The simplest Regge calculus model in the canonical form

TL;DR

This paper introduces a simplified 3D Regge calculus model in canonical form, derived from 4D Regge calculus, with a finite degrees of freedom and a novel canonical structure ensuring consistency with discrete general relativity.

Contribution

It presents the first canonical formulation of a simple 3D Regge model derived from 4D Regge calculus, highlighting new relations with time derivatives for correct degrees of freedom.

Findings

The model has a finite number of degrees of freedom.

New relations with time derivatives are essential for the model.

The canonical structure aligns with discrete general relativity principles.

Abstract

Dynamics of a Regge three-dimensional (3D) manifold in a continuous time is considered. The manifold is closed consisting of the two tetrahedrons with identified corresponding vertices. The action of the model is that obtained via limiting procedure from the general relativity (GR) action for the completely discrete 4D Regge calculus. It closely resembles the continuous general relativity action in the Hilbert-Palatini (HP) form but possesses finite number of the degrees of freedom. The canonical structure of the theory is described. Central point is appearance of the new relations with time derivatives not following from the Lagrangian but serving to ensure completely discrete 4D Regge calculus origin of the system. In particular, taking these into account turns out to be necessary to obtain the true number of the degrees of freedom being the number of linklengths of the 3D Regge manifold at a given moment of time.