# Spectroscopy of Extremal (and Near-Extremal) Kerr Black Holes

## TL;DR

This paper analyzes the linear perturbations of extremal Kerr black holes across the complex frequency spectrum, revealing the formation of a branch cut at the superradiant frequency and providing evidence for their mode stability.

## Contribution

It introduces a novel MST-based method to compute quasi-normal modes of extremal Kerr black holes and explores the spectral features in the extremal limit.

## Key findings

- Formation of a branch cut at the superradiant frequency in extremal Kerr
- Largest superradiant amplification factor at extremality with a discontinuity at $_{SR}$
- No evidence of exponentially-growing modes, indicating mode stability

## Abstract

We investigate linear, spin-field perturbations of Kerr black holes in the extremal limit throughout the complex-frequency domain. We calculate quasi-normal modes of extremal Kerr, as well as of near-extremal Kerr, via a novel approach: using the method of Mano, Suzuki and Takasugi (MST). We also show how, in the extremal limit, a branch cut is formed at the superradiant-bound frequency, $\omega_{SR}$, via a simultaneous accumulation of quasi-normal modes and totally-reflected modes. For real frequencies, we calculate the superradiant amplification factor, which yields the amount of rotational energy that can be extracted from a black hole. In the extremal limit, this factor is the largest and it displays a discontinuity at $\omega_{SR}$ for some modes. Finally, we find no exponentially-growing modes nor branch points on the upper-frequency plane in extremal Kerr after a numerical investigation, thus providing evidence of the mode-stability of this space-time away from the horizon.