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

Chapter 3
Geometry of foliations
Contents
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Globally hyperbolic spacetimes and foliations . . . . . . . . . . . . . 39
3.3 Foliation kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Last part of the 3+1 decomposition of the Riemann tensor . . . . . 47
3.1 Introduction
In the previous chapter, we have studied a single hypersurface Σ embedded in the spacetime
(M,g). At present, we consider a continuous set of hypersurfaces (Σt) t∈R that covers the manifold
M. This is possible for a wide class of spacetimes to which we shall restrict ourselves: the
so-called globally hyperbolic spacetimes. Actually the latter ones cover most of the spacetimes
of astrophysical or cosmological interest. Again the title of this chapter is “Geometry...”, since
as in Chap. 2, all the results are independent of the Einstein equation.
3.2 Globally hyperbolic spacetimes and foliations
3.2.1 Globally hyperbolic spacetimes
A Cauchy surface is a spacelike hypersurface Σ in M such that each causal (i.e. timelike or
null) curve without end point intersects Σ once and only once [156]. Equivalently, Σ is a Cauchy
surface iff its domain of dependence is the whole spacetime M. Not all spacetimes admit a
Cauchy surface. For instance spacetimes with closed timelike curves do not. Other examples
are provided in Ref. [131]. A spacetime (M,g) that admits a Cauchy surface Σ is said to be
globally hyperbolic. The name globally hyperbolic stems from the fact that the scalar wave
equation is well posed,