Regge calculus as a fourth-order method in numerical relativity

Abstract

The convergence properties of numerical Regge calculus as an approximation to continuum vacuum General Relativity is studied, both analytically and numerically. The Regge equations are evaluated on continuum spacetimes by assigning squared geodesic distances in the continuum manifold to the squared edge lengths in the simplicial manifold. It is found analytically that, individually, the Regge equations converge to zero as the second power of the lattice spacing, but that an average over local Regge equations converges to zero as (at the very least) the third power of the lattice spacing. Numerical studies using analytic solutions to the Einstein equations show that these averages actually converge to zero as the fourth power of the lattice spacing.

A fully -dimensional Regge calculus model of the Kasner cosmology

“…Our numerical studies, together with analytic results [4] should put to rest some of the recent concerns about Regge calculus [5,6,7] -it does appear to be a viable approximation to general relativity.…”

On the convergence of Regge calculus to general relativity

“…Brewin [4] and M. Miller [5] have both developed their own schemes for averaging the Regge equations. A direct assault on the convergence properties of Regge calculus is therefore likely to be applicable only on a specific lattice, with a particular choice of averaging in the continuum limit.…”

“…It is this residual, using an appropriately chosen norm, which both Brewin and M. Miller considered [4,5]. Brewin observed that the residual of the simplicial equations remained roughly constant as the lattice was refined on a fixed region of spacetime.…”

“…After the recent papers by Brewin [4] and M. Miller [5], and despite the proven track record of the Regge calculus, a lively debate arose [6] as to whether or not solutions of the Regge equations would converge to solutions of the Einstein equations in some suitable limit. Neither Brewin or M. Miller directly computed solutions of the Regge equations.…”

End of a dark age?

“…In practice we imagine modified Regge calculus graphs with greatly simplifying assumptions would be used as approximations of the exact Regge calculus graph (also true of standard Regge calculus, obviously). When all link lengths are small, i.e., in the absence of disordered locality, these approximate Regge calculus solutions would then correspond to GR solutions and we have standard Regge calculus [48][49][50]. As with GR solutions, Regge calculus solutions are nontrivial and there is no reason to believe that finding extrema of a Regge graphical action modified per disordered locality would be any easier.…”