Asymptotic Spectrum of Kerr Black Holes in the Small Angular Momentum Limit

TL;DR

This paper investigates the highly damped quasinormal modes of Kerr black holes in the small angular momentum limit, revealing a continuous transition of mode frequencies from Kerr to Schwarzschild black holes and analyzing extremal cases.

Contribution

It provides a detailed analytical and numerical analysis of the transition of quasinormal mode frequencies as Kerr black holes approach zero angular momentum, clarifying previous discrepancies.

Findings

The real part of the quasinormal mode frequency approaches the Schwarzschild value as angular momentum decreases.

The transition of mode frequencies is continuous from Kerr to Schwarzschild black holes.

Extremal Kerr black holes exhibit different topological features affecting quasinormal modes.

Abstract

We study analytically the highly damped quasinormal modes of Kerr black holes in the small angular momentum limit. To check the previous analytic calculations in the literature, which use a combination of radial and tortoise coordinates, we reproduce all the results using the radial coordinate only. According to the earlier calculations, the real part of the highly damped quasinormal mode frequency of Kerr black holes approaches zero in the limit where the angular momentum goes to zero. This result is not consistent with the Schwarzschild limit where the real part of the highly damped quasinormal mode frequency is equal to c^3 ln(3)/(8 pi G M). In this paper, our calculations suggest that the highly damped quasinormal modes of Kerr black holes in the zero angular momentum limit make a continuous transition from the Kerr value to the Schwarzschild value. We explore the nature of this transition using a combination of analytical and numerical techniques. Finally, we calculate the highly damped quasinormal modes of the extremal case in which the topology of Stokes/anti-Stokes lines takes a different form.