Mode stability and shallow quasinormal modes of Kerr–de Sitter black holes away from extremality
Peter Hintz
ETH Zürich, Zürich, Switzerland
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Abstract
A Kerr–de Sitter black hole is a solution of the Einstein vacuum equations with cosmological constant . It describes a black hole with mass and specific angular momentum . We show that for any there exists so that mode stability holds for the linear scalar wave equation when and . In fact, we show that all quasinormal modes in any fixed half-space are equal to or , , as . We give an analogous description of quasinormal modes for the Klein–Gordon equation. We regard a Kerr–de Sitter black hole with small as a singular perturbation either of a Kerr black hole with the same angular momentum-to-mass ratio, or of de Sitter spacetime without any black hole present. We use the mode stability of subextremal Kerr black holes, proved by Whiting and Shlapentokh-Rothman, as a black box; the quasinormal modes described by our main result are perturbations of those of de Sitter space. Our proof is based on careful uniform a priori estimates, in a variety of asymptotic regimes, for the spectral family and its de Sitter and Kerr model problems in the singular limit .
Cite this article
Peter Hintz, Mode stability and shallow quasinormal modes of Kerr–de Sitter black holes away from extremality. J. Eur. Math. Soc. (2024), published online first
DOI 10.4171/JEMS/1463