First class constraints in Regge calculus

TL;DR

This paper investigates I class constraints in Regge calculus's tetrad-connection formulation, identifying known and new constraints that ensure geometric consistency and relate areas and link lengths.

Contribution

It introduces two new types of I class constraints specific to Regge calculus, not present in continuum general relativity, enhancing understanding of its geometric structure.

Findings

Gauss law generates rotations in local frames.

New constraints relate triangle areas and link lengths.

Constraints ensure bivector properties of tensors.

Abstract

Considered are I class constraints in the tetrad-connection formulation of Regge calculus. One of these is well-known Gauss law which generates rotations in the local frames associated with tetrahedrons in the continuous time 3D section. Another two types of these are new, satisfied by definition of Regge manifold and having no I class analogs in the continuum general relativity. Constraints of the first type express vanishing of the dual squares of antisymmetric tensors of the triangles in the 3D section thus ensuring each such tensor being a bivector. Constraints of the second type are trigonometric relations between areas of triangles of 3D section caused by that the set of areas is redundant as compared to the set of linklengts.