Asymptotic Spectroscopy of Rotating Black Holes

TL;DR

This work analytically derives the transmission ${\mathcal{T}}$ and reflection ${\mathcal{R}}$ amplitudes for waves scattering off a four-dimensional rotating black hole in the highly damped, nearly imaginary-frequency regime, and identifies a spectrum of resonances (QNMs, TTMs, TRMs, IHMs) with semiclassical bound states along complex contours. By recasting the problem in terms of a one-dimensional inverted-potential Schrödinger equation and employing WKB, Stokes phenomena, and monodromy, the authors obtain complex Bohr-Sommerfeld conditions that yield resonant frequencies $\omega(n)=\widetilde{\omega}_j+4\pi iT_j(n+\mu_j/4)$. The resonances organize into three Boltzmann-like sectors with characteristic temperatures $T_i$, $T_o$, and horizon-related scales, linking the scattering data to Hawking radiation via exact cancellations and suggesting a two-sector microscopic interpretation (QNMs vs TTMs) in a dual description. The results establish a robust, largely universal framework that connects complex geodesics, semiclassical excitations, and the analytically continued decay spectrum, offering clues toward the quantum structure of rotating black holes and potential holographic dualities in this regime.

Abstract

We calculate analytically the transmission and reflection amplitudes for waves incident on a rotating black hole in d=4, analytically continued to asymptotically large, nearly imaginary frequency. These amplitudes determine the asymptotic resonant frequencies of the black hole, including quasinormal modes, total-transmission modes and total-reflection modes. We identify these modes with semiclassical bound states of a one-dimensional Schrodinger equation, localized along contours in the complexified r-plane which connect turning points of corresponding null geodesics. Each family of modes has a characteristic temperature and chemical potential. The relations between them provide hints about the microscopic description of the black hole in this asymptotic regime.

Figures (3)

• Figure 1: Illustration of perturbation analysis for a rotating black hole in the highly-damped regime. Anti-Stokes (solid) lines, Stokes (dashed) lines and turning points $r_i$ (disks) are shown in the complex $r$-plane for the case $a=0.3$ and $Q=0$. Arrows along anti-Stokes lines point in the direction of increasing $\mathop{\mathrm{Im}}(z)$. Inner and outer horizon radii (diamonds) are also shown.
• Figure 2: Illustration of WKB computation for the case $\omega_R<m\Omega$. The directed contour (solid line with arrows) encloses only the outer horizon. Bracketed triplets denote the values of $\{c_+,c_-;r'\}$, as defined in Eqs. (\ref{['eq:WKBdefinition']})-(\ref{['eq:WKBNotation']}). Other parameters and symbols are defined as in Figure \ref{['fig:Stokes']}.
• Figure 3: The effective temperature $|T_o|$ as a function of $a$ for an uncharged rotating black hole, normalized by $T_H(a=Q=0)$.