On the linearization of Regge calculus

TL;DR

This paper analyzes the linearization of 3D Regge calculus around Euclidean space, providing explicit formulas, relating it to the curlTcurl operator, and demonstrating convergence of eigenpairs as a non-conforming finite element method.

Contribution

It introduces an explicit quadratic form for linearized Regge calculus, relates it to the curlTcurl operator, and shows convergence of eigenpairs in a discrete complex framework.

Findings

Explicit quadratic form for linearized Regge calculus.

Relation of the quadratic form to the curlTcurl operator.

Eigenpair convergence as a non-conforming finite element method.

Abstract

We study the linearization of three dimensional Regge calculus around Euclidean metric. We provide an explicit formula for the corresponding quadratic form and relate it to the curlTcurl operator which appears in the quadratic part of the Einstein-Hilbert action and also in the linear elasticity complex. We insert Regge metrics in a discrete version of this complex, equipped with densely defined and commuting interpolators. We show that the eigenpairs of the curlTcurl operator, approximated using the quadratic part of the Regge action on Regge metrics, converge to their continuous counterparts, interpreting the computation as a non-conforming finite element method.