Note on the resonant frequencies of rapidly rotating black holes

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 give 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.