A fully (3+1)-D Regge calculus model of the Kasner cosmology

TL;DR

This paper presents the first fully 4D numerical Regge calculus model of Kasner cosmology, demonstrating stable, accurate evolution and convergence, with explicit handling of diffeomorphism freedom.

Contribution

It introduces a novel initial-data prescription and an implicit evolution scheme for 4D Regge calculus, applied to Kasner cosmology, with detailed benchmarking and convergence analysis.

Findings

Reproduces continuum Kasner solution with high accuracy

Demonstrates stable and convergent evolution over many steps

Shows preservation of spatial homogeneity and diffeomorphism freedom

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.