Bartnik's splitting conjecture and the cosmological wall: a Cauchy Busemann function approach

Abstract

Bartnik's conjecture states that a timelike geodesically complete cosmological spacetime satisfying the timelike convergence condition splits as a static product. We show that any counterexample exhibits a peculiar global causal structure: two disjoint open regions, the future and the past cosmological developments, separated by a non-empty closed causally convex set: the cosmological wall. The developments are bounded by $C^0$ null hypersurfaces, the cosmological horizons, and a component of each horizon contains a lightlike line. These objects are natural and hence carry physical significance, as they do not depend on the choice of Cauchy hypersurface. They can be defined through the Cauchy Busemann functions $b^\pm$ of a compact Cauchy hypersurface $Σ$, the developments being their finiteness domains. Using a product trick we show that $b^+$ is the graph function of a Cauchy horizon in $M\times \mathbb{R}$, and so inherits the regularity and dynamical properties of horizons: $b^+$ is continuous, locally Lipschitz and locally semi-concave on its finiteness domain, and its calibrated curves are timelike rays. We prove that if the two cosmological developments intersect, which happens whenever $b^+$ is $C^1$ on its finiteness domain, then the spacetime contains a timelike line and hence splits, a result which recovers several positive results of the literature. We also obtain a localized rigidity version of Hawking's singularity theorem: if some compact Cauchy hypersurface has non-positive mean curvature on the future development, the spacetime splits. Hence, in a counterexample, every compact Cauchy hypersurface expands somewhere in the future development and contracts somewhere in the past one. If the wall has empty interior, it coincides with both horizons and is ruled by lightlike lines, and under null completeness its components are smooth totally geodesic null hypersurfaces.