Table of contents

Geometric inequalities holding for any convex body and connecting the perimeter of plane curves associated with this body to the total mean curvature of its boundary are proved. Their application to some analytic models of nonspherical collapse yields a more precise formulation of the hoop conjecture.

Using the Hamilton-Jacobi formalism the authors analyse the motion of test particles around local and global cosmic strings. In both cases, they determine the trajectories and they show that the particles cannot be trapped by the cosmic strings.

The expansion of an outgoing light-cone evolving according to the Einstein equations is found as an integral of the energy density of the infalling matter and gravitational radiation. A condition on the energy density is derived which is necessary and sufficient for the cone to be trapped, in which case it recollapses to a lower dimensional caustic. A similar condition on the energy density is derived which is necessary and sufficient for the cone to be free of caustics, and therefore complete.

General relativity (GR) has previously been extended to incorporate degenerate metrics using Ashtekar's Hamiltonian formulation of the theory. The authors show that a natural alternative choice for the form of the Hamiltonian constraints leads to a theory which agrees with GR for nondegenerate metrics, but differs in the degenerate sector from Ashtekar's original degenerate extension. The Poisson bracket algebra of the alternative constraints closes in the nondegenerate sector, with structure functions that involve the inverse of the spatial triad. Thus, the algebra does not close in the degenerate sector. They find that it must be supplemented by an infinite number of secondary constraints, which are shown to be first class (although their explicit form is not worked out in detail). All of the constraints taken together are implied by, but do not imply, Ashtekar's original form of constraints. Thus, the alternative constraints give rise to a different degenerate extension of GR. In the corresponding quantum theory, the single loop and intersecting loop holonomy states found in the connection representation satisfy all of the constraints. These states are therefore exact (formal) solutions to this alternative degenerate extension of quantum gravity, even though they are not solutions to the usual vector constraint.

The innermost stable circular orbits of binary systems of nonrotating, compact objects of arbitrary mass ratio are studied in the absence of gravitational radiation reaction. The method uses two-body equations of motion that are valid through (post)²-Newtonian order (order ((Gm/rc²)²)), but that also include test body effects in the Schwarzschild geometry exactly. The harmonic coordinate separation of the innermost stable orbit varies from the test body value of 5m to about 6m for equal masses, where m is the total mass. The orbital energy, angular momentum and frequency of the innermost stable orbit are also determined as a function of the ratio of the two masses. Implications of the results for detection of gravitational radiation by laser-interferometer gravitational observatories are discussed.

Renormalizability of a quantum gravity model with the local GL(4,R) symmetry is investigated. The renormalization conditions based on the action which consists of curvature-squared terms only do not exclude the appearance of fourth-derivative terms in the metric as counterterms. With the addition of all possible, fourth-derivative terms in the metric to the most general curvature-squared Lagrangian, the renormalization becomes complete. The proof is done using the Slavnov-Taylor identities and the power counting argument. Here, a higher-derivative gauge fixing for the general coordinate transformation group is used. Unitarity of the model is not shown.

The author considers the string in a constant profile plane fronted wave background, which corresponds to a noninertial frame in flat spacetime, and studies at the linearized level the bosonic massive spin-2 state propagation; he gives the exact string coordinates solution and derives consistent field equations and the relative Lagrangian for the spin-2 field.

The cosmological model with n (n>1) spaces of constant curvature has been investigated in the case when only one of them is not Ricci flat. The spacetime is minimally coupled to a massless scalar field. This model is an integrable one and the Wheeler-De Witt equation takes the form of the equation for the scalar filed in the Milne universe. The solutions of the WDW equation were obtained. Two ground-state wavefunctions related to the Hartle-Hawking and Vilenkin boundary conditions were found. The procedure of the third quantization was performed and two nonequivalent third quantized Fock spaces were obtained.

