A fully -dimensional Regge calculus model of the Kasner cosmology

Abstract

Abstract. We describe the first discrete-time 4-dimensional numerical application of Regge calculus. The spacetime is represented as a complex of 4-dimensional simplices, and the geometry interior to each 4-simplex is flat Minkowski spacetime. This simplicial spacetime is constructed so as to be foliated with a one parameter family of spacelike hypersurfaces built of tetrahedra. We implement a novel two-surface initial-data prescription for Regge calculus, and provide the first fully 4-dimensional application of an implicit decoupled evolution scheme (the "Sorkin evolution scheme"). We benchmark this code on the Kasner cosmology -a cosmology which embodies generic features of the collapse of many cosmological models. We (1) reproduce the continuum solution with a fractional error in the 3-volume of 10 −5 after 10000 evolution steps, (2) demonstrate stable evolution, (3) preserve the standard deviation of spatial homogeneity to less than 10 −10 and (4) explicitly display the existence of diffeomorphism freedom in Regge calculus. We also present the second-order convergence properties of the solution to the continuum.PACS numbers: 04.20.-q, 04.25.Dm, 04.60.Nc. Regge calculus as an independent tool in general relativityIn this paper we describe the first fully (3 + 1)-dimensional application of Regge calculus [1, 2] to general relativity. We develop an initial-value prescription based on the standard York formalism, and implement a 4-stage parallel evolution algorithm. We benchmark these on the Kasner cosmological model.We present three findings. First, that the Regge solution exhibits second-order convergence of the physical variables to the continuum Kasner solution. Secondly, ‡ Permanent address:

On the convergence of Regge calculus to general relativity

“…These oscillations do not appear to induce instabilities in the numerical evolution of simplicial lattices. Indeed, previous studies have indicated that the oscillations gradually decay as the evolution proceeds [2]. However, M. Miller [8] has reported instabilities arising in linearised Regge calculus on asymmetrical grids.…”

“…In this section we solve the Regge equations for the vacuum Kasner cosmology using a (3 + 1)-dimensional formulation of Regge calculus described elsewhere [2]. The initial value problem is solved and the lattice is evolved subject to simplicial lapse and shift conditions.…”

“…Although the lattice approach appears well suited to numerical applications, progress in the field has been slow. Only recently have the first completely generic four-dimensional numerical simulations been performed [2,3].…”

“…The situation became even more confused after the work of Gentle and W. Miller [2], who demonstrated explicit quadratic convergence of particular solutions of Regge calculus to a solution, the Kasner spacetime, of general relativity. This spacetime was one of the test cases used by Brewin and M. Miller.…”

Deriving the ADM3+1evolution equations from the second variation of arclength

“…Attempts have been made to adapt the ADM 3+1 equations to the Regge calculus [11,12] but progress has been slow. A much more promising scheme, for the Regge calculus, is due to Sorkin [13] with later development by Barrett et al [14] and Gentle and Miller [15].…”