Table of contents

The author discusses a new method of integration over matrix variables based on a suitable gauge choice in which the angular variables decouple from the eigenvalues at least for a class of two-matrix models. The calculation of correlation functions involving angular variables is simple in this gauge. Where the method is applicable it also gives an extremely simple proof of the classical integration formula used to reduce multi-matrix models to an integral over the eigenvalues.

Goldberg, Lewandowski and Stornaiolo (1992) recently investigated the question raised by Rovelli and Smolin (1990), if the loop variables are good coordinates on the Ashtekar phase space of general relativity. The author considers the influence of a cosmological constant on their results.

The full metric representing a non-linear superposition of the Kerr solution with a massless magnetic dipole is obtained. This metric is asymptotically flat and describes correctly the exterior gravitational field of a magnetized spinning mass.

The author considers stationary-axisymmetric vacuum spacetimes possessing an additional 'inside' symmetry property; namely he assumes that the projected Riemannian 3-space induced by the stationary Killing field can be filled in with a one-parameter congruence of confocal ellipsoids. He presents a family of exact solutions of vacuum Einstein's equations possessing these symmetries, and it is shown that these spacetimes consist of a three-parameter family of solutions which coincide with the subfamily of the Kerr-NUT spacetimes with vanishing electric (and magnetic) charge. Hence, a simple geometrical characterization can be given for this subfamily of algebraically specialized Petrov type D solutions which resulted from the application of completely different spin coefficient techniques.

The interpretation of recent numerical results of Shapiro and Teukolsky (1991) concerning the Vlasov equation in general relativity is discussed. The conclusion is reached that, while their computations may well accurately describe the collapse of dust, there are serious grounds to doubt that they are representative of the behaviour of more general solutions of the Vlasov equation. The circumstances under which solutions of the Vlasov-Einstein system may develop singularities are examined and it is shown that there exist initial data not containing trapped surfaces for which this happens. Some comments are made on the significance of these considerations for the cosmic censorship hypothesis.

The authors discuss some aspects of the generalization of Fujikawa's method for computing the trace anomaly when an external field is present. This is done explicitly for the case of a scalar theory in four dimensions. They also use the idea of a compensating field in order to construct a general conformal invariant action for a scalar theory in two dimensions. This is done after a suitable choice of the integration variable in the functional measure. The corresponding trace anomaly is computed and the result obtained is more general than the usual one.

This work gives various classical results of relevance to quantum cosmology. The junction conditions are derived for matching a region of Lorentz-signature spacetime to a region of Riemannian-signature space across a spatial surface according to the vacuum Einstein or Einstein-Klein-Gordon equations. In particular, the spatial metric and the Klein-Gordon field must be instantaneously stationary at the junction. Real, globally valid fields and equations are found for the general Cauchy problem where signature change is allowed. An affine parameter is defined for geodesics crossing the junction orthogonally. For homogeneous and isotropic cosmological-constant cosmologies, the signature-changing solution is unique up to a constant, non-singular and inflationary, with a flatness parameter which is ultimately unity. The hypothesis of initially Riemannian signature gives a prediction of a highly restricted class of observable cosmologies for a given matter field. In particular, inflation, initially zero entropy and partial isotropy are predicted.

The author discusses (1+1)-gravity in the form proposed by Mann et al. (1990) and gives explicit solutions for the Cauchy problem in the cases of dust and pure radiation. Two singularity theorems are also derived.

A large N-matrix model of a general complex matrix generates dynamical triangulations in which the triangles can be chequered (i.e. coloured so that neighbours are opposite colours). Gravity and multicritical matter in such triangulations are described by the KdV-type equations found in the Hermitian matrix model but without the doubling of degrees of freedom. However, by tuning further couplings it is possible to obtain a series of more general string equations with redoubled degrees of freedom.

The authors suggest a model of induced gravity in which th fundamental object is a relativistic membrane minimally coupled to a background metric and to an external three-index gauge potential. They compute the low-energy limit of the two-loop effective action as a power expansion in the surface tension. A generalized bootstrap hypothesis is made in order to identify the physical metric and gauge field with the lowest-order terms in the expansion of the vacuum average of the composite operators conjugate to the background fields. They find that the large-distance behaviour of these classical fields is described by the Einstein action with a cosmological term plus a Maxwell-type action for the gauge potential. The Maxwell term enables one to apply the Hawking-Baum (1983,1984) argument to show that the physical cosmological constant is 'probably' zero.

