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

TL;DR

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 is studied using numerical solutions of the full Einstein-Proca equations.

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.

Figures (6)

• Figure 1: The energy (top) and angular momentum (bottom) in the Proca field as a function of time (solid lines), along with the loss in mass (top) and angular momentum (bottom) of the BH (dashed lines).
• Figure 2: The BH horizon frequency $\Omega_{\rm BH}$, as calculated from the BH's mass and angular momentum, and the ratio of the flux of Proca field energy and angular momentum $\dot{E}^H/\dot{J}^H$ through the BH horizon, as a function of time.
• Figure 3: The final BH irreducible mass, total mass, and angular momentum after saturation of the $m=1$ superradiant instability for a BH with $a=0.99$ ($M_{{\rm irr},0}/M_0\approx 0.76$) initially. The lines show the prediction obtained with the assumption that the BH will lose energy and angular momentum in fixed proportion $\omega$, until $\Omega_{\rm BH}=\omega$, while the points show the measured values from the simulations.
• Figure 4: The energy (left) and angular momentum density (right) of the Proca field in the final state with $\tilde{\mu}=0.25$ (top) and $\tilde{\mu}=0.5$ (bottom) for a slice containing the BH spin axis (the $z$ axis) and perpendicular to the equatorial plane ($z=0$); note the different scales for the two cases.
• Figure 5: The energy in the Proca field for simulations performed at three different resolutions, and the difference in this quantity with resolution, scaled assuming fourth-order convergence for $\tilde{\mu}=0.4$ (top) and $\tilde{\mu}=0.5$ (bottom).