# Regge calculus: Questions of consistency resolved?

Hello,

Regge calculus is an approximation scheme for General Relativity, which has been introduced in early-sixties and has been adopted both in numerical relativity and numerical quantum relativity. In contrast to its widespread use in computational science, there does not seem to exist much theory on whether the Regge calculus is actually consistent - i.e. whether there is some degree of exactness (like mesh width) we can adapt arbitrarly to obtain a solution with arbitrarly small error (say in some Sobolev-norm).

In fact there has been debate about this:

The question of consistency is a purely mathematical one, and therefore I do not expect it to be "debatable". I will have to work with this theory and hence I do wonder in how far there is a theoretical basis to tell apart "It works" and "It does not work".

Thank you very much!

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Have you read the papers you linked to? The first paper makes a pretty strong case for inconsistency. But the second paper says RC is nevertheless convergent. It is possible for a numerical method to be inconsistent but convergent. – timur Jun 1 '11 at 3:01
I first read "Reggae calculus". I'm somewhat disappointed. – Gunnar Þór Magnússon Dec 15 '11 at 10:00
@Gunnar: What is Reggae calculus? – timur Aug 24 '12 at 0:12
@timur: what Bob Marley gets from not brushing his teeth. – Ryan Thorngren Aug 24 '12 at 0:53
@Ryan: That's great! Thanks. – timur Aug 24 '12 at 1:14

## 1 Answer

The consistency is proved by Cheeger, M\"uller and Schrader in 1984, "On the Curvature of Piecewise Flat Sapces". Roughly speaking, given a smooth Riemannian manifold with a smooth metric, there exists a sequence of triangulation, on which Regge's definition converges to the smooth curvature as a measure.

At the linearized level, there is also a recent paper on the consistency: Christiansen 2011, "On The Linearization of Regge Calculus". One of the theorems in the paper is that we have consistency between linearized Regge and linearized Einstein equation as well.

That said, when you talk about the convergence of a numerical algorithm, it depends on a lot of other things as well, such as your formulation of the Einstein's equation. You will also need some form of stability to ensure convergence. Those questions remain to be solved (hopefully in my thesis:-).

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