Superradiance initiated inside the ergoregion

TL;DR

This paper demonstrates superradiance inside the ergoregion of stationary black hole metrics, showing how initial oscillatory data can split into parts with negative and positive energies, with implications for black hole physics.

Contribution

It provides a rigorous analysis of superradiance phenomena for solutions supported inside the ergoregion, including the Kerr metric, revealing finite-time behaviors and energy splitting.

Findings

Solutions split into negative and positive energy parts over time.

Negative energy solutions reach the event horizon in finite time.

Positive energy solutions can escape to infinity, demonstrating superradiance.

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 $E_0$. We prove that when the time variable $x_0$ increases this solution splits into two parts: one with the negative energy $-E_1$ ending at the event horizon in a finite time, and the second part, with the energy $E_2=E_0+E_1>E_0$, escaping, under some conditions, to the infinity when $x_0\rightarrow +\infty$. 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 $O(\frac{1}{k})$) where $k$ 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.