Regge Calculus Research Articles

Phases of four-dimensional simplicial quantum gravity.

The phase diagram and critical exponents for pure simplicial quantum gravity (Regge calculus) in four dimensions are discussed. In the small-$G$ phase, where $G$ is the bare Newton's constant, the simplices are collapsed and no continuum limit exists. In the large-$G$ phase the ground state appears to be well behaved, and the curvature goes to zero continuously as the critical value of $G$ is approached. Fluctuations in the curvature diverge at the critical point, while volume fluctuations remain finite. The critical exponents at the transition are estimated, and appear to be independent of the strength of the higher-derivative coupling $a$. With the lattice analogue of the DeWitt gravitational measure and for large enough $G$, the lattice higher-derivative theories ($a>0$) and the reflection-positive pure Regge theory ($a=0$) appear to belong to the same phase for large enough $G$, which would suggest a common, unitary quantum continuum limit.

Phases of simplicial quantum gravity

Recent results for simplicial quantum gravity are reviewed. Models are considered in which the topology is fixed, and the edge lengths are varied while the coordination number is held fixed (‘Regge calculus’). As expected on the basis of universality, in two dimensions the results for pure gravity critical exponents are found to be in agreement with the conformal field theory predictions. The effects of both higher derivative terms and gravitational measure contributions are investigated in detail, as well as the inclusion of a D-component scalar field. For sufficiently large D a phase transition is found, leading from the Liouville phase into a branched polymer phase. The results suggest universal critical behavior, and are in good agreement with recent results obtained with the dynamical triangulation models. While no phase transition is found for pure gravity in two dimensions, a phase transition between a ‘rough’ and a ‘smooth’ phase of spacetime is found in both three and four dimensions, in agreement with the results of the 2+ c-expansion in the continuum. In both cases the transition appears to be continuous (suggesting therefore the existence of a lattice continuum limit), and the critical exponents can be estimated. While fluctuations in the local volumes are responsible for critical behavior in two dimensions, in three and four dimensions it is instead the fluctuations in the local curvature that diverge at the critical point.

Quantum gravity and random surfaces

Abstract We review recent progress in lattice quantum gravity and random surfaces with a particular emphasis given to discussion of two dimensional gravity, dynamical triangulation, matrix models and Regge calculus. We also discuss the possibility of constructing the theory of quantum gravity as an ordinary field theory in four dimensions.

Exploration of simplicial quantum gravity in four dimensions

Euclidean quantum gravity in four dimensions is investigated within the Regge calculus by computer simulations. The action is bounded without dimensional terms by fixing the average lattice spacing. We set the scale by a parateter and consider two different pressures. In the low β region we observe a phase with negative curvature and a homogenous distribution of the link lengths independence of the model. The large β region is characterized by inhomogenous link lengths distribution with spikes and positive curvature depending on the measure.

Regge calculus in anisotropic quantum cosmology

The authors investigate the no-boundary path integral prescription in a Regge calculus model in three spacetime dimensions. The simplicial manifold is topologically the 'doughnut' D*S1, where D is the two-dimensional closed disc. The discretization and the edge lengths are chosen symmetrically so that the boundary 2-metric is completely described by two independent edge lengths, and the interior 3-metric is completely described by these plus two further independent edge lengths. They pass to a limit in which the discretization on the boundary becomes infinitely dense. The classical solutions in the resulting semi-continuum model are in qualitative agreement with the solutions to the continuum Einstein equations with the same boundary data. Convergent contours for the no-boundary integral in the semi-continuum model are sought in analogy with the conformal rotation prescription of Gibbons et al (1978) and using steepest-descent techniques to analyse the contours for the conformal factor. A convergent prescription is found in the case of a vanishing cosmological constant, and the resulting wavefunction is shown to have the expected semiclassical behaviour. Possible reasons for the authors not being able to find a similar prescription with a non-vanishing cosmological constant are discussed.

