Instabilities of extremal rotating black holes in higher dimensions

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.