Null distance on cosmological spacetimes and monotone convergence

The metric theory of spacetimes studies Lorentzian manifolds using tools of metric geometry.

This is achieved via the null distance, which is a definite distance constructed from a time function on a spacetime.

This enables the study of Gromov-Hausdorff-type convergence of spacetimes, a program recently initiated by Sakovich and Sormani.

In this paper we study such notions of convergence for cosmological spacetimes with compact slices, i.e., $(a,b)\times M$ endowed with a Lorentzian metric $-dt^2+h_t$, where $h_t$ is a family of Riemannian metrics on the compact manifold $M$.

Assuming mild extension properties of $h_t$, we first establish that these spacetimes are causally-null compactifiable and future developed.

We then study monotone sequences with a uniform upper bound on the spatial diameter, obtaining uniform convergence of the null distances, as well as convergence of the associated timed metric spaces in the future developed Gromov-Hausdorff sense.

Finally, we prove that causally-null compactifiable spacetimes satisfying a mild causal accessibility condition are causally-null, and relate the causally-null distance induced by the limit distance with the null distance induced by the (possibly non-smooth) limit metric tensor.

Examples are provided to motivate the necessity of our hypotheses.