A Kirchhoff-like conservation law in Regge calculus

TL;DR

This paper introduces a Kirchhoff-like conservation law in Regge calculus, extending the framework to non-vacuum spacetimes and enhancing understanding of simplicial diffeomorphism invariance.

Contribution

It presents a simplicial form of the contracted Bianchi identity based on the Cartan moment, enabling non-vacuum Regge calculus and deepening the understanding of discrete diffeomorphisms.

Findings

Derivation of a Kirchhoff-like conservation law in Regge calculus.

Extension of Regge calculus to non-vacuum spacetimes.

Enhanced understanding of simplicial diffeomorphism group.

Abstract

Simplicial lattices provide an elegant framework for discrete spacetimes. The inherent orthogonality between a simplicial lattice and its circumcentric dual yields an austere representation of spacetime which provides a conceptually simple form of Einstein's geometric theory of gravitation. A sufficient understanding of simplicial spacetimes has been demonstrated in the literature for spacetimes devoid of all non-gravitational sources. However, this understanding has not been adequately extended to non-vacuum spacetime models. Consequently, a deep understanding of the diffeomorphic structure of the discrete theory is lacking. Conservation laws and symmetry properties are attractive starting points for coupling matter with the lattice. We present a simplicial form of the contracted Bianchi identity which is based on the E. Cartan moment of rotation operator. This identity manifests itself in the conceptually-simple form of a Kirchhoff-like conservation law. This conservation law enables one to extend Regge Calculus to non-vacuum spacetimes and provides a deeper understanding of the simplicial diffeomorphism group.