Ruth M. Williams

QUANTUM REGGE CALCULUS

Abstract We consider the quantization of Regges discrete description of gravity using functional methods. We show that in the weak field limit the standard continuum theory emerges.

Regge calculus: a brief review and bibliography

Presents a brief review of the theory and applications of Regge calculus in classical and quantum gravity, followed by a comprehensive bibliography which the author hopes will be of use to workers in the subject.

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.

Simplicial quantum gravity with higher derivative terms: Formalism and numerical results in four dimensions☆

Abstract Higher derivative terms for Regges formulation of lattice gravity are discussed. The analytic weak-field expansion for the regular tessellation α5 of the four-sphere is presented. Preliminary numerical results for some computations in four dimensions are also discussed.

Nonlocal effective gravitational field equations and the running of Newton's constant G

Non-perturbative studies of quantum gravity have recently suggested the possibility that the strength of gravitational interactions might slowly increase with distance. Here a set of generally covariant effective field equations are proposed, which are intended to incorporate the gravitational, vacuum-polarization induced, running of Newton’s constant G. One attractive feature of this approach is that, from an underlying quantum gravity perspective, the resulting long distance (or large time) effective gravitational action inherits only one adjustable parameter ξ, having the units of a length, arising from dimensional transmutation in the gravitational sector. Assuming the above scenario to be correct, some simple predictions for the long distance corrections to the classical standard model Robertson-Walker metric are worked out in detail, with the results formulated as much as possible in a model-independent framework. It is found that the theory, even in the limit of vanishing renormalized cosmological constant, generally predicts an accelerated power-law expansion at later times t ∼ ξ ∼ 1/H.

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.

ON THE MEASURE IN SIMPLICIAL GRAVITY

Functional measures for lattice quantum gravity should agree with their continuum counterparts in the weak field, low momentum limit. After showing that the standard simplicial measure satisfies the above requirement, we prove that a class of recently proposed non-local measures for lattice gravity do not satisfy such a criterion, already to lowest order in the weak field expansion. We argue therefore that the latter cannot represent acceptable discrete functional measures for simplicial geometries.

Recent progress in Regge calculus

Abstract While there has been some advanced 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.

Gluing 4 simplices: A Derivation of the Barrett-Crane spin foam model for Euclidean quantum gravity

We derive the the Barrett-Crane spin foam model for Euclidean 4dimensional quantum gravity from a discretized BF theory, imposing the constraints that reduce it to gravity at the quantum level. We obtain in this way a precise prescription of the form of the Barrett-Crane state sum, in the general case of an arbitrary manifold with boundary. In particular we derive the amplitude for the edges of the spin foam from a natural procedure of gluing different 4-simplices along a common tetrahedron. The generalization of our results to higher dimensions is also shown.

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 nonadjacent 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.