Table of contents

Domain walls for a simple model of a scalar field coupled to gravity are investigated in the thin-wall approximation. When gravity is not present it is known that spherically symmetric domain walls minimise the Euclidean action. A corresponding result for gravitational interactions has not been found. Here some conditions sufficient to guarantee spherically symmetric configurations are established. Relaxing them could lead to toroidal bubbles, which are probably unstable.

Using the Szekeres family of solutions of Einstein's equations, exact models describing the collision of plane gravitational waves with planar shells of null dust are constructed.

All perfect fluid spacetimes with a purely electric Weyl tensor are shown to have an alignment between the fluid 4-velocity and a canonical null tetrad determined by the Weyl tensor. If, in addition, it is assumed that the flow is irrotational, the eigenframes of the shear and Weyl tensors coincide. In all but two rather special cases, it is proved that the vectors of this eigenframe are hypersurface orthogonal and consequently that a coordinate system exists in which the metric, shear and E_ab (the electric part of the Weyl tensor) are all diagonal. Geodesic Petrov type D spacetimes are shown to be either Bianchi type 1 or to belong to the class of solutions considered by Szekeres (1975) and Szafron (1977). The Allnutt solutions (1982) are shown to be the only purely electric type D fields in which the shear is non-degenerate and in which the acceleration vector lies in the plane spanned by the principal null vectors. The field equations are partially integrated in two classes where no solutions are yet known.

The authors give necessary and sufficient conditions for the existence of trapped surfaces in spherically symmetric initial data sets. The conditions they obtain show explicitly that the formation of trapped surfaces depends on the degree of concentration and on the average flow of matter. Thus they give an explicit realisation of the so-called 'trapped surface conjecture'.

The authors analyse exact static solutions of the vacuum Einstein equations obtained from the Appell ring solution to the Laplace equation. They first perform a Newtonian analysis and show that the Appell solution may be interpreted as the field of a shell of infinite negative total mass bordered by a ring of infinite positive total mass. This situation can be reproduced in general relativity, but a different interpretation, not requiring the shell as a source, is also possible pasting two spacetimes through the ring. The superposition of a particle with an Appell ring is also considered and in this case the shell interpretation is found to be inconsistent.

The author studies static solutions of the vacuum Einstein equations in D=m+n+2 dimensions which have the following symmetries: the solutions are spherically symmetric in (m+2) dimensions, (or more generally the m-sphere is replace by an arbitrary m-dimensional Einstein space), while the internal space is any arbitrary n-dimensional Einstein space. The global properties of all such solutions are derived by considering the equivalent dimensionally reduced system in (m+2) dimensions, and by using techniques from the theory of dynamical systems after a judicious choice of variables. Apart from the trivial case of the Schwarzschild solution with constant scalar field, all solutions are found to contain naked singularities or else not to be asymptotically flat, as would be expected from the 'no hair' theorems.

The field equations for a spherically symmetric perfect fluid in non-symmetric gravitational theory are cast as a set of first-order differential equations suitable for numerical integration. An analytic series solution is presented as an expansion around r=0. It is shows how interior solutions match with the exterior one and how, at the boundary, the Euler equation for the fluid becomes the equation of motion of a test particle in the exterior metric. An expression is derived from a conserved pseudotensor for the total mass-energy of a static body in terms of its interior matter parameters.

A particular colliding plane gravitational wave solution is, in the interaction region, a portion of a covering space of the Schwarzschild solution inside the white hole, excluding the singularity. A time-reversed extension exists including covering spaces of the Schwarzschild exteriors and part of the black hole, and ultimately giving two receding waves with flat space between. Equivalently, the normal Schwarzschild exterior may be extended inside the black and white holes to compactified plane-wave regions with flat space between. This extension compares well with the usual Kruskal extension in that it is almost complete and without curvature singularities apart from shock waves. Similar extensions exist for the general Kerr-Newman case.

The authors present a general discussion of Kaluza-Klein theory where the coordinates for the extra dimensions are taken to be anticommuting. The field equations are derived and their consistency with the metric ansatz is discussed. The connection with previous work is mentioned.

The authors discuss the response of inertial and uniformly accelerated monopole particle detectors in the presence of static boundaries. They give a simple formula relating their response to the energy density of the state relative to the state without boundaries and illustrate it in a number of examples. They provide a physical interpretation of the energy spectrum derived by Ford (1988) in the case of a pair of parallel plates (Casimir system).

It is shown that in the five-dimensional Kaluza-Klein theory with torsion, a severe constraint on the Maxwell fields arises when the torsion potential is related to the usual scalar field. It is further pointed out that the Kaluza-Klein scheme provides a natural explanation of the fact that torsion does not couple to spin-1 gauge fields.

Presents a discussion of the literature on spin-weighted spheroidal functions. There are many investigations of these functions in the literature, but these papers contain errors or misprints in quantities which are crucial to the study of perturbations of rotating black holes. Using a method presented in one of the investigations, the author presents the correct formula for the eigenvalues of the differential equation satisfied by the spin-weighted spheroidal functions.