Perfect fluid generalizations of the parabolic case of Szekeres class I solutions are interpreted as spacetimes whose matter source is a mixture of adiabatically interacting dust (with density rho ) and a homogeneous perfect fluid (with density and pressure mu and p). Simple analytical solutions emerge when the dynamics of mu and p are governed by a Friedmann equation and the equation of state p=( gamma -1) mu holds. The Szafron models correspond to the dynamics of a spatially flat FRW cosmology, admitting a smooth matching with the latter along a suitable timelike hypersurface. New solutions arise for a homogeneous fluid satisfying spatially curved FRW dynamics. For slowly varying p and mu satisfying an arbitrary equation of state, approximate WKB type solutions are derived. Geometric properties and singularities are examined. All solutions are compatible with the existence of a 'regular centre' defined as the regular worldline of privileged isotropic observers. Some of the solutions admit evolution branches along with mu and rho are everywhere positive. These latter cases, under appropriate restrictions, could provide an adequate framework with which to model inhomogeneities in a cosmological background and other situations of astrophysical interest.

This work deals with problems and properties of definitions of mass in general relativity. The author proves that it is sufficient for the spinor field determining the Nester-Witten 2-form to satisfy only one complex scalar equation in order that the integral over a convex 2-surface is positive when the dominant energy condition holds. As corollaries he shows that the quasilocal momentum of Ludvigsen and Vickers (1982) is future-pointing, and obtains a proof of the positivity of the Bondi mass which uses a different spinor propagation equation to previous proofs. As another corollary, he shows that there are infinitely many ways of defining a quasilocal 4-momentum which is future-pointing, allows definition of a corresponding positive mass, gives the Bondi and ADM momenta at infinity, is zero in flat spacetime and is correct in the linearized theory and in the spherically symmetric case.

The space-space field equations of the Jordan Brans-Dicke theory describing a spatially homogeneous vacuum spacetime of the Bianchi type can be brought under transformation into a general set of nonlinear differential equations, an all-encompassing general equation implicitly contained in each of the cosmological models studied, most of whose analytic solutions the authors have found. The scalar field displays a considerable influence in the cosmological context during the early, strongly relativistic stage of the universe's expansion where the main difference between the cosmologies of Einstein general relativity and Jordan Brans-Dicke theories lies in the possible existence of a sourceless phi field which, since it is not generated by matter, is contrary to Mach's principle. Several solutions presented are reported for the first time.

The author derives a new generally covariant action for the minimally coupled massless Klein-Gordon field, in a pure spin connection formulation. It is a natural extension of the action for vacuum complex general relativity in a pure connection formulation to include a massless scalar field coupling.

A nonlinear perturbative calculational scheme for solving the Hamilton-Jacobi equation for general relativity with matter fields is described. By taking advantage of the invariance of the Hamilton-Jacobi equation under spatial reparametrizations, the generating functional may be written in terms of a spatial gradient expansion. The solution in superspace, which describes an ensemble of inhomogeneous universes, may be reduced to a solution of a finite dimensional field space at each order of the expansion. Exact solutions of the separated Hamilton-Jacobi equations of order zero and two are given for gravity interacting with dust and scalar fields. Dust fields allow for a quantum formulation with a positive-conserved probability density. Contrary to what is usually required, the new Hamiltonian density does not vanish strongly, but may be written in a form which is quadratic in weakly vanishing terms. The authors' approach is covariant, and they show how to introduce an arbitrary time parameter into the Hamilton-Jacobi formalism. This formalism may be applied with profit to the nonlinear evolution of dust fields in matter-dominated as well as vacuum-dominated universes (inflationary cosmologies).

A Weyl spinor of Petrov type I has four different principal null directions (PND) at any point-these can be represented by four points on S⁺, the sphere of intersection of a spacelike plane 'T=1' with the cone of null directions at that point. Penrose and Rindler (1986) have shown that a frame can be chosen so that these four points form the vertices of a disphenoid (a tetrahedron with opposite edges equal in pairs). This disphenoid has degenerate properties when the Weyl spinor (and its invariants I and J) satisfy certain conditions which are given mainly in terms of restrictions on eigenvalues. However, this approach is not well suited to algebraic computing. This presents a simple computer-algebra-adapted method, based on an invariant, M, formed from the Weyl spinor, for finding and analysing degenerate cases; this is an extension of an earlier work by McIntosh and Arianrhod (1990). It also extends results of Penrose and Rindler concerning the relationship between these degenerate cases and the corresponding eigenvalues of the matrix Psi of Weyl spinor components. In addition, it examines some relationships between these degeneracies and the structure of some exact spacetime metrics, in particular vacuum metrics of the Kasner type. The results of this work have been used elsewhere by the authors and Fletcher to investigate the geometrical structure of the Curzon metric by studying the nature of its PND; they are also being used to study Segre types of the Plebanski spinor formed from the trace-free Ricci spinor.

An asymptotic expansion for the heat kernel G_Delta (x,x'; tau ) corresponding to a second order elliptic differential operator Delta acting on fields over manifolds with boundary is given. This expansion takes the form of an extended DeWitt ansatz satisfying mixed Dirichlet and Neumann boundary conditions. The functional trace, calculated directly using this expansion, agrees with more indirect derivations. The results are also represented as a distribution for general x,x' which, when integrated over the manifold, agree with other calculations carried out in the pure Dirichlet limit.