Table of contents

We present a covariant definition of inertial forces in general relativity (gravitational, centrifugal, Euler and Coriolis-Lense-Thirring) which is valid in all spacetimes including ones with no symmetry.

The causal boundary of string propagation-defined as the hypersurface in loop space bordering the timelike (spacelike) domains in which two successive measurements of the string field do (do not) interfere with one another-is argued to be O= integral d sigma ( delta X( sigma ))²= Sigma ^infinity _{l=- infinity} delta x_-l^mu delta x_{mu l}. Some possible consequences are discussed.

N=4 SU(2) conformal invariance is studied in harmonic superspace. Two examples of N=4 conformal models are constructed together with their supercurrent generating the (4,0) conformal symmetry of these models. The Feigin-Fuchs representation of the N=4 conserved current is described and the N=4 conformal anomaly is given. Other features are also studied.

In analogy to what happens with the chiral anomaly, we show that the trace (conformal) anomaly can also be understood as a discrepancy between classical and quantum field equations. We work out explicitly a simple example using a regularization similar to that employed by Tsutsui. We also show how to generalize this kind of regularization in order to establish a connection with other ones, like the generalized zeta -function method.

Operator realization of quantum constraints and the lowest-order structure functions is found in the 1-loop (linear in h) approximation of the Dirac and BFV quantization for the general gauge theory subject to first-class constraints. Semiclassical equivalence of the Dirac and BFV methods to the reduced quantization of physical variables is established in terms of the conserved inner product in the space of physical states. The global obstructions to this result are briefly discussed in connection with the ideas of the secondary (third) quantization of constrained systems.

Covariant operator realization of Dirac constraints is proposed in theories of the gravitational type, subject to Hamiltonian and momentum constraints. This realization is based on their metric and Killing configuration-space structures, has Hermitian conjugation properties following from the BFV quantization method and generalizes to the level of the exact theory (apart from the problem of the ultraviolet regularization and anomalies) the recently proposed semiclassical algorithms in models with arbitrary first-class constraints.

We apply the Stueckelberg formalism to describe massive and massless quantum vector fields propagating in the static metric of a (1+1) black hole. We find exact solutions of the field equations and compute the expectation value of the energy-momentum tensor. In the massless case, the renormalized stress-energy tensor vanishes identically. In the massive one, it agrees with the results obtained for scalar and Dirac fields in the region near the event horizon, yielding a temperature of M/2 pi for the black hole.

We study the classical and quantum dynamics of strings near spacetime singularities. We deal with singular gravitational plane waves of arbitrary polarization and arbitrary profile functions W₁(U) and W₂(U), whose behaviour for U to 0 is W₁(U)= alpha ₁/ mod U mod ( beta ₁), W₂(U)= alpha ₂/ mod U mod ( beta ₂) (U being a null variable). In these spacetimes, the string dynamics is exactly (and explicitly) solvable even at the spacetime singularities. The string behaviour depends crucially on whether both parameters beta ₁ and beta ₂ are smaller or larger than two. When beta ₁>or=2 and/or beta ₂>or=2, the string does not cross the singularity U = 0 but goes off to infinity in a given direction alpha depending on the polarization of the gravitational wave. The string time evolution is fully determined by the spacetime geometry, whereas the overall sigma -dependence is fixed by the initial string state. The proper length at fixed tau to 0 (U to 0) stretches infinitely. For beta ₁<2 and beta ₂<2, the string passes smoothly through the spacetime singularity and reaches the U>0 region. In this case, outgoing operators make sense and we find the explicit transformation relating in and out operators. For the quantum string states, this implies spin polarization rotations and particle transmutations. The expectation values of the outgoing mass and mode number operators are computed and the excitation of the string modes after the crossing of the spacetime singularity is analysed.

The one-loop renormalized effective potentials for the massive phi ⁴ theory on the spatially homogeneous models of Bianchi type I and Kantowski-Sachs type are evaluated. It is used to see how the quantum field affects the cosmological phase transition in the anisotropic spacetimes. For reasons of the mathematical technique it is assumed that the spacetimes are slowly varying or have specially metric forms. We obtain the analytic results and present detailed discussions about the quantum field corrections to the symmetry breaking or symmetry restoration in the model spacetimes.

In Riemannian geometry the only ambiguity in a metric that is compatible with a symmetric connection arises from scaling by constants and de Rham decompositions. The situation for pseudo-Riemannian metrics is, however, quite different. In this paper this phenomenon is studied. One of the main conclusions reached is that there is an intimate link between the problem of the existence of alternative metrics for a symmetric connection of higher order tangent bundle structures. Such alternative metrics are provides a solution to the problem in principle, the difficulties of which derive from the plethora of Jordan canonical forms of a nilpotent matrix. The paper concludes with a description of all alternative metrics for manifolds of dimension five and less.

We investigate the behaviour of the solutions to the Einstein equations with a causal viscous fluid source in the case of the spatially flat Robertson-Walker metric. In our model, the bulk viscosity coefficient is related to the energy density as zeta = alpha rho ^m, and the relaxation time is given by zeta / rho . We find the exact solutions when m = 1/2 , and we study analytically the asymptotic stability of the distinct families of solutions for arbitrary m. We find that the qualitative asymptotic behaviour in the far future is not altered by relaxation processes, but they change the behaviour in the past, either excluding deflationary evolutions or making the Universe bounce due to the violation of the energy conditions.

