Unstable fields in Kerr spacetimes

TL;DR

This paper demonstrates the existence of unstable solutions to the Teukolsky equation in the interior and over-extremal Kerr spacetimes, indicating potential instabilities in these exotic regions of black hole and naked singularity models.

Contribution

It identifies and analyzes unstable modes in Kerr black hole interiors and naked singularities, linking these instabilities to the change in the Teukolsky equation's character in certain regions.

Findings

Unstable solutions exist for all harmonic modes in these regions.

Axially symmetric modes grow exponentially over time.

The instability is related to the 'time machine' region where the equation's nature changes.

Abstract

We show that both the interior region $r<M-\sqrt{M^2-a^2}$ of a Kerr black hole and the $a^2>M^2$ Kerr naked singularity admit unstable solutions of the Teukolsky equation for any value of the spin weight. For every harmonic number there is at least one axially symmetric mode that grows exponentially in time and decays properly in the radial directions. These can be used as Debye potentials to generate solutions for the scalar, Weyl spinor, Maxwell and linearized gravity field equations on these backgrounds, satisfying appropriate spatial boundary conditions and growing exponentially in time, as shown in detail for the Maxwell case. It is suggested that the existence of the unstable modes is related to the so called "time machine" region, where the axial Killing vector field is time-like, and the Teukolsky equation, restricted to axially symmetric fields, changes its character from hyperbolic to elliptic.