Schmidt boundaries of foliated space-times

Abstract

For every -dimensional foliated Lorentzian manifold , where is a codimension q space-like foliation, we build its Q-completion and Q-boundary . These are analogs, within transverse Lorentzian geometry of foliated manifolds, to the b-completion and b-boundary (due to (Schmidt 1971 Gen. Relativ. Gravit. 1 269–80)). The bundle morphism (mapping the -component of the Levi–Civita connection 1-form of into the unique torsion-free adapted connection on the bundle of Lorentzian transverse orthonormal frames) is shown to induce a surjective continuous map of the adapted boundary () of onto its Q-boundary. Map is used to characterize as the set of end points , in the topology of , of all Q-incomplete curves . As an application we determine a class of b-boundary points, where , g is Schwartzschildʼs metric, and is the codimension two foliation tangent to the Killing vector fields and .