GL(4,R) representation of the gravity action on the piecewise flat spacetime

TL;DR

This paper formulates a discrete gravity action on piecewise flat spacetime using GL(4,R) matrices as connection variables, connecting affine connections with Regge calculus and discussing diffeomorphisms and path integrals.

Contribution

It introduces a novel discrete formulation of gravity employing GL(4,R) matrices on 3-simplices, bridging affine connections with Regge calculus.

Findings

Discrete gravity action expressed via GL(4,R) matrices.

Connection between affine connections and Regge action.

Brief discussion on diffeomorphisms and path integral in discrete setting.

Abstract

The gravity action on the piecewise flat Riemannian manifold is formulated using the discrete set of the nondegenerate 4$\times$4 matrices on the 3-simplices as some connection type variables. These variables are the discrete counterpart of the affine (Christoffel) connection used as independent variables in the Palatini form of the Einstein gravity action. Excluding these with the help of the equations of motion we get the original discrete gravity action on the piecewise flat spacetime (Regge action). The discrete version of the diffeomorphisms and path integral are briefly discussed.