Mode stability and shallow quasinormal modes of Kerr–de Sitter black holes away from extremality

Abstract

A Kerr–de Sitter black hole is a solution (M,g_{\Lambda,\mathfrak{m},\mathfrak{a}}) of the Einstein vacuum equations with cosmological constant \Lambda>0 . It describes a black hole with mass \mathfrak{m}>0 and specific angular momentum \mathfrak{a}\in\mathbb{R} . We show that for any \varepsilon>0 there exists \delta>0 so that mode stability holds for the linear scalar wave equation \Box_{g_{\Lambda,\mathfrak{m},\mathfrak{a}}}\phi=0 when |\mathfrak{a}/\mathfrak{m}|\in[0,1-\varepsilon] and \Lambda\mathfrak{m}^{2}<\delta . In fact, we show that all quasinormal modes \sigma in any fixed half-space \operatorname{Im}\sigma>-C\sqrt{\Lambda} are equal to 0 or -i\sqrt{\Lambda/3}(n+o(1)) , n\in\mathbb{N} , as \Lambda\mathfrak{m}^{2}\searrow 0 . We give an analogous description of quasinormal modes for the Klein–Gordon equation. We regard a Kerr–de Sitter black hole with small \Lambda\mathfrak{m}^{2} 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 \Lambda\mathfrak{m}^{2}\searrow 0 .