We analyse a solution of the Einstein field equations which describe a finite incompressible and static spherical shell of matter as a source of the vacuum Schwarzschild spacetime. The outer boundary is at a coordinate radius less than 9M/4 but the inner surface is singular as a consequence of the assumed uniformity of the matter density. The presence of a curvature singularity at the inner surface allows the outer boundary of the shell to be arbitrarily close to the Schwarzschild event horizon. Since the spacetime of the shell interior satisfies the conditions for the Lynden-Bell and Katz definition of matter and gravitational energies, we analyse how these quantities behave in the limit to a black hole.

Using a simple model problem we show that the w-modes found by Kojima (1988) and by Kokkotas and Schutz (1986) appear in a very natural way in systems of coupled wave equations. As has been shown by Kind (1992) such wave equations govern relativistic star oscillations.

The conditions of equilibrium are obtained for a charged test particle in the field of a spherical charged mass. Einstein-Maxwell theory is used. The classical condition q₁q₂=Gm₁m₂ is neither necessary nor sufficient, whereas q₁=+or-G¹2/m₁, q₂=+or-G¹2/m₂ is sufficient but not necessary. There are also separation-dependent equilibrium positions, and some of these allow the charges to have opposite signs. Stability to radial displacements is investigated.

An approach to a canonical formulation of three-dimensional topologically massive gravity is developed. The constraints are calculated in an explicit form both for this theory and for pure Chern-Simons gravity. The local measure in the corresponding path integrals is found. It is shown that owing to the local measure factors the one-loop volume divergences are absent in these theories.

A theorem is proved which reduces the problem of completeness of orbits of Killing vector fields in maximal globally hyperbolic, say, vacuum spacetimes to some properties of the orbits near the Cauchy surface. In particular it is shown that all Killing orbits are complete in maximal developments of asymptotically flat Cauchy data, or of Cauchy data prescribed on a compact manifold.

In a spacetime fibred by spacelike hypersurfaces (called 'space') such that light travels on geodesics in space, i.e. light geodesics have geodesic space projections, the space geometry must be fixed for all times up to rescaling, and the time scale is spatially independent.

The Hamiltonian and energy of the closed Universe with respect to a physical system of reference, which is related to the gas of clocks, are defined.

We study the change of signature of the metric in a five-dimensional Kaluza-Klein theory applied to the description of a primordial cosmology. A model is constructed in which the one-dimensional internal space with the topology of S¹ is created in the Euclidean region. We also consider the modified case, when the Gauss-Bonnet term is added to the Lagrangian. It produces the inflation of the four-dimensional Robertson-Walker metric, impossible to obtain with the Einstein-Hilbert Lagrangian.

It is shown that for a given spherically symmetric distribution of a perfect fluid on a spacelike hypersurface with a boundary and a given, time-dependent boundary pressure, there exists a unique, local-in-time solution of the Einstein equations. A Schwarzchild spacetime can be attached to the fluid body if and only if the boundary pressure vanishes. We assume a smooth equation of state for which the density and the speed of sound remain positive for vanishing pressure.

The linearized Einstein equations for a static, spherically symmetric fluid ball and its empty surroundings are considered. It is shown that, given initial data obeying the constraints, there exists a unique solution, which describes the motion of the perturbed fluid and the gravitational waves propagating inside and outside the fluid ball. The physical junction conditions for the boundary of the ball suffice to determine the evolution inside and outside of the ball in terms of initial values. The equation of state is assumed smooth and such that the density and the speed of sound remain positive for vanishing pressure.

The Lanczos spin tensor is obtained for all empty type D spacetimes.

A modification of Kaluza-Klein theory has been proposed in which the internal space is replaced by an algebraic structure described by a non-commutative geometry based on the algebra M_n of n * n matrices. It is shown that this theory is identical to traditional Kaluza-Klein theory with the group SU_n as the internal space provided that in the latter one makes the assumption that only SU_n-invariant modes are used. In the algebraic version there is an added freedom in that the 'curvature' of the internal structure can take arbitrary values. Some cosmological implications of the resulting Einstein-Hilbert action are mentioned.

Adopting a simple ansatz, we find exact solutions to the Einstein-Maxwell-dilaton equations, which stand for the multi-black hole configuration with maximal charge in a cosmological metric and dilaton field background driven by a cosmological term.

Spacetimes admitting non-null valence two Killing spinors are discussed. An explicit example of such a spacetime is given which shows that, contrary to claims appearing in the literature, not all of these spacetimes have been listed exhaustively. We also give conditions for determining whether a given spacetime is conformally related to a Killing-Yano spacetime.

The field equations are linearized and solved. Some comments are made on the implications of the unavoidable occurrence of singularities in these solutions.

Using a technique proposed by Horsky and Mitskievic(1989) a perfect fluid solution may be generalized to a charged perfect fluid solution with a vanishing Lorentz force if a particular condition is fulfilled. We give this condition in the present work.