Scalar Curvature in Discrete Gravity

TL;DR

This paper numerically investigates scalar curvature in a discretized 2D space, comparing lattice models with continuous sphere metrics to understand their convergence as lattice size increases.

Contribution

It introduces two lattice discretizations of a 2-sphere and compares their scalar curvature results with the continuous case.

Findings

Numerical results approach continuous scalar curvature as lattice size increases.

Different lattice slicings yield consistent curvature estimates in the large limit.

The study validates discrete models for approximating continuous curvature in 2D.

Abstract

We focus on studying, numerically, the scalar curvature tensor in a two-dimensional discrete space. The continuous metric of a two-sphere is transformed into that of a lattice using two possible slicings. In the first, we use two integers, while in the second we consider the case where one of the coordinates is ignorable. The numerical results of both cases are then compared with the expected values in the continuous limit as the number of cells of the lattice becomes very large.