A Lorentzian Gromov-Hausdoff notion of distance

TL;DR

This paper introduces a Lorentzian Gromov-Hausdorff distance to define a metric on the space of compact globally hyperbolic spacetimes, facilitating mathematical and physical analysis of their moduli space.

Contribution

It proposes the first Lorentzian Gromov-Hausdorff distance, establishing a metric structure on the moduli space of globally hyperbolic spacetimes with boundary.

Findings

Defines a Lorentzian Gromov-Hausdorff distance

Establishes the moduli space as a metric space

Lays groundwork for further geometric analysis

Abstract

This paper is the first of three in which I study the moduli space of isometry classes of (compact) globally hyperbolic spacetimes (with boundary). I introduce a notion of Gromov-Hausdorff distance which makes this moduli space into a metric space. Further properties of this metric space are studied in the next papers. The importance of the work can be situated in fields such as cosmology, quantum gravity and - for the mathematicians - global Lorentzian geometry.