On the definition of curvature in Regge calculus

Abstract

Abstract Regge calculus defines a curvature for piecewise constant metrics on simplicial complexes subject to a partial continuity requirement. We prove that if a part of the complex is embedded in a Euclidean space, by a piecewise affine map, and we perform smoothing by convolution there, then the smoothed metrics have a densitized scalar curvature that converges, in the sense of measures, to that defined by Regge, as the smoothing parameter goes to zero.