Table of contents

Both the Ernst equation and the six Painlevé transcendental equations are reductions of the self-dual Yang - Mills (SDYM) equations. We show how this link can be used to find exact solutions to Einstein's equations and to understand aspects of the integrability of the Ernst equation.

The two independent `plus' and `cross' polarization waveforms associated with the gravitational waves emitted by inspiralling, non-spinning, compact binaries are presented, ready for use in the data analysis of signals received by future laser interferometer gravitational-wave detectors such as LIGO and VIRGO. The computation is based on a recently derived expression of the gravitational field at the second-post-Newtonian approximation of general relativity beyond the dominant (Newtonian) quadrupolar field. The use of these theoretical waveforms to make measurements of astrophysical parameters and to test the nature of relativistic gravity is discussed.

Techniques are presented for calculating directly the scalar functional determinant on the Euclidean d-ball. General formulae are given for Dirichlet and Robin boundary conditions when d is even and the field is massless. The method involves a large mass asymptotic limit which is carried out in detail for d = 2 as an exercise, incidentally producing some specific summations and identities. Extensive use is made of the Watson - Kober summation formula. The calculation of the finite-mass determinant is also briefly considered.

We study the Hilbert bundle description of stochastic quantum mechanics in curved spacetime developed by Prugovecki, which gives a powerful new framework for exploring the quantum mechanical propagation of states in curved spacetime. We concentrate on the quantum transport law in the bundle, specifically on the information which can be obtained from the flat space limit. We give a detailed proof that quantum transport coincides with parallel transport in the bundle in this limit, confirming statements of Prugovecki. Furthermore, we show that the quantum-geometric propagator in curved spacetime proposed by Prugovecki, yielding a Feynman path integral-like formula involving integrations over intermediate phase-space variables, is Poincaré gauge-covariant (i.e. is gauge-invariant except for transformations at the endpoints of the path) provided the integration measure is interpreted as a `contact point measure' in the soldered stochastic phase-space bundle raised over curved spacetime.

The zeta function associated with higher-spin fields on the Euclidean 4-ball is investigated. The leading coefficients of the corresponding heat-kernel expansion are given explicitly and the zeta functional determinant is calculated. For fermionic fields the determinant is shown to differ for local and spectral boundary conditions.

We suggest a method of construction of general diffeomorphism-invariant boundary conditions for metric fluctuations. The case of the (d + 1)-dimensional Euclidean disc is studied in detail. The eigenvalue problem for the Laplace operator on metric perturbations is reduced to that on d-dimensional vector, tensor and scalar fields. The explicit form of the eigenfunctions of the Laplace operator is derived. We also study restrictions on boundary conditions which are imposed by the symmetry of the Laplace operator.

In this paper we consider (2 + 1)-dimensional gravity coupled to N point particles. We introduce a gauge in which the z and components of the dreibein field become holomorphic and anti-holomorphic, respectively. As a result we can restrict ourselves to the complex plane. Next we show that solving the dreibein field is equivalent to solving the Riemann - Hilbert problem for the group SO(2,1). We give the explicit solution for two particles in terms of hypergeometric functions. In the N-particle case we give a representation in terms of conformal field theory. The dreibeins are expressed as correlators of two free fermion fields and twist-operators at the position of the particles.

A formulation of Poincaré symmetry as an inner symmetry of field theories defined on a fixed Minkowski spacetime is given. Local P gauge transformations and the corresponding covariant derivative with P gauge fields are introduced. The renormalization properties of scalar, spinor and vector fields in P gauge field backgrounds are determined. A minimal gauge field dynamics consistent with the renormalization constraints is given.

Using a shooting routine calculation, Basu and Vilenkin introduced a critical value of topological defects during inflation. In this paper, we prove analytically the existence of the Basu - Vilenkin critical value. The thicker defects are smeared out by the expansion.

Using the long-wave perturbation scheme (gradient expansion), the effect of inhomogeneity on the inflationary phase is investigated. We have solved the perturbation equation whose source term comes from inhomogeneity of a scalar field and a seed metric. The result indicates that a sub-horizon scale inhomogeneity strongly affects the onset of inflation.

Gravitational and massless particle radiation from infinitely long cosmic strings with finite thickness is studied by constructing analytic solutions to the Einstein field equations with certain forms of energy - momentum tensor. Using Thorne's -energy arguments, it is shown that the radiation always carries away energy. As a result, the angle defect outside the string is less than that in the non-radiation case. Since the solutions are exact, the back reaction of the radiation is automatically taken into account. In fact, it is shown that because of this back reaction the spacetime is always singular on the symmetry axis. The singularity is a scalar one and not removable.

We generalize the classification of (non-vacuum) pp-waves [1] based on the Killing algebra of the spacetime by admitting distribution-valued profile functions. Our approach is based on the analysis of the (infinite-dimensional) group of `normal-form-preserving' diffeomorphisms.

We construct ultrarelativistic Kerr geometries from their distributional energy - momentum tensors. The latter are obtained by boosting Kerr's distributional energy - momentum tensor in arbitrary directions, thereby generalizing previous work by the authors.

We show that one can construct initial data for the Einstein equations which satisfy the vacuum constraints. This initial data are defined on a manifold with topology with a regular centre and are asymptotically flat. Furthermore, these initial data will contain a shell-like region which is foliated by two-surfaces of topology . These two-surfaces are future trapped in the language of Penrose. The Penrose singularity theorem guarantees that the vacuum spacetime which evolves from this initial data is future null incomplete.

A new method is derived for constructing electromagnetic surface sources for stationary axisymmetric electrovac spacetimes endowed with non-smooth or even discontinuous Ernst potentials. This can be viewed as a generalization of some classical potential theory results, since lack of continuity of the potential is related to dipole density and lack of smoothness, to monopole density. In particular, this approach is useful for constructing the dipole source for the magnetic field. This formalism involves solving a linear elliptic differential equation with boundary conditions at infinity. As an example, two different models of surface densities for the Kerr - Newman electrovac spacetime are derived.

A manifestly diffeomorphism invariant extension of Einstein gravity is constructed, which includes certain kinds of singular metrics, and whose ADM formulation is Ashtekar's gravity. The latter is shown to be locally equivalent to the covariant theory. It turns out that exactly those kinds of degenerate four-dimensional metrics are allowed which do not destroy the local causal structure of spacetime. It is also shown that Ashtekar's gravity possesses an extension that provides a local invariance, without complexifying or changing the signature of the metric.

We show how to treat the constraints and reality conditions in the SO(3) - ADM (Ashtekar) formulation of general relativity, for the case of a vacuum spacetime with a cosmological constant. We clarify the difference between the reality conditions on the metric and on the triad. Assuming the triad reality condition, we find a new variable, allowing us to solve the gauge constraint equations and the reality conditions simultaneously.

In a recent paper we obtained exact exterior and matching interior stationary axially symmetric solutions of the Einstein - Maxwell field equations for a rigidly rotating charged dust with vanishing Lorentz force. The solutions were expressed in terms of only one of the zeros of the Bessel function . The main aims of this work are to base the exterior and interior solutions on all the infinite number of zeros of , prove the non-existence of timelike hypersurface-orthogonal Killing vectors and that, under a certain condition, the electrovacuum exteriors contain regions with closed timelike lines.