Gravitational instabilities in Kerr space-times

TL;DR

This paper demonstrates that Kerr space-times, including regions beyond the inner horizon and naked singularities, are unstable under gravitational linear perturbations, through analysis of the Teukolsky equation and spheroidal harmonics.

Contribution

It introduces a transformation of the radial Teukolsky equation into Schrödinger form and proves the existence of infinite unstable modes in Kerr space-times.

Findings

Unstable axisymmetric modes exist in Kerr space-times.

Regions beyond the inner horizon are gravitationally unstable.

Naked singularities in Kerr space-times are also unstable.

Abstract

In this paper we consider the possible existence of unstable axisymmetric modes in Kerr space times, resulting from exponentially growing solutions of the Teukolsky equation. We describe a transformation that casts the radial equation that results upon separation of variables in the Teukolsky equation, in the form of a Schr\"odinger equation, and combine the properties of the solutions of this equations with some recent results on the asymptotic behaviour of spin weighted spheroidal harmonics to prove the existence of an infinite family of unstable modes. Thus we prove that the stationary region beyond a Kerr black hole inner horizon is unstable under gravitational linear perturbations. We also prove that Kerr space-time with angular momentum larger than its square mass, which has a naked singularity, is unstable.