Black-hole bomb and superradiant instabilities

A wave impinging on a Kerr black hole can be amplified as it scatters off the hole if certain conditions are satisfied, giving rise to superradiant scattering. By placing a mirror around the black hole one can make the system unstable. This is the black-hole bomb of Press and Teukolsky. We investigate in detail this process and compute the growing time scales and oscillation frequencies as a function of the mirror's location. It is found that in order for the system black hole plus mirror to become unstable there is a minimum distance at which the mirror must be located. We also give an explicit example showing that such a bomb can be built. In addition, our arguments enable us to justify why large Kerr-AdS black holes are stable and small Kerr-AdS black holes should be unstable.