Table of contents

The author presents a careful study of the interacting bosonic string, in the Polyakov formalism, in the on-shell case. The interaction changes the conformal properties of the world sheet. The author's analysis is carried out for arbitrary closed oriented Riemann surfaces. In the on-shell case, the zero-loop and one-loop cases are explicitly reconsidered from a conformal geometrical point of view which allows for better understanding of the nature of the interaction.

Earlier results of Higgs (1959), Stelle (1978) and Whitt (1984) on the dynamical equivalence between Einstein's theory and a class of quadratic theories of gravitation are analysed in view of a more general result, implying the same conclusion for a much larger class of theories (essentially all those which depend arbitrarily on the Ricci tensor). It is shown that all previously known cases of conformal equivalence follow from the prescription of a general Legendre transform, which is in fact suggested by an earlier idea of Einstein (1923) and Eddington (1924). The extension to cover also the dependence on Weyl's tensor (1950) is shortly discussed.

The motion of free test particles in the flat Friedmann and de Sitter spacetimes is investigated. The authors find that the world lines of the particles converge toward the geodesic congruence normal to the homogeneous 3-spaces of these models. The relevance of this result to the physical significance of various proofs of the so-called 'no-hair theorem' is pointed out.

A fairly large class of Lorentz manifolds is defined, called WH normal manifolds, which are approximately those for which timelike infinity is a single point. It is shown that, in such a space, an immersed spacelike hypersurface which is complete must, if it is self-intersecting, not achronal or proper, satisfy strong topological conditions; in particular, if the immersion is surjective in the fundamental group, then the hypersurface must be closed, embedded and achronal (i.e. a partial Cauchy surface). WH normal spaces include products of any Riemannian manifold with Minkowski 1-space; in such a space, a complete immersed spacelike hypersurface must be a immersed as a covering space for the Riemannian factor.

A complete continuous time formulation of the Regge calculus is presented by developing the associated continuous time Regge action. It will be shown that the time constraint is, by way of the Bianchi identities (which will also be developed in detail), conserved by the evolution equations. This analysis will also lead to an explicit first integral for each of the evolution equations. The dynamical equations of the theory will therefore be reduced to a set of first-order differential equations. In this formalism the time constraints will reduce to a simple sum of the integration constants. This result is unique to the Regge calculus-there does not appear to be a complete set of first integrals available for the vacuum Einstein equations.

Smooth distributions of self-gravitating strings and membranes are modelled by foliating chains. A stress tensor is devised that ensures each leaf is an extremal immersion in a spacetime metric determined self-consistently from Einstein's equations. Cosmological solutions are described leading to anisotropies that offer a mechanism of dynamical dimensional reduction.

A proof is given that, when a spacetime admits an invariant timelike congruence orthogonal to the orbits of a non-Abelian two-parameter group of isometries, the given congruence is vorticity-free provided the group is maximal. The result is used to derive a canonical coordinate form for perfect-fluid solutions satisfying the above condition. It is also shown that such a group of isometries cannot be orthogonally transitive (in agreement with a result by Bugalho (1987)) and a brief discussion is given of the self-similar case.

The D-dimensional Schwarzschild-de Sitter metric and Reissner-Nordstrom-de Sitter metric are derived directly by solving the Einstein and Einstein-Maxwell equations. The D-dimensional Kerr metric is rederived by using the complex coordinate transformation method and the D-dimensional Kerr-de Sitter metric is also given. The conjecture about the D-dimensional metric of a rotating, charged mass is given at the end of this paper.

The authors present a family of exact solutions of the four-dimensional Einstein field equations obtained by means of the application of the inverse scattering technique followed by a Kaluza-Klein dimensional reduction. They describe the collision of gravitational solitons on four-dimensional Friedmann-Robertson-Walker backgrounds with a material content consisting of a perfect fluid with a barotropic equation of state. They investigate the behaviour of the stress-energy tensor and show that the collision can form voids and halos. Applications to the Brans-Dicke theory and the problem of geodesic focusing are also considered.

It is usually stated that, if a universe undergoes a period of significant exponential expansion ('inflation'), then necessarily at the present time we live in a high-density universe with the density parameter Omega very close to the critical value of 1. However, this is not necessarily so; there exist inflationary universes leading to any desired value whatever for the present-day value of Omega , and which furthermore also solve both the horizon and monopole problems. Thus inflation is compatible both with very high-density and with very low-density universes. Furthermore, different approaches to looking at the probability of inflationary universe models suggest it may not even be particularly probable that inflation leads to a universe which is presently near the critical density.

This paper exploits an analogy with a time-independent Schrodinger equation to propose a concrete interpretation of a 'wavefunction of the universe' psi which solves the Wheeler-DeWitt equation for a minisuperspace quantum cosmology, and then uses this interpretation to define information theoretic notions of entropy. Given a wavefunction psi depending, in a configuration space representation, on some set of variables q⁰, . . ., q^N, one is instructed to pick one of the q^A, say q⁰ identical to a, as a 'time' and then view psi (Q; a) as an unnormalised conditional probability amplitude for finding the values Q identical to (q¹, . . ., q^N) at 'time' a. By identifying the 'natural' measure dQ on an a=constant hypersurface in minisuperspace, rho (Q, Q'; a)dQ varies as psi *(Q; a) psi (Q'; a)dQ is interpreted as representing a density matrix in the Q-representation; and, by viewing the Q as abstract operators for fixed a, rho is interpreted as an abstract density matrix. This interpretation admits no a priori notion of unitary evolution with respect to a 'translation' da, unitarity being viewed instead as a derived concept arising only in a semiclassical limit. The object then is to use rho to define reasonable measures of 'coarse-grained entropy' in accord with physical intuition which could serve as a diagnostic for an 'arrow of time'. Thus, e.g., by interpreting the diagonal components rho ( lambda _i, lambda _i; a) in some O_a-representation as conditional probabilities p_i for measuring the eigenvalues lambda _i of some operator O_a, one is led to consider an observable dependent entropy S₀(a)=- Sigma _ip_i log p_i. And similarly, at least when the a=constant hypersurfaces are flat, one can map psi into a 'Wigner function' f_w and then use that f_w to define a 'Wigner function entropy' S_w.

Ashtekar's reformulation (1987) of canonical general relativity is extended to simple supergravity in four dimensions. The key idea is to use the left-handed spin connection as a complex coordinate on the phase space. The constraints are simple polynomial functions of degree four or less when expressed in terms of the new conjugate variables. The reformulation is derived starting from a complex first-order supergravity action. The action is invariant under left- and right-handed supersymmetry transformations which are not conjugate, since the connection varies in the latter case but not in the former.