Superradiant instabilities of rotating black holes in the time domain

TL;DR

This paper investigates superradiant instabilities of scalar fields around rotating black holes using long-time numerical simulations, revealing growth rates and mode structures consistent with prior frequency-domain analyses.

Contribution

It introduces an efficient time-domain method to study superradiant instabilities over ultra-long timescales, including for massive fields and mirror boundary conditions.

Findings

Confirmed the maximum growth rate of superradiant instability matches previous frequency-domain results.

Demonstrated the method's ability to resolve complex mode interactions and beating effects.

Provided detailed mode structure and growth rates for various parameters.

Abstract

Bosonic fields on rotating black hole spacetimes are subject to amplification by superradiance, which induces exponentially-growing instabilities (the `black hole bomb') in two scenarios: if the black hole is enclosed by a mirror, or if the bosonic field has rest mass. Here we present a time-domain study of the scalar field on Kerr spacetime which probes ultra-long timescales up to $t \lesssim 5 \times 10^6 M$, to reveal the growth of the instability. We describe an highly-efficient method for evolving the field, based on a spectral decomposition into a coupled set of 1+1D equations, and an absorbing boundary condition inspired by the `perfectly-matched layers' paradigm. First, we examine the mirror case to study how the instability timescale and mode structure depend on mirror radius. Next, we examine the massive-field, whose rich spectrum (revealed through Fourier analysis) generates `beating' effects which disguise the instability. We show that the instability is clearly revealed by tracking the stress-energy of the field in the exterior spacetime. We calculate the growth rate for a range of mass couplings, by applying a frequency-filer to isolate individual modal contributions to the time-domain signal. Our results are in accord with previous frequency-domain studies which put the maximum growth rate at $\tau^{-1} \approx 1.72 \times 10^{-7} (GM/c^3)^{-1}$ for the massive scalar field on Kerr spacetime.