Recent Progress in Regge Calculus

TL;DR

This paper reviews recent advances in Regge calculus, focusing on its applications in quantum gravity, the definition of gauge transformations, and the analysis of the supermetric's signature through analytic and numerical methods.

Contribution

It investigates the signature of the simplicial supermetric in Regge calculus and explores gauge transformations, providing new insights into the structure of simplicial configuration space.

Findings

Numerical results confirm analytic predictions of supermetric signature change.

Degeneracy and signature change are observed in the supermetric for specific topologies.

Gauge-related directions in configuration space are identified for the three-torus.

Abstract

While there has been some advance in the use of Regge calculus as a tool in numerical relativity, the main progress in Regge calculus recently has been in quantum gravity. After a brief discussion of this progress, attention is focussed on two particular, related aspects. Firstly, the possible definitions of diffeomorphisms or gauge transformations in Regge calculus are examined and examples are given. Secondly, an investigation of the signature of the simplicial supermetric is described. This is the Lund-Regge metric on simplicial configuration space and defines the distance between simplicial three-geometries. Information on its signature can be used to extend the rather limited results on the signature of the supermetric in the continuum case. This information is obtained by a combination of analytic and numerical techniques. For the three-sphere and the three-torus, the numerical results agree with the analytic ones and show the existence of degeneracy and signature change. Some ``vertical'' directions in simplicial configuration space, corresponding to simplicial metrics related by gauge transformations, are found for the three-torus.