On the stability of the massive scalar field in Kerr space-time

The current early stage in the investigation of the stability of the Kerr metric is characterized by the study of appropriate model problems. Particularly interesting is the problem of the stability of the solutions of the Klein-Gordon equation, describing the propagation of a scalar field in the background of a rotating (Kerr-) black hole. Results suggest that the stability of the field depends crucially on its mass $\mu$. Among others, the paper provides an improved bound for $\mu$ above which the solutions of the reduced, by separation in the azimuth angle in Boyer-Lindquist coordinates, Klein-Gordon equation are stable. Finally, it gives new formulations of the reduced equation, in particular, in form of a time-dependent wave equation that is governed by a family of unitarily equivalent positive self-adjoint operators. The latter formulation might turn out useful for further investigation. On the other hand, it is proved that from the abstract properties of this family alone it cannot be concluded that the corresponding solutions are stable

Quantifying excitations of quasinormal mode systems

Computations of the strong field generation of gravitational waves by black hole processes produce waveforms that are dominated by quasinormal (QN) ringing, a damped oscillation characteristic of the black hole. We describe here the mathematical problem of quantifying the QN content of the waveforms generated. This is done in several steps: (i) We develop the mathematics of QN systems that are complete (in a sense to be defined) and show that there is a quantity, the ``excitation coefficient,'' that appears to have the properties needed to quantify QN content. (ii) We show that incomplete systems can (at least sometimes) be converted to physically equivalent complete systems. Most notably, we give a rigorous proof of completeness for a specific modified model problem. (iii) We evaluate the excitation coefficient for the model problem, and demonstrate that the excitation coefficient is of limited utility. We finish by discussing the general question of quantification of QN excitations, and offer a few speculations about unavoidable differences between normal mode and QN systems.Comment: 27 pages, 14 figures. To be published in: J. Math. Phys. (1999

On the r-mode spectrum of relativistic stars

We present a mathematically rigorous proof that the r-mode spectrum of relativistic stars to the rotational lowest order has a continuous part. A rigorous definition of this spectrum is given in terms of the spectrum of a continuous linear operator. This study verifies earlier results by Kojima (1998) about the nature of the r-mode spectrum.Comment: 6 pages, no figure

n Voorlopige model vir die sistematiese beskrywing van gebruikersvriendelikheid in woordeboeke

Die gebruikersperspektief staan tans sentraal in die leksikografiese gesprek. Dit spruit uit die beginsel dat enige woordeboek in die eerste plek vir gebruik deur 'n bepaalde teikengebruikersgroep bedoel moet wees, en as sodanig moet die woordeboek op daardie gebrui-kersgroep afgestem wees — dit moet gebruikersvriendelik wees. In hierdie artikel word van die standpunt uitgegaan dat die gebruik van 'n woordeboek as 'n kommunikatiewe handeling begryp moet word, wat die geleentheid bied om insigte uit die kommunikasiewetenskap in die metalek-sikografie te benut. Na aanleiding van die algemene kommunikasiemodel vir interpersoonlike kommunikasie wat in die literatuur oor kommunikasiewetenskap voorkom, word 'n leksikogra-fiese kommunikasiemodel voorgestel. Uit hierdie model kan minstens elf leksikografiese para-meters afgelei word aan die hand waarvan gebruikersvriendelikheid in woordeboeke sistematies beskryf kan word. Die struktuur wat hierdie parameters aan 'n gesprek oor gebruikersvriendelik-heid in woordeboeke kan gee, bemiddel die formulering van die ware doel van 'n gebruikers-vriendelike woordeboek. Sleutelwoorde: gebruik, gebruiker, gebruikersperspektief, gebruikersver-wysingsraamwerk, gebruikersvriendelikheid, kanaal, kommunikasiemodel, konteks, leksikograaf, leksikografiese kommunikasiemodel, medium, meta-teks, stylgids, terugvoer, werklike doel, woordeboek, woordeboekpla

Standaardwoordeboek en kern-woordeboek: 'n Vergelykende evaluering

Hierdie artikel probeer om 'n evaluering te gee van die vierde uitgawe van die Verklarende Handwoordeboek van die Afrikaanse Taal wat in 2000 verskyn het, in vergelyking met Kernwoordeboek, waarvan die eerste uitgawe in 2000 gepubliseer is. Die ooreenkomste en verskille tussen die eksponente van hierdie twee woordeboektipes in die Afrikaanse leksikografie word uitgelig deur te fokus op die voorwerk, die makrostruktuur, die toegangstruktuur en aspekte van die mikrostruktuur in die sentrale teks. Gebruikersvriendelikheid en die leksikograaf se plig om verantwoording aan die gebruiker te doen, geld as belangrike kriteria. Die konsep kernwoordeskat vir leksikografiese doeleindes word ondersoek en metodes om die kernwoordeskat as makro-struktuur van 'n kernwoordeboek te bepaal, word krities beskou. Ander Afrikaanse woordeboeke word vergelykend betrek ten einde bepaalde aspekte van heersende leksikografiese praktyk uit te wys. Sleutelwoorde: eksterne toegangstruktuur, frekwensie, gebruikers-vriendelik, interne toegangstruktuur, kernwoordeboek, kernwoordeskat, leksikale item, leksikograaf, leksikografie, leksikon, lemma, lemmateken, lemmatisering, makrostruktuur, metateks, mikrostruktuur, nesting, nisting, seleksie, sentrale teks, standaardtaal, standaardwoordeboek, sublemmatisering, teksverdigting, tipologie, toegangstruktuur, toelig-ting, verantwoording, voorwerk, woordeska

On the stability of the Kerr metric

The reduced (in the angular coordinate phi) wave equation and Klein-Gordon equation are considered on a Kerr background and in the framework of C-0-semigroup theory. Each equation is shown to have a well-posed initial value problem, i.e., to have a unique solution depending continuously on the data. Further, it is shown that the spectrum of the semigroup's generator coincides with the spectrum of an operator polynomial whose coefficients can be read off from the equation. In this way the problem of deciding stability is reduced to a spectral problem and a mathematical basis is provided for mode considerations. For the wave equation it is shown that the resolvent of the semigroup's generator and the corresponding Green's functions can be computed using spheroidal functions. It is to be expected that, analogous to the case of a Schwarzschild background, the quasinormal frequencies of the Kerr black hole appear as resonances, i.e., poles of the analytic continuation of this resolvent. Finally, stability of the solutions of the reduced Klein-Gordon equation is proven for large enough masses

A new result on the Klein-Gordon equation in the background of a rotating black hole

This short paper should serve as basis for further analysis of a previously found new symmetry of the solutions of the wave equation in the gravitational field of a Kerr black hole. Its main new result is the proof of essential self-adjointness of the spatial part of a reduced normalized wave operator of the Kerr metric in a weighted L^2-space. As a consequence, it leads to a purely operator theoretic proof of the well-posedness of the initial value problem of the reduced Klein-Gordon equation in that field in that L^2-space and in this way generalizes a corresponding result of Kay (1985) in the case of the Schwarzschild black hole. It is believed that the employed methods are applicable to other separable wave equations