Properties of the Null Distance and Spacetime Convergence

TL;DR

This paper explores the properties of the null distance in Lorentzian manifolds, establishing conditions for convergence of warped spacetimes and analyzing how warping function variations affect limits.

Contribution

It introduces conditions under which warped product spacetimes with null distance are integral current spaces and proves a general convergence theorem relating different notions of geometric convergence.

Findings

Warped product spacetimes can be integral current spaces under certain conditions.

A convergence theorem relates uniform, Gromov--Hausdorff, and intrinsic flat convergence.

Non-uniform warping functions can lead to different limiting behaviors.

Abstract

The null distance for Lorentzian manifolds was recently introduced by Sormani and Vega. Under mild assumptions on the time function of the spacetime, the null distance gives rise to an intrinsic, conformally invariant metric that induces the manifold topology. We show when warped products of low regularity and globally hyperbolic spacetimes endowed with the null distance are (local) integral current spaces. This metric and integral current structure sets the stage for investigating convergence analogous to Riemannian geometry. Our main theorem is a general convergence result for warped product spacetimes relating uniform, Gromov--Hausdorff and Sormani--Wenger intrinsic flat convergence of the corresponding null distances. In addition, we show that non-uniform convergence of warping functions in general leads to distinct limiting behavior, such as limits that disagree.