
Σt. (3.3)
t∈R