The Regge Calculus as a second order accurate approximation to smooth metrics

By a simple argument it will be shown that for any positive definite smooth metric there exists a quadratically convergent discrete metric. That is, the fractional difference between the estimates of the length for a short path, calculated according to the smooth and discrete metrics, will be shown to satisfy ∆L/L = O(K∆) where K is the maximum Gaussian curvature on the coordinate hyperplanes and ∆ is a length scale of the largest simplex in the discrete space. This result applies only to algebraic functions of the metric, such as length, area and volumes. No attempt will be made in this paper to extend the result to a statement about the curvature tensors for the two metrics. It is also argued that this result can be used as a possible basis for an adaptive refinement scheme in the Regge calculus.