Linearised regge calculus

Here is a paper that I wrote a draft of back in 1987 but never made it to publication: Linearised Regge Calculus.

A long long time ago, I worked on a discrete version of general relativity called Regge Calculus. This is a rather elegant theory invented by the late great Tullio Regge using simplicial manifolds with Riemannian (or Lorentzian) metrics that are flat on each simplex (“piecewise flat”). I was particularly interested in how this could mimic general relativity on large scales, i.e., I wanted to show that if you worked to a finite resolution so that the (smaller) simplicial structure wouldn’t be apparent, then the metrics approximated nice smooth ones obeying Einstein’s equations.

This is a hard problem, but can be simplified by looking a weak gravitational fields, those that are close to flat space. By linearising the equations, one gets something that can be analysed. I got my result in a paper I was rather proud of, A convergence result for linearised Regge calculus. From the abstract: “…solutions of the linearised Regge equations converge to analytic solutions of the linearised Einstein equations.”

However it needed a technical result that is sort of “obvious” but actually highly technical, that the calculation of the curvature from the metric has the same properties in the linearised Regge case as it does for linearised Einstein. I worked it all out, with heavy use of the simplicial approach to algebraic topology which I had learnt by giving supervisions (tutorials to students) on C.R.F Maunder’s course in Cambridge, following his book Algebraic Topology. The results are summarised very briefly in the paper The fundamental theorem of Regge calculus. I called it that because it seemed pretty fundamental, and probably at that age I didn’t care whether anyone else agreed or not (but obviously the editor of the journal didn’t object).

The actual proof is in the preprint posted above. I don’t know where I sent it, but it seems it didn’t make the CERN or the KEK preprint libraries (and the arXiv didn’t exist then, of course). I always meant to revise the presentation a little before publishing. But I never got around to it; there must have been something more important to do. And now I no longer have the tex file I can’t upload it to arXiv (can I?) People keep asking me for it, so here it is!