Area Regge Calculus and Discontinuous Metrics

TL;DR

This paper explores how using triangle areas as variables in Regge calculus can lead to metric discontinuities, especially on null hypersurfaces, resulting in refractive wave solutions.

Contribution

It constructs solutions to area Regge calculus on a triangulated lattice, highlighting the conditions under which metric discontinuities occur.

Findings

Discontinuities are absent on spacelike hypersurfaces.

Discontinuities can occur on null hypersurfaces, interpreted as refractive waves.

The approach links area variables to metric properties in Regge calculus.

Abstract

Taking the triangle areas as independent variables in the theory of Regge calculus can lead to ambiguities in the edge lengths, which can be interpreted as discontinuities in the metric. We construct solutions to area Regge calculus using a triangulated lattice and find that on a spacelike hypersurface no such discontinuity can arise. On a null hypersurface however, we can have such a situation and the resulting metric can be interpreted as a so-called refractive wave.