Statistical Lorentzian geometry and the closeness of Lorentzian manifolds

TL;DR

The paper introduces a novel way to measure the similarity between Lorentzian manifolds based on causal structures derived from randomly scattered points, with implications for quantum gravity.

Contribution

It proposes a new family of closeness functions for Lorentzian geometries using causal partial orders and probabilistic comparisons, extending to a true distance with infinite sampling density.

Findings

Defined a pseudo-distance for finite point density comparing manifolds.

Extended to a true distance on the space of Lorentzian geometries with infinite density.

Illustrated the approach with examples involving 2D manifolds of different topology.

Abstract

I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume element, some information on the topology and metric is encoded in the partial order that the causal structure induces among those points; one can then define closeness between Lorentzian geometries by comparing the sets of probabilities they give for obtaining the same posets. If the density of points is finite, one gets a pseudo-distance, which only compares the manifolds down to a finite volume scale, as illustrated here by a fully worked out example of two 2-dimensional manifolds of different topology; if the density is allowed to become infinite, a true distance can be defined on the space of all Lorentzian geometries. The introductory and concluding sections include some remarks on the motivation for this definition and its applications to quantum gravity.