A Closeness Function on Coarse Grained Lorentzian Geometries

TL;DR

The paper addresses quantifying how close Lorentzian geometries are within causal set theory by exploiting interval abundances in random causal sets. It introduces the $n$-interval spectrum $\mathcal{S}_n(M,g)$ and a family of $L^r$ closeness functions $\mathcal{D}^{(r)}_n$ based on the abundances $N_m^{(n)}$, offering a computable, scale-aware metric for large $n$ and a weak convergence notion on the quotient space $\tilde{\mathcal{L}}$. Although $\mathcal{D}^{(r)}_n$ is weaker than a full Lorentzian Gromov-Hausdorff distance due to interval isospectral degeneracies, it yields practical discrimination between spacetimes and supports a notion of interval convergence, demonstrated via simulations across Minkowski diamonds, thickened diamonds, hypercubes, and FRW regions. The work also analyzes degeneracies and discusses incorporating additional invariants to lift them, outlining a path toward stronger convergence criteria and broader applicability to non-flat cosmological spacetimes.

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.

Figures (20)

• Figure 1: Causal sets generated by Poisson sprinklings into various regions of spacetime.
• Figure 2: We show 30 different realisations of $\mathcal{S}_n(\mathbb D^d)$ for $n=10,000$, $d=2,\ldots 6$ plotted with the mean and standard error. One can make out the spread, which is large for any given $m$, but is relatively small as far as the entire spectrum is concerned. Compare with Fig. \ref{['Fig1.fig']} below.
• Figure 3: $\mathcal{S}_n(\mathbb D^d)$ for $d=2, \ldots 8$ for a single realisation with $n=20,000$.
• Figure 4: $\mathcal{S}_n(\mathbb D^4)$ for a range of $n$ values. While the number of intervals increases with the size of the causal set, the overall shape of the function does not change.
• Figure 5: (a) $\mathcal{S}_n(\mathbb D^d\times \mathbb I_t)$ for $t=0.15$ and $d=2,\ldots, 8$. (b) $\mathcal{S}_n(\mathbb D^4\times \mathbb I_t)$ for $n=10,000, \ldots 20,000$. The qualitative behaviour is very similar to that of $\mathbb D^d$.

Theorems & Definitions (1)

• Definition