Instabilities of extremal rotating black holes in higher dimensions

TL;DR

This paper proves a conjecture linking the eigenvalues of a horizon operator to the linear stability of extremal rotating black holes in higher dimensions, extending stability criteria and revealing new instabilities.

Contribution

It provides a rigorous proof of the instability criterion for extremal rotating black holes, extending previous numerical evidence and methods to a broader class of black holes.

Findings

Eigenvalues below -1/4 indicate instability

Extended the conjecture to include Anti-deSitter black holes

Identified additional instabilities in ultra-spinning black holes

Abstract

Recently, Durkee and Reall have conjectured a criterion for linear instability of rotating, extremal, asymptotically Minkowskian black holes in $d\ge 4$ dimensions, such as the Myers-Perry black holes. They considered a certain elliptic operator, $\cA$, acting on symmetric trace-free tensors intrinsic to the horizon. Based in part on numerical evidence, they suggested that if the lowest eigenvalue of this operator is less than the critical value $-1/4$ ( called "effective BF-bound"), then the black hole is linearly unstable. In this paper, we prove an extended version of their conjecture. Our proof uses a combination of methods such as (i) the "canonical energy method" of Hollands-Wald, (ii) algebraically special properties of the near horizon geometries associated with the black hole, (iii) the Corvino-Schoen technique, and (iv) semiclassical analysis. Our method of proof is also applicable to rotating, extremal asymptotically Anti-deSitter black holes. In that case, we find additional instabilities for ultra-spinning black holes. Although we explicitly discuss in this paper only extremal black holes, we argue that our results can be generalized to {\em near} extremal black holes.