Regular Series


Vol. 36 (2005), No. 1, pp. 5 – 151


An Unusual Eigenvalue Problem

abstract

We discuss an eigenvalue problem which arises in the studies of asymptotic stability of a self-similar attractor in the sigma model. This problem is rather unusual from the viewpoint of the spectral theory of linear operators and requires special methods to solve it. One of such methods based on continued fractions is presented in detail and applied to determine the eigenvalues.


On Analyticity of Static Vacuum Metrics at Non-Degenerate Horizons

abstract

We show that static metrics solving vacuum Einstein equations (possibly with a cosmological constant) are one-sided analytic at non-degenerate Killing horizons. We further prove analyticity in a two-sided neighborhood of “bifurcate horizons”.


Cotton Blend Gravity pp Waves

abstract

We study conformal gravity in \(d=2+1\), where the Cotton tensor is equated to a — necessarily traceless — matter stress tensor, for us that of the improved scalar field. We first solve this system exactly in the pp wave regime, then show it to be equivalent to topologically massive gravity.


Asymptotic Algebra of Quantum Electrodynamics

abstract

The Staruszkiewicz quantum model of the long-range structure in electrodynamics is reviewed in the form of a Weyl algebra. This is followed by a personal view on the asymptotic structure of quantum electrodynamics.


The General Penrose Inequality: Lessons from Numerical Evidence

abstract

Formulation of the Penrose inequality becomes ambiguous when the past and future apparent horizons do cross. We test numerically several natural possibilities of stating the inequality in punctured and boosted single- and double-black holes, in a Dain–Friedrich class of initial data and in conformally flat spheroidal data. The Penrose inequality holds true in vacuum configurations for the outermost element amongst the set of disjoint future and past apparent horizons (as expected) and (unexpectedly) for each of the outermost past and future apparent horizons, whenever these two bifurcate from an outermost minimal surface, regardless of whether they intersect or remain disjoint. In systems with matter the conjecture breaks down only if matter does not obey the dominant energy condition.


On Stability of Renormalised Classical Electrodynamics

abstract

It is shown that the total energy of the static “field + particle” system, defined in the framework of classical, renormalised electrodynamics of particles and fields, depends in an unstable way upon the field boundary data. It is argued that this phenomenon may be also an origin of the unstable dynamical behaviour of the system (i.e.  existence of “runaway solutions”). It is proved that a suitable polarisation mechanism of the particle restores the stability, at least on the level of statics. Whether or not it restores also the full, dynamical stability of the theory is still an open question.


Readings of the Lichnerowicz–York Equation

abstract

James York, in a major extension of André Lichnerowicz’s work, showed how to construct solutions to the constraint equations of general relativity. The York method consists of choosing a 3-metric on a given manifold; a divergence-free, tracefree (\(TT\)) symmetric 2-tensor wrt this metric; and a single number, the trace of the extrinsic curvature. One then obtains a quasi-linear elliptic equation for a scalar function, the Lichnerowicz–York (L–Y) equation. The solution of this equation is used as a conformal factor to transform the data into a set that satisfies the constraints. If the manifold is compact and without boundary, one quantity that emerges is the volume of the physical space. This article reinterprets the L–Y equation as an eigenvalue equation so as to get a set of data with a preset physical volume. One chooses the conformal metric, the \(TT\) tensor, and the physical volume, while regarding the trace of the extrinsic curvature as a free parameter. The resulting equation has extremely nice uniqueness and existence properties. A even more radical approach would be to fix the base (conformal) metric, the physical volume, and the trace. One also selects a \(TT\) tensor, but one is free to multiply it by a constant(unspecified). One then solves the L–Y equation as an eigenvalue equation for this constant. A third choice would be to fix the \(TT\) tensor and multiply the base metric by a constant. Each of these three formulations has good uniqueness and existence properties.


On Eight Kinds of Spinors

abstract

The eight inequivalent spinorial double covers of the full Lorentz group \(\mathbb {L}\) are described explicitly. Two among them include the antilinear representations of space and time reflections on two-component spinors, discovered in 1976 by Staruszkiewicz. The group of all inequivalent central extensions of \(\mathbb {L}\) by \(\mathbb {Z}_2\) has 16 elements and contains an eight-element subgroup of ‘vectorial’ double covers, characterized by the property of being trivial when restricted to the proper Lorentz group.


Quantum Mechanics of the Electric Charge in a Cut Fock Space

abstract

Assumption that the phase of the Coulomb field is a dynamical degree of freedom, conjugate to the charge operator, leads to the consistent, Lorentz invariant field theory of photons at spatial infinity, which depends parametrically on the value of the fine structure constant. We confirm existence of the normalizable bound state in the spectrum of the first Casimir operator of the Lorentz group. The state exists only for \(e^2 \lt \pi \) indicating that \(e^2=\pi \) is the singular point of the theory. We also show that the theory has an essential singularity at the origin of the complex \(e^2\) plane.


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