The gravitational interaction of conical strings

Up to now calculations of the interaction of cosmic strings have neglected gravity. We consider the purely gravitational interactions that occur at large distances, using the conical line singularity for the gravitational field of a string. We construct spaces with multiple intersecting conical strings, that are exactly consistent with General Relativity, and which can be covered in a single Minkowski coordinate patch, using a Regge calculus type construction. We show that after two such strings pass through each other they remain connected by another string, and we derive the branching rules which govern the junction of three strings. These rules apply to conical type strings in any smoothly curved background, whether they are straight or curved, moving or stationary, and they show that, at the junction, the three strings must be as coplanar as is possible in such a space. For these results to be matched onto the short range results of Field Theory calculations, it is suggested that gravitational radiation must be introduced. This would mean that gravitation is not negligible in these interactions.

Continuous time Regge gravity in the tetrad-connection variables

The continuous time limit of the tetrad-connection formulation of Regge calculus is found. The shift and lapse functions take on their values loosely. In the continuous space limit the action obtained is reduced to the continuous general relativity action.

Lattice gauge gravity.

We propose a lattice gravity action based on the gauge-theory point of view. We avoid a problem, the action potentially going to infinity by taking the link length to a constant $a$ (the lattice constant). By fixing the local Lorentz gauge freedom, we obtain a gauge-fixed version of the $d$-dimensional action as a sum of ${a}^{d\ensuremath{-}2}sin\ensuremath{\delta}$ ($\ensuremath{\delta}=\mathrm{deficit}\mathrm{angle}$) which is a compactified version of Regge calculus. The fermion gravity coupling term can be constructed in a similar way. We point out the possibility that this lattice gravity action approaches the Einstein action in the long-range limit if the action has an ultraviolet fixed point.

3-dimensional Gravity and Regge Calculus

3-dimensional Gravity and Regge Calculus

Conserved constraints in three-plus-one Regge calculus

The problem of formulating a consistent set of constraints for discrete versions of general relativity is discussed. The application of the Dirac formalism to a slightly modified form of the 3+1 version of Regge calculus developed by Piran and Williams (1986) is described, and illustrated by consideration of a simple model universe.

Null-strut calculus. II. Dynamics.

In this paper, we continue from the preceding paper to develop a fully functional Regge calculus geometrodynamic algorithm from the null-strut-calculus construction. The developments discussed include (a) the identification of the Regge calculus analogue of the constraint and evolution equations on the null-strut lattice, (b) a description of the Minkowski solid geometry for the simplicial blocks of the null-strut lattice, (c) a description of the evolution algorithm for the geometrodynamic scheme and an analysis of its consistency, and (d) a presentation of the dynamical degrees of freedom for a simplicial hypersurface and the description of an initial-value prescription. To demonstrate qualitatively this new approach to geometrodynamics, we present the most simple application of null-strut calculus that we know of---the Friedmann cosmology using the three-boundary of a 600-cell simplicial polytope to model the simplicial hypersurface.

Null-strut calculus. I. Kinematics.

This paper describes the kinematics of null-strut calculus---a 3+1 Regge calculus approach to general relativity. We show how to model the geometry of spacetime with simplicial spacelike three-geometries (TET's) linked to "earlier" and "later" momentumlike lattice surfaces (TET*) entirely by light rays or "null struts." These three-layered lattice spacetime geometries are defined and analyzed using combinatorial formulas for the structure of polytopes. The following paper in this series describes how these three-layered spacetime lattices are used to model spacetimes in full conformity with Einstein's theory of gravity.

