Discrete Gravity as a Local Theory of the Poincar\'e Group in the First Order Formalism

TL;DR

This paper develops a discrete, Poincaré-invariant gravity theory using a first order formalism on a Voronoi complex, connecting it to Regge Calculus and enabling quantum and matter coupling.

Contribution

It introduces a novel first order discrete gravity framework on Voronoi complexes, aligning with continuum Cartan formalism and improving upon Regge Calculus for quantum and matter interactions.

Findings

Equations resemble first order Einstein equations in the continuum.

In the small deficit angle limit, solutions reduce to Regge Calculus.

A well-defined quantum measure free from Regge Calculus ambiguities.

Abstract

A discrete theory of gravity locally invariant under the Poincar\'e group is considered as in a companion paper. We define a first order theory, in the sense of Palatini, on the metric-dual Voronoi complex of a simplicial complex. We follow the same spirit of the continuum theory of General Relativity in the Cartan formalism. The field equations are carefully derived taking in account the constraints of the theory. They look very similar to first order Einstein equations in the Cartan formalism. It is shown that in the limit of {\it small deficit angles} these equations have Regge Calculus, locally, as the only solution. A quantum measure is easly defined which does not suffer the ambiguities of Regge Calculus, and a coupling with fermionic matter is easily introduced