Comment on "The extremal black hole bomb"

TL;DR

This paper defends the validity of an analytical approach to black-hole bomb resonances, demonstrating that the Gamma functions involved are well-behaved and addressing concerns about potential poles raised by other researchers.

Contribution

It provides explicit calculations confirming the robustness of the previous analytical method for black-hole bomb resonances and clarifies the regime of validity of the approximations used.

Findings

Gamma functions are well-behaved near resonance peaks

Supports the validity of the previous analytical treatment

Clarifies the regime of applicability of the approximations

Abstract

Recently, we have provided an analytical treatment of the phenomena known as the 'black-hole bomb' (arXiv:0910.0734). In particular, we have determined analytically the unstable growing resonances of a massive scalar field in the rotating Kerr black hole spacetime. It was later claimed by J. G. Rosa (arXiv:0912.1780) that the analytic procedure may fail for some values of the field's mass. This claim was based on the concern that some of the Gamma functions that are involved in the analysis may develop poles. In this comment we show by explicit calculations that the Gamma functions which are used are all well behaved near the peak of the black-hole resonances. This fact supports the validity of our previous analytical treatment. We further comment on the regime of validity of some of the approximations used by Rosa.