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:

The application of Regge calculus, a lattice formulation of general relativity, is reviewed in the context of numerical relativity. Particular emphasis is placed on problems of current computational interest, and the strengths and weaknesses of the lattice approach are highlighted. Several new and illustrative applications are presented, including initial data for the head on collision of two black holes, and the time evolution of vacuum axisymmetric Brill waves. I. NUMERICAL RELATIVITYThe complexity of the Einstein equations, combined with the sparsity of relevant analytic solutions, necessitates a range of other tools with which to explore complex physical scenarios. These include series expansions and perturbation techniques, together with numerical solutions of the fully non-linear equations. Unfortunately, the numerical solution of Einstein's equations has proved to be an exceedingly difficult problem. Over the last three decades numerous schemes have been developed, or adapted, to tackle the vast range of problems which fall within the purview of numerical relativity.The classic approach, involving a three-plus-one dimensional split of space and time, was first expounded by Arnowitt, Deser and Misner (ADM) [1]. This approach is natural in the context of our Newtonian intuition, and is also directly applicable to computer simulations. Whilst the traditional ADM approach has dominated numerical relativity, questions regarding its long-term stability have lead to the development of many other formulations of the Einstein equations in three-plus-one dimensions.Techniques of current interest include those developed by Sasaki and Nakamura, and later by Baumgarte and Shapiro, known generically as Conformal ADM (CADM) formulations. Incorporating insights from York's analysis of the initial value problem, these algorithms have shown superior stability properties compared with ADM in some applications. Other recent formulations are based on symmetric-hyperbolic forms of the Einstein equations, where it is hoped that the mathematical proofs of stability and well-posedness confer numerical advantages compared with techniques whose mathematical structures are more uncertain. The recent review by Lehner [2] discusses many of these issues.Despite the enormous effort invested in these techniques (and many others, including characteristic formulations), most modern numerical relativity codes continue to suffer from problems with long-term stability and lack of accuracy. Lack of resolution is only a partial solution; new insights or techniques seem to be required to overcome many of these problems.

Abstract. We construct initial data for a particular class of Brill wave metrics using Regge calculus, and compare the results to a corresponding continuum solution, finding excellent agreement. We then search for trapped surfaces in both sets of initial data, and provide an independent verification of the existence of an apparent horizon once a critical gravitational wave amplitude is passed. Our estimate of this critical value, using both the Regge and continuum solutions, supports other recent findings.

We reinvestigate the utility of time-independent constant mean curvature foliations for the numerical simulation of a single spherically-symmetric black hole. Each spacelike hypersurface of such a foliation is endowed with the same constant value of the trace of the extrinsic curvature tensor, K. Of the three families of K-constant surfaces possible (classified according to their asymptotic behaviors), we single out a sub-family of singularity-avoiding surfaces that may be particularly useful, and provide an analytic expression for the closest approach such surfaces make to the singularity. We then utilize a non-zero shift to yield families of K-constant surfaces which (1) avoid the black hole singularity, and thus the need to excise the singularity, (2) are asymptotically null, aiding in gravity wave extraction, (3) cover the physically relevant part of the spacetime, (4) are well behaved (regular) across the horizon, and (5) are static under evolution, and therefore have no "grid stretching/sucking" pathologies. Preliminary numerical runs demonstrate that we can stably evolve a single spherically-symmetric static black hole using this foliation. We wish to emphasize that this coordinatization produces K-constant surfaces for a single black hole spacetime that are regular, static and stable throughout their evolution. I. CONSTANT CRUNCH SURFACESIn this paper, we address a single question: Is there a numerically-viable coordinatization of a Schwarzschild black hole spacetime foliated by hypersurfaces of constant (not necessarily zero) mean extrinsic curvature? In other words, can we coordinatize the Schwarzschild spacetime with constant mean extrinsic curvature (T r(K) = constant) hypersurfaces so as to bound the growth of metric components and their gradients? We demonstrate here that the single shift freedom yields a spacetime metric that is static, and therefore bounds the growth in time of such gradients. A more complete analysis of the stability of our coordinatization, and a more thorough canvassing of the parameter space, will appear elsewhere.[1] Our foliation is consistent with that of Iriondo et al. [2], who provided a generic constant mean curvature (CMC) foliation of the Reissner-Nordström spacetime for the purpose of finding trapped surfaces. In this paper we focus on the utility of CMC slicings for the numerical simulation of black holes, in support of the emerging field of gravity-wave astrophysics.The trace of the extrinsic curvature tensor (T r(K) = K a a = K) at a point on a spacelike hypersurface measures the fractional rate of contraction of 3-volume along a unit normal to the surface. It represents the amount of "crunch" the 3-surface is experiencing at the point, at a given time. If all the observers throughout a spacelike hypersurface moving in time orthogonal to the surface experience the same amount of contraction per unit proper time, we say that the surface is a K-surface or a "constant crunch" surface. In this paper we examine foliations of a single sphericallysymmetric, static black h...

Abstract.Simplicial lattices provide an elegant framework for discrete spacetimes. The inherent orthogonality between a simplicial lattice and its circumcentric dual yields an austere representation of spacetime which provides a conceptually simple form of Einstein's geometric theory of gravitation. A sufficient understanding of simplicial spacetimes has been demonstrated in the literature for spacetimes devoid of all non-gravitational sources. However, this understanding has not been adequately extended to non-vacuum spacetime models. Consequently, a deep understanding of the diffeomorphic structure of the discrete theory is lacking. Conservation laws and symmetry properties are attractive starting points for coupling matter with the lattice. We present a simplicial form of the contracted Bianchi identity which is based on the E. Cartan moment of rotation operator. This identity manifest itself in the conceptuallysimple form of a Kirchhoff-like conservation law. This conservation law enables one to extend Regge Calculus to non-vacuum spacetimes and provides a deeper understanding of the simplicial diffeomorphism group.