Table of contents

For any asymptotically flat spacetime with a suitable causal structure obeying (a weak form of) Penrose's cosmic censorship conjecture and satisfying conditions guaranteeing focusing of complete null geodesics, we prove that active topological censorship holds. We do not assume global hyperbolicity, and therefore make no use of Cauchy surfaces and their topology. Instead, we replace this with two underlying assumptions concerning the causal structure: that no compact set can signal to arbitrarily small neighbourhoods of spatial infinity (`-avoidance'), and that no future incomplete null geodesic is visible from future null infinity. We show that these and the focusing condition together imply that the domain of outer communications is simply connected. Furthermore, we prove lemmas that have as a consequence that if a future incomplete null geodesic were visible from infinity, then, given our -avoidance assumption, it would also be visible from points of spacetime that can communicate with infinity, and so would signify a true naked singularity.

A 2p + 5 parametric family of black holes is constructed in dilaton - axion gravity with p vector fields using a holomorphic representation of U-duality in three dimensions. The metric of the non-extremal black holes has a Reissner - Nordström-type structure and generically possesses an internal horizon. However, in the extremal limit the generic solution exhibits a dilatonic-type null singularity. The solution thus interpolates between the well known Reissner - Nordström and charged dilatonic spacetimes. Only in the case of the orthogonal electric and magnetic charges (if p > 1) may the extremal solution have a non-singular event horizon.

The approximate stress tensor of a massless, conformally invariant scalar field in the Unruh vacuum in the Schwarzschild spacetime is constructed. It satisfies all regularity and consistency conditions and properly reproduces exact numerical calculations.

By writing the complete set of 3 + 1 (ADM) equations for linearized waves, we are able to demonstrate the properties of the initial data and of the evolution of a wave problem set by Alcubierre and Schutz. We show that the gauge modes and constraint error modes arise in a straightforward way in the analysis, and are of a form which will be controlled in any well specified convergent computational discretization of the differential equations.

Topological Yang - Mills theory with the Belavin - Polyakov - Schwarz - Tyupkin SU(2) instanton is solved completely, revealing an underlying multi-link intersection theory. Link invariants are also shown to survive the coupling to a certain kind of matter (hyperinstantons). The physical relevance of topological field theory and its invariants is discussed. By embedding topological Yang - Mills theory into pure Yang - Mills theory, it is shown that the topological version TQFT of a quantum field theory QFT allows us to formulate consistently the perturbative expansion of QFT in the topologically nontrivial sectors. In particular, TQFT classifies the set of good measures over the instanton moduli space and solves the inconsistency problems of the previous approaches. The qualitatively new physical implications are pointed out. Link numbers in QCD are related to a non-Abelian analogue of the Aharonov - Bohm effect.

A general structure of effective action in a new chiral superfield model associated with N = 1, D = 4 supergravity is investigated. This model corresponds to finite quantum field theory and does not demand the regularization and renormalization of the effective action calculation. It is shown that in a local approximation the effective action is defined by two objects called the general superfield effective Lagrangian and the chiral superfield effective Lagrangian. A proper-time method is generalized for calculation of these two effective Lagrangians in a superfield manner. A power expansion of the effective action in supercovariant derivatives is formulated and the lower terms of such an expansion are calculated in explicit superfield form.

The presence of Killing - Yano tensors implies the existence of non-standard supersymmetries in point-particle theories on curved backgrounds. In a string theoretical context, these are symmetries of the modes describing the particle-like behaviour of the string. In the presence of isometries we show that, in addition to these, one can also define a new type of non-standard supersymmetry among a mixture of particle and winding modes. The interplay with T-duality is also examined and illustrated by explicit examples.

We show that the anomalous couplings of D-brane gauge and gravitational fields to Ramond - Ramond tensor potentials can be deduced by a simple anomaly inflow argument applied to intersecting D-branes and use this to determine the 8-form gravitational coupling.

Using Penrose's binor calculus for SU(2) (SL(2,C)) tensor expressions, a graphical method for the connection representation of Euclidean quantum gravity (real connection) is constructed. It is explicitly shown that: (i) the recently proposed scalar product in the loop-representation coincides with the Ashtekar - Lewandowski cylindrical measure in the space of connections; (ii) it is possible to establish a correspondence between the operators in the connection representation and those in the loop representation. The construction is based on an embedded spin network, the Penrose's graphical method of SU(2) calculus, and the existence of a generalized measure on the space of connections modulo gauge transformations.

