Modes of the Kerr geometry with purely imaginary frequencies

TL;DR

This paper investigates the properties of Kerr black hole modes with purely imaginary frequencies, revealing polynomial nature, new solution branches, and correcting previous misconceptions through analytical and numerical analysis.

Contribution

It introduces a method to compute purely imaginary frequency modes, clarifies their nature, and uncovers new solution branches in Kerr black hole perturbations.

Findings

Purely imaginary modes are polynomial in nature.

Some polynomial modes are both quasinormal and total transmission modes.

New solution branches for algebraically special modes with m=0.

Abstract

In this paper, we examine the behavior of modes of the Kerr geometry when the mode's frequency is purely imaginary. We demonstrate that quasinormal modes must be polynomial in nature if their frequency is purely imaginary, and present a method for computing such modes. The nature of these modes, however, is not always easy to determine. Some of the polynomial modes we compute are quasinormal modes. However, some are simultaneously quasinormal modes and total transmission modes, while others fail to satisfy the requisite boundary conditions for either. This analysis is, in part, an extension of the results known for Schwarzschild black holes, but clarifies misconceptions for the behavior of modes when the black hole has angular momentum. We also show that the algebraically special modes of Kerr with m=0 have an additional branch of solutions not seen before in the literature. All of these results are in precise agreement with new numerical solutions for sequences of gravitational quasinormal modes of Kerr. However, we show that some prior numerical and analytic results concerning the existence of quasinormal modes of Kerr with purely imaginary frequencies were incorrect.