A fully -dimensional Regge calculus model of the Kasner cosmology

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Abstract
We describe the first discrete-time four-dimensional numerical application of Regge calculus. The spacetime is represented as a complex of four-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 from tetrahedra. We implement a novel 2-surface initial-data prescription for Regge calculus, and provide the first fully four-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 (i) reproduce the continuum solution with a fractional error in the 3-volume of after evolution steps; (ii) demonstrate stable evolution; (iii) preserve the standard deviation of spatial homogeneity to less than and (iv) explicitly display the existence of diffeomorphism freedom in Regge calculus. We also present the second-order convergence properties of the solution to the continuum.

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