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Ruth M. Williams

Publications and source records attributed to Ruth M. Williams.

32 records · Page 2Linked to original sources
arXiv2000

Constraints on Area Variables in Regge Calculus

We describe a general method of obtaining the constraints between area variables in one approach to area Regge calculus, and illustrate it with a simple example. The simplicial complex is the simplest tessellation of the 4-sphere. The number of independent constraints on the variations of the triangle areas is shown to equal the difference between the numbers of triangles and edges, and a general method of choosing independent constraints is described. The constraints chosen by using our method are shown to imply the Regge equations of motion in our example.

gr-qc↗
arXiv2000

Wormholes in Simplicial Minisuperspace

Using simplicial minisuperspace techniques we obtain a wormhole configuration, capable of modelling both a throat between two equal Euclidean and Lorentzian universes. Computing the wavefunction associated to such configuration we conclude that for Planck-size Euclidean wormholes there is a prediction of a finite non-vanishing radius of their throat, even in the absence of matter. The wavefunction associated to their Lorentzian counterparts predicts the growth of the wormhole's throat as the two Lorentzian universes expand.

gr-qc↗
arXiv1999

Anisotropic simplicial minisuperspace model

The computation of the simplicial minisuperspace wavefunction in the case of anisotropic universes with a scalar matter field predicts the existence of a large classical Lorentzian universe like our own at late times

gr-qc↗
arXiv1999

The asymptotics of an amplitude for the 4-simplex

An expression for the oscillatory part of an asymptotic formula for the relativistic spin network amplitude for a 4-simplex is given. The amplitude depends on specified areas for each two-dimensional face in the 4-simplex. The asymptotic formula has a contribution from each flat Euclidean metric on the 4-simplex which agrees with the given areas. The oscillatory part of each contribution is determined by the Regge calculus Einstein action for that geometry.

gr-qc↗
arXiv1999

A note on area variables in Regge calculus

We consider the possibility of setting up a new version of Regge calculus in four dimensions with areas of triangles as the basic variables rather than the edge-lengths. The difficulties and restrictions of this approach are discussed.

gr-qc↗
arXiv1998

't Hooft's Polygon Approach Hyperbolically Revisited

The initial data in the polygon approach to (2+1)D gravity coupled to point particles are constrained by the vertex equations and the particle equations. We establish the hyperbolic nature of the vertex equations and derive some consequences. In particular we are able to identify the hyperbolic group of motions as discrete analogues of the diffeomorphisms in the continuum theory. We show that particles can be included ``hyperbolically'' as well, but they spoil the gauge invariance. Finally we derive consistent sets of initial data.

gr-qc↗
arXiv1998

Lens Spaces and Handlebodies in 3D Quantum Gravity

We calculate partition functions for lens spaces L_{p,q} up to p=8 and for genus 1 and 2 handlebodies H_1, H_2 in the Turaev-Viro framework. These can be interpreted as transition amplitudes in 3D quantum gravity. In the case of lens spaces L_{p,q} these are vacuum-to-vacuum amplitudes $Ø-> Ø$, whereas for the 1- and 2-handlebodies H_1, H_2 they represent genuinely topological transition amplitudes $Ø-> T^2$ and $Ø-> T^2 # T^2$, respectively.

gr-qc↗
arXiv1997

Signature of the Simplicial Supermetric

We investigate the signature of the Lund-Regge metric on spaces of simplicial three-geometries which are important in some formulations of quantum gravity. Tetrahedra can be joined together to make a three-dimensional piecewise linear manifold. A metric on this manifold is specified by assigning a flat metric to the interior of the tetrahedra and values to their squared edge-lengths. The subset of the space of squared edge-lengths obeying triangle and analogous inequalities is simplicial configuration space. We derive the Lund-Regge metric on simplicial configuration space and show how it provides the shortest distance between simplicial three-geometries among all choices of gauge inside the simplices for defining this metric (Regge gauge freedom). We show analytically that there is always at least one physical timelike direction in simplicial configuration space and provide a lower bound on the number of spacelike directions. We show that in the neighborhood of points in this space corresponding to flat metrics there are spacelike directions corresponding to gauge freedom in assigning the edge-lengths. We evaluate the signature numerically for the simplicial configuration spaces based on some simple triangulations of the three-sphere (S^3) and three-torus (T^3). For the surface of a four-simplex triangulation of S^3 we find one timelike direction and all the rest spacelike over all of the simplicial configuration space. For the triangulation of T^3 around flat space we find degeneracies in the simplicial supermetric as well as a few gauge modes corresponding to a positive eigenvalue. Moreover, we have determined that some of the negative eigenvalues are physical, i.e. the corresponding eigenvectors are not generators of diffeomorphisms. We compare our results with the known properties of continuum superspace.

