The moduli space of isometry classes of globally hyperbolic spacetimes

TL;DR

This paper investigates the structure and properties of the moduli space of globally hyperbolic spacetimes, comparing different uniformities, analyzing limit spaces, and establishing conditions for precompactness in Lorentzian geometry.

Contribution

It demonstrates the differences between GH and GGH uniformities, analyzes limit spaces, and provides a Gromov-like criterion for precompactness of Lorentz spaces.

Findings

GH and GGH uniformities differ but share the same Cauchy sequences

Limit spaces of Cauchy sequences are well-defined and retain causality properties

A Gromov-like condition characterizes precompact classes of Lorentz spaces

Abstract

This is the last article in a series of three initiated by the second author. We elaborate on the concepts and theorems constructed in the previous articles. In particular, we prove that the GH and the GGH uniformities previously introduced on the moduli space of isometry classes of globally hyperbolic spacetimes are different, but the Cauchy sequences which give rise to well-defined limit spaces coincide. We then examine properties of the strong metric introduced earlier on each spacetime, and answer some questions concerning causality of limit spaces. Progress is made towards a general definition of causality, and it is proven that the GGH limit of a Cauchy sequence of $\mathcal{C}^{\pm}_{\alpha}$, path metric Lorentz spaces is again a $\mathcal{C}^{\pm}_{\alpha}$, path metric Lorentz space. Finally, we give a necessary and sufficient condition, similar to the one of Gromov for the Riemannian case, for a class of Lorentz spaces to be precompact.