The differential structure of the Ashtekar - Isham quantum configuration space of canonical gravity allows the expression of the operators, representing various geometrical objects, by compact analytic formulae. In this paper such a formula is presented for the Rovelli - Smolin volume operator. This operator is compared with the quantum volume defined by Ashtekar and Lewandowski and a difference is indicated.

By means of a detailed analysis we corroborate the widespread belief that absorption of gravitational radiation by the cosmic fluid is very small, and obtain upper bounds for different epochs of the expansion of the universe (namely the quark - gluon plasma, the electron - neutrino mixture, the matter - radiation interaction and a gas of WIMPs).

Self-similarity in general relativity is briefly reviewed and the differences between self-similarity of the first kind (which can be obtained from dimensional considerations and is invariantly characterized by the existence of a homothetic vector in perfect-fluid spacetimes) and generalized self-similarity are discussed. The covariant notion of a kinematic self-similarity in the context of relativistic fluid mechanics is defined. It has been argued that kinematic self-similarity is an appropriate generalization of homothety and is the natural relativistic counterpart of self-similarity of the more general second (and zeroth) kind. Various mathematical and physical properties of spacetimes admitting a kinematic self-similarity are discussed. The governing equations for perfect-fluid cosmological models are introduced and a set of integrability conditions for the existence of a proper kinematic self-similarity in these models is derived. Exact solutions of the irrotational perfect-fluid Einstein field equations admitting a kinematic self-similarity are then sought in a number of special cases, and it is found that: (i) in the geodesic case the 3-spaces orthogonal to the fluid velocity vector are necessarily Ricci-flat; (ii) in the further specialization to dust (i.e. zero pressure) the differential equation governing the expansion can be completely integrated and the asymptotic properties of these solutions can be determined; (iii) the solutions in the case of zero expansion consist of a class of shear-free and static models and a class of stiff perfect-fluid (and non-static) models; and (iv) solutions in which the kinematic self-similar vector is parallel to the fluid velocity vector are necessarily Friedmann - Robertson - Walker (FRW) models. Solutions in which the kinematic self-similarity is orthogonal to the velocity vector are also considered. In addition, the existence of kinematic self-similarities in FRW spacetimes is comprehensively studied. It is known that there are a variety of circumstances in general relativity in which self-similar models act as asymptotic states of more general models. Finally, the questions of under what conditions are models which admit a proper kinematic self-similarity asymptotic to an exact homothetic solution and under what conditions are the asymptotic states of cosmological models represented by exact solutions of Einstein's field equations which admit a generalized self-similarity are addressed.

Assuming a particular form of metric containing seven constant parameters, Einstein's equations are solved for a perfect fluid in steady cylindrically symmetric rigid-body rotation about a regular axis, the equation of state being where . The resultant metric contains two free parameters. The angular velocity of the vortex is finite, and the motion in the radially infinite cylinder is without shear. The pressure p and density are also finite, vanishing at infinity. There are no singularities in the spacetime, which is of Petrov type I except for the case which is of type D. Some other cases, falling outside the range of given above, are discussed briefly.

A covariant algorithm is given to obtain principal 2-forms, Debever null directions and canonical frames associated with Petrov type I Weyl tensors. The relationship between these Weyl elements is explained, and their explicit expressions depending on Weyl invariants are obtained. These results are used to determine a cosmological observer in type I universes, and their usefulness in spacetime intrinsic characterization is shown.

What restrictions are there on a spacetime for which the Ricci curvature is such as to produce convergence of geodesics (such as the preconditions for the singularity theorems) but for which there are no singularities? We answer this question for a restricted class of spacetimes: static or stationary, geodesically complete, and globally hyperbolic. The answer is that, in at least one spacelike direction, the Ricci curvature must fall off (in a generalized manner of speaking) at a rate inversely quadratic in a naturally-occurring Riemannian metric on the space of stationary observers. Along the way, we establish some global results on the stationary observer space, regarding its completeness and its behaviour with respect to universal covering spaces; we also define a new geometric invariant for stationary spacetimes, related to causal behaviour (among other things).

