Table of contents

A phase-integral condition determining frequencies for which a gravitational wave would pass through the curvature potential that surrounds a Schwarzschild black hole is derived. Numerical results obtained from this condition agree well with the eigenfrequencies that distinguish the so-called `algebraically special' perturbations of the hole. The results suggest that the `special' modes should perhaps be identified as total transmission solutions.

The main goal of this letter is to give a new perfect fluid solution endowed with differential rotation within the class of metric reported by Senovilla. The structural functions h, m and s are determined as functions of the variable x to certain exponents depending on one arbitrary constant.

We give a straightforward generalization of the uniqueness proof for static electrovac black holes including the case of non-vanishing magnetic charge. This also establishes the non-existence of multiple black hole solutions of the Einstein--Maxwell system with electric and magnetic fields in a static, asymptotically flat spacetime.

We consider the compactification of the dual form of N=1, D=10 supergravity on a six-dimensional Calabi--Yau manifold. An N=1 off-shell supergravity effective Lagrangian in four dimensions can be constructed in a dual version of the gravitational sector (new-minimal supergravity form). Superspace duality has a simple interpretation in terms of Poincaré duality of two-form cohomology. The resulting 4D Lagrangian may describe the low-energy point-field limit of a five-brane theory, dual to string theory, provided Calabi--Yau spaces are consistent vacua of such dual theory.

In order to solve the super-Beltrami equations (SBE) on the supertorus, we construct the quasielliptic Weierstrass -function as the -Cauchy kernel thereon. Using this solution we compute the stress--energy tensor and Green functions corresponding to induced supergravity in Polyakov's path integral formalism. This allows us to recover the corresponding results on the supercomplex plane and the torus. Finally, we discuss generalizations to super-Riemann surfaces of higher genus.

We examine some of the implications of implementing the usual boundary conditions on the closed bosonic string in the Hamiltonian framework. Using the Krichever--Novikov formalism, it is shown that at the quantum level, the resulting constraints lead to relations among the periods of the basis 1-forms. These are compared with those of Riemann which arise from a different consideration.

We determine the general scalar potential consistent with (p,q) supersymmetry in two-dimensional non-linear sigma-models with torsion, generalizing previous results for special cases. We thereby find many new supersymmetric sigma-models with potentials, including new (2,2) and (4,4) models.

A new parametrization of the solutions of Toda field theory is introduced. In this parametrization, the solutions of the field equations are real, well-defined functions on spacetime, which is taken to be two-dimensional Minkowski space or a cylinder. The global structure of the covariant phase space of Toda theory is examined and it is shown that it is isomorphic to the Hamiltonian phase space. The Poisson brackets of Toda theory are then calculated. Finally, using the methods developed to study the Toda theory, we extend these results to the non-Abelian Toda field theories.

We consider the simplicial state sum model of Ponzano and Regge as a path integral for quantum gravity in three dimensions.

We examine the Lorentzian geometry of a single 3-simplex and of a simplicial manifold, and interpret an asymptotic formula for 6j-symbols in terms of this geometry. This extends Ponzano and Regge's similar interpretation for Euclidian geometry.

We give a geometric interpretation of the stationary points of this state sum, by showing that, at these points, the simplicial manifold may be mapped locally into flat Lorentzian or Euclidian space. This lends weight to the interpretation of the state sum as a path integral, which has solutions corresponding to both Lorentzian and Euclidian gravity in three dimensions.

The standard formula for the change in the effective action under a conformal transformation is extended to the case when the boundary is piecewise smooth. We then determine the functional determinants on regions of the plane obtained by stereographic projection of the fundamental domains on the orbifolded 2-sphere. Examples treated are the sector of a disk and a circular crescent. The effective action on a spherical cap is also found.

The geometric interpretation of the Batalin--Vilkovisky anti-bracket as the Schouten bracket of functional multivectors is examined in detail. The identification is achieved by the process of repeated contraction of even functional multivectors with fermionic functional 1-forms. The classical master equation may then be considered as a generalization of the Jacobi identity for Poisson brackets, and the cohomology of a nilpotent even functional multivector is identified with the BRST cohomology. As an example, the BRST--BV formulation of gauge fixing in theories with gauge symmetries is reformulated in the jet bundle formalism.

We carry out the non-perturbative canonical quantization of several types of cosmological models that have already been studied in the geometrodynamic formulation using the complex path-integral approach. We establish a relation between the choices of complex contours in the path integral and the sets of reality conditions for which the metric representation is well defined, proving that the ambiguity in the selection of complex contours disappears when one imposes suitable reality conditions. In most of the cases, the wavefunctions defined by means of the path integral turn out to be non-normalizable and cannot be accepted as proper quantum states. Moreover, the wavefunctions of the Universe picked out in quantum cosmology by the no-boundary condition and the tunneling proposals do not belong, in general, to the Hilbert space of quantum states. Finally, we show that different sets of reality conditions can lead to equivalent quantum theories. This fact enables us to extract physical predictions corresponding to Lorentzian gravity from quantum theories constructed with other than Lorentzian reality conditions.

