Is the Regge Calculus a consistent approximation to General Relativity?

TL;DR

This paper investigates whether Regge calculus accurately approximates General Relativity by analyzing residual errors on solutions of Einstein's equations, questioning its consistency as a discretization method.

Contribution

It critically examines the validity of residual errors as a criterion for the consistency of Regge calculus with Einstein's equations.

Findings

Residual errors cannot reliably distinguish Einstein solutions from non-solutions on generic lattices.

Either Regge calculus is inconsistent with General Relativity or residual errors are not suitable for validation.

The study challenges the use of residual errors as a sole measure of approximation accuracy.

Abstract

We will ask the question of whether or not the Regge calculus (and two related simplicial formulations) is a consistent approximation to General Relativity. Our criteria will be based on the behaviour of residual errors in the discrete equations when evaluated on solutions of the Einstein equations. We will show that for generic simplicial lattices the residual errors can not be used to distinguish metrics which are solutions of Einstein's equations from those that are not. We will conclude that either the Regge calculus is an inconsistent approximation to General Relativity or that it is incorrect to use residual errors in the discrete equations as a criteria to judge the discrete equations.