Bonn 2001 – wissenschaftliches Programm

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GR: Gravitation und Relativitätstheorie

GR V: HV V

GR V.1: Hauptvortrag

Donnerstag, 29. März 2001, 11:00–11:45, VII

On the stability of the Kerr black hole — •Horst Beyer — MPI fuer Gravitationsphysik (AEI), Am Muehlenberg 1, D-14476 Golm b. Potsdam

The Kerr metric describes the gravitational field exterior to a rotating black hole and is therefore of high astrophysical importance. Different from its non-rotating counterpart, the Schwarzschild metric, the question of its stability is still largely open and challenging. Previously known results are discussed in the first part of the talk. These suggest that the outcome to the question might depend on the fact whether the mass of the perturbing field is zero or non-zero. The remainder of the talk presents recently obtained new results. In particular the wave equation and the Klein-Gordon are considered on a Kerr background and in the framework of semigroup theory. Among others it will be shown that these equations have a well-posed initial value problem, i.e., have a unique solution for all times which depends continuously on the data. Further, it is shown for both cases that the spectrum of the semigroup’s generator governing time evolution coincides with the spectrum of an operator polynomial whose coefficients can be ‘read off’ from the equation. From this the stability of the solutions of the Klein-Gordon equation is proven for masses exceeding a given bound. Finally, it is shown for the wave equation how 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 Schwarzschild case the so called ‘quasinormal frequencies’ of the Kerr black hole appear as ‘resonances’, i.e., as poles of the analytic continuation of this resolvent.