Table of contents

General solutions to the conformally self-dual, scalar flat equations for Riemannian Bianchi type IX diagonal metrics are given in terms of the Painlevé VI transcendents. The Lax pair and Hamiltonian structure of these equations are also discussed briefly.

A technical point regarding the invariance of Polyakov's non-local form of the effective action under uniform rescalings is addressed.

In contrast to other approaches to (2+1)-dimensional quantum gravity, the Wheeler--DeWitt equation appears to be too complicated to solve explicitly, even for simple spacetime topologies. Nevertheless, it is possible to obtain a good deal of information about solutions and their interpretation. In particular, strong evidence is presented that Wheeler--DeWitt quantization is not equivalent to reduced phase space quantization.

We obtain exact exterior and matching interior stationary axially symmetric solutions of the Einstein--Maxwell field equations, for rigidly rotating charged dust with vanishing Lorentz force. The solutions generate two sources, an infinitely long cylinder of charged dust rigidly rotating about its axis, and a surface layer with surface 4-current located on a singular hypersurface perpendicular to the axis of the cylinder at the origin of the coordinate system. The surface layer extends from the interior of the cylinder to the exterior, but the physical components of its surface stress--energy tensor and surface 4-current, vanish at infinite radial distances on the hypersurface, in any direction away from the origin. The mass and charge densities of the cylinder, also vanish at infinite distances away from either side of the hypersurface. The solutions are physically significant, because both sources of spacetime represent physically reasonable matter with well defined matter, electromagnetic and surface stress--energy tensors, whose physical components vanish at spatial infinity. The junction conditions on the hypersurface separating the exterior from the interior spacetime, and the more complicated set of junction conditions on the singular hypersurface, are satisfied. An analysis by means of the physical components of the Riemann curvature tensor, shows that there are no singularities on the rotation axis and that spacetime is asymptotically flat everywhere at spatial infinity.

An addendum to this article has been published in 1996 Class. Quantum Grav.13 791-7.

The Ashtekar--Renteln ansatz gives the self-dual solutions to the Einstein equation. A direct generalization of the Ashtekar--Renteln ansatz to N=1 supergravity is given both in the canonical and in the covariant formulation and a geometrical property of the solutions is pointed out.

The finite form of the N=2 super-Weyl transformations in the chiral and twisted-chiral irreducible formulations of the two-dimensional N=2 superfield supergravity are found in N=2 superspace. The super-Weyl anomaly of the N=2 extended fermionic string theory is computed in terms of the N=2 superfields, by using a short-time expansion of the N=2 chiral heat kernel. The super-Weyl invariant N=2 superconformal structure is introduced, and a new definition of the N=2 super-Riemannian surfaces is proposed.

We define a pure state associated with every sufficiently regular state of linear quantum field theory. For a scalar field on a globally hyperbolic spacetime, the energy relative to a timelike vector field defines a quasifree state, whose associated pure state is a natural vacuum of the scalar field. If the energy state is defined relative to a globally timelike complete Killing field, its associated pure state is the well known `frequency splitting' stationary vacuum state. It is proved that such vacua corresponding to two commuting Killing fields are equal. It is conjectured that the vacua are equal even if the Killing fields do not commute. Implications for the particle interpretation of quantum field theory in curved spacetimes are discussed.

We consider an Abelian gauge field theory on the partially compactified spacetime , where is the N-dimensional torus and is a n-dimensional Riemannian manifold. The mass of the gauge field generated by quantum fluctuations of a massive or massless scalar field, minimally coupled to a constant background gauge potential, is the quantity of our main interest. As long as the eigenvalue spectrum of is positive (where is the Laplace--Beltrami operator of the manifold , R the Riemann scalar curvature and m the mass of the quantum field), it is found that the topologically generated mass is real for arbitrary N. If, however, the spectrum has zero eigenvalues we found that, depending on the compactification lengths of the torus, the generated mass may also be imaginary.

We give two constructions that relate the self-duality condition on the Weyl curvature of a spacetime with Bianchi IX symmetry to the Painlevé equations, the first by way of the self-dual Yang--Mills equation, the second by way of the isomonodromy problem. We explain how the generic self-dual conformal structure with this symmetry can be obtained from the general solution of the sixth Painlevé equation.

We consider the effect of quantum corrections to a conformally invariant field theory for a scalar field in curved three-dimensional spacetime. In addition to the conformally invariant kinetic action there is a interaction term. The analysis is most easily performed in a space of constant curvature using a formalism of Drummond. Explicit calculation to order shows that all divergences can be removed by renormalizing the cosmological constant and wavefunction; no divergences appear that require the introduction of terms proportionsl to or .

In this paper we shall examine the effects of both bulk and shear viscosities upon a variety of cosmological models. We assume that the viscous terms can be modelled by dimensionless equations of state, and this allows us to write the Einstein equations as a system of autonomous differential equations.

After briefly discussing the Friedmann--Robertson--Walker and Bianchi I models, we discuss in some detail the Bianchi V and Kantowski--Sachs models. In all cases we find the critical points of the flow and elucidate their nature, making use of the energy conditions. Because of our choice of dimensionless variables, (almost) all critial points represent self-similar solutions to the field equations. We also study the case of a Bianchi V perfect fluid model with a non-linear equation of state.

We find that the models are structurally stable under the addition of shear viscosity, whereas they are structurally unstable under the introduction of bulk viscosity. Almost all of the Bianchi V models examined have initial singularities where the matter is dynamically unimportant; the Kantowski--Sachs models have final singularities where the matter is dynamically important.

