Superradiant Instability and Backreaction of Massive Vector Fields around Kerr Black Holes

Abstract

We study the growth and saturation of the superradiant instability of a\ncomplex, massive vector (Proca) field as it extracts energy and angular\nmomentum from a spinning black hole, using numerical solutions of the full\nEinstein-Proca equations. We concentrate on a rapidly spinning black hole\n($a=0.99$) and the dominant $m=1$ azimuthal mode of the Proca field, with real\nand imaginary components of the field chosen to yield an axisymmetric\nstress-energy tensor and, hence, spacetime. We find that in excess of $9\\%$ of\nthe black hole's mass can be transferred into the field. In all cases studied,\nthe superradiant instability smoothly saturates when the black hole's horizon\nfrequency decreases to match the frequency of the Proca cloud that\nspontaneously forms around the black hole.\n