Table of contents

We give a formulation of the vacuum Einstein equations in terms of a set of volume-preserving vector fields on a 4-manifold . These vectors satisfy a set of equations which are a generalization of the Yang - Mills equations for a constant connection on flat spacetime.

Gedankenexperiment with the entropy and energy exchange between a thick box and a black hole is considered. It is shown that validity of the generalized second law follows from the ordinary second one. In particular, it manifests itself in positivity of the heat capacity of a box. Our treatment does not use either the Unruh - Wald or Bekenstein entropy bound. It is demonstrated that the latter itself proves to be the consequence of laws of themodynamics.

Two-dimensional models of black-hole evaporation processes hold the potential of settling the question of whether black-holes have a consistent quantum mechanics. Various proposals are surveyed and appraised.

The theory of vectors and spinors in (9 + 1)-dimensional spacetime is introduced in a completely octonionic formalism based on an octonionic representation of the Clifford algebra . The general solution of the classical equations of motion of the CBS superparticle is given to all orders of the Grassmann hierarchy. A spinor and a vector are combined into a Grassmann, octonionic, Jordan matrix in order to construct a superspace variable to describe the superparticle. The combined Lorentz and supersymmetry transformations of the fermionic and bosonic variables are expressed in terms of Jordan products.

In this paper we discuss the quantum potential approach of Bohm in the context of a quantum cosmological model. This approach makes it possible to convert the wavefunction of the universe into a set of equations describing the time evolution of the universe. Following Ashtekar et al we make use of a quantum canonical transformation to cast a class of a quantum cosmological models into a simple form in which they can be solved explicitly, and then we use the solutions to recover the time evolution.

We study real linear scalar field theory on two simple non-globally hyperbolic spacetimes containing closed timelike curves within the framework proposed by Kay for algebraic quantum field theory on non-globally hyperbolic spacetimes. In this context, a spacetime (M,g) is said to be `F-quantum compatible' with a field theory if it admits a *-algebra of local observables for that theory which satisfies a locality condition known as `F-locality'. Kay's proposal is that, in formulating algebraic quantum field theory on (M,g), F-locality should be imposed as a necessary condition on the *-algebra of observables.

The spacetimes studied are the two- and four-dimensional spacelike cylinders (Minkowski space quotiented by a timelike translation). Kay has shown that the four-dimensional spacelike cylinder is F-quantum compatible with massless fields. We prove that it is also F-quantum compatible with massive fields and prove the F-quantum compatibility of the two-dimensional spacelike cylinder with both massive and massless fields. In each case, F-quantum compatibility is proved by constructing a suitable F-local algebra.

We attempt to understand the fate of spacelike gravitational singularities in string theory via the quantum stress tensor for string matter in a fixed background. We first approximate the singularity with a homogeneous anisotropic background and review the minisuperspace equations describing the evolution of the scale factors and the dilaton. We then review and discuss the behaviour of large strings in such models. In a simple model which expands isotropically for a finite period of time we compute the number density of strings produced by quantum pair production and find that this number, and thus the stress tensor, becomes infinite when the Hubble volume of the expansion exceeds the string scale, in a manner reminiscent of the Hagedorn transition. Based on this calculation we argue that either the region near the singularity undergoes a phase transition when the density reaches the order of a string mass per string volume, or that the backreaction of the produced string matter dramatically modifies the geometry.

Within the framework of the quantum field theory at finite temperature on a conical space, we determine the Euclidean thermal spinor Green's function for a massless spinor field. We then calculate the thermal average of the energy - momentum tensor of a thermal bath of massless fermions. In the high-temperature limit, we find that the straight cosmic string does not perturb the thermal bath.

In this work, the behaviour of the system with N massive parallel rigid wires is analysed. The aim is to explore its resemblance to a system of multiple cosmic strings. Assuming that it behaves like a `gas' of massive rigid wires, we use a thermodynamics approach to describe this system. We obtain a constraint relating the linear mass density of the massive wires, the number of the massive wires in the system and the dispersion velocity of the system.

We prove that the maximal development of any spherically symmetric spacetime with collisionless matter (obeying the Vlasov equation) or a massless scalar field (obeying the massless wave equation) and possessing a constant mean curvature Cauchy surface also contains a maximal Cauchy surface. Combining this with previous results establishes that the spacetime can be foliated by constant mean curvature Cauchy surfaces with the mean curvature taking on all real values, thereby showing that these spacetimes satisfy the closed-universe recollapse conjecture. A key element of the proof, of interest in itself, is a bound for the volume of any Cauchy surface in any spacetime satisfying the timelike convergence condition in terms of the volume and mean curvature of a fixed Cauchy surface and the maximal distance between and . In particular, this shows that any globally hyperbolic spacetime having a finite lifetime and obeying the timelike-convergence condition cannot attain an arbitrarily large spatial volume.

Three-dimensional black-hole solutions of a generalized dilaton gravity theory are analysed. The theory is specified by two fields, the dilaton and the graviton , and two parameters, the cosmological constant and the Brans - Dicke parameter . It contains seven different cases, of which one distinguishes as special cases, string theory, general relativity and a theory equivalent to four-dimensional general relativity with one Killing vector. We study the causal structure and geodesic motion of null and timelike particles in the black-hole geometries and find the ADM masses of the different solutions.

A recently suggested procedure to define inertial forces in a general spacetime is considered in the Newtonian limit, and found to be in contradiction with Newtonian mechanics when the gravitational field is time dependent. A slight modification of the fundamental equation of the formalism, however, restores the validity of the correspondence principle.

We discuss the definitions of standard clocks in theories of gravitation. These definitions are motivated by the invariance of actions under different gauge symmetries. We contrast the definition of a standard Weyl clock with that of a clock in general relativity and argue that the historical criticisms of theories based on non-metric compatible connections by Einstein, Pauli and others must be considered in the context of Weyl's original gauge symmetry. We argue that standard Einsteinian clocks can be defined in non-Riemannian theories of gravitation by adopting the Weyl group as a local gauge symmetry that preserves the metric and discuss the hypothesis that atomic clocks may be adopted to measure proper time in the presence of non-Riemannian gravitational fields. These ideas are illustrated in terms of a recently developed model of gravitation based on a non-Riemannian spacetime geometry.

Exact singularity-free Senovilla-type solutions are obtained in five-dimensional spacetime for a fluid with or without heat flow. The energy density and the pressure of the fluid satisfy a barotropic equation of state and remain finite throughout the evolution in each case. It is possible to find exact solutions in both the cases - one with the anisotropic pressure and the other with isotropic pressure where . The latter includes, in a special case, one of the vacuum solutions obtained earlier.