A geometric construction of the Riemann scalar curvature in Regge calculus

TL;DR

This paper introduces a new geometric method to measure the Riemann scalar curvature at a point in discrete spacetime using only clocks and photons, relevant for quantum gravity models.

Contribution

It presents a novel construction of scalar curvature in Regge calculus using a combined lattice approach, enabling measurements with finite observers.

Findings

Derived a vertex-based scalar curvature expression similar to hinge-based Regge calculus

Showed scalar curvature as a weighted average of deficits and dual areas

Provided a method compatible with discrete quantum gravity models

Abstract

The Riemann scalar curvature plays a central role in Einstein's geometric theory of gravity. We describe a new geometric construction of this scalar curvature invariant at an event (vertex) in a discrete spacetime geometry. This allows one to constructively measure the scalar curvature using only clocks and photons. Given recent interest in discrete pre-geometric models of quantum gravity, we believe is it ever so important to reconstruct the curvature scalar with respect to a finite number of communicating observers. This derivation makes use of a new fundamental lattice cell built from elements inherited from both the original simplicial (Delaunay) spacetime and its circumcentric dual (Voronoi) lattice. The orthogonality properties between these two lattices yield an expression for the vertex-based scalar curvature which is strikingly similar to the corresponding hinge-based expression in Regge calculus (deficit angle per unit Voronoi dual area). In particular, we show that the scalar curvature is simply a vertex-based weighted average of deficits per weighted average of dual areas.