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

TL;DR

This paper analyzes the special frequency solutions of black-hole wave equations, revealing unique properties and their implications for understanding black-hole perturbations and mode behavior.

Contribution

It provides an exact solution at the algebraically special frequency, clarifies the physical significance of SUSY relationships, and explains the nature of modes and transmission at this frequency.

Findings

No quasinormal or total-transmission modes at the special frequency for RWE.

The special frequency exhibits a cancellation of divergences and branch-cut discontinuity vanishing.

Finite rotation makes modes totally transmitting, clarifying Schwarzschild limit singularity.

Abstract

We study the Regge-Wheeler and Zerilli equations (RWE and ZE) at the `algebraically special frequency' $\Omega$, 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 $\Omega$ in general is elucidated as follows. The RWE has no (quasinormal or total-transmission) modes at all; however, $\Omega$ 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 $\omega=\Omega$ 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 $\omega$-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].