Recent Progress in Regge Calculus

TL;DR

This work analyzes Regge calculus within quantum gravity by examining two core aspects: how to define gauge transformations (diffeomorphisms) in a discrete setting and the structure of the simplicial supermetric on simplicial configuration space, known as the Lund-Regge metric. It combines analytic results from the continuum with discrete, numerical investigations on $S^3$ and $T^3$ to determine the signature and degeneracies of the simplicial supermetric, revealing a timelike conformal direction, multiple spacelike directions, and gauge-related (approximate) diffeomorphisms. On $S^3$, the signature is $(-,+,+,+,+,+,+,+,+,+)$ with a single negative mode, while on $T^3$ the flat limit shows 13 negative and 176 positive directions with zero modes; gauge modes appear as eigenvectors with near-diffeomorphism behavior, and degeneracies reflect lattice symmetry. The results illuminate how discrete Regge approaches encode continuum diffeomorphism invariance within a finite-dimensional superspace, providing a path to finite, gauge-consistent formulations of quantum gravity and guiding future studies of the gauge content and signature evolution in larger triangulations.

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.