Exterior differentiation in the Regge calculus

Abstract

Regge manifolds are piecewise continuous manifolds constructed from a finite number of basic building blocks. On such manifolds piecewise continuous forms can be defined in a way similar to differential forms on a differentiable manifold. Regge manifolds are used extensively in the construction of space-times in numerical general relativity. In this paper a definition of exterior differentiation suitable for use on piecewise continuous forms on a Regge manifold is presented. It is shown that this definition leads to a version of Stokes’ theorem and also to the usual result that d2=0. This is preceded by a discussion of certain geometrical properties of the Regge manifolds. It is shown that the version of Stokes’ theorem presented here coincides with the usual definition when the Regge manifold is refined, by increasing the number of cells while keeping the total volume constant, to a smooth manifold.