The Hilbert Action in Regge Calculus

TL;DR

This paper derives the Hilbert action within simplicial geometry, showing that Regge calculus can be obtained exactly from the dual lattice structure without averaging, highlighting the unity of Delaunay and Voronoi lattices.

Contribution

It provides an exact derivation of the Regge action from simplicial geometry using dual lattices, avoiding averaging procedures common in prior approaches.

Findings

Exact expression for the Regge action derived from simplicial geometry.

Demonstrates the unity of Delaunay and Voronoi lattices in Regge calculus.

No averaging or continuum limit needed for the derivation.

Abstract

The Hilbert action is derived for a simplicial geometry. I recover the usual Regge calculus action by way of a decomposition of the simplicial geometry into 4-dimensional cells defined by the simplicial (Delaunay) lattice as well as its dual (Voronoi) lattice. Within the simplicial geometry, the Riemann scalar curvature, the proper 4-volume, and hence, the Regge action is shown to be exact, in the sense that the definition of the action does not require one to introduce an averaging procedure, or a sequence of continuum metrics which were common in all previous derivations. It appears that the unity of these two dual lattice geometries is a salient feature of Regge calculus.