Table of contents

According to Penrose's Weyl tensor hypothesis (1979), an appropriate boundary condition to impose at any initial singularity is that the Weyl tensor should vanish there. Cosmological models that admit isotropic singularities provide a mathematical framework for studying this hypothesis. Tod (1957) has conjectured that, within the isotropic singularity framework, the only models that satisfy the Penrose boundary condition are the exact FRW models. The author discusses the relationship of this conjecture to an earlier one of Goode and Wainwright (1985), and reduces the proof of Tod's conjecture to establishing uniqueness of solution to an appropriate initial value problem.

For colliding null fields Taub has shown that the energy-momentum tensor in the interaction region may take one of three forms. In fact, this is true only on the assumption that the spacetime is the product of two 2-spaces. A counterexample not satisfying this condition is pointed out.

The twistor equation is studied in a four-real-dimensional spacetime. All the metric tensors which locally admit a solution are found. They either belong to the Fefferman class or are given by the Trautman-Kerr-Schild anzatz by using a non-twisting null conformal Killing vector field in the Minkowski spacetime. The corresponding solutions are derived.

A unique orthonormal 3-frame for each spatial metric is selected using new rotational gauge conditions. In terms of these special frames the Hamiltonian is essentially non-negative and local, thus permitting more than a positive energy proof: a true 'localization' of energy. As these special orthonormal frames are also suitable variables for the initial value constraints they are good parameters for the space of solutions and well represent the true physical degrees of freedom for Einstein's theory of gravity.

The author gives a complete description of the phase space in the Kaluza-Klein theory with asymptotic behaviour inspired from that of the Kaluza-Klein monopole solution. This is done by exhibiting the explicit asymptotic and parity conditions that the metrics and momenta must satisfy. This allows one to give the definitions of the Hamiltonian, the linear and angular momenta, the generators of asymptotic boosts and the electrical charge.

It is possible to modify the spin-coefficient operators eth and thorn in such a way that they are conformally invariant. The resulting calculus is very useful, but appears somewhat ad hoc and it is not clear how to apply the operators to indexed quantities. The author shows that these operators are the directional derivatives associated with a preferred Weyl connection that exists in structures related to almost conformal Hermitian manifolds and extends the calculus to act on indexed quantities.

Gravitational fields are ordinarily measured by observing their effects on test particles, from whose trajectories one can reconstruct the geometry of spacetime. In 2+1 dimensions, spacetime geometry is encoded in the holonomies of a flat ISO(2,1) connection, and the quantum mechanical observables that correspond most closely to particle trajectories are Wilson lines. The author discusses the extraction of geometric information from such Wilson line observables, and shows that a quantum description of geometry can be recovered. In the process, the gravitational Hilbert space is investigated in some detail, and an interesting relationship to Thurston's theory of geometric structures (1979) is discovered.

The authors present a simple d=1 model of N=4 supersymmetric quantum mechanics (SQM) in which N=4 supersymmetry can be spontaneously broken down to N=2. When both SU(2) automorphism symmetries of N=4 1D supertranslation algebra are explicitly broken, this superalgebra acquires a constant central charge proportional to the product of two SU(2) breaking parameters. Due to this crucial property, no contradiction arises with Witten's theorem (1989) forbidding the partial spontaneous supersymmetry breaking in SQM models based on standard (having no central charges) 1D supertranslation algebras. For the partial breaking to really come about, it also proves necessary to consider the most general class of N=4, d=1 SQM Hamiltonians involving the terms quartic in fermion fields. They show that after performing a duality transformation these models can be interpreted as resulting from certain U(1) invariant N=2 2D Kahler sigma models via the Scherk-Schwartz-type dimensional reduction (1979).

A Palatini-type formulation of gravity coupled to matter and supergravity is given, in which the gravitational variables are a trio of self-dual 2-forms and an SL(2,C) connection. The action is polynomial in all the fields. This framework is shown to be the natural covariantization of Ashtekar's canonical formalism (1988), and is used to find the general vacuum solution of the four initial value constraints associated with spacetime diffeomorphisms in that formalism.

A new action principle, in which the only gravitational variables are an SL(2,C) connection and a scalar density, is derived for general relativity (GR) coupled to matter and for supergravity. In this form, GR appears as a non-metric, generally covariant gauge theory, the metric being reconstructed from the other fields in a solution. A similar non-metric action with a real SO(3,1) connection is also derived, however it involves an independent fourth rank tensor field representing the curvature.

The authors study toy p-branes from old string action, and regularize their Casimir energies by the Epstein zeta function. They obtain the special dimensions for the open (p=2, 3, 4) and closed (p=2, 3, 4, 6, 8) branes with equal sides. The upper bound of p is obtained. They also discuss the special dimensions for open and closed membranes with unequal sides. The resulting dimensions may be integers for the case of unequal sides.

The 1-loop renormalized effective potential for a phi ⁴ theory in the early Universe with a small inhomogeneity is evaluated under the adiabatic approximation. It is used to investigate the Coleman-Weinberg symmetry breaking processes usually leading to inflationary cosmologies in the presence of an inhomogeneity. The result shows that whether the symmetry is radiatively broken or not will crucially depend upon the scalar-gravitational coupling xi and the magnitude of the scalar curvature R. In particular, the author shows how the mode-mixing behaviour resulting from the inhomogeneity of spacetime is naturally replaced by introducing the non-local term into the field equation. This property allows the analysis of other quantum field effects in the inhomogenous spacetime, which will be presented in later papers.

