Table of contents

We establish the equality of the ADM mass and the total electric charge for asymptotically flat, static electrovacuum black hole spacetimes with completely degenerate, not necessarily connected, horizons.

An essential step towards the identification of a fermion mass generation mechanism at the Planck scale is to analyse massive fermions in a given quantum gravity framework. In this letter the two mass terms entering the Hamiltonian constraint for the Einstein - Majorana system are studied in the loop representation of quantum gravity and fermions. One term resembles a bare mass gap because it is not zero for states with zero (fermion) kinetic energy, unlike the other term which is interpreted as `dressing' the mass. The former contribution originates from (at least) triple intersections of the loop states acted on whilst the latter is traced back to every pair of coinciding end points, where fermions are located. Thus, fermion mass terms get encoded in the combinatorics of loop states. Finally, the possibility is discussed of relating fermion masses to the topology of space.

We discuss the quantized, D = 4, non-Abelian BF theory within the superfibre bundle formalism. By introducing an even pseudotensorial 2-superform over a principal superfibre bundle with superconnection, we obtain the geometrical BRST and anti-BRST transformations of the fields occurring in such a theory. These are the closed modulo classical equations of motion. Then, giving a prescription to construct the gauge-fixing action through a modified BRST operator, we determine the on-shell nilpotent BRST symmetry and the quantum action. We also show how this quantum action is invariant under the on-shell nilpotent anti-BRST symmetry obtained by modifying the geometrical anti-BRST operator. However, we find that, in relation to the anti-BRST symmetry, the quantum action also possesses a vector supersymmetry of ghost number (+1).

We propose a generalized action for N = 1 superparticles in D = 3,4,6 and 10 spacetime dimensions in terms of the extrinsic geometry of the worldline superspace. The superfield equations of motion are derived. The off-shell superdiffeomorphism invariance (in the rheonomic sense) of the superparticle generalized action is demonstrated. We show that half of the fermionic and one bosonic (super)fields disappear from the generalized action in the analytical basis. We also derive the equations of motion for a superparticle interacting with an Abelian gauge field.

We discuss the following aspects of two-dimensional N = 2 supersymmetric theories defined on compact super Riemann surfaces: parametrization of (2,0) and (2,2) superconformal structures in terms of Beltrami coefficients and formulation of superconformal models on such surfaces (invariant actions, anomalies and compensating actions, Ward identities).

We give a class of exact solutions of the five-dimensional (Kaluza - Klein) equations of general relativity, identify the mass and electric charge of the source and solve the geodesic equation for the motion of a charged test particle. There are interesting differences between charged black holes in four and five dimensions that could, in principle, be used to test the dimensionality of the world.

A canonical quantization is developed for massive vector fields on ultrastatic spacetimes with compact Cauchy surfaces. At the classical level, the field equation is recast as an abstract Cauchy problem subject to a subsidiary condition. Time evolution is obtained as a unitary mapping of Cauchy data, and the subsidiary condition is satisfied by restricting the data. At the quantum level, field operators are constructed and a representation of the CCRs is obtained on a Fock space.

We show some general properties of the conformal Yano - Killing tensor on a Riemannian manifold. Several differential and algebraic equations are derived.

The asymptotic conformal Yano - Killing tensor proposed earlier by the author is analysed for the Schwarzschild metric and the tensor equations defining this object are given. The result shows that the Schwarzschild metric (and other metrics which are asymptotically `Schwarzschildean' up to at spatial infinity) is among the metrics fulfilling stronger asymptotic conditions and supertranslation ambiguities disappear. It is also clear from the result that 14 asymptotic gravitational charges are well defined on the `Schwarzschildean' background.

All global solutions of arbitrary topology of the most general (1 + 1)-dimensional dilaton gravity models are obtained. We show that for a generic model there are globally smooth solutions on any non-compact 2-surface. The solution space is parametrized explicitly and the geometrical significance of continuous and discrete labels is elucidated. As a corollary we gain insight into the (in general non-trivial) topology of the reduced phase space.

The classification covers basically all two-dimensional metrics of Lorentzian signature with a (local) Killing symmetry.

We investigate some properties of geometric operators in canonical quantum gravity in the connection approach à la Ashtekar, which are associated with the volume, area and length of spatial regions. We give the motivations for the construction of analogous discretized lattice quantities, compute various quantum commutators of the type [area, volume], [area, length] and [volume, length], and find that they are generally non-vanishing.

Although our calculations are performed mostly within a lattice-regularized approach, some are - for special, fixed spin-network configurations - identical to corresponding continuum computations. Comparison with the structure of the discretized theory leads us to conclude that anomalous commutators may be a general feature of operators constructed along similar lines within a continuum loop representation of quantum general relativity; the validity of the lattice approach remains unaffected.

