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.
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Explore related subjectsDiscover the latest articles and news from researchers in related subjects, suggested using machine learning. Author information Authors and AffiliationsMax Planck Institute for Gravitational Physics, Albert Einstein Institute, 14476 Golm, Germany, , , , , , US
Horst R. Beyer
Received: 28 August 2000 / Accepted: 4 April 2001
About this article Cite this articleBeyer, H. On the Stability of the Kerr Metric. Commun. Math. Phys. 221, 659–676 (2001). https://doi.org/10.1007/s002200100494
Issue Date: August 2001
DOI: https://doi.org/10.1007/s002200100494
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