We study irrotational, three-dimensional dust spacetimes and show that their only dynamical singularities are shell singularities in the sense of Kriele, Lim and Scott (1994). These singularities are weak and generically naked.

Some geometrical features of shear-free congruences in (2 + 1)-dimensional general spacetimes are studied. The object is to analyse the condition of vanishing shear as a possible relativistic extension of the rigidity condition in non-relativistic kinematics. As an example, the congruence modelling of a shear-free rotating disc in (2 + 1) Minkowski spacetime is determined.

We investigate topology change in (1 + 1) dimensions by analysing the scalar-curvature action at the points of metric-degeneration that (with minor exceptions) any non-trivial Lorentzian cobordism necessarily possesses. In two dimensions any cobordism can be built up as a combination of only two elementary types, the `yarmulke' and the `trousers.' For each of these elementary cobordisms, we consider a family of Morse-theory inspired Lorentzian metrics that vanish smoothly at a single point, resulting in a conical-type singularity there. In the yarmulke case, the distinguished point is analogous to a cosmological initial (or final) singularity, with the spacetime as a whole being obtained from one causal region of Misner space by adjoining a single point. In the trousers case, the distinguished point is a `crotch singularity' that signals a change in the spacetime topology (this being also the fundamental vertex of string theory, if one makes that interpretation). We regularize the metrics by adding a small imaginary part, whose sign is fixed to be positive by the condition that it lead to a convergent scalar field path integral on the regularized spacetime. As the regulator is removed, the scalar density approaches a delta-function, whose strength is complex: for the yarmulke family the strength is , where is the rapidity parameter of the associated Misner space; for the trousers family it is simply . This implies that in the path integral over spacetime metrics for Einstein gravity in three or more spacetime dimensions, topology change via a crotch singularity is exponentially suppressed, whereas appearance or disappearance of a universe via a yarmulke singularity is exponentially enhanced. We also contrast these results with the situation in a vielbein-cum-connection formulation of Einstein gravity.

We consider non-diagonal inhomogeneous cosmologies admitting an Abelian orthogonally transitive two-dimensional group of isometries under the assumption of separation of variables. We find all the sets of compatible systems of ordinary equations which give rise to reasonable solutions (satisfying energy conditions) and study them. It turns out that five of these families exist. Two of them represent stiff fluids with the density equal to the pressure. A third family satisfies an equation of state of the linear type . Regarding the fourth family we can integrate the field equations so that there only remains an ordinary differential equation for an unknown to be solved. We also give some explicit particular solutions in this case. The fifth family contains an explicit singularity-free stiff fluid solution which has already been studied by Mars in 1995 and an explicit singularity-free family satisfying the dominant energy condition (but not the strong energy condition) and which is eternally expanding and contracting without passing any big bang or big crunch.

We define a new parameter `cumulative drag index' for a particle in circular orbit in a stationary, axisymmetric gravitational field and study its behaviour for the two well known solutions of general relativity, namely the Kerr spacetime and the Gödel spacetime, wherein the inertial frame dragging has an important role. As it shows similar behaviour for both co- and counter-rotating particles, it may indeed be an indication of the influence of the faraway universe on local physics and thus Machian.

Gravitational radiation can be expressed in terms of an infinite series of radiative, symmetric trace-free (STF) multipole moments which can be connected to the behaviour of the source. We consider a truncated model for gravitational radiation from binary systems in which each STF mass and current moment of order l is given by the lowest-order, Newtonian-like l-pole moment of the orbiting masses; we neglect post-Newtonian corrections to each STF moment. Specializing to orbits which are circular (apart from the radiation-induced inspiral), we find an explicit infinite series for the energy flux in powers of v/c, where v is the orbital velocity. We show that the series converges for all values v/c < 2/e when one mass is much smaller than the other, and v/c < 4/e for equal masses, where e is the base of natural logarithms. These values include all physically relevant values for a compact binary inspiral. This convergence cannot indicate whether or not the full series (obtained from the exact moments) will converge. But if the full series does not converge, our analysis shows that this failure to converge does not originate from summing over the Newtonian part of the multipole moments.