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

## TL;DR

This paper investigates the superradiant instability of massive vector fields around rapidly spinning black holes, demonstrating significant energy transfer and the formation of a stable Proca cloud through numerical solutions of Einstein-Proca equations.

## Contribution

It provides the first detailed numerical analysis of superradiant instability for massive vector fields around Kerr black holes, including backreaction effects.

## Key findings

- Over 9% of black hole mass transferred to the Proca field
- Instability saturates when black hole's horizon frequency matches the Proca cloud frequency
- Formation of a stable, axisymmetric Proca cloud around the black hole

## Abstract

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