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

Alec Maassen van den Brink
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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].

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Alec Maassen van den Brink. 2000-01-11. Analytic treatment of black-hole gravitational waves at the algebraically special frequency. https://doi.org/10.1103/physrevd.62.064009

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