On the convergence of Regge calculus to general relativity

TL;DR

This paper investigates how Regge calculus approximates general relativity, demonstrating that solutions can converge with second-order accuracy despite residual discrepancies, through explicit Kasner cosmology constructions.

Contribution

It introduces a mechanism showing convergence of Regge calculus solutions to Einstein solutions even when residuals oscillate, resolving previous conflicting results.

Findings

Regge solutions can converge to Einstein solutions with second-order accuracy.

Oscillatory residuals do not prevent convergence.

Explicit Kasner cosmology solutions illustrate the convergence mechanism.

Abstract

Motivated by a recent study casting doubt on the correspondence between Regge calculus and general relativity in the continuum limit, we explore a mechanism by which the simplicial solutions can converge whilst the residual of the Regge equations evaluated on the continuum solutions does not. By directly constructing simplicial solutions for the Kasner cosmology we show that the oscillatory behaviour of the discrepancy between the Einstein and Regge solutions reconciles the apparent conflict between the results of Brewin and those of previous studies. We conclude that solutions of Regge calculus are, in general, expected to be second order accurate approximations to the corresponding continuum solutions.