Affine connection form of Regge calculus

TL;DR

This paper reformulates Regge calculus using an affine connection approach, representing the action in a way that is invariant under coordinate changes and general linear transformations, connecting it to the traditional Regge action.

Contribution

It introduces an affine connection form of Regge calculus with invariance under GL(4,R) transformations, extending previous Euclidean-based formulations.

Findings

Derived a GL(4,R)-invariant action for Regge calculus.

Showed that eliminating the connection yields the standard Regge action.

Extended the geometric framework of Regge calculus to affine connections.

Abstract

Regge action is represented analogously to how the Palatini action for general relativity (GR) as some functional of the metric and a general connection as independent variables represents the Einstein-Hilbert action. The piecewise flat (or simplicial) spacetime of Regge calculus is equipped with some world coordinates and some piecewise affine metric which is completely defined by the set of edge lengths and the world coordinates of the vertices. The conjugate variables are the general nondegenerate matrices on the 3-simplices which play a role of a general discrete connection. Our previous result on some representation of the Regge calculus action in terms of the local Euclidean (Minkowsky) frame vectors and orthogonal connection matrices as independent variables is somewhat modified for the considered case of the general linear group GL(4,R) of the connection matrices. As a result, we have some action invariant w. r. t. arbitrary change of coordinates of the vertices (and related GL(4,R) transformations in the 4-simplices). Excluding GL(4,R) connection from this action via the equations of motion we have exactly the Regge action for the considered spacetime.