Dynamics of the Friedmann Universe Using Regge Calculus

Collins, P.; Williams, Ruth M.

doi:10.1103/physrevd.7.965

Cited by 62 publications

(62 citation statements)

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“…All previous initial data constructed in Regge calculus has been for the special case of a moment of time symmetry, where the problem reduces to the requirement that the scalar curvature of the initial three geometry is zero [8]. Successful time-symmetric Regge calculations include initial data for single and multiple black holes [17,18], Friedmann-Robertson-Walker and Taub cosmologies [19,20,21], and Brill waves on both flat [22] and black hole [23] backgrounds.…”

Section: A York-type Initial-value Prescription For Regge Calculusmentioning

confidence: 99%

A fully -dimensional Regge calculus model of the Kasner cosmology

Gentle

Miller

1998

Class. Quantum Grav.

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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:

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Section: A York-type Initial-value Prescription For Regge Calculusmentioning

confidence: 99%

A fully -dimensional Regge calculus model of the Kasner cosmology

Gentle

Miller

1998

Class. Quantum Grav.

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“…The shape of a 4-prism is not uniquely determined by its edge lengths, so it is necessary to give further information to eliminate the floppiness. In the study of the Friedmann universe by Collins and Williams,11 there is sufficient symmetry to determine the shapes of the prisms. In the work of Porter 13,14 and later of Dubal, 28 angles in the faces of the prisms are introduced as extra variables related to the extrinsic curvature.…”

Section: Portermentioning

confidence: 99%

“…2 a 3 = 600 l 11) and this is compared with the scale factor of the Friedmann universe as a function of proper time. (The other variable one might want to plot would be the diagonal length d, but in the present case d is virtually identical in magnitude to l, due to the small value chosen for the lapse parameter v.) Note that the proper time elapsed between two consecutive simplicial hypersurfaces may be defined in different ways.…”

Section: πmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Numerical applications in 3+1 dimensions have been mainly to problems with symmetry and no general code has been developed. [11][12][13][14][15]37 This is also the case in the alternative approach known as null-strut calculus, which builds a spacelike-foliated spacetime with the maximal number of null edges.16−20 Although null-strut calculus was used first to demonstrate numerically the approximate diffeomorphism freedom in Regge calculus, 18 and, except for the work described in ref. 30, was the first fully 3+1 dimensional numerical scheme implemented 21 , we have not found a way to adapt it to the evolution scheme described in this paper.…”

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Parallelizable implicit evolution scheme for Regge calculus

et al. 1997

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The role of Regge calculus as a tool for numerical relativity is discussed, and a parallelizable implicit evolution scheme described. Because of the structure of the Regge equations, it is possible to advance the vertices of a triangulated spacelike hypersurface in isolation, solving at each vertex a purely local system of implicit equations for the new edge-lengths involved. (In particular, equations of global "elliptic-type" do not arise.) Consequently, there exists a parallel evolution scheme which divides the vertices into families of non-adjacent elements and advances all the vertices of a family simultaneously. The relation between the structure of the equations of motion and the Bianchi identities is also considered. The method is illustrated by a preliminary application to a 600-cell Friedmann cosmology. The parallelizable evolution algorithm described in this paper should enable Regge calculus to be a viable discretization technique in numerical relativity. Much current activity in numerical relativity is centered around making predictions which can be tested by the proposed Laser Interferometry Gravitational Observatory (LIGO).1,2 There is a need to solve Einstein's equations numerically for many physical situations which could give rise to gravitational waves, so that data from LIGO can be interpreted, and used, if appropriate, as evidence for the existence of black holes. More generally, numerical solutions of Einstein's equations are invaluable for the understanding of astrophysical data, and for guidance as to what experiments to undertake.Methods of solving Einstein's equations numerically include finite difference schemes and finite element schemes. Regge calculus is a type of finite element method, and in this paper we shall describe a way of casting it into the form of a highly efficient tool of numerical relativity.The basic idea of Regge calculus is the division of spacetime into simplicial cells with flat interior geometry.3 The dynamical variables are the edge-lengths of the simplices, and the curvature, which is restricted to the "hinge simplices" of codimension two, can be expressed in terms of the defect angles at these hinges, where the flat cells meet. Variation of the action leads to the simplicial form of Einstein's equations. The convergence of the Regge action and equations to the corresponding continuum quantities has been investigated thoroughly and has been shown to be satisfactory under quite general conditions. [4][5][6][7][8][9]22,30 Regge calculus has been applied to a large variety of problems in classical and quantum gravity (see Ref. 10 for a review). Numerical applications in 3+1 dimensions have been mainly to problems with symmetry and no general code has been developed. [11][12][13][14][15]37 This is also the case in the alternative approach known as null-strut calculus, which builds a spacelike-foliated spacetime with the maximal number of null edges.16−20 Although null-strut calculus was used first to demonstrate numerically the approximate diffeomorphism fr...

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Computational Relativity: Numerical and Algebraic Approaches Report of Workshop A4

Smarr

1984

General Relativity and Gravitation

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Dynamics of the Friedmann Universe Using Regge Calculus

Cited by 62 publications

References 7 publications

A fully -dimensional Regge calculus model of the Kasner cosmology

A fully -dimensional Regge calculus model of the Kasner cosmology

Parallelizable implicit evolution scheme for Regge calculus

Computational Relativity: Numerical and Algebraic Approaches Report of Workshop A4

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