Regge Calculus in Teleparallel Gravity

TL;DR

This paper develops a discrete lattice approach to teleparallel gravity using Regge calculus, representing torsion through dislocation in simplicial complexes and deriving the discrete action and field equations.

Contribution

It introduces a novel discrete lattice formulation of teleparallel gravity based on Regge calculus, with torsion localized on simplicial hinges and a derived discrete action.

Findings

Discrete teleparallel action derived from lattice dislocations

Field equations formulated for simplicial teleparallel gravity

Torsion localized on two-dimensional hinges in the lattice

Abstract

In the context of the teleparallel equivalent of general relativity, the Weitzenbock manifold is considered as the limit of a suitable sequence of discrete lattices composed of an increasing number of smaller an smaller simplices, where the interior of each simplex (Delaunay lattice) is assumed to be flat. The link lengths between any pair of vertices serve as independent variables, so that torsion turns out to be localized in the two dimensional hypersurfaces (dislocation triangle, or hinge) of the lattice. Assuming that a vector undergoes a dislocation in relation to its initial position as it is parallel transported along the perimeter of the dual lattice (Voronoi polygon), we obtain the discrete analogue of the teleparallel action, as well as the corresponding simplicial vacuum field equations.