Discrete geometry of a small causal diamond

Abstract

We study the discrete causal set geometry of a small causal diamond in a curved spacetime using the average abundance hCki 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 hCki 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 hCki. We also find a new dimension estimator for C which replaces the flat spacetime Myrheim-Meyer estimator in generic curved spacetimes.