The Discrete Geometry of a Small Causal Diamond

TL;DR

This paper investigates the discrete geometry of small causal diamonds in curved spacetime using causal set theory, deriving curvature corrections, scalar curvature, Ricci tensor components, and a new dimension estimator based on chain abundances.

Contribution

It introduces first-order curvature corrections to flat spacetime chain abundances and proposes a new discrete scalar curvature, Ricci tensor component, and dimension estimator for causal sets in curved spacetime.

Findings

Derived first-order curvature corrections to chain abundances.

Formulated a new discrete scalar curvature expression.

Developed a new dimension estimator for curved spacetimes.

Abstract

We study the discrete causal set geometry of a small causal diamond in a curved spacetime using the average abundance of k-element chains or total orders in the underlying causal set C. We begin by obtaining the first order curvature corrections to the flat spacetime expression for the abundance using Riemann normal coordinates. For fixed spacetime dimension this allows us to find a new expression for the discrete scalar curvature of C as well as the time-time component of its Ricci tensor in terms of the abundances of k-chains. We also find a new dimension estimator for C which replaces the flat spacetime Myrheim-Meyer estimator in generic curved spacetimes.