Complete quasinormal modes of Type-D black holes

TL;DR

This work delivers the first complete QNM spectra for Type-D black holes by marrying analytic continuation with confluent Heun function methods, thereby resolving two long-standing open problems: the apparent Kerr-Schwarzschild discontinuity as $a\to0$ and the proximity of QNMs to algebraically special frequencies. The approach eliminates auxiliary parameters like the renormalized angular momentum and robustly handles branch cuts on the negative imaginary axis, providing complete coverage of QNMs and TTMs for Schwarzschild and Kerr spacetimes (massless and massive perturbations) with high precision ($<10^{-10}$ error) and broad overtones ($0\le n\le 41$, $2\le \ell\le16$). It reveals new phenomena such as zero-damped mode behavior near extremality, Zeeman-like splitting into $2\ell+1$ branches for Kerr $ ext{l}$-modes, and resonant interactions between overtone families, refining the understanding of AS frequencies and unconventional modes. The results offer a computationally superior framework relative to Cook’s continued-fraction and isomonodromic methods, enabling comprehensive GW-ringdown modeling and precise tests of GR in the strong-field regime. The methodology also generalizes to RN and modified gravity settings, providing a powerful tool for testing gravity with current and future gravitational-wave observations.

Abstract

Quasinormal mode (QNM) spectra of black holes exhibit two open problems [Conf. Proc. C 0405132, 145 (2004); CQG 26, 163001 (2009)]: (i) the discontinuity in highly damped QNMs between Schwarzschild and Kerr solutions as $a \to 0$, and (ii) the unexplained spectral proximity between QNMs and algebraically special (AS) frequencies, particularly the anomalous multiplet splitting for Kerr $\ell=2$, $m \geq 0$ modes. We develop a novel method to compute complete QNM spectra for Type-D black holes, solving both problems and establishing a mathematical framework for boundary value problems of dissipative systems. Using analytic continuation of radial eigenvalue equations, our method eliminates the dependence on auxiliary parameters in the connection formulas for confluent Heun solutions. This breakthrough overcomes the long-standing challenge of calculating QNMs that cross or lie on the negative imaginary axis (NIA). For Schwarzschild and Kerr spacetimes ($0 \leq {n} \leq 41$, $2 \leq \ell \leq 16$), we present complete spectra validated through scattering amplitudes with errors $<10^{-10}$. The results provide definitive solutions to both open problems: (i) The inability of conventional methods to compute QNMs crossing or residing on the NIA leads to apparent discontinuities in Kerr spectra as $a \to 0$. (ii) When a QNM exactly coincides with the AS frequency, an additional QNM (unconventional mode) appears nearby. For the Kerr case with $\ell=2$, overtone sequences from both unconventional and AS modes exhibit precisely $2\ell+1$ branches, without multiplets or supersymmetry breaking at AS frequencies. Our method confirms Leung's conjecture of high-$\ell$ unconventional mode deviations from the NIA through the first $\ell=3$ calculation. Moreover, this paradigm surpasses the state-of-the-art Cook's continued fraction and isomonodromic methods in computational efficiency.

Figures (13)

• Figure 1: The first 42 QNMs for the gravitational perturbations of the Schwarzschild BH.
• Figure 2: QNM spectrum for $\ell= 2$ and $\hat{n} = 8$ originating from the unconventional mode \ref{['eq:SchNewModesL2']}.
• Figure 3: QNM spectrum for $\ell= 2$ and $\hat{n} = 9$ originating from the AS mode \ref{['eq:ASmode']}.
• Figure 4: QNM spectrum for $\ell= 3, \hat{n} = 40$ originating from the unconventional mode \ref{['eq:SchNewModesL3']}.
• Figure 5: Kerr algebraically special (TTM) mode sequences for $\ell =2,\,m=\{0,1,2\}$ and $\ell =3,\,m=\{0,1,2,3\}$ with $a\in \hbox{[}0,1\hbox{]}$.