Table of contents

It is shown how, within the framework of general relativity and without the introduction of wormholes, it is possible to modify a spacetime in a way that allows a spaceship to travel with an arbitrarily large speed. By a purely local expansion of spacetime behind the spaceship and an opposite contraction in front of it, motion faster than the speed of light as seen by observers outside the disturbed region is possible. The resulting distortion is reminiscent of the `warp drive' of science fiction. However, just as happens with wormholes, exotic matter will be needed in order to generate a distortion of spacetime like the one discussed here.

We derive a number of classically equivalent actions for p-branes and discuss their tensionless limit. We then use one of these actions to show that the standard Kaluza--Klein ansatz does not yield a tensile string from a tensionless one.

The search for a theory of quantum gravity has for a long time been almost fruitless. A few years ago, however, Ashtekar found a reformulation of Hamiltonian gravity, which thereafter has given rise to a new promising quantization project: the canonical Dirac quantization of Einstein gravity in terms of Ahtekar's new variables. This project has already produced interesting results, although many important ingredients are still needed before we can say that the quantization has been successful.

Related to the classical Ashtekar-Hamiltonian, there have been discoveries regarding new classical actions for gravity in (2+1) and (3+1) dimensions, and also generalizations of Einstein's theory of gravity. In the first type of generalization, one introduces infinitely many new parameters, similar to the conventional Einstein cosmological constant, into the theory. These generalizations are called `neighbours of Einstein's theory' or `cosmological constants generalizations', and the theory has the same number of degrees of freedom, per point in spacetime, as the conventional Einstein theory. The second type is a gauge group generalization of Ashtekar's Hamiltonian, and this theory has the correct number of degrees of freedom to function as a theory for a unification of gravity and Yang--Mills theory. In both types of generalizations, there are still important problems that are unresolved: e.g. the reality conditions, the metric-signature condition, the interpretation, etc.

In this review, I will try to clarify the relations between the new and old actions for gravity, and also give a short introduction to the new generalizations. The new results/treatments in this review are: (1) a more detailed constraint analysis of the Hamiltonian formulation of the Hilbert--Palatini Lagrangian in (3+1) dimensions; (2) the canonical transformation relating the Ashtekar- and the ADM-Hamiltonian in (2+1) dimensions is given; (3) there is a discussion regarding the possibility of finding a higher-dimensional Ashtekar formulation.

There are also two clarifying figures (at the beginning of sections 2 and 3, respectively) showing the relations between different action-formulations for Einstein gravity in (2+1) and (3+1) dimensions.

In the same spirit as for N=2 and N=4 supersymmetric nonlinear models in two spacetime dimensions by Zumino and by Alvarez-Gaumé and Freedman, we analyse the (2,0) and (4,0) heterotic geometry in holomorphic coordinates. We study the properties of the torsion tensor and give the conditions under which (2,0) geometry is conformally equivalent to a (2,2) one. Using additional isometries, we show that it is difficult to equip a manifold with a closed torsion tensor, but for the real four-dimensional case where we exhibit new examples. We show that, contrarily to Callan et al's claim for real four-dimensional manifolds, (4,0) heterotic geometry is not necessarily conformally equivalent to a (4,4) Kähler--Ricci flat geometry. We rather prove that, whatever the real dimension is, they are special quasi-Ricci flat spaces, and we exemplify our results on Eguchi--Hanson and Taub-NUT metrics with torsion.

The Hamiltonian structure and the residual symmetries of the Liouville theory in the lightcone gauge are studied. The generators of the re-parametrization symmetry are constructed and used to analyse the symmetry of the theory in the lightcone gauge, whilst the symmetry of the gauge-fixed action is studied in the light front formalism. The difference between the symmetry of the gauge-fixed theory and that of the gauge-fixed action is clarified, and the relation to the Nöther current method is established.

We report a series of exact cosmological solutions of the gravitational (1+1)-dimensional equations with a classical scalar field and an isotropic perfect fluid source. We consider a Robertson--Walker universe and obtain the scale factor and the field for several physically interesting potentials. In the case of a non-vanishing positive cosmological constant , we have found asymptotically de Sitter solutions which resemble the (3+1)-dimensional results. When we have to take restrictive conditions on the sign of the potential function in order to obtain exactly solvable models which include exponential inflationary processes.

The divergences that arise in the regularized partition function for closed bosonic string theory in flat space lead to three types of perturbation series expansions, distinguished by their genus dependence. This classification of infinities can be traced to geometrical characteristics of the string worldsheet. Some categories of divergences may be eliminated in string theories formulated on compact curved manifolds.

We consider a macroscopic charge-current carrying (cosmic) string in the background of a Schwarzschild black hole. The string is taken to be circular and is allowed to oscillate and to propagate in the direction perpendicular to its plane (that is parallel to the equatorial plane of the black hole). Numerical investigations indicate that the system is non-integrable, but the interaction with the gravitational field of the black hole still gives rise to various qualitatively simple processes like `adiabatic capture' and `string transmutation'.

The Wheeler--DeWitt equation is derived from the bosonic sector of the heterotic string effective action assuming a toroidal compactification. The spatially closed, higher-dimensional Friedmann--Robertson--Walker (FRW) cosmology is investigated and a suitable change of variables rewrites the equation in a canonical form. Real- and imaginary-phase exact solutions are found and a method of successive approximations is employed to find more general power-series solutions. The quantum cosmology of the Bianchi IX universe is also investigated and a class of exact solutions is found.

The BRST transformations for gravity with torsion are discussed by using the Maurer--Cartan horizontality conditions. With the help of an operator which allows the decomposition of the exterior spacetime derivative as a BRST commutator we solve the Wess--Zumino consistency condition corresponding to invariant Lagrangians and anomalies.

