Complete total-transmission modes of Kerr black holes

Abstract

We construct the complete spectrum of gravitational total-transmission modes (TTMs) of Kerr black holes and find four globally continuous families, $n_{\infty }=1,2,3,4$, each with two complex-conjugate frequency branches. Compared with the three-family classification of Cook and Lu, our global continuation resolves the mirror-related sectors of their $n=2$ family into separate families without introducing additional symmetry-unrelated roots. This organization distinguishes complex conjugation at fixed $m$ from the mirror symmetry connecting the $m$ and $-m$ spectra. The $n_{\infty }=3$ family approaches the Schwarzschild algebraically special frequency, whereas the $n_{\infty }=1,2,$ and $4$ families diverge as $\omega \propto a^{-4/3}$ along lower-half-plane directions $-150^{\circ }$, $-90^{\circ }$, and $-30^{\circ }$, respectively. High-precision data up to $\ell =32$ confirm the Cook--Lu small-spin asymptotics and reveal how the divergent branches are embedded in the global four-family spectrum, with a family-dependent relation between the spherical-limit angular labels and the large-$|a\omega |$ ordering. For axisymmetric perturbations, the $n_{\infty }=2$ and $n_{\infty }=3$ branches coalesce at genuine exceptional points, where the scattering spectral function develops a second-order zero and the TTM excitation factors are strongly enhanced, offering a different interpretation of the structure previously described by Cook and Lu as an overtone-multiplet splitting. Finally, we identify a previously unreported anomalous proximity between the $n_{\infty }=3$ TTM family and an unconventional Kerr quasinormal-mode sequence, with frequency separations reaching order $10^{-8}$ in the representative case studied.

arXiv comment24 pages,13 figures