Null distance and convergence of Lorentzian length spaces

TL;DR

This paper extends the null distance concept to Lorentzian length spaces, exploring their convergence properties and compatibility with synthetic curvature bounds, advancing the understanding of Lorentzian geometry beyond smooth manifolds.

Contribution

It generalizes null distance to Lorentzian length spaces and studies their Gromov-Hausdorff convergence, linking causality, topology, and curvature in a synthetic setting.

Findings

Null distance encodes topology and causality in Lorentzian length spaces.

Gromov-Hausdorff convergence is studied in warped product Lorentzian length spaces.

Initial results show compatibility with synthetic curvature bounds.

Abstract

The null distance of Sormani and Vega encodes the manifold topology as well as the causality structure of a (smooth) spacetime. We extend this concept to Lorentzian length spaces, the analog of (metric) length spaces, which generalize Lorentzian causality theory beyond the manifold level. We then study Gromov-Hausdorff convergence based on the null distance in warped product Lorentzian length spaces and prove first results on its compatibility with synthetic curvature bounds.