Quasi-Normal Modes and Exponential Energy Decay for the Kerr-de Sitter Black Hole

Abstract

We provide a rigorous definition of quasi-normal modes for a rotating black hole. They are given by the poles of a certain meromorphic family of operators and agree with the heuristic definition in the physics literature. If the black hole rotates slowly enough, we show that these poles form a discrete subset of C. As an application we prove that the local energy of linear waves in that background decays exponentially once orthogonality to the zero resonance is imposed.

Asymptotic Distribution of Quasi-Normal Modes for Kerr–de Sitter Black Holes

“…The integral in (1.18) is holomorphic and bounded polynomially in h, by the bounds for R r given by Proposition 1.5, together with the estimates in the proof of [26,Proposition 3.4]. Now, the poles of R θ in U λ are given by (1.16); let λ θ l (ω, k) be the pole corresponding to h −2 F θ (hl,ω,ν,k; h) and Π θ l /(λ − λ θ l ) be the principal part of the corresponding meromorphic decomposition.…”

“…We now review how the construction of R g (ω) in [26] works and reduce Proposition 1.3 to two separate spectral problems in the radial and the angular variables. For the convenience of reader, we include the simpler separation of variables procedure for the case a = 0 at the end of this section.…”

“…Moreover, [26,Proposition 3.4] suggests that under these assumptions, all values of λ for which (ω, λ, k) is a pole of both R r and R θ have to satisfy |λ|, |μ| ≤ C λ , for some constant C λ .…”

“…The next theorem presents an expansion of waves for Kerr-de Sitter black holes in the same style as the Bony-Häfner expansion: The proofs start with the Teukolsky separation of variables used in [26], which reduces our problem to obtaining quantization conditions and resolvent estimates for certain radial and angular operators (Propositions 1.5 and 1.6). These conditions are stated and used to obtain Theorems 1 and 2 in Sect.…”

“…Near the event horizons, we rely on [26,Section 6], which uses a perturbation argument (thus smallness of a) and analyticity of the metric near the event horizons. Same applies to Sect.…”