Null Distance and Gromov-Hausdorff Convergence of Warped Product Spacetimes

Abstract

What is the analogous notion of Gromov-Hausdorff convergence for sequences of spacetimes? Since a Lorentzian manifold is not inherently a metric space, one cannot simply use the traditional definition. One approach offered by Sormani and Vega \cite{SV} is to define a metric space structure on a spacetime by means of the null distance. Then one can define convergence of spacetimes using the usual definition of Gromov-Hausdorff convergence. In this paper we explore this approach by giving many examples of sequences of warped product spacetimes with the null distance converging in the Gromov-Hausdorff sense. In addition, we give an optimal convergence theorem which shows that under natural geometric hypotheses a sequence of warped product spacetimes converge to a specific limiting warped product spacetime. The examples given further serve to show that the hypotheses of this convergence theorem are optimal.