Geodesic Deviation in Regge Calculus

TL;DR

This paper explores geodesic deviation in Regge calculus, comparing it with continuous general relativity, and establishes a quantitative relation between curvature and deficit angles, aiding future matter inclusion methods.

Contribution

It provides a detailed comparison between Regge calculus and continuous GR for geodesic deviation, establishing a quantitative link between curvature and deficit angles.

Findings

Continuum and simplicial descriptions coincide with cumulative Regge contributions.

A quantitative relation between continuous curvature and Regge deficit angles is established.

Results may facilitate incorporating matter into Regge calculus.

Abstract

Geodesic deviation is the most basic manifestation of the influence of gravitational fields on matter. We investigate geodesic deviation within the framework of Regge calculus, and compare the results with the continuous formulation of general relativity on two different levels. We show that the continuum and simplicial descriptions coincide when the cumulative effect of the Regge contributions over an infinitesimal element of area is considered. This comparison provides a quantitative relation between the curvature of the continuous description and the deficit angles of Regge calculus. The results presented might also be of help in developing generic ways of including matter terms in the Regge equations.