Adrian P. Gentle

Regge calculus: A Unique tool for numerical relativity

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.

A fully -dimensional Regge calculus model of the Kasner cosmology

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. 1. Regge calculus as an independent tool in general relativity In 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: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia. Email: adrian@newton.maths.monash.edu.au

A Kirchhoff-like conservation law in Regge calculus

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 Einsteins 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 manifests itself in the conceptually simple 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.

On the convergence of Regge calculus to general relativity

Motivated by a recent study which cast 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.

Apparent horizons in simplicial Brill wave initial data

We use Regge calculus to construct initial data for a particular class of Brill wave metrics and compare the results with 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 recent findings.

The constraints as evolution equations for numerical relativity

The Einstein equations have proved surprisingly difficult to solve numerically. A standard diagnostic of the problems which plague the field is the failure of computational schemes to satisfy the constraints, which are known to be mathematically conserved by the evolution equations. We describe a new approach to rewriting the constraints as first-order evolution equations, thereby guaranteeing that they are satisfied to a chosen accuracy by any discretization scheme. This introduces a set of four subsidiary constraints which are far simpler than the standard constraint equations and which should be more easily conserved in computational applications. We explore the manner in which the momentum constraints are already incorporated in several existing formulations of the Einstein equations, and demonstrate the ease with which our new constraint-conserving approach can be incorporated into these schemes.

The BSSN formulation is a partially constrained evolution system

Relativistic simulations in 3+1 dimensions typically monitor the Hamiltonian and momentum constraints during evolution, with significant violations of these constraints indicating the presence of instabilities. In this paper we rewrite the momentum constraints as first-order evolution equations, and show that the popular BSSN formulation of the Einstein equations explicitly uses the momentum constraints as evolution equations. We conjecture that this feature is a key reason for the relative success of the BSSN formulation in numerical relativity.

Geodesic deviation in Regge calculus

Geodesic deviation is the most basic manifestation of the influence of gravitational fields on matter. We investigate geodesic deviation within the framework of Regge calculus, and compare the results with the continuous formulation of general relativity on two different levels. We show that the continuum and simplicial descriptions coincide when the cumulative effect of the Regge contributions over an infinitesimal element of area is considered. This comparison provides a quantitative relation between the curvature of the continuous description and the deficit angles of Regge calculus. The results presented might also be of help in developing generic ways of including matter terms in the Regge equations.

CONSTRAINTS IN QUANTUM GEOMETRODYNAMICS

We compare different treatments of the constraints in canonical quantum gravity. The standard approach on the superspace of 3-geometries treats the constraints as the sole carriers of the dynamic content of the theory, thus rendering the traditional dynamical equations obsolete. Quantization of the constraints in both the Dirac and ADM square root Hamiltonian approaches leads to the well known problems of time evolution. These problems of time are of both an interpretational and technical nature. In contrast, the geometrodynamic quantization procedure on the superspace of the true dynamical variables separates the issues of quantization from the enforcement of the constraints. The resulting theory takes into account states that are off-shell with respect to the constraints, and thus avoids the problems of time. We develop, for the first time, the geometrodynamic quantization formalism in a general setting and show that it retains all essential features previously illustrated in the context of homogeneous cosmologies.