Time foliations in Lorentzian manifolds (Enrico Trebeschi)

Abstract: Lorentzian geometry the mathematical model for general relativity. It generalizes Riemannian geometry for the presence of isotropic directions, namely it admits curves of null lenght, which are the paths that light follows.
The requirement for a Lorentzian manifold to be physically meaningful are quite strict, and are summarized on the definition of globally hyperbolic. As a consequence, the topology of spacetimes is not rich: every globally hyperbolic Lorentzian (n+1)-manifold consists in the product between a n-manifold and the real line.
Nonethless, the choice of such parameterization is physically meaningful: it consists in setting a privileged time in the universe. This choice gives rise to a foliation of the manifold, whose leaves correspond to events happening at the same time.
Our goal will be to explain these concepts with the aid of explicit example coming from negatively curved spacetimes.