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-se migroup theory. Each equation is shown to have a well-posed initial value p roblem, i.e., to have a unique solution depending continuously on the data. Further, it is shown that the spectrum of the semigroup's generator coinci des with the spectrum of an operator polynomial whose coefficients can be r ead 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 mod e considerations. For the wave equation it is shown that the resolvent of t he semigroup's generator and the corresponding Green's functions can be com puted using spheroidal functions. It is to be expected that, analogous to t he case of a Schwarzschild background, the quasinormal frequencies of the K err black hole appear as resonances, i.e., poles of the analytic continuati on of this resolvent. Finally, stability of the solutions of the reduced Kl ein-Gordon equation is proven for large enough masses.