International Journal of Modern Physics D

The basic features of the complex canonical formulation of general relativity and the recent developments in the quantum gravity program based on it are reviewed. The exposition is intended to be complementary to the review articles already available and some original arguments are included. In particular the conventional treatment of the Hamiltonian constraint and quantum states in the canonical approach to quantum gravity is criticized and a new formulation is proposed.

We briefly discuss the nature of spacetimes after two impulsive gravitational waves collide. The discussion is based mainly on recent works of Chandrasekhar and Xanthopoulos.

We investigate the physical interpretation of the loop states that appear in the loop representation of quantum gravity. By utilizing the “weave” state, which has been recently introduced as a quantum description of the microstructure of flat space, we analyze the relation between loop states and graviton states. This relation determines a linear map M from the state-space of the nonperturbative theory (loop space) into the state-space of the linearized theory (Fock space). We present an explicit form of this map, and a preliminary investigation of its properties. The existence of such a map indicates that the full nonperturbative quantum theory includes a sector that describes the same physics as (the low energy regimes of) the linearized theory, namely gravitons on flat space.

Ashtekar’s formulation of canonical gravity is extended to N=2 supergravity. The action has a complex first-order chiral form and is invariant under two local supersymmetry transformations. Since we adopt the chiral formulation, left- and right-handed supersymmetries appear in an asymmetric way. The supersymmetry algebra is not closed on auxiliary fields, which is a peculiar result of the chiral formulation, where the right-handed supersymmetries are realized by constrained parameters. By means of the canonical formalism, simple polynomial constraints are derived and we can show that the Poisson brackets of supersymmetry currents are actually closed. This guarantees the consistency of our supersymmetry transformations, which is unclear at the level of the algebra on the fields.

We develop the perturbative analysis of two-dimensional quantum gravity with local action. The general renormalization structure of the theory is discussed. The one-loop divergences of the effective action in the covariant gauge of the general type are found. The technique for the calculation of the one-loop divergences for nonminimal operators in two dimensions is presented. The gauge dependence of the counterterms is discussed. It is also presumed that there are no one-loop divergences in some particular covariant gauge (finiteness.)

The renormalization group analysis is given. The addition of the cosmological constant and Liouville type term is considered.

We extend the three-dimensional stringy black hole of Horne and Horowitz to four dimensions. After a brief discussion of the global properties of the metric, we discuss the stability of the background with respect to small perturbations, following the methods of Gilbert and of Chandrasekhar. The potential for axial perturbations is found to be positive definite.

COBE has provided preliminary results on the spectral distribution of both Cosmic Background Radiation (CBR) brightness and dipole anisotropy. We combine these data in order to compare COBRA results with those of COBE and to obtain more stringent upper limits to CBR spectral distortions. The possibility of having detected extragalactic diffuse lines is discussed.

The phase space portrait of the cosmological models deduced from fourth-order gravity theories is discussed with the analytical and numerical methods of our previous paper. A comparison is carried out between models inferred from Lagrangian densities containing powers higher than two in the Ricci scalar of curvature, and the Starobinsky model. Some peculiar structures, such as attractors and singular points, emerging neatly from both theories, have a close physical affinity, in addition to the mathematical one. Trajectories of interest in both scenarios are those undergoing an inflationary expansion and then reaching a Friedmannian asymptotic stable phase. These features are moreover discussed through a potential U(R) in R^N-models. Three kinds of potential regions are recognized. They are the allowed regions (a-regions), in which trajectories can reach the Friedmannian phase after possibly undergoing an inflationary period, the disconnected regions (d-regions), in which trajectories, although physical, never reach the Friedmannian stage and the forbidden regions (f-regions), in which there are no physical solutions. A general survey of the global phase space for Starobinsky and R^N-models is given via Poincaré projections of suitable variables. a-, d-, and f-regions are represented.

A nonsingular cosmological scenario is presented in which the currently observed Riemannian configuration of spacetime originates from a primordial Weylian “stiff matter” phase driven by a geometrized inflaton-like field. In the particular case of a Friedman-Robertson-Walker line element, it is shown that an unstable original Minkowski vacuum starts to collapse adiabatically at a remote past and bounces when a minimum radius is attained. Throughout the collapsing stage, this matter-free Universe is accelerated (or “inflationary”). As in models of spontaneous quantum creation, the bouncing phase can be shown to correspond to the propagation of a Weyl instanton in an Euclideanized, classically forbidden region. In the course of the bouncing period, concurrent to the maximal deviation of the Riemannian configuration, a geometry-driven amplification mechanism operates causing the exponential increase of entropy and matter fluctuations. Once the environment temperature is always bounded, this nonadiabatical process stands for a “ Big — but finite — Bang” creation event, followed by a standard radiation-dominated era. The model describes an eternal, Friedman-like open Universe, free of many problems that hinder standard cosmology; the observed existence of a baryon excess also fits naturally within the proposed scheme.