Gromov's reconstruction theorem and measured Gromov-Hausdorff convergence in Lorentzian geometry

TL;DR

This paper extends Gromov's reconstruction theorem to Lorentzian geometry, introduces a new concept of isomorphism for Lorentzian metric measure spaces, and explores convergence notions with applications to spacetime reconstruction from causal set theory.

Contribution

It generalizes Gromov's theorem to Lorentzian spaces, defines isomorphy for Lorentzian metric measure spaces, and proposes convergence concepts with foundational properties.

Findings

Established Gromov's reconstruction theorem in Lorentzian geometry

Introduced a natural concept of isomorphy for Lorentzian metric measure spaces

Proposed and analyzed three notions of convergence with fundamental properties

Abstract

We establish Gromov's celebrated reconstruction theorem in Lorentzian geometry. Alongside this result, we introduce and study a natural concept of isomorphy of normalized bounded Lorentzian metric measure spaces. We outline applications to the spacetime reconstruction problem from causal set theory. Lastly, we propose three notions of convergence of (isomorphism classes of) normalized bounded Lorentzian metric measure spaces, for which we prove several fundamental properties.