The Geometry of Small Causal Cones

TL;DR

This paper derives a formula for the volume of small causal cones and applies it within causal set theory to express geometric quantities and scalar field derivatives related to a spacelike hypersurface.

Contribution

It introduces a new formula for small causal cone volumes and uses it to connect causal set expressions with spacetime geometry and scalar field derivatives.

Findings

Derived a formula for small causal cone volume

Constructed causal set expressions for geometric quantities

Linked scalar field derivatives to causal set representations

Abstract

We derive a formula for the spacetime volume of a small causal cone. We use this formula within the context of causal set theory to construct causal set expressions for certain geometric quantities relating to a spacetime with a spacelike hypersurface. We also consider a scalar field on the causal set, and obtain causal set expressions relating to its normal derivatives with respect to the hypersurface.