Independent 4-tetrahedra connection representation of Regge calculus

TL;DR

This paper introduces a novel connection-based representation of Regge calculus using independent 4-tetrahedra, enabling a sum of independent actions and potentially simplifying the analysis of piecewise flat manifolds in quantum gravity.

Contribution

It develops a self-connection representation for Regge calculus with independent 4-tetrahedra, allowing tetrahedra to remain independent in the connection sector.

Findings

Representation of each bisimplex action via rotation matrices.

Independent 4-tetrahedra in connection variables.

Action expressed as a sum of independent tetrahedral terms.

Abstract

We consider simplest piecewise flat manifold consisting of two identical 4-tetrahedra (call it bisimplex). General relativity action for arbitrary piecewise flat manifold can be expressed in terms of sum of the (half of) bisimplex actions. We use representation of each bisimplex action in terms of certain rotation matrices (connections). This gives representation of any minisuperspace piecewise flat gravity system in terms of connections which do not connect neighboring 4-tetrahedra (more appropriate would be call these self-connections). If Regge calculus with independent 4-tetrahedra is considered, i. e. when the length of an edge is not constrained to be the same for all the 4-tetrahedra containing this edge, self-connection representation leaves 4-tetrahedra independent also in connection matrices sector. Action remains sum of independent 4-tetrahedra terms.