Horizon instability of extremal Kerr black holes: Nonaxisymmetric modes and enhanced growth rate

We show that the horizon instability of the extremal Kerr black hole is associated with a singular branch point in the Green function at the superradiant bound frequency. We study generic initial data supported away from the horizon and find an enhanced growth rate due to nonaxisymmetric modes. The growth is controlled by the conformal weight $h$ of each mode. We speculate on connections to near-extremal black holes and holographic duality.

Figure 1

An example of the inverse Laplace transform (8). We plot the real part of $v^{5/2}{L}^{-1}[s^{-q}/(p+s^{ia})]$ at late times using the parameters $\{p=1.2568474\times {10}^{-4}–4.7826013\times {10}^{-5}i,q=7/2-2.9459584i,\alpha =1.8919169\}$ corresponding to the complex conformal weight associated with the $\ell =m=2$ mode and the third derivative. The dots are numerical calculations using the Mathematica package numericallaplaceinversion based on [22], while the line is a fit to $A\cos (\gamma +\beta \log v)$ (the real part of $v^{5/2}$ times the right-hand side of (8) with $A=0.25641175$, $\beta =1.05403570$ and $\gamma =-0.746571461$.