Table of contents

In the phase space of a complex SO(3) Yang-Mills theory, one may define the Yang-Mills Hamiltonian H_YM, which gives Yang-Mills theory, and Ashtekar's constraints, which give general relativity. The author looks for points on Ashtekar's constraint surface which stay on that surface under the evolution generated by H_YM. Such points exist for non-zero values of the cosmological constant; in general relativity, they correspond to self-dual spacetimes and, in Yang-Mills theory, to Yang-Mills fields which are self-dual with respect to a metric constructed algebraically from the Yang-Mills electric field.

A Bondi form of the C-metric is obtained. It has an accelerated origin at which there is a particle. Another singularity, presumably representing a string attached to the particle, is present.

The strong cosmic censorship conjecture says that 'most' spacetimes developed as solutions of Einstein's equations from prescribed initial data cannot be extended outside of their domains of dependence. Here, the authors discuss results which show that if they restrict attention to the polarised Gowdy spacetimes, strong cosmic censorship holds. More specifically, these results show that in the space of Cauchy data for polarised Gowdy spacetimes, there is an open dense subset for which the maximal globally hyperbolic development is inextendible. Among the Gowdy spacetimes which can be extended, the authors find a set of them which admit a countable infinity of inequivalent classes of extensions. They also find that a polarised Gowdy spacetime (T³ or S²*S¹) may be extended as a solution of Einstein's equations (with the Gowdy isometries) across a compact Cauchy horizon only if it is analytic.

The compacted spin-coefficient formalism of Geroch, Held and Penrose (1973), which is based on the structure given by a pair of null directions, is developed from the point of view of a connection on a principal fibre bundle. Standard geometric structures implicit in the formalism are thus made clear. In particular, the torsion of the connection is shown to be quite useful for geometric interpretations.

The equations of Euler type used by Belinskii et al. (1978) to generate Ricci-flat Kahler metrics on 4-space are generalized to be a system of equations that generates Kahler metrics with vanishing scalar curvature. This family of new metrics also contains a family of scalar-flat Kahler surfaces constructed by LeBrun (1988).

The existence of Killing vectors in conformally flat perfect fluid spacetimes in general relativity is considered. In particular Killing vectors which are neither orthogonal nor parallel to the fluid velocity vector are considered and stationary fields in which the fluid velocity vector is not parallel to the timelike Killing vector field are shown to exist. This class of solutions is shown to include several stationary (but non-static) axisymmetric fields, thus providing counter-examples to a theorem of Collinson (1976). In the case when the fluid is non-expanding, the number of spacelike Killing vectors is shown to depend on the rank of four functions of time which appear in the metric. Some examples of stationary but non-static fields are presented in closed form.

In order to give a precise and general formulation to the strong equivalence principle (SEP) the authors define, in an appropriate inertial frame and in the slow-motion approximation, a local gravitational system. They say that the SEP is fulfilled if, when the size r of the system is sufficiently small, its dynamical behaviour, to a given accuracy, is universal and not affected by the external world. In the theory of general relativity, along with the Newtonian tidal force F_tid varies as r, there exists a non-linear, relativistic contribution F_nl which, in order of magnitude, is independent of r; this seems to leave a trace of the external world in an arbitrarily small gravitating system and threatens to violate the principle. It is shown, however, that F_nl is always smaller than F_tid, at least in the weak-field slow-motion approximation and hence, in this approximation, the SEP survives. In other metric theories of gravity, however, violations of the SEP occur. A notion of universal gravitational clocks is briefly discussed.

Starting from a variational principle for perfect fluids, the authors develop a Hamiltonian formulation for perfect fluids coupled to gravity expressed in Ashtekar's spinorial variables. The constraint and evolution equations for the gravitational variables are at most quadratic in these variables, as in the vacuum case and in the coupling of gravity to other matter fields, while some of the matter evolution equations are in general non-polynomial. They specialize the formalism to barotropic fluids and spherically symmetric spacetimes, and, within this class, to Kantowski-Sachs spacetimes. They find explicitly the Kantowski-Sachs solutions corresponding to 'stiff matter', which they use as examples to look at the behaviour of the Ashtekar variables when the spatial metric becomes degenerate on one hypersurface. They find that in these solutions the coordinate time arising in the present treatment is singularly related to proper time, and the singularities are only reached at infinite values of the former. They obtain some simple necessary conditions that have to be satisfied if one wants to evolve data past singularities of this kind. None of the barotropic-fluid-filled Kantowski-Sachs spacetimes satisfy these conditions.

Presents a detailed geometric derivation of N=2, 4D supergravity coupled to Abelian vector multiplets (1,1/2,0). The theory is covariant under reparametrisations of the manifold M spanned by the scalar fields. By requiring the closure of the supersymmetry transformation laws, the authors find a coordinate-free characterisation of M. The geometry of M, called 'special geometry', is relevant to compactified string theories, since it is common to the moduli spaces of Calabi-Yau threefolds and c=9 (2,2) superconformal field theories.

