A Closeness Function on Coarse Grained Lorentzian Geometries

TL;DR

This paper introduces a numerically computable closeness function for finite volume Lorentzian geometries based on causal sets, aiding the analysis of continuum-like behavior in causal set theory.

Contribution

It presents a new closeness measure for Lorentzian geometries derived from causal sets, which is easier to compute than traditional geometric distances.

Findings

The closeness function is numerically feasible for large causal sets.

It provides a quantitative measure of continuum-like behavior.

The function is weaker than the Lorentzian Gromov-Hausdorff distance.

Abstract

We construct a family of closeness functions on the space of finite volume Lorentzian geometries using the abundance of discrete intervals in the underlying random causal sets. Although strictly weaker than a Lorentzian Gromov-Hausdorff distance function, it has the advantage of being numerically calculable for large causal sets. It thus provides a concrete and quantitative measure of continuumlike behaviour in causal set theory and can be used to define a weak convergence condition for Lorentzian geometries.