The authors investigate the quantum cosmology of spatially locally homogeneous cosmological models with flat compact spatial surfaces in three and four spacetime dimensions. In (2+1) dimensions the only possible spatial surfaces are the 2-torus and the Klein bottle, whereas in (3+1) dimensions there are ten possible spatial surfaces. The metric ansatz is kept in its most general form compatible with Hamiltonian mini-superspace dynamics. They find that the no-boundary proposal of Hartle and Hawking (1983) is applicable, at the semiclassical level, in both of the (2+1)-dimensional models but in only six of the (3+1)-dimensional models.

Customarily the dynamical behaviour of the Bianchi IX model is studied either directly, i.e. numerically, or indirectly by approximating it to a map which is conducive to analytical treatment. These studies have given rise to results that appear somewhat incongruent. The authors clarify the relationship between these two treatments by firstly clearing the status of chaos in each setting and secondly by studying the possibility of the increase in entropy in each case.

The problem of equilibrium in a stationary gravitational (and electromagnetic) background is formulated for a macroscopic string model of the general kind applicable to phenomena ranging from ordinary elastic strings to superconducting cosmic strings. A key role is found to be played by the tangent vector given in terms of the Killing vector and the string surface stress momentum-energy tensor. The resulting system is worked out explicitly for string models of the special non-dispersive type that has been shown to take account of the averaged effect of small-scale oscillations in a simple Goto-Nambu model, and that also arises in the more artificial case of string models obtained by dimensional reduction.

For pt.I see ibid., vol.7, p.517, (1990). This is the second paper in a series of research started by Bagrov et al. The Abelian complete sets of matrix first-order differential symmetry operators for the Dirac equation are used for the separation of variables in the Dirac equation in Stackel spaces with external gauge fields. The permissible space metrics and gauge field potentials are not limited by field equations.

Some exact solutions of the general covariant Dirac equation in gravitational fields, namely in constant curvature de Sitter spacetime and in Milne's Universe are obtained. The tetrads for both cases are constructed on the basis of global quasi-Cartesian coordinates that allow one to exclude coordinate effects connected with the rotation of the local frame under the transition of the triad from one spacetime point to another. Solutions in the form of a spherical bispinor wave with time amplitude modulation are demonstrated for both cases of gravitational fields and the possibility of analogical solutions in the form of cylindrical and plane spinor waves for the de Sitter spacetime is discussed. All the solutions are obtained by means of the algebraic method of separation of variables.

The fermion-graviton system is studied at the linearized level in a (2+1)-dimensional spacetime with the gravitational Chern-Simons term. In this approximation it is shown that this system presents anomalous rotational properties and spin, in analogy with the gauge-field-matter system.

Classical evolution of density perturbations, while the scale remains in the large scale limit, is clarified in the following generalized gravity theories: f(R) gravity, generalized scalar tensor theory, gravity with non-minimally coupled scalar field and induced gravity. The author uses gauge-invariant potential and field variables and shows that the result can be expressed in a similar form to a conventional result derived in Einstein gravity with a minimally coupled scalar field. The author rederives the solutions in the large scale in a simple way using the conformal transformation properties of these theories. The author gives a general prescription for calculating the inflationary spectrum in these gravity theories which can be applicable to specific models.

The authors consider the five-dimensional Kaluza-Klein theory with a Dirac field coupled to the metric field. As a result, the authors obtain a four-dimensional Einstein-Maxwell-Dirac theory with an extra CP violating term. The resultant anomalous electromagnetic dipole moment of the electron is of the order of magnitude of about 10^-31 e cm, and may not be detectable by present experimental techniques. However, the mass of the charged Dirac particle is found to be of the order of the Planck mass (10^-7 g), so that the Kaluza-Klein theory containing spin-1/2 particles must be rejected.

According to the Maxwell equations in an arbitrary reference frame in general relativity the electric charge distribution may be completely compensated by a magnetic field in the presence of the frame's rotation. Exact self-consistent examples of systems containing a charged fluid, purely magnetic field and gravitation, are constructed forming a generalization of the Godel solution. The absence of an electric field in these systems in the co-moving (with matter) reference frame is in agreement with the above property of the Maxwell equations resulting in no repulsion between similar charges. In the limiting case when the charge density of the fluid vanishes, the Godel solution is recovered. Geometric, hydrodynamic and electromagnetic properties of the new solutions are studied.

Various dynamical aspects of a non-trivial theory of gravitation in (1+1) dimensions are investigated. It is shown that many features of Einstein's general relativity for (3+1) dimensions are qualitatively and to some extent quantitatively duplicated in this theory, including gravitational collapse, cosmological solutions and gravitational waves. Contact with Newton's theory is made and post-Newtonian expansions of the theory give relativistic corrections, similar to those of general relativity. Therefore this theory is presented as viable for both pedagogical purposes as well as a laboratory for the testing of new theoretical ideas.

For four-dimensional spacetimes with 2-spaces of constant curvature, the authors present a complete list of solutions of Einstein's vacuum field equations with nonzero cosmological constant. All solutions requiring a regular extension are given in manifestly regular double-null coordinates.

The authors analyse all presently known exact solutions of the vacuum Einstein and Einstein-Maxwell equations sharing the properties of being of Petrov type D, non-stationary, cylindrically symmetric, asymptotically flat and regular everywhere. All these solutions are particular subfamilies of a general one that describes the evolution of a single solitonic perturbation travelling on a flat spacetime background. They also investigate the relationship between these solitonic-type families and analytic extensions from stationary families. In particular they reobtain the Piran, Safier and Katz solution (1986) by performing an analytic extension from the massless Kerr-NUT spacetime.