Improved and Perfect Actions in Discrete Gravity

TL;DR

This paper develops improved and perfect actions in Regge calculus that exactly reproduce continuum General Relativity dynamics, capturing gauge symmetries, with explicit constructions in 3D and 4D cases.

Contribution

It introduces a method to construct perfect actions in Regge calculus that replicate continuum dynamics and gauge symmetries, including curved simplices.

Findings

Constructed perfect action in 3D with cosmological constant.

Developed perfect action in 4D for a single simplex.

Discussed Regge calculus with curved simplices.

Abstract

We consider the notion of improved and perfect actions within Regge calculus. These actions are constructed in such a way that they - although being defined on a triangulation - reproduce the continuum dynamics exactly, and therefore capture the gauge symmetries of General Relativity. We construct the perfect action in three dimensions with cosmological constant, and in four dimensions for one simplex. We conclude with a discussion about Regge Calculus with curved simplices, which arises naturally in this context.