Manifold Properties from Causal Sets using Chains

TL;DR

This paper investigates how chains in causal sets can be used to estimate key continuum spacetime properties like curvature and dimension, demonstrating successful application in specific spacetimes and proposing tests for manifoldlikeness.

Contribution

It extends the formalism for estimating continuum properties from causal sets, showing its effectiveness in certain spacetimes and proposing a method to test manifoldlikeness.

Findings

Accurately estimates curvature, proper time, and dimension in $ ext{dS}_2$ and $ ext{FLRW}_3$ spacetimes.

Demonstrates the formalism's applicability with slight modifications in specific models.

Proposes a test for manifoldlikeness using non-manifoldlike causal set models.

Abstract

We study the utility of chains defined on causal sets in estimating continuum properties like the curvature, the proper time and the spacetime dimension through a numerical analysis. In particular, we show that in $\text{dS}_2$ and $\text{FLRW}_3$ spacetimes the formalism of arXiv:1212.0631 with slight modifications gives the right continuum properties. We also discuss a possible test of manifoldlikeness using this formalism by considering two models of non-manifoldlike causal sets. This is a part of a broader idea of the geometrical reconstruction of continuum properties given a discrete sub structure, in this case the causal set.