We analyse the classical and quantum theory of a scalar field interacting with gravitation in two dimensions. We describe a class of analytic solutions to the Wheeler--DeWitt equation from which we are able to synthesize states that give prominence to a set of classical cosmologies. These states relate in a remarkable way to the general solution of the classical field equations. We express these relations, without approximation, in terms of a metric and a closed form on the domain of quantum states.

An exact one parameter () family of solutions representing scalar field collapse is shown to exhibit a type of critical behaviour which has been discussed by Choptuik. The three possible evolutions are outlined. For supercritical evolution (when black holes form) it is shown that a quantity related to the mass of the black hole exhibits a power law dependence on , for near critical evolution . The solution supports the conjecture that black hole formation initially occurs at infinitesimal mass.

A comprehensive analysis of general relativistic spacetimes which admit a shear-free, irrotational and geodesic time-like congruence is presented. The equations governing the models for a general energy--momentum tensor are written down. Coordinates in which the metric of such spacetimes takes on a simplified form are established. The general subcases of `zero anisotropic stress', `zero heat-flux vector' and `two-component fluids' are investigated. In particular, perfect-fluid Friedmann--Robertson--Walker models and spatially homogeneous models are discussed. Models with a variety of physically relevant energy--momentum tensors are considered. Anisotropic fluid models and viscous fluid models with heat conduction are examined. Also, models with a perfect fluid plus a magnetic field or with pure radiation, and models with two non-collinear perfect fluids (satisfying a variety of physical conditions) are investigated. In particular, models with a (single) perfect fluid which is tilting with respect to the shear-free, vorticity-free and acceleration-free time-like congruence are discussed.

There exist mechanical effects which allow an observer on circular orbits around a metric source to recognize the likely closeness of an event horizon. When these effects manifest themselves the spacetime is said to be in a pre-horizon regime. Here this concept will be made more precise in the case of the Kerr metric.

A kind of massless free field, a `symmetric recurrent' spinor field is defined. The principal spinors of such a field define shear-free ray congruences. A vacuum solution of Einstein's equations is of type or conformally flat iff its Weyl spinor is symmetric recurrent. The massless free fields of the Robinson--Sommers theorem are symmetric recurrent. A spacetime with a certain kind of symmetric recurrent spinor admits a Killing spinor. A massless free field associated with a Killing spinor is symmetric recurrent. Symmetric recurrent fields are constructed for spacetimes with certain types of Killing spinor, essentially one per Killing spinor.

The renormalization and triviality of theory in the Bianchi type I and Kantowski--Sachs type of spacetimes are investigated within the variational Gaussian approach. It is used to see how the space curvature will affect the triviality of the quantum field. For reasons of mathematical technique it is assumed that the spacetimes are slowly varying, and the adiabatic approximation is adopted in the calculations. Our analysis shows that in some situations, which depend upon the metric function, the space anisotropy and magnitude of curvature coupling, Stevenson's precarious phase will disappear and only the trivial phase will remain.

We continue the investigation of the formation of trapped surfaces in strongly curved, conformally flat geometries. Initial data in quasi-polar gauges, rather than in maximal ones, are considered. This implies that apparent horizons coincide with minimal surfaces. Necessary and sufficient conditions for the formation of trapped surfaces are given. These results can be generalized to include the case with gravitational radiation. We have found that the mass of a body inside a fixed volume is bounded from above if the geometry of a Cauchy slice is smooth.

Signature (type) changing spacetimes are the classical limit of a variant of the no-boundary proposal of Hartle and Hawking (1983). I study spacetimes where the Einstein equation is coupled to a scalar field and give a heuristic, semiclassical argument which suggests that there is no quantum mechanical obstruction against switching the signature where it is classically admissible.

Various metrics with different kinds of conical singularities, including one which does not seem to have appeared in the literature previously, are obtained by identification of flat space. A consideration of holonomy, both linear and affine in the terminology of Kobayashi and Nomizu, suggests an intepretation of the metrics in terms of torsion.

We take the interpretation of Kaluza--Klein theory where curvature in 4+1 dimensions induces effective matter in 3+1 dimensions, and calculate the physical properties of solitons. These are exact solutions of the 5D equations that are static and have spherical symmetry in ordinary (3D) space. We find that their effective 4D matter properties obey the equation of state of ultra-relativistic particles or radiation, and that the matter is highly concentrated towards the centre. The choice of parameters that gives back the conventional Schwarzschild solution is somewhat anomalous, but we gain new insight into this solution. For the general class of solitons, the conventionally defined gravitational mass goes to zero at the centre. However, that point is a singularity where the 4+1 and 3+1 scalars defined by the geometry diverge. We conclude that Kaluza--Klein solitons are naked singularities in ordinary space, surrounded by highly compressed clouds of hot matter that are analogous to point sources in Einstein theory.

Using the gravitational wave generation formalism developed by Blanchet, Damour and Iyer, this paper computes the post-Newtonian-accurate spin octupole moment. The result is checked by explicitly verifying the transformation of the radiative spin octupole moment under shifts of the spatial origin.

The gravitation created by a magnetic field enclosed in a circularly symmetric shell is found and studied in three dimensions. Formulae for the acceleration, time delay and light bending are obtained.

A claim that the model can be used to justify the Hartle--Hawking proposal assumed that quantum cosmology should predict `final conditions'. If the more natural `initial conditions for classical behaviour' are of concern, then the tunnelling boundary condition seems preferable: it predicts an inflationary epoch. This is provided the unbounded scalar potential for negative field can be somehow excluded.

The claim that the model of inflationary cosmology satisfies the Hartle--Hawking boundary condition instead of the tunnelling bounday condition is justified, as was described in our original paper.

The solutions recently published by Santos were found by this author in 1975.