Omniscient foliations and the geometry of cosmological spacetimes

TL;DR

This paper explores geometric conditions on foliations of cosmological spacetimes that prevent null geodesic lines, providing insights into the spacetime's structure and supporting aspects of Bartnik's splitting conjecture.

Contribution

It introduces specific geometric conditions on foliations that inhibit null geodesic lines, linking foliation properties to the global geometry of cosmological spacetimes.

Findings

Certain foliation conditions prevent null geodesic lines.

Absence of null lines constrains spacetime geometry.

Conditions are conformally invariant.

Abstract

We identify certain general geometric conditions on a foliation of a spacetime (M,g) by timelike curves that will impede the existence of null geodesic lines, especially if (M,g) possesses a compact Cauchy hypersurface. The absence of such lines, in turn, yields well-known restrictions on the geometry of cosmological spacetimes, in the context of Bartnik's splitting conjecture. Since the (non)existence of null lines is actually a conformally invariant property, such conditions only need to apply for some suitable conformal rescaling of g.