A Closeness Function on Coarse Grained Lorentzian Geometries

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.