A complex matrix WKB approach is applied to study the black hole normal modes and quasibound states problems. In contrast to the standard WKB scheme no asymptotic expansions are used at any stage of the calculations. New quantization formulae are derived which allow for subsequent perturbative calculations to be made in terms of convergent expansions with controlled accuracy. Earlier WKB results are obtained in the lower order approximations of the present approach.

Spacetimes with Riemann-Christoffel curvature tensor characterized by the sum of directional derivatives of Dirac distributions with support in a finite number of parallel lines are studied. The energy-momentum tensor associated with these spaces is similar to the one associated with the usual cosmic strings except for its singular structure.

The author investigates the parameter space of the Einstein-axion axisymmetric black hole solutions. He finds that each point in the parameter space leads to either a naked singularity or to a black hole with a surface gravity that vanishes everywhere except on the symmetry axis. He therefore concludes that such black holes are physically unrealistic and do not violate the 'no-hair' conjecture.

A three-parameter family of solutions for perfect fluid cylinders is used as interior to the vacuum Levi-Civita metric for an infinite line mass. Matching is possible for 0<or= sigma <1/2, where sigma is the mass per unit proper length. This extends the range of sigma previously believed possible for an infinite line mass.

New algebraically special solutions of the Einstein-Maxwell equations are constructed. Among them are the first examples of solutions with twisting rays and a purely radiative Maxwell field.

A pure connection formulation of (2+1)-dimensional gravity, with a cosmological constant and coupling to a scalar field, is found using a Legendre transform from Ashtekar's Hamiltonian. The metric formulae, in the connection formulation, are shown to be a reformulation of Einstein's equations. It is shown that the infinite number of parameters used to generalize general relativity in 3+1 dimensions, have no counterpart in 2+1 dimensions. Topologically massive gravity is formulated in Ashtekar's variables, and in the pure connection Lagrangian form.

A generic class of theories where gravity is mediated by one tensor field together with an arbitrary number of scalar fields is considered. The predictions of these theories are worked out in four different observationally relevant regimes: (i) quasi-stationary weak fields (solar system conditions); (ii) rapidly varying weak fields (gravitational wave experiments); (iii) quasi-stationary strong fields (motion of systems of compact bodies, i.e. neutron stars or black holes); and (iv) the mixing of strong and radiative field effects in the gravitational radiation of N-compact-body systems. Moreover, the authors derive several significant relations between the theoretical quantities entering these predictions. They show how strong-field-gravity effects in the motion and gravitational radiation of N-compact-body systems can be parametrized by a set of theory parameters that generalize the usual post-Newtonian parameters ( gamma , beta ,. . .) introduced in the context of quasi-stationary weak gravitational fields. These new parameters ( beta ₂, beta ', beta ₃, beta ",. . .) provide a chart for the yet essentially unexplored domain of strong-gravitational-field effects, and thereby suggest new directions for testing relativistic gravity. This is illustrated by studying in detail a specific two-parameter tensor-bi-scalar theory T( beta ', beta ") which has the same post-Newtonian limit as general relativity but leads to new nonEinsteinian predictions for the various observables that can be extracted from binary pulsar data.

Scaling arguments are generalized to a class of self-gravitating matter models with spherically symmetric field configurations. These include the Einstein-Yang-Mills system for any gauge group and also the Einstein-sigma model, without or with an additional Skyrme term. For the considered class it is possible to express the Schwarzschild mass as a relatively simple nonlocal functional of the matter variables alone. The behaviour of this functional with respect to scaling transformations leads to interesting consequences. In some cases the authors can exclude particle-like solutions, whereas for other models they find some virial relations including gravity. The method is also extended to black holes. As an application they give some necessary existence conditions for solutions of various matter models on a fixed black hole background. Finally they apply the arguments to self-coupled black hole solutions.

In a recent paper devoted to the linearized field equations in R+R² gravitational theories, Xu and Ellis (1991) have attempted to determine the metric of a static spherically symmetric body in harmonic coordinates. The author points out that their result is erroneous and he exhibits the correct solution. The form of this solution shows that the harmonic coordinate condition is not convenient in linearized R+R² gravity.