A note on the resonant frequencies of rapidly rotating black holes

TL;DR

This paper clarifies the valid range of Detweiler's formula for Kerr black hole resonant frequencies, showing it only accurately describes a few modes near extremality, contrary to previous assumptions of broad applicability.

Contribution

It demonstrates that Detweiler's formula is only valid for a limited number of modes in near-extremal black holes, correcting a common misconception and aligning with numerical data.

Findings

Detweiler's formula accurately describes only the first few modes near extremality.

The number of well-described modes is limited to approximately one or two for astrophysical black holes.

Numerical data supports the limited validity range of the formula.

Abstract

I discuss the range of validity of Detweiler's formula for the resonant frequencies of rapidly rotating Kerr black holes. While his formula is correct for extremal black holes, it has also been commonly accepted that it describes very well the resonant frequencies of near extremal black holes, and that therefore there is a large number of modes clustering on the real axis as the black hole becomes extremal. I will show that this last statement is not only incorrect, but that it also does not follow from Detweiler's formula, provided it is handled with due care. It turns out that only the first n << -log{(r_+-r_-)/r_+} modes are well described by that formula, which translates, for any astrophysical black hole, into one or two modes only. All existing numerical data gives further support to this claim. I also discuss some implications of this result for recent investigations on the late-time dynamics of rapidly rotating black holes.