An ADM 3+1 formulation for Smooth Lattice General Relativity

A new hybrid scheme for numerical relativity will be presented. The scheme will employ a 3-dimensional spacelike lattice to record the 3-metric while using the standard 3+1 ADM equations to evolve the lattice. Each time step will involve three basic steps. First, the coordinate quantities such as the Riemann and extrinsic curvatures are extracted from the lattice. Second, the 3+1 ADM equations are used to evolve the coordinate data, and finally, the coordinate data is used to update the scalar data on the lattice (such as the leg lengths). The scheme will be presented only for the case of vacuum spacetime though there is no reason why it could not be extended to non-vacuum spacetimes. The scheme allows any choice for the lapse function and shift vectors. An example for the Kasner $T^3$ cosmology will be presented and it will be shown that the method has, for this simple example, zero discretisation error.Comment: 18 pages, plain TeX, 5 epsf figues, gzipped ps file also available at http://newton.maths.monash.edu.au:8000/preprints/3+1-slgr.ps.g

Is the Regge Calculus a consistent approximation to General Relativity?

We will ask the question of whether or not the Regge calculus (and two related simplicial formulations) is a consistent approximation to General Relativity. Our criteria will be based on the behaviour of residual errors in the discrete equations when evaluated on solutions of the Einstein equations. We will show that for generic simplicial lattices the residual errors can not be used to distinguish metrics which are solutions of Einstein's equations from those that are not. We will conclude that either the Regge calculus is an inconsistent approximation to General Relativity or that it is incorrect to use residual errors in the discrete equations as a criteria to judge the discrete equations.Comment: 27 pages, plain TeX, very belated update to match journal articl

Long term stable integration of a maximally sliced Schwarzschild black hole using a smooth lattice method

We will present results of a numerical integration of a maximally sliced Schwarzschild black hole using a smooth lattice method. The results show no signs of any instability forming during the evolutions to t=1000m. The principle features of our method are i) the use of a lattice to record the geometry, ii) the use of local Riemann normal coordinates to apply the 1+1 ADM equations to the lattice and iii) the use of the Bianchi identities to assist in the computation of the curvatures. No other special techniques are used. The evolution is unconstrained and the ADM equations are used in their standard form.Comment: 47 pages including 26 figures, plain TeX, also available at http://www.maths.monash.edu.au/~leo/preprint

A Smooth Lattice construction of the Oppenheimer-Snyder spacetime

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime. The results are in excellent agreement with theory and numerical results from other authors.Comment: 60 pages, 28 figure

On the convergence of Regge calculus to general relativity

Motivated by a recent study casting doubt on the correspondence between Regge calculus and general relativity in the continuum limit, we explore a mechanism by which the simplicial solutions can converge whilst the residual of the Regge equations evaluated on the continuum solutions does not. By directly constructing simplicial solutions for the Kasner cosmology we show that the oscillatory behaviour of the discrepancy between the Einstein and Regge solutions reconciles the apparent conflict between the results of Brewin and those of previous studies. We conclude that solutions of Regge calculus are, in general, expected to be second order accurate approximations to the corresponding continuum solutions.Comment: Updated to match published version. Details of numerical calculations added, several sections rewritten. 9 pages, 4 EPS figure