Table of contents

The author analyses the short-distance structure of the full non-linear equations of topologically massive gravity in the vicinity of a point source. Self-consistent stationary solutions are possible only if the mass and spin of the sources are constrained by m+ mu sigma =0.

The authors prove an estimation for the binding energy for a spherical distribution of charged matter momentarily at rest in general relativity. It is of the same form as the Newtonian bound for a charged spherical shell. This generalises an earlier result of Bizon, Malec and O'Murchadha (1990) that was proven for neutral matter, and confirms an old conjecture of Arnowitt, Deser and Misner (1965).

A solution to the Einstein equation that represents several parallel-plane topological defects is presented. The motion of a particle bouncing between two parallel domain walls is studied. The solution is generalised to include an arbitrary number of cosmic strings crossing the walls.

A new exact asymptotically flat solution of Einstein's vacuum equations representing the exterior gravitational field of a static deformed mass with the entire set of multipole moments is obtained in the explicit form. It possesses the event horizon which is singular only on the equator (x=1, y=0).

In the Petrov classification (1969) of the Weyl tensor (or spinor), the type I or (1111) case is referred to in the literature as non-degenerate; there is, however, a 'degenerate' class of type I metrics in which the four distinct principal null directions (PND) only span a 3-space at each point; the degeneracy refers to the dimension of the space of PND. This subcase is shown to exist when I³/J²)6. Metrics of the Kasner type provide an important example of the two type I cases, and an illustration of the kind of geometrical insight into the structure of spacetime metrics which is afforded by analysis of the space of PND.

The author deals with gauge theories in Rⁿ, in which the gauge algebra is the (infinite-dimensional) algebra of vector fields on some m-dimensional manifold M. In effect, this leads to (curved) metrics on Rⁿ*M. Several different examples are reviewed, concentrating on cases where n+m=4 and the metric is a self-dual solution of Einstein's equations. In particular, one example with n=m=2 corresponds to the SU( infinity ) chiral model on R².

The author investigates completely aligned, Petrov type D solutions of the Einstein-Maxwell field equations, which have a perfect fluid and an (non-null) electromagnetic field as source, subject to the additional assumption that the magnetic part of the Weyl tensor relative to the fluid 4-velocity, is zero. The author indicates how all such solutions may be naturally classified into two classes, depending on whether or not the invariants associated with the Weyl, trace-free Ricci and Maxwell tensors are independent. A detailed study, centred about conditions for zero vorticity in the fluid flow, is carried out on the functionally independent class. Further, we present new solutions, which belong to this class, some of which may be of interest in the study of gravitational collapse as well as possibly providing suitable cosmological models for early epochs of our Universe.

Exact solutions to the semiclassical backreaction equations with an unquantised Dirac field are found when the ratio of the renormalisation parameters beta and alpha is 9/4. For alpha (0 there is a one-parameter family of solutions which do not have particle horizons and a two-parameter family of solutions which oscillate at about the Planck frequency and, on average, behave asymptotically as a matter-dominated Friedmann solution. There is a two-parameter family with an asymptotically de Sitter stage, which is stable for any alpha )0 and beta )0.

The authors apply the method of harmonic superspace for solving locally the torsion constraints for the manifolds of (4, 0) supersymmetric sigma -models. The solution is given in terms of an unconstrained harmonic-analytic prepotential with a curved vector index. This prepotential also determines the form of the most general off-shell (4, 0) sigma -model action. The torsion-free (hyper-Kahler) case is recovered if one takes the gradient of a scalar analytic superfield as the vector potential.

The Lagrangian describing the interactions of gravity, electromagnetism and the Jordan-Thiry-Kaluza-Klein scalar field is quantised by using the background field method. The authors compute all the divergences appearing at the 1-loop order. The theory turns out to be not renormalisable, even on-shell.

The authors obtain explicitly the general Bianchi type-I solution of the field equations derived from the R² gravitational Lagrangian in arbitrary dimension.

Topologically massive gravity is shown to be the unique local dynamical and unitary gravity model which is naive power counting renormalisable. This ultraviolet behaviour is established by an unusual parametrisation of the metric. However, neither covariant nor dimensional regularisation can be implemented. BRST regularisation reveals the possibility of an anomaly term: although it is absent at 1-loop order, where everything is finite, it would require higher-loop calculation to rule it out in general. The viability of this theory is therefore still unclear.

The author presents a simple way of deriving cosmological perturbation equations in generalised gravity theories which accounts for metric perturbations in a gauge-invariant way. The author uses an imperfect fluid formulation of the perturbation equations developed in Einstein gravity and absorb all new contributions as effective fluid quantities. The author applies this approach to the f( phi , R)- omega ( phi ) phi ,_c phi ^c Lagrangian which includes most of the gravity theories employing a scalar field and scalar curvature. The relation between the proposed method and the conformal transformation method is discussed. Background and perturbation equations are displayed for specific gravity theories which can be recovered as special cases from the above general Lagrangian.

Rovelli and Smolin (1989) have developed a non-perturbative quantisation of canonical gravity based on the Ashtekar variables (1988) in terms of which the constraints of canonical quantum gravity become polynomials. The author proposes alternative loop variables which are not invariant under the discrete transformations which cause the problems for the Rovelli-Smolin observables and yet which still form a closed Poisson bracket algebra. The author first discusses the case of (2+1)-gravity and then defines similar variables for complexified (3+1)-gravity. Expressions are obtained for the diffeomorphism and Hamiltonian constraints in terms of the new loop variables and the author gives an interpretation of them within the framework of the differential geometry of both the loop space, and the space of SL(2, C) connections, of the spatial three manifold.

The author proposes an approach to string theory where the zero-order theory is the null string. An explicit form of the propagator for the null string in the momentum space is found. Considering the tension as a perturbative parameter, the perturbative series is completely summable and the propagator of the bosonic open string with tension T is found.

The Chern-Simons action for three-dimensional gauge theories is a special case of a particle mechanics 'sigma model Chern-Simons' action with a sympletic target space. The authors discuss the classical and quantum mechanics of this action and its (N-extended) supersymmetric generalisations. They show further that the N=1 supersymmetric Chern-Simons term can be used to cancel the global anomaly that arises in the standard supersymmetric sigma model when the target space is not a spin manifold. A new global anomaly arises, however, if the target space does not admit a spin^c structure.