A cosmological solution of Regge calculus

TL;DR

This paper corrects and improves a Regge calculus model of Kasner cosmology, demonstrating that with proper discretization, the lattice model converges rapidly to the continuum Einstein equations, supported by numerical analysis.

Contribution

It provides a corrected and more accurate Regge calculus model for Kasner cosmology that converges to Einstein's equations with finer discretization.

Findings

The corrected model converges quickly to the full Kasner-Einstein equations.

Numerical solutions support the rapid convergence of the lattice model.

The approach clarifies the relationship between discrete and continuum cosmological models.

Abstract

We revisit the Regge calculus model of the Kasner cosmology first considered by S. Lewis. One of the most highly symmetric applications of lattice gravity in the literature, Lewis' discrete model closely matched the degrees of freedom of the Kasner cosmology. As such, it was surprising that Lewis was unable to obtain the full set of Kasner-Einstein equations in the continuum limit. Indeed, an averaging procedure was required to ensure that the lattice equations were even consistent with the exact solution in this limit. We correct Lewis' calculations and show that the resulting Regge model converges quickly to the full set of Kasner-Einstein equations in the limit of very fine discretization. Numerical solutions to the discrete and continuous-time lattice equations are also considered.