The Einstein equations for a spacetime of the form can be reduced to a Hamiltonian system on the Teichmüller space. However, the resulting Hamiltonian is the square root of a quadratic form in the momenta. Trying to make sense of this as a quantum operator is problematic since the Hamiltonian operator would be non-polynomial and non-local. In the first half of this article, I will examine the eigenfunctions of this operator. These go under the name of Maass functions and have been studied extensively by number theorists. In the second half, I will show that the quantum evolution due to this Hamiltonian does not cause the spacetime to collapse in a finite lapse of mean curvature time. Because of the difficult number theory involved with the Maass functions, I will actually perform the calculation for a simplified problem---a model of Teichmüller space---and argue that the answer should not be sensitive to the simplifications made in my model.

An approach to black hole quantization is proposed wherein it is assumed that quantum coherence is preserved. After giving our motivations for such a quantization procedure we formulate the background field approximation, in which particles are divided into `hard' particles and `soft' particles. The background spacetime metric depends both on the in-states and on the out-states. A consequence of our approach is that four-geometries describing gravitational collapse will show the same topological structure as flat Minkowski space. We present some model calculations and extensive discussions. In particular, we show, in the context of a toy model, that the S-matrix describing soft particles in the hard particle background of a collapsing star is unitary; nevertheless, part of the spectrum of particles is shown to be approximately thermal. We also conclude that there is an interesting topological (and signature) constraint on manifolds underlying conventional functional integrals.

We show how free equations of massless and massive fields in 2 + 1-dimensional anti-de Sitter space can be reformulated in a form of certain zero-curvature conditions. This `unfolded formulation' makes it trivial to construct explicit solutions of field equations as well as evaluation of relevant Casimir operators of the anti-de Sitter group, in a coordinate-independent way. The three-dimensional theories under consideration exemplify in the simplest fashion general features of the unfolded formulations of relativistic equations developed previously for massless higher-spin theories in 3 + 1 dimensions, which are expected to have a wide area of applicability.

In this work we investigate the possibility of inflation and dynamical reduction in a multi-dimensional cosmology. General considerations lead us to consider the most natural spatially flat and initially isotropic hydrodynamical model with two phenomenological phase transitions. We find that the baryonic asymmetry generation never occurs in such models. We conclude that extra dimensions probably never evolved in a classical way, being always quantum in nature.

We give a detailed study of the various aspects of the mixing effects due to geodesic instability in compact negative curvature FLRW models, as an explicit function of the cosmological parameters and the topological compactification scale. Our results show that the effect is: (i) much less pronounced than some of the claims made previously; (ii) strongly dependent upon the current value of the cosmological density parameter . Nevertheless it can, for reasonable values of the density parameter, produce appreciable mixing with a consequent reduction in the measured CMWBR anisotropy on a range of angular scales. This effect is expressible in terms of the curvature scalar and is therefore gauge invariant. Furthermore, we show: (a) the usual observer area distance routinely employed in cosmological studies also encodes all the dynamical information relating to the geodesic instability; (b) the estimates of the length scales on the surface of the last scattering depend significantly on the density parameter, and could be significantly different in low density universes compared with a critical density universe.

We present a derivation of the general Smarr formula and the general mass variation formula for stationary axisymmetric axion--dilaton black holes.

We calculate the Euclidean propagator for a conformally coupled massless scalar field in the background of the three-dimensional black hole. The expectation value in the Hartle--Hawking state is obtained in the spacetime.

Some one-parameter families of time-symmetric Cauchy hypersurfaces were investigated. All of them have such a property that for some `critical' value of the parameter an apparent horizon appears. It turns out that for parameter values sufficiently close to the critical one, numerous properties of the horizon are `universa' (i.e. independent of details of geometry and distribution of matter). It was conjectured that the above `universality' property occurs in the case of typical one-parameter time-symmetric families of Cauchy hypersurfaces.

We give a Hamiltonian formulation of relativistic superfluids free of any constraints, starting from an action principle based on four scalar fields. The main physical objects are two conserved currents---one associated with the flow of particles and the other with the flow of entropy, or heat. The superfluid state is characterized by the condition that the fluid flow be irrotational. There are three implicit relations which must be solved in order to pass from the Lagrangian to the Hamiltonian form of the equations of motion. We consider the weak coupling limit where, the particle and heat components of the fluid behave independently, to find explicit solutions of these implicit relations. We also examine the gauge invariance of the theory.

We investigate rotation and rotating structures in (2 + 1)-dimensional Einstein gravity. We show that rotation generally leads to pathological physical situations such as closed time-like curves and singular shear.

We compute the motions of null infinity to which the components of the angular momentum of the gravitational field, as defined by Penrose, are conjugate. We find that the boosts are supplemented by anomalous translations. If is the skew bivector determining the component of the angular momentum in question, the anomaly is proportional to , where is a unit vector in the direction of the Bondi--Sachs momentum and is the quadrupole moment of the shear of the cut at which the angular momentum is evaluated. This effect persists in the weak field limit. This surprising result is a general consequence of the requirement that angular momentum be super-translation--invariant in a quiescent regime, and not some essentially twistorial peculiarity.

We discuss earlier unsuccessful attempts to formulate a positive gravitational energy proof in terms of the `new variables' of Ashtekar. We also point out the difficulties of a Witten spinor type of proof. We then use the special orthonormal frame gauge conditions to obtain a locally positive expression for the `new variables' Hamiltonian, and thereby a `localization' of gravitational energy as well as a positive energy proof.