We will apply the quantum-inequality-type restrictions to Alcubierre's warp drive metric on a scale in which a local region of spacetime can be considered `flat'. These are inequalities that restrict the magnitude and extent of the negative energy which is needed to form the warp drive metric. From this we are able to place limits on the parameters of the `warp bubble'. It will be shown that the bubble wall thickness is on the order of only a few hundred Planck lengths. Then we will show that the total integrated energy density needed to maintain the warp metric with such thin walls is physically unattainable.

An action for simplicial Euclidean general relativity involving only left-handed fields is presented. The simplicial theory is shown to converge to continuum general relativity in the Plebanski formulation as the simplicial complex is refined. This contrasts with the Regge model for which Miller and Brewin have shown that the full field equations are much more restrictive than Einstein's in the continuum limit. The action and field equations of the proposed model are also significantly simpler than those of the Regge model when written directly in terms of their fundamental variables.

An entirely analogous hypercubic lattice theory, which approximates Plebanski's form of general relativity, is also presented.

A completely integrable two-dimensional model of gravity with dynamical metric and torsion coupled to 2D massless Dirac matter is extended by inclusion of a nonlinear fermionic interaction. The classical equations of motion of this model are solved by considering perturbations up to second order. The Green function for the massless Dirac equation in this model is also calculated.

I introduce the spacetime foam structure by briefly reviewing the ideas of Wheeler, topological fluctuations and the possibility of virtual black holes. The contribution of Jacobson (the equation of state of the foam) is recalled.

In the second part, I introduce a model of spacetime foam at the surface of the event horizon of a black hole. I apply these ideas to the calculus of the number of states of a black hole, of its entropy and of other thermodynamical properties. A formula for the number of micro-holes on the surface of the event horizon is derived.

Subsequently, I extend the thermodynamical properties of the event horizon to thermodynamical properties of the space. Here I face the problem of the maximum entropy contained in a space region of a given volume.

Finally, on the basis of the results obtained previously, I briefly treat the possibility of micro-black-hole creation by the Unruh effect.

We derive a first-order formalism for solving (2 + 1) gravity on Riemann surfaces, analogous to the recently discovered classical solutions for N moving particles. We choose the York time gauge and the conformal gauge for the spatial metric. We show that Moncrief's equations of motion can be generally solved by the solution f of a -model. We build out of f a mapping from a regular coordinate system to a Minkowskian multivalued coordinate system. The polydromy is in correspondence with the branch cuts on the complex plane representing the Riemann surface. The Poincaré holonomies, which define the coupling of Riemann surfaces to gravity, simply describe the Minkowskian free motion of the branch points. By solving f we can find the dynamics of the branch points in the physical coordinate system. We check this formalism in some cases, i.e. for the torus and for every Riemann surface with SO(2,1) holonomies.

The exact general solution to the Einstein equations in a homogeneous Universe with a full causal viscous fluid source for the bulk viscosity index is found. We have investigated the asymptotic stability of Friedmann and de Sitter solutions, the former is stable for and the latter for . The comparison with results of the truncated theory is made. For , it was found that families of solutions with extrema no longer remain in the full case and they are replaced by asymptotically Minkowski evolutions. These solutions are monotonic.

We find a static, anisotropic, non-supersymmetric generalization of the extreme supersymmetric domain walls of simple non-dilatonic supergravity theory. As opposed to the time-dependent isotropic non- and ultra-extreme domain walls, the anisotropic non-extreme wall has the same spatial topology as the extreme wall. The solution has naked singularities which vanish in the extreme limit.

The method of averaging is used to investigate the phenomenon of capture into resonance for a model that describes a Keplerian binary system influenced by radiation damping and external normally incident periodic gravitational radiation. The dynamical evolution of the binary orbit while trapped in resonance is elucidated using a second-order partially averaged system. This method provides a theoretical framework which can be used to explain the main evolutionary dynamics of a physical system that has been trapped in resonance.

It is shown via the principle of path independence that the (time gauge) constraint algebra derived by Charap et al for vielbein general relativity is a generic feature of any covariant theory formulated in a vielbein frame. In the process of doing so, the relationship between the coordinate and orthonormal frame algebera is made explicit.

We study motion in the field of two fixed centres described by a family of Einstein - Maxwell - dilaton theories. Transitions between regular and chaotic motion are observed as the dilaton coupling is varied.