Relativistic collapse using Regge calculus. II. Spherical collapse results

In a previous paper (see ibid., vol.6, p.1925 (1989)) the Regge-Einstein equations for the collapse of polytropic spheres were derived. The solutions of these equations are considered using a computer code. Firstly initial data for three representative stellar models are constructed, these being equilibrium configurations which are stable, marginally stable and unstable to radial perturbations. Then various time evolutions of the model stars and pressure reduced versions of them are presented, some of these evolutions involving core bounce or collapse to a black hole. As a check on the reliability of the Regge calculus solutions the author compares them with solutions of the equivalent continuum Einstein equations evolving identical initial data. It is found that, in general, the Regge calculus and continuum codes produce solutions in good quantitative agreement, except in the case of a bounce, when the solutions begin to diverge after the bounce has occurred. The results described, however, show that 3+1 Regge calculus can accurately model dynamical spacetimes containing a perfect fluid.

Relativistic collapse using Regge calculus. I. Spherical collapse equations

Following work by Porter (1987), Regge calculus is used to simulate the dynamical collapse of model stars. In this paper the author describes the general methodology of including a perfect fluid in dynamical Regge calculus spacetimes. The Regge-Einstein equations for spherical collapse are obtained and are then specialised to mimic a particular continuum gauge. The equivalent continuum problem is also set up. This is to be solved using standard numerical techniques (i.e. the method of finite difference).

Regge calculus as a local theory of the Poincaré group

We reformulate Regge calculus in terms of dynamical variables belonging to the Poincaré group. Our formulation uses the dual lattice, is naturally related in the continuum limit to the Einstein action written in terms of differential forms, and retains and possibility of choosing between the first order and the second order formalism.

Nonrigid motion and Regge calculus

We study the problem of recovering the structure from the motion of figures that are permitted to perform a controlled nonrigid motion. We use Regge calculus to approximate a general surface by a net of triangles. The nonrigid flexing motion that we examine corresponds to the triangles remaining rigid and to bending occurring only at the joints between the triangles. We show that depth information of the vertices of the triangles can be obtained by using a modified version of the incremental rigidity scheme. In cases in which the motion of the figure displays fundamentally different views at each frame presentation, the algorithm works well, not only for strictly rigid motion but also for a limited amount of bending deformation. We modify this scheme to allow for flexing motion,

On the continuum limit of curvature squared actions in the Regge calculus

We evaluate the continuum limit of a family of curvature squared actions for the Regge calculus proposed by Hamber and Williams. The answers depend on how the continuum limit is defined. When the link lengths are defined as the distance in an embedding space between the endpoints of the link, we find that no member of this family approaches the continuum limit correctly. Defining the link lenghts as the length of a geodesic between the endpoints of the link, we find that a unique member is selected, and we prove for the general two dimensional compact manifold that this Regge calculus action converges to ∫ R 2√ gd 2 x.

Equivalence of the Regge and Einstein equations for almost flat simplicial space-times

The theory of distributions is applied to almost flat simplicial space-times. Explicit expressions are given for the first-order defects. It is shown explicitly that the Riemann tensor for an almost flat simplicial space-time contains delta-functions on the bones and derivatives of delta-functions on the 3-dimensional faces of the boundary of the space-time. The latter terms have not previously been seen in the Regge calculus. It is shown that the Regge and Hilbert actions have equal values on almost fiat simplicial space-times and that the Einstein equations lead directly to the Regge field equations.

Initial data for time-symmetric gravitational radiation using Regge calculus

The author applies Regge calculus to the construction of initial data for Brill waves: axisymmetric nonrotating vacuum solutions of Einstein's equation. The Regge calculus solutions are compared with those of the continuum theory, with encouraging results.

Independent variables in 3+1 Regge calculus

The space of metrics in (discrete time) 3+1 Regge calculus is discussed, and the problems of counting its dimensions, and of finding independent variables to parametrise the space, are addressed. The most general natural class of metrics is considered first, and bounds on its dimension are obtained, although no good parametrisations are found. The relationship between these metrics and those used in canonical Regge calculus is shown, and this leads to an interesting result via the Bianchi identities. A restricted class of metrics is then considered and independent variables, which parametrise these metrics and which may be computationally convenient, are given. The dimension of this space of metrics gives an improved lower bound for the dimension of the general space.