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

Abstract

Abstract. We establish a Bohr-Sommerfeld type condition for quasinormal modes of a slowly rotating Kerr-de Sitter black hole, providing their full asymptotic description in any strip of fixed width. In particular, we observe a Zeeman-like splitting of the high multiplicity modes at a = 0 (Schwarzschild-de Sitter), once spherical symmetry is broken. The numerical results presented in Appendix B show that the asymptotics are in fact accurate at very low energies and agree with the numerical results established by other methods in the physics literature. We also prove that solutions of the wave equation can be asymptotically expanded in terms of quasi-normal modes; this confirms the validity of the interpretation of their real parts as frequencies of oscillations, and imaginary parts as decay rates of gravitational waves.Quasi-normal modes (QNMs) of black holes are a topic of continued interest in theoretical physics: from the classical interpretation as ringdown of gravitational waves [11] to the recent investigations in the context of string theory [33]. The ringdown 1 plays a role in experimental projects aimed at the detection of gravitational waves, such as LIGO [1]. See [35] for an overview of the vast physics literature on the topic and [6,36,37,53] for some more recent developments.In this paper we consider the Kerr-de Sitter model of a rotating black hole and assume that the speed of rotation a is small; for a = 0, one gets the stationary Schwarzschild-de Sitter black hole. The de Sitter model corresponds to assuming that the cosmological constant Λ is positive, which is consistent with the current Lambda-CDM standard model of cosmology.

Asymptotics of Linear Waves and Resonances with Applications to Black Holes

“…The general set of Λ and a for which ∆ r has all real roots as above was studied numerically in [AkMa,§3], and is pictured on Figure 1(a) in the introduction. Note that in [AkMa], the roots are labeled r −− < r − < r + < r C ; we do not adopt this (perhaps more standard) convention in favor of the notation of [Dy11a,Dy12,Va10], and since the roots r 1 , r 2 are irrelevant in our analysis.…”

“…Theorems 1-3 are related to the resonance expansion and quantization condition proved for the slowly rotating Kerr-de Sitter in [Dy12]. In this paper, however, we use dynamical assumptions stable under perturbations, rather than complete integrability of geodesic flow on Kerr(-de Sitter), and do not recover the precise results of [Dy12].…”

“…In this paper, however, we use dynamical assumptions stable under perturbations, rather than complete integrability of geodesic flow on Kerr(-de Sitter), and do not recover the precise results of [Dy12].…”

“…Similarly to the quantization condition of [Dy12], these resonances ω mlk are indexed by three integer parameters m ≥ 0 (depth), l ≥ 0 (angular energy), and k ∈ [−l, l] (angular momentum). The parameter l roughly corresponds to the real part of the resonance and the parameter m, to its imaginary part.…”

A Possible Failure of Determinism in General Relativity

“…Here, null particles can be trapped on circular unstable trajectories, defining the photon sphere. This region has a strong pull in the control of the decay of fluctuations and the spacetime's QNMs which have large frequency (i.e., large jReωj) [11,[31][32][33]. For instance, the decay time scale is related to the instability time scale of null geodesics near the photon sphere.…”