The global structure of McVittie's solution representing a point mass embedded in a spatially flat Robertson-Walker universe is investigated. The scalar curvature singularity at proper radius R=2m, where m (constant) is the Schwarzschild mass, and the apparent horizon which surrounds it are studied. The conformal diagram for the spacetime is obtained via a qualitative analysis of the radial null geodesics. Particular attention is paid to the physical interpretation of this spacetime; previous work on this issue is reviewed, and to how recent quasi-local definitions of black and white holes relate to this spacetime.

Criteria which a space-time must satisfy to represent a point mass embedded in an open Robertson-Walker ͑RW͒ universe are given. It is shown that McVittie's solution in the case kϭ0 satisfies these criteria, but does not in the case kϭϪ1. The existence of a solution for the case kϭϪ1 is proven and its representation in terms of an elliptic integral is given. The following properties of this and McVittie's kϭ0 solution are studied; uniqueness, the behavior at future null infinity, the recovery of the RW and Schwarzschild limits, the compliance with energy conditions, and the occurrence of singularities. The existence of solutions representing more general spherical objects embedded in a RW universe is also proven.

Covariant equations characterizing the strength of a singularity in spherical symmetry are derived and several models are investigated. The difference between central and non-central singularities is emphasized. A slight modification to the definition of singularity strength is suggested. The gravitational weakness of shell crossing singularities in collapsing spherical dust is proven for timelike geodesics, closing a gap in the proof.

We study a quantum fermion field on a background non-extremal Kerr black hole. We discuss the definition of the standard black hole quantum states (Boulware, Unruh and Hartle-Hawking), focussing particularly on the differences between fermionic and bosonic quantum field theory. Since all fermion modes (both particle and anti-particle) have positive norm, there is much greater flexibility in how quantum states are defined compared with the bosonic case. In particular, we are able to define a candidate 'Boulware'-like state, empty at both past and future null infinity; and a candidate 'Hartle-Hawking'-like equilibrium state, representing a thermal bath of fermions surrounding the black hole. Neither of these states have analogues for bosons on a non-extremal Kerr black hole and both have physically attractive regularity properties. We also define a number of other quantum states, numerically compute differences in expectation values of the fermion current and stress-energy tensor between two states, and discuss their physical properties.

We study dyonic soliton and black hole solutions of the su(2) Einstein-Yang-Mills equations in asymptotically anti-de Sitter space. We prove the existence of non-trivial dyonic soliton and black hole solutions in a neighbourhood of the trivial solution. For these solutions the magnetic gauge field function has no zeros and we conjecture that at least some of these non-trivial solutions will be stable. The global existence proof uses local existence results and a non-linear perturbation argument based on the (Banach space) implicit function theorem.