A Kirchhoff-like conservation law in Regge calculus

Gentle, Adrian P.; Kheyfets, Arkady; McDonald, Jonathan R.; Miller, Warner A.

A Kirchhoff-like conservation law in Regge calculus

Authors

Gentle, Adrian P.

Kheyfets, Arkady

McDonald, Jonathan R.

Miller, Warner A.

Description

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.
13 pages, 4 figures