Table of contents

We find all three-dimensional Einstein-Weyl spaces with the vanishing scalar curvature.

The properties of generalized p-forms, first introduced by Sparling, are discussed and developed. Generalized Cartan structure equations for generalized affine connections are introduced. A new representation of Einstein's equations, using generalized forms, is given.

In this paper we apply the methods outlined in the previous paper of this series to the particular set of states obtained by choosing the complexifier to be a Laplace operator for each edge of a graph. The corresponding coherent state transform was introduced by Hall for one edge and generalized by Ashtekar, Lewandowski, Marolf, Mourão and Thiemann to arbitrary, finite, piecewise-analytic graphs.

However, both of these works were incomplete with respect to the following two issues.

The focus was on the unitarity of the transform and left the properties of the corresponding coherent states themselves untouched.

While these states depend in some sense on complexified connections, it remained unclear what the complexification was in terms of the coordinates of the underlying real phase space.

In this paper we complement these results: first, we explicitly derive the complexification of the configuration space underlying these heat kernel coherent states and, secondly, prove that this family of states satisfies all the usual properties.

(i) Peakedness in the configuration, momentum and phase space (or Bargmann-Segal) representation.

(ii) Saturation of the unquenched Heisenberg uncertainty bound.

(iii) (Over)completeness.

These states therefore comprise a candidate family for the semiclassical analysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity. They also enable error-controlled approximations to difficult analytical calculations and therefore set a new starting point for numerical, semiclassical canonical quantum general relativity and gauge theory.

The text is supplemented by an appendix which contains extensive graphics in order to give a feeling for the so far unknown peakedness properties of the states constructed.

Within the framework of higher dimensions the mass of a uniform density star is evaluated. The four-dimensional upper bound for the mass-to-radius ratio obtained by Schwarzschild is generalized within the framework of higher-dimensional spacetime. It is found that the analogue upper bound for the mass-to-radius ratio in higher dimensions tends to increase at first as the number of dimensions of spacetime increases, it attains a maximum at nine dimensions and thereafter decreases. It is found that D = 4 is the lowest number of spacetime dimensions for which the mass-to-radius ratio of a uniform density star can be derived.

A class of spherically symmetric Stephani cosmological models is examined in the context of evolution type. It is assumed that the equation of state at the symmetry centre of the models is barotropic (p(t) = αρ(t)) and the function k(t) playing the role of spatial curvature is proportional to the Stephani version of the Friedmann-Robertson-Lemaître-Walker scale factor R(t) (k(t) = βR(t)).

A classification of the cosmological models is performed depending on different values and signs of parameters α and β. It is shown that for β<0 (hyperbolic geometry) a dust-like (α = 0) cosmological model exhibits accelerated expansion at later stages of evolution.

The Hubble and deceleration parameters are defined in the model and it is shown that the deceleration parameter decreases with the distance becoming negative for sufficiently distant galaxies.

The redshift-magnitude relation m(z) is calculated and discussed in the context of SnIa observational data. It is noted that the most distant supernovae of type Ia fit quite well to the relation m(z) calculated in the considered model (H₀ = 65 km s^-1 Mpc^-1, Ω₀⩽0.3) without introducing the cosmological constant.

It is also shown that the age of the universe in the model is longer than in the Friedmann model corresponding to the same H₀ and Ω₀ parameters.

The Casimir stress on two parallel plates in a de Sitter background for a massless scalar field satisfying Robin boundary conditions on the plates is calculated. The metric is written in conformally flat form to make maximum use of the Minkowski space calculations. We have also considered the case of different cosmological constants for the space between and outside of the plates to make it applicable to the case of domain wall formations in the early universe.

We study a bosonic string with one end free and the other confined to a D0-brane. Only the odd oscillator modes are allowed, which leads to a Virasoro algebra of even Virasoro modes only. The theory is quantized in a gauge where worldsheet time and ordinary time are identified. There are no negative or null norm states, and no tachyon. The Regge slope is twice that of the open string; this can serve as a test of the usefulness of the model as a semi-quantitative description of mesons with one light and one extremely heavy quark when such higher-spin mesons are found. The Virasoro conditions select specific SO(D-1) irreps. The asymptotic density of states can be estimated by adapting the Hardy-Ramanujan analysis to a partition of odd integers; the estimate becomes exact as D goes to infinity.

We present a strategy for a statistically rigorous Bayesian approach to the problem of determining cosmological parameters from the results of observations of anisotropies in the cosmic microwave background. Our strategy relies on Markov chain Monte Carlo methods, specifically the Metropolis-Hastings algorithm, to perform the necessary high-dimensional integrals. We describe the Metropolis-Hastings algorithm in detail and discuss the results of our test on simulated data.

