Torsion Degrees of Freedom in the Regge Calculus as Dislocations on the Simplicial Lattice

TL;DR

This paper extends Regge Calculus by incorporating dislocations to model torsion, enabling discretization of gravitational theories like Einstein-Cartan with torsion degrees of freedom.

Contribution

It introduces a novel discretization method for gravitational theories with torsion using dislocations on the simplicial lattice.

Findings

Derived a discrete Einstein-Cartan action.

Formulated field equations for the discretized theory.

Provided a framework for including torsion in Regge Calculus.

Abstract

Using the notion of a general conical defect, the Regge Calculus is generalized by allowing for dislocations on the simplicial lattice in addition to the usual disclinations. Since disclinations and dislocations correspond to curvature and torsion singularities, respectively, the method we propose provides a natural way of discretizing gravitational theories with torsion degrees of freedom like the Einstein-Cartan theory. A discrete version of the Einstein-Cartan action is given and field equations are derived, demanding stationarity of the action with respect to the discrete variables of the theory.