On the Stability of the Kerr Metric

Abstract:

The reduced (in the angular coordinate ϕ) 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.