Analytic treatment of black-hole gravitational waves at the algebraically special frequency

TL;DR

The paper analyzes the Regge–Wheeler and Zerilli equations at the algebraically special frequency $Ω=-\frac{iN}{2}$, uncovering a nontrivial SUSY structure that links the two equations and dictates their spectral properties. It proves that the RW equation has no QNM/TTM at $Ω$, while the ZE possesses a single mode that is both a QNM and a TTM$_\mathrm{L}$, with further implications for Kerr rotation and the Schwarzschild limit. The work develops a rigorous framework for handling long-range potentials, exploits exact RW solutions, and connects to Leaver-series techniques, culminating in a detailed picture of mode structure, boundary behavior, and discontinuities near the NIA. These results resolve longstanding ambiguities about the nature of algebraically special modes and provide a robust foundation for future numerical and analytic studies of black-hole perturbations. The findings have implications for theoretical understanding of black-hole spectroscopy and potential interpretation of gravitational-wave signals in the algebraically special regime.

Abstract

We study the Regge-Wheeler and Zerilli equations (RWE and ZE) at the `algebraically special frequency' $Ω$, where these equations admit an exact solution (elaborated here), generating the SUSY relationship between them. The physical significance of the SUSY generator and of the solutions at $Ω$ in general is elucidated as follows. The RWE has no (quasinormal or total-transmission) modes at all; however, $Ω$ is nonetheless `special' in that (a) for the outgoing wave into the horizon one has a `miraculous' cancellation of a divergence expected due to the exponential potential tail, and (b) the branch-cut discontinuity at $ω=Ω$ vanishes in the outgoing wave to infinity. Moreover, (a) and (b) are related. For the ZE, its only mode is the-inverse-SUSY generator, which is at the same time a quasinormal mode_and_ a total-transmission mode propagating to infinity. The subtlety of these findings (of general relevance for future study of the equations on or near the negative imaginary $ω$-axis) may help explain why the situation has sometimes been controversial. For finite black-hole rotation, the algebraically special modes are shown to be totally transmitting, and the implied singular nature of the Schwarzschild limit is clarified. The analysis draws on a recent detailed investigation of SUSY in open systems [math-ph/9909030].