Exact Bianchi Identity in Regge Gravity

TL;DR

This paper derives an exact form of the Bianchi identity within Regge calculus, establishing a fundamental algebraic relationship between deficit angles in discrete gravity that holds for arbitrary curvature levels.

Contribution

It introduces a novel, exact Bianchi identity in Regge gravity, extending the continuum concept to discrete manifolds without small curvature restrictions.

Findings

Provides an algebraic relationship between deficit angles of neighboring hinges.

Valid for arbitrarily curved manifolds, not limited to weak fields.

Identity is nonlinear in curvatures, generalizing previous approximations.

Abstract

In the continuum the Bianchi identity implies a relationship between different components of the curvature tensor, thus ensuring the internal consistency of the gravitational field equations. In this paper an exact form for the Bianchi identity in Regge's discrete formulation of gravity is derived, by considering appropriate products of rotation matrices constructed around null-homotopic paths. It implies an algebraic relationship between deficit angles belonging to neighboring hinges. As in the continuum, the derived identity is valid for arbitrarily curved manifolds, without a restriction to the weak field, small curvature limit, but is in general not linear in the curvatures.