gr-qc↗
arXiv1997

Recent Progress in Regge Calculus

While there has been some advance in the use of Regge calculus as a tool in numerical relativity, the main progress in Regge calculus recently has been in quantum gravity. After a brief discussion of this progress, attention is focussed on two particular, related aspects. Firstly, the possible definitions of diffeomorphisms or gauge transformations in Regge calculus are examined and examples are given. Secondly, an investigation of the signature of the simplicial supermetric is described. This is the Lund-Regge metric on simplicial configuration space and defines the distance between simplicial three-geometries. Information on its signature can be used to extend the rather limited results on the signature of the supermetric in the continuum case. This information is obtained by a combination of analytic and numerical techniques. For the three-sphere and the three-torus, the numerical results agree with the analytic ones and show the existence of degeneracy and signature change. Some ``vertical'' directions in simplicial configuration space, corresponding to simplicial metrics related by gauge transformations, are found for the three-torus.

gr-qc↗
arXiv1994

Discrete Gravity in One Dimension

A model for quantum gravity in one (time) dimension is discussed, based on Regge's discrete formulation of gravity. The nature of exact continuous lattice diffeomorphisms and the implications for a regularized gravitational measure are examined. After introducing a massless scalar field coupled to the edge lengths, the scalar functional integral is performed exactly on a finite lattice, and the ensuing change in the measure is determined. It is found that the renormalization of the cosmological constant due to the scalar field fluctuations vanishes identically in one dimension. A simple decimation renormalization group transformation is performed on the partition function and the results are compared with the exact solution. Finally the properties of the spectrum of the scalar Laplacian are compared with results obtained for a Poissonian distribution of edge lengths.

hep-th↗
arXiv1994

A Parallelizable Implicit Evolution Scheme for Regge Calculus

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.

gr-qc↗
arXiv1994

Newtonian Potential in Quantum Regge Gravity

We show how the Newtonian potential between two heavy masses can be computed in simplicial quantum gravity. On the lattice we compute correlations between Wilson lines associated with the heavy particles and which are closed by the lattice periodicity. We check that the continuum analog of this quantity reproduces the Newtonian potential in the weak field expansion. In the smooth anti-de Sitter-like phase, which is the only phase where a sensible lattice continuum limit can be constructed in this model, we attempt to determine the shape and mass dependence of the attractive potential close to the critical point in $G$. It is found that non-linear graviton interactions give rise to a potential which is Yukawa-like, with a mass parameter that decreases towards the critical point where the average curvature vanishes. In the vicinity of the critical point we give an estimate for the effective Newton constant.

hep-th↗
arXiv1993

Simplicial Gravity Coupled to Scalar Matter

A model for quantized gravity coupled to matter in the form of a single scalar field is investigated in four dimensions. For the metric degrees of freedom we employ Regge's simplicial discretization, with the scalar fields defined at the vertices of the four-simplices. We examine how the continuous phase transition found earlier, separating the smooth from the rough phase of quantized gravity, is influenced by the presence of scalar matter. A determination of the critical exponents seems to indicate that the effects of matter are rather small, unless the number of scalar flavors is large. Close to the critical point where the average curvature approaches zero, the coupling of matter to gravity is found to be weak. The nature of the phase diagram and the values for the critical exponents suggest that gravitational interactions increase with distance. \vspace{24pt} \vfill

hep-th↗