In this paper we present a class of metrics to be considered as new possible sources for the Kerr metric. These new solutions are generated by applying the Newman - Janis algorithm (NJA) to any static spherically symmetric (SSS) `seed' metric. The continuity conditions for joining any two of these new metrics is presented. A specific analysis of the joining of interior solutions to the Kerr exterior is made. The boundary conditions used are those first developed by Dormois and Israel. We find that the NJA can be used to generate new physically allowable interior solutions. These new solutions can be matched smoothly to the Kerr metric. We present a general method for finding such solutions with oblate spheroidal boundary surfaces. Finally, a trial solution is found and presented.

This work extends the ideas developed in two previous papers by the authors. First- and second-order perturbation solutions of Einstein's equations (in Newman - Penrose form) for the Bondi - Sachs metric are found on a background Minkowski manifold. These solutions allow a tensorial calculation of the Bondi mass using the Taub superpotential.

Using second-order black-hole perturbation theory, we show that the difference between the ADM mass and the final black-hole mass, computed to the lowest significant order, is equal, to the same order, to the total gravitational radiation energy, obtained by applying the Landau - Lifschitz (pseudotensor) equation to the first-order perturbation. This result may be considered as a consistency check for the theory.

It has been recently pointed out that a definition of the geometric entropy using the partition function in a conical space does not in general lead to a positive-definite quantity. For a scalar field model with a non-minimal coupling we clarify the origin of the anomalous behaviour from the viewpoint of the canonical formulation.

In a cosmological context, the electric and magnetic parts of the Weyl tensor, and , represent the locally free curvature, i.e. they are not pointwise determined by the matter fields. By performing a complete covariant decomposition of and , we show that the parts of the derivative of the curvature which are locally free (i.e. not pointwise determined by the matter via the Bianchi identities) are exactly the symmetrized trace-free spatial derivatives of and together with their spatial curls. These parts of the derivatives are shown to be crucial for the existence of gravitational waves.

The gravitational dynamics of anisotropic elastic spheres supported only by tangential stresses and satisfying an equation of state is analysed, and a fairly large class of non-static, spherically symmetric solutions of the Einstein field equations is found by quadratures. The solutions contain three arbitrary functions. Two such functions are immediately recognized as the initial distributions of mass and energy, familiar from the Tolman - Bondi (dust) models, while the third is the elastic internal energy per unit volume. If this function is a constant, the energy density becomes proportional to the matter density and therefore the metric reduces to the Tolman - Bondi one. In the general case, however, the solutions contain oscillating models as well as finite-bouncing models.

Schmutzer studied a homogeneous and isotropic dust model of the universe and gave some special exact solutions of it. In the present paper we obtain more general exact solutions of the above model.

Dynamical systems for hypersurface homogeneous and hypersurface self-similar models with non-null symmetry surfaces are derived. The equations are cast in a geometric form based on properties of the symmetry surfaces that emphasize the close connection between the various models. Perfect fluid models are discussed in particular. It is shown how the models form a hierarchical structure where simpler models act as building blocks for more complicated ones. Expressions for the fluid's kinematic properties are given and discussed in the context of various special cases.

A broad class of generalized gravity theories can be cast into Einstein gravity with a minimally coupled scalar field using a suitable conformal rescaling of the metric. Using this conformal equivalence between the theories, we derived the equations for the background and the perturbations, and the general asymptotic solutions for the perturbations in the generalized gravity from the simple results known in the minimally coupled scalar field. Results for the scalar and tensor perturbations can be presented in unified forms. The large-scale evolutions for both perturbations are characterized by corresponding conserved quantities. The simple result for the scalar perturbation is possible mainly due to our proper choice of a gauge-invariant combination which corresponds to the perturbed scalar field in the uniform-curvature gauge.

In this comment we bring attention to the fact that when we apply the ontological interpretation of quantum mechanics, we must be sure to use it in the representation in which we want to express the physical results. This is particularly important when canonical transformations that mix momenta and coordinates are present. This implies that some of the results obtained by Baut and Kowalski-Glikman are incorrect.

The properties of a transformation previously considered for generating new perfect-fluid solutions from known ones are further investigated. It is assumed that the 4-velocity of the fluid is parallel to the stationary Killing field, and also that the norm and the twist potential of the stationary Killing field are functionally related. This case is complementary to the case studied in our previous paper. The transformation can be applied to generate possibly new perfect-fluid solutions from known ones only for the case of the barotropic equation of state or, alternatively, for the case of a static spacetime. For static spacetimes our method recovers the Buchdahl transformation. It is demonstrated, moreover, that Herlt's technique for constructing stationary perfect-fluid solutions from static ones is, actually, a special case of the method considered in this paper.