Superradiant instabilities of rotating black holes in the time domain

TL;DR

This work develops a time-domain framework to study superradiant instabilities of rotating black holes, focusing on a massive scalar field on Kerr spacetime and exploring confinement via a mirror or field mass. By performing a 2+1D reduction followed by a coupled 1+1D harmonic decomposition and employing perfectly matched layers, the authors simulate ultra-long evolutions up to $t \sim 10^6 M$ and monitor instability growth through exterior energy flux. They show that the growth rates and quasi-bound-state frequencies extracted from time-domain data quantitatively agree with established frequency-domain results, and they introduce a frequency-filtering technique to resolve individual modes even when beating obscures the envelope. The approach provides a robust tool for analyzing non-separable or nonlinear regimes (e.g., Proca fields, axion-like self-interactions) and offers insights into how superradiant instabilities evolve and potentially saturate, with implications for astrophysical black hole constraints on ultra-light bosons.

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 $τ^{-1} \approx 1.72 \times 10^{-7} (GM/c^3)^{-1}$ for the massive scalar field on Kerr spacetime.

Figures (13)

• Figure 1: The growth rate $M \nu$ of the superradiant instability in the dominant mode of the scalar field ($l=m=1$, $n=0$), for rapidly-rotating black holes $0.99 \le a/M \le 1$, as a function of mass coupling $M\mu$. The growth rates for $a < M$ were found by computing the quasi-bound state spectrum with the continued-fraction method of Ref. Dolan:2007. The growth rates for the extremal case $a=M$ were found by direct integration of the radial equations. The plot shows that a maximal growth rate of $M \nu \approx 1.72 \times 10^{-7}$ occurs for $a \approx 0.997 M$ and $M \mu \approx 0.45$.
• Figure 2: Evolution of black hole parameters under superradiant instability induced by a scalar field of mass $\mu$, in the linear approximation. In the superradiant regime, the black hole loses mass $M$ and angular momentum $J$ into the scalar field (predominantly emitting into the $m=l=1$, $n=0$ quasi-bound state). The BH parameters evolve along the coloured curves, from left to right (whereas the field mass $\mu$ is assumed to be constant). The instability ends at the marked points, which lie just above the (dotted) line $\mu = \Omega$. The inset, which shows $\mu^2 A / (2 \pi)$ as a function of $J / M^2$, illustrates that the horizon area $A$increases in this process, up to a maximum of $A \approx 4 \pi a \Omega / \mu^2$ (dotted line). [N.B. Units $G=c=\hbar=1$ are used].
• Figure 3: Validation of the Perfectly-Matched Layer method. Plot (a) compares snapshots taken at $t=140M$, from two evolutions of the same initial data (a time-symmetric Gaussian of width $\sigma_x = 5M$ centred around $x_0 = 10M$ in mode $l=m=1$) for a BH of spin $a=0.99M$, using (i) [red, solid] no PML, and (ii) [blue, dashed] a PML centred around $x = -100M$ of amplitude $\gamma(x)$ (with $\gamma / 10$ shown as dashed magenta line). Outside the PML region, (i) and (ii) are in good agreement. Inside the PML region, incident waves are smoothly attenuated without reflection. Plot (b) shows the same data on a logarithmic scale. Plot (c), showing data extracted at $x=x_1=0$ as a function of time, illustrates that, to the right of the PML, data sets (i) and (ii) are virtually indistinguishable.
• Figure 4: Time evolution of field on black hole background with 'mirror' at $r_m = 20M$. These plots show snapshots of $\psi_l$ [Eq. (\ref{['eq:lmode']})] for the dipole mode $l = m = 1$ of the scalar field, as a function of tortoise coordinate $x/M$, at $t = 0$ (top left), $10M$, $100M$, $10^3M$, $10^4M$ and $10^5M$ (bottom right), for the Schwarzschild ($a=0$, solid line) and fast-rotating Kerr ($a = 0.99M$, broken lines) BHs. Here the initial data is a time-symmetric Gaussian (\ref{['eq:gaussian']}) of width $\sigma_x = 2M$ centred about $x_0 = 10M$. By late times, the field has been absorbed by the Schwarzschild BH, and amplified by the Kerr BH.
• Figure 5: Time evolution of energy density $\mathcal{E}$ on black hole spacetime with 'mirror' at $r_m = 20M$. These plots show snapshots of the energy density obtained from the field shown in Fig. \ref{['fig:psi-snaps']} (see caption) via Eq. (\ref{['eq:Edensity']}). The last plot (note the scale) shows that the energy of the field on the rotating BH (broken line) has increased by a factor of $\sim 100$ over a timescale of $\sim 10^5M$ via the superradiant mechanism; whereas the field on the non-rotating spacetime ($a=0$) has drained away through the horizon.