We consider a class of exact solutions which represent non-expanding impulsive waves in backgrounds with a non-zero cosmological constant. Using a convenient five-dimensional formalism it is shown that these spacetimes admit at least three global Killing vector fields. The same geometrical approach enables us to find all geodesics in a simple explicit form and describe the effect of impulsive waves on test particles. Timelike geodesics in the axially symmetric Hotta-Tanaka spacetime are studied in detail. It is also demonstrated that for a vanishing cosmological constant, the symmetries and geodesics reduce to those for well known impulsive pp-waves.

Over the past decade there has been an increasing interest in the study of black holes, and related objects, in higher (and lower) dimensions, motivated to a large extent by developments in string theory. The aim of the present paper is to obtain higher-dimensional analogues of some well known results for black holes in 3 + 1 dimensions. More precisely, we obtain extensions to higher dimensions of Hawking's black hole topology theorem for asymptotically flat (Λ = 0) black hole spacetimes, and Gibbons' and Woolgar's genus-dependent, lower entropy bound for topological black holes in asymptotically locally anti-de Sitter (Λ<0) spacetimes. In higher dimensions the genus is replaced by the so-called σ-constant, or Yamabe invariant, which is a fundamental topological invariant of smooth compact manifolds.

It is known that a flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime can be joined smoothly to a particular case of the Kasner vacuum spacetime. We apply the Darmois junction conditions to show that such a matching can be extended to an open FLRW spacetime. We also investigate the effect of a non-zero cosmological constant Λ on the matchings and conclude that the presence of the constant has no significant effect.

We study a system of two pointlike particles coupled to three-dimensional Einstein gravity. The reduced phase space can be considered as a deformed version of the phase space of two special-relativistic point particles in the centre-of-mass frame. When the system is quantized, we find some possibly general effects of quantum gravity, such as minimal distances and foaminess of the spacetime at the order of the Planck length. We also obtain a quantization of geometry, which restricts the possible asymptotic geometries of the universe.

For causal graphs we propose a definition of proper time which for small scales is based on the concept of volume, while for large scales the usual definition of length is applied. The scale where the change from `volume' to `length' occurs is related to the size of a dynamical clock and defines a natural cut-off for this type of clock. By changing the cut-off volume we may probe the geometry of the causal graph on different scales and thereby define a continuum limit. This provides an alternative to the standard coarse-graining procedures. For regular causal lattices (such as, for example, the two-dimensional lightcone lattice) this concept can be proven to lead to a Minkowski structure. An illustrative example of this approach is provided by the breather solutions of the sine-Gordon model on a two-dimensional lightcone lattice.

In this paper we derive, by means of harmonic analysis, the complete spectrum of Osp(2|4)×SU(2)×SU(2)×SU(2) multiplets from which one obtains compactifying D = 11 supergravity on the homogeneous space Q¹¹¹. In particular, we analyse the structure of the short multiplets and compare them to the corresponding composite operators of the = 2 conformal field theory dual to such a compactification, found in a previous publication. We find complete agreement between the quantum numbers of the supergravity multiplets on one hand and those of the conformal operators on the other hand, confirming the structure of the conjectured SCFT. However, the determination of the actual spectrum by harmonic analysis teaches us a lot more: indeed we find out which multiplets are present for each representation of the isometry group, how many there are, the exact values of the hypercharge and of the `energy' for each multiplet.

We study the transition amplitudes in the state-sum models of quantum gravity in D = 2-4 spacetime dimensions by using the field theory over the G^D formulation, where G is the relevant Lie group. By promoting the group theory Fourier modes into creation and annihilation operators we construct a Fock space for the quantum field theory whose Feynman diagrams give the transition amplitudes. By making products of the Fourier modes we construct operators and states representing the spin networks associated with triangulations of spatial boundaries of a triangulated spacetime manifold. The corresponding spin network amplitudes give the state-sum amplitudes for triangulated manifolds with boundaries. We also show that one can introduce a discrete time evolution operator, where the time is given by the number of D-simplices in a triangulation, or equivalently by the number of vertices of the Feynman diagram. The corresponding transition amplitude is a finite sum of Feynman diagrams, and in this way one avoids the problem of infinite amplitudes caused by summing over all possible triangulations.

The objective of this brief comment is to point out several problems associated with the general framework underpinning the letter `Scalar fields as dark matter in spiral galaxies' (Matos T and Guzman S 2000 Class. Quantum Grav.17 L9) and with its application in this work in particular. Concretely we will focus on three points, namely, the singularity of the scalar field configuration, the analysis of the geodesics and the way in which the authors `add' velocities in trying to account for the effects of the luminous mass on the motion of the test particles.