Superradiant instability of massive vector fields around spinning black holes in the relativistic regime

We study the superradiant instability of massive vector fields, i.e. Proca fields, around spinning black holes in the test field limit. This is motivated by the possibility that observations of astrophysical black holes can probe the existence of ultralight bosons subject to this mechanism. By making use of time-domain simulations, we characterize the growth rate, frequency, spatial distribution, and other properties of the unstable modes, including in the regime where the black hole is rapidly spinning and the Compton wavelength of the Proca field is comparable to the black hole radius. We find that relativistic effects in this regime increase the range of Proca masses that are unstable, as well as the maximum instability rate. We also study the gravitational waves that can be sourced by such an instability, finding that they can be significantly stronger than in the massive scalar field case.

Figure 1

The energy (top) and angular momentum (bottom) in the Proca field as a function of time for cases with $m=1$ and a black hole with spin $a=0.99$. After a short transient period, the evolution is dominated by an exponentially growing mode.

Figure 2

The ratio of the energy to angular momentum in the Proca field $E/J$, and the ratio of the energy flux to angular momentum flux ${\dot{E}}^H/{\dot{J}}^H$ through the black horizon, as a function of time for cases with $\tilde{\mu }=0.3$, 0.4, and 0.5, $m=1$, and a black hole with spin $a=0.99$. After the onset of the instability, this ratio settles to a roughly constant value which can be used as a measure of the frequency ${\omega }_R$ of the dominant unstable mode.

Figure 3

Measured values of the frequency of the dominant unstable mode for $m=1$, $a=0.99$, and various values of $\tilde{\mu }$. Top panel: The real part of the frequency as measured from the ratio of the energy to angular momentum along with the fitting function given by Eq. (14). Bottom panel: The growth (black points) or decay (red points) rate of the dominant mode of the Proca field along with the fitting function given by Eq. (15).

Figure 4

The $m=1$ instability growth rate as a function of black hole spin for several values of the Proca mass. The dotted blue line shows the dependence ${\omega }_I\propto r_{+}(1-{\omega }_R/{\mathrm{\Omega }}_H)$ with the amplitude fit to the $a\le 0.9$ points.

Figure 5

The same as Fig. 3, but for a black hole with spin $a=0.7$. The dotted blue line in the top panel (${\omega }_{\text{Fit}}$) is the fit from the $a=0.99$ data in Fig. 3 (not the data shown in this figure).

Figure 6

Similar to Fig. 3, but for modes with $m=2$ (and $a=0.99$). Top panel: The real part of the frequency as measured from ${\omega }_R=2E/J$, along with Eq. (14) (which was obtained by fitting to the data in Fig. 3). Bottom panel: The growth (black points) or decay (red points) rate of the dominant mode of the Proca field along with the fitting function.

Figure 7

The falloff of energy density along the equator for cases with $m=1$, $a=0.99$ and various values of $\mu$. The equatorial radius is multiplied by $M{\mu }^2$ which sets the length scale, and for smaller values of $\mu$ the falloff of energy goes like $\exp (-2M{\mu }^2{r}_{eq})$.

Figure 8

Energy (left) and angular momentum density (right) for cases with a black hole with $a=0.99$ and (from top to bottom) $\tilde{\mu }=0.2$, 0.5, and 0.95, and $m=1$, 1, and 2, respectively. The $z$ axis is the spin axis of the black hole and $z=0$ is the equatorial plane. Note the difference in spatial scales between the top and bottom two cases.

Figure 9

Energy density near the black hole for cases with $m=1$, $a=0.99$, and $\tilde{\mu }=0.3$, 0.4, and 0.5 (left to right). The $z$ axis is the black hole spin axis.

Figure 10

Streamlines of the real and imaginary parts of ${\chi }_i$ in the equatorial plane ($z=0$; top), and in the $y=0$ plane (bottom) for snapshots from $m=1$, $a=0.99$, and $\tilde{\mu }=0.2$ (left) and 0.5 (right). Note the difference in scales in the two cases.

Figure 11

The GW power as a function of $\tilde{\mu }$ for the dominant $m=1$ unstable mode consisting of a single real-valued Proca field around an $a=0.99$ black hole (black points). In addition to the total GW power, which predominately comes from the $(\ell ,m)=(2,\pm 2)$ components, we also show the contributions from the $(\ell ,m)=(3,\pm 2)$ (red Xs) and $(\ell ,m)=(4,\pm 2)$ (magenta diamonds) spin-2 spherical harmonics. The GW power has been divided by the square of the Proca field energy so as to be independent of the magnitude of the mode, in the test field limit. For comparison we also show the superradiant instability growth rate of energy, at a reference energy of $E=M$ (blue squares). The ratio of GW luminosity to the superradiant growth rate for a Proca cloud of energy $E$ would then be given by multiplying the ratio of the black points to the blue squares by $(E/M)$.

Figure 12

A comparison of the instability growth rate for the dominant $m=1$ mode obtained here [black dots which are fit by the dotted curve, Eq. (15)] to the nonrelativistic approximation of [14] (blue curves) and the low-spin approximation of [10] (red curve) at two values of the black hole spin $a=0.7$ and 0.99.

Figure 13

Top: The energy in the Proca field as a function of time for a case with $\tilde{\mu }=0.3$, $m=1$ and a black hole with spin $a=0.9$, at three different resolutions. Bottom: The difference in this quantity between resolutions, scaled assuming fourth-order convergence. The feature at early times is just truncation error from the numerical integration of the Proca energy, zeroed inside the black hole horizon, while a sizable fraction of the initial perturbation is falling into the black hole.