Synge's world function applied to causal diamonds and causal sets

Abstract: One of the major tasks in discrete theories of gravity, including causal set theory, is to discover how the combinatorics of the underlying discrete structure recovers various geometric aspects of the emergent spacetime manifold. In this paper, I develop a new covariant approach to connect the combinatorics of a Poisson sprinkled causal set to the geometry of spacetime, using the so-called Synge's world function. The Poisson sprinkling depends crucially on the volume of a causal interval. I expand this volume, in 2 dimensions, in powers of Synge's world function and Ricci scalar. Other geometric properties of Synge's world function are well-known, making it easy to connect to the curvature of spacetime. I use this connection to provide a straightforward proof that the BDG action of causal sets gives the Einstein-Hilbert action in the continuum limit, without having to work in a particular coordinate system like Riemann normal coordinates.