Adopting the method described in previous papers, we search for Nöther's symmetries in point-like Friedman--Robertson--Walker (FRW) Lagrangians derived for general non-minimally coupled gravitational theories. We obtain exact solutions for flat models capable of producing inflation and recovering the Einstein regime at the present time (i.e. the scalar field and the coupling become constants directly related to the Newton constant ).

We study the spectrum of physical states for higher-spin generalizations of string theory, based on two-dimensional theories with local spin-2 and spin-s symmetries. We explore the relation of the resulting effective Virasoro string theories to certain W minimal models. In particular, we show how the highest-weight states of the W minimal models decompose into Virasoro primaries.

The metric surrounding a cosmic string is examined in the generic case for which its stress--energy tensor is non-degenerate, i.e. its energy per unit length U is not equal to its tension T, as will be the case when allowance is made for microscopic wiggles or superconducting currents. The deflection angle is shown to be given to leading order in Newton's constant G by , so that the actual value of the tension does not alter light deflection, whereas the `missing angle' is estimated as .

A formalism suggested by Stewart for the study of perturbations of cosmological spacetimes is applied to the case of isentropic perfect fluid perturbations. Autonomous systems of differential equations for distinct types of perturbations, including gravitational waves, vorticity perturbations, and waves of density perturbations, arise in a natural way.

Several different `answers' to the question of the best form of a third-order ODE for twisting, type-N, vacuum spacetimes which admit two homothetic motions have already been presented in the literature. The problem of course is quite difficult and these equations have yet to be solved. Therefore, on the basis of our (earlier) understanding of the maximal group of symmetries of the defining equations, we present yet another. We believe that coming to this problem via several different formalisms, arriving at several different forms, will eventually lead to a solution of the problem.

In addition, we give a simple presentation of the extension of that maximal group of symmetries to the case where the vacuum has been generalized to allow for a pressureless, null fluid, which introduces an arbitrary analytic function into the parameter space for the group.

A conformal definition of time-like infinity, , is presented which is based on differential structures and direction-dependent metrics. We show that the asymptotic structure of a spacetime M satisfying our definition may be used to construct a new spacetime, , which is automatically stationary and which, in a well defined sense, is the stationary asymptote of M. In the case of a black hole, we see (through the no-hair theorem) that is of Kerr type.

A class of spherically symmetric perfect fluid solutions with shear, generalizing that of Gutman and Bespal'ko, is obtained in exact form using the author's previous works. Amongst the solutions, those which are of imbedding class one are determined. The relations of the solutions obtained here to those already known by Sussman, Coley, Tupper, van den Bergh and others are investigated. Their plane-symmetric version is touched upon briefly.

We give a review of exact results in non-equilibrium general relativistic kinetic theory, and a comprehensive analysis of the relaxation-time model of collisions. We find the conditions imposed by conservation of particle number and energy--momentum, and by the H-theorem. The exact truncated Boltzmann solution is shown to obey exact thermodynamic laws similar to the approximate laws of Israel and Stewart, and to be subject to the consistency conditions of shear-free flow and a restriction on the anisotropic stress. The Einstein--Boltzmann equations with the relaxation-time model of collisions are solved in FRW spacetime. The solution has vanishing bulk viscous and anisotropic pressures and zero energy and particle fluxes, but is nevertheless a non-equilibrium solution provided m=0.

We examine motion of test particles with various masses, electric charges and dilatonic charges in a background metric and fields of a charged dilatonic black hole.

An algorithm is described for the construction of actions for scalar, spinor, and vector gauge fields that remains well defined when the metric is degenerate and that involves no contravariant tensor fields. These actions produce the standard matter dynamics and coupling to gravity when the tetrad is non-degenerate, but have the property that all fields on a spacetime manifold M that appear in them can be pulled back to any smooth manifold through an arbitrary map , and that this pullback leaves the action invariant when has degree one.

The Kaluza--Klein mechanism is introduced into three-dimensional Regge calculus. After compactification of one direction the three-dimensional Regge action splits into two parts: one is the two-dimensional Regge action and the other is the action of the discretized electromagnetic field on the two-dimensional lattice, which respectively yield the Einstein action and the electromagnetic action in the continuum limit.

I discuss the relation between arbitrarily high-order theories of gravity and scalar--tensor gravity at the level of the field equations and the action. I show that (2n+4)-order gravity is dynamically equivalent to Brans--Dicke gravity with an interaction potential for the Brans--Dicke field and n further scalar fields. This scalar--tensor action is then conformally equivalent to the Einstein--Hilbert action with n+1 scalar fields. This clarifies the nature and extent of the conformal equivalence between extended gravity theories and general relativity with many scalar fields.

We correct the statement, contained in a recent letter, that the equations of motion of a physical theory have different physical content from the corresponding Lagrangian (when it exists).

The unphysical nature of the limits used in a recent paper by Huang calls into question the validity of certain of his results. In addition, Huang neglects to mention an alternative method of calculating quantum effects of this type, which leads to results in agreement with the classical model.

General relativity in three or more dimensions can be obtained by taking the limit in the Brans--Dicke theory. In two dimensions general relativity is an unacceptable theory. We show that the two-dimensional closest analogue of general relativity is a theory that also arises in the limit of the two-dimensional Brans--Dicke theory.

The meaning of the Weyl curvature hypothesis as a condition to prevent vacuum instability and infinite particle creation at the cosmological singularity is discussed. The restriction on the isotropic expansion law is obtained from the same reasoning which leads to the hypothesis that the Ricci scalar curvature must tend to zero when the singularity is approached.