Superradiance initiated inside the ergoregion

Abstract

We consider the stationary metrics that have both the black hole and the ergoregion. The class of such metric contains, in particular, the Kerr metric. We study the Cauchy problem with highly oscillatory initial data supported in a neighborhood inside the ergoregion with some initial energy [Formula: see text]. We prove that when the time variable [Formula: see text] increases this solution splits into two parts: one with the negative energy [Formula: see text] ending at the event horizon in a finite time, and the second part, with the energy [Formula: see text], escaping, under some conditions, to the infinity when [Formula: see text]. Thus we get the superradiance phenomenon. In the case of the Kerr metric the superradiance phenomenon is “short-lived”, since both the solutions with positive and negative energies cross the outer event horizon in a finite time (modulo [Formula: see text]) where [Formula: see text] is a large parameter. We show that these solutions end on the singularity ring in a finite time. We study also the case of naked singularity.