# Synchronous frequencies of extremal Kerr black holes: resonances, scattering and stability

## TL;DR

This paper investigates the nature of zero damped modes in extremal Kerr black holes, concluding that these resonances are unphysical and providing new insights into the transition to extremality and related instabilities.

## Contribution

The study demonstrates that zero damped modes at the extremal limit are unphysical and links their properties to the Aretakis instability, offering a new understanding of extremal black hole behavior.

## Key findings

- Zero damped modes are unphysical in extremal black holes.
- Resonances at the real axis are associated with scattering modes.
- The analysis offers a new perspective on the Aretakis instability.

## Abstract

The characteristic damping times of the natural oscillations of a Kerr black hole become arbitrarily large as the extremal limit is approached. This behavior is associated with the so-called zero damped modes (ZDMs), and suggests that extremal black holes are characterized by quasinormal modes whose frequencies are purely real. Since these frequencies correspond to oscillations whose angular phase velocity matches the horizon angular velocity of the black hole, they are sometimes called "synchronous frequencies". Several authors have studied the ZDMs for near-extremal black holes. Recently, their correspondence to branch points of the Green's function of the wave equation was linked to the Aretakis instability of extremal black holes. Here we investigate the existence of ZDMs for extremal black holes, showing that these real-axis resonances of the field are unphysical as natural black hole oscillations: the corresponding frequency is always associated with a scattering mode. By analyzing the behavior of these modes near the event horizon we obtain new insight into the transition to extremality, including a simple way to understand the Aretakis instability.