The authors present new results that indicate a form of 'time shift' in the non-linear (self) interaction of cylindrical solitonic gravitational waves on a flat background as they are reflected off the symmetry axis. The authors introduce appropriate timing procedures to show that this time shift increases monotonically with the energy of the wave as measured, for instance, by the C energy of the system, being negligible for solitons of negligible amplitude ('test' solitons) and diverging together with the C energy. They thus find for the electro-gravitational solitons a behaviour that has so far been observed only for non-linear systems such as the Kortweg-de Vries equation.

In the context of a recently proposed GL(4)-invariant formulation of general relativity the authors discuss a generalized Palatini formalism in which the conditions of metricity and torsionlessness are both obtained as dynamical equations. This theory can be reformulated in terms of new variables which, in a special gauge, reduce to the ones introduced by Ashtekar (1987,8). They describe the emergence of the new variables both in Lagrangian and in Hamiltonian language.

The electromagnetic and gravitational multipole moments of stationary sources in general relativity are obtained in the conformal picture in terms of the expansion coefficients of the electrovacuum potentials on the axis of symmetry. Generic expressions for the values of the first five multipole moments are given, and the field of a pure pole-dipole source is investigated.

Applies the infinitesimal Hou-Li transformations delta (u) to an arbitrary Weyl solution, and finds that they do not preserve asymptotic flatness. However, the author finds a closely related set of new transformations delta (u) that do have this desirable property. He determines the action of these new transformations on the multipole moments of the solution.

Homogeneous solutions to topologically massive gravity have been previously found for spacetimes in 2+1 dimensions admitting a simply transitive isometry group. The author discusses the case of homogeneous spacetimes in 2+1 dimensions with isotropy, reviews a technique for classifying them and examines in each case whether the isometry group has a simply transitive subgroup. He further shows that none of these cases correspond to new solutions to the Einstein-Cotton equations of topologically massive gravity, and conclude that all homogeneous solutions have simply transitive subgroups.

The Hartle-Hawking no-boundary proposal (1983) for the wavefunction of the Universe suggests, via a minisuperspace calculation, that the Universe may have had an infinite period of inflation. If the classical spacetime is said to have an edge or singularity where the semiclassical approximation breaks down, that would be at an infinite amount of comoving time in the past, though at only a finite proper time for geodesics with non-zero momentum, so the spacetime would still be singular even if infinitely old.

For pt.I see ibid., vol.7, p.1319 (1990). On a conformally stationary spacetime, Fermat's principle, and hence the equations of motion of light rays, can be formulated in three-dimensional (purely spatial) terms. This reduction from four-dimensional spacetime to three-dimensional space is very similar to the well known reduction formalism of Kaluza-Klein theory. The resulting equation of motion of light rays in 3-space is formally identical with the Lorentz force equation for charged particles of unit specific charge and fixed energy in a magnetostatic field on a Riemannian 3-manifold. This analogy, in which the magnetostatic field corresponds to the rotation of the timelike conformal Killing vector field, has the following consequence. To every magnetostatic electron lens there can be constructed mathematically an analogous gravitational lens.

Exact solutions of the Einstein-Liouville equations in a universe with a Robertson-Walker k=0 geometry at early times (when the matter content is effectively a distribution of massless particles) can have an anisotropic distribution function. In general this will force the geometry to evolve away from isotropy and homogeneity at late times, when the particles effectively become massive as the universe cools.

The authors introduce an observer-dependent quantum vacuum for the electromagnetic field in curved space via the diagonalisation of a Hamiltonian. This method works in foliable spacetimes and for irrotational observers. This paper generalises the results of a previous one where the separation of variables was possible. The authors also show a mechanism to remove the longitudinal photons in curved space.

The divergence in the bosonic string partition function found by Gross and Periwal (1988) is analysed using the Schottky parametrisation of the Polyakov measure for moduli space. It is show how the problem of the integration region can be solved and how lower limits on the measure may be obtained in this domain. The factorial growth of the bound on the partition function with respect to the genus can thus be derived by translating the cut-off for closed geodesic lengths to a cut-off for Schottky group parameters.

As shown by Schwinger (1963), the special relativistic equal time commutator of the (symmetric) Rosenfeld energy-momentum tensor has a particularly simple form for free matter fields of lower spins. The author derives a corresponding classical relation for the (non-symmetric) canonical, dynamical and modified energy-momentum tensors of the general local Poincare theory, of gravity interacting with the matter field, assuming the minimal number of constraints in the theory. This relation holds also for the matter field and the gravitational field separately.

The recent proof by Debever, Van den Bergh and Leroy (1989) that diverging null Einstein-Maxwell fields of Petrov type D are necessarily non-twisting, can simply be extended to aligned pure radiation fields.