Resonance spectrum of near-extremal Kerr black holes in the eikonal limit
The fundamental resonances of rapidly rotating Kerr black holes in the eikonal limit are derived analytically. We show that there exists a critical value, $\mu_c=\sqrt{{{15-\sqrt{193}}\over{2}}}$, for the dimensionless ratio $\mu\equiv m/l$ between the azimuthal harmonic index $m$ and the spheroidal harmonic index $l$ of the perturbation mode, above which the perturbations become long lived. In particular, it is proved that above $\mu_c$ the imaginary parts of the quasinormal frequencies scale like the black-hole temperature: $\omega_I(n;\mu>\mu_c)=2\pi T_{BH}(n+{1\over 2})$. This implies that for perturbations modes in the interval $\mu_c<\mu\leq 1$, the relaxation period $\tau\sim 1/\omega_I$ of the black hole becomes extremely long as the extremal limit $T_{BH}\to 0$ is approached. A generalization of the results to the case of scalar quasinormal resonances of near-extremal Kerr-Newman black holes is also provided. In particular, we prove that only black holes that rotate fast enough (with $M\Omega\geq {2\over 5}$, where $M$ and $\Omega$ are the black-hole mass and angular velocity, respectively) possess this family of remarkably long-lived perturbation modes.
💡 Research Summary
The paper presents an analytic treatment of the quasinormal mode (QNM) spectrum of rapidly rotating (near‑extremal) Kerr black holes in the eikonal (large‑l) limit. Starting from the Teukolsky master equation, the authors separate variables into angular and radial parts and focus on the regime where the spheroidal harmonic index l≫1 and the spin parameter a≈M (the black‑hole mass). In this limit the angular eigenvalue can be expressed in terms of the dimensionless ratio μ≡m/l, where m is the azimuthal number. By employing matched‑asymptotic expansions—using the near‑horizon geometry, which reduces to an AdS₂×S² throat, to solve the radial equation close to the event horizon, and a WKB solution far from the horizon—the authors derive a quantization condition for the complex frequency ω=ω_R+iω_I.
A central result is the identification of a critical value of the ratio μ,
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