João Rosa

University of Oxford

The extremal black hole bomb

We analyze the spectrum of massive scalar bound states in the background of extremal Kerr black holes, focusing on modes in the superradiant regime, which grow exponentially in time and quickly deplete the black hole's mass and spin. Previous analytical estimates for the growth rate of this instability were limited to the small and large scalar mass regimes. In this work, we discuss an analytical method to compute the superradiant spectrum in general, namely in the phenomenologically interesting regime where the horizon radius and the Compton wavelength of the massive scalar particles are comparable. To do this, we solve the radial mode equation in two overlapping regions and match the solutions in their common domain of validity. We show that matching the functional forms of these functions involves approximations that are not valid for the whole range of scalar masses, exhibiting unphysical poles that produce a large enhancement of the growth rate. Alternatively, we match the functions at a single point and show that, despite the uncertainty in the choice of the match point, this method eliminates the spurious poles and agrees with previous numerical computations of the spectrum using a continued-fraction method.