Superradiant instability of the Kerr-like black hole in Einstein-bumblebee gravity

An exact Kerr-like solution has been obtained recently in Einstein-bumblebee gravity model where Lorentz symmetry is spontaneously broken. In this paper, we investigate the superradiant instability of the Kerr-like black hole under the perturbation of a massive scalar field. We find the Lorentz breaking parameter $L$ does not affect the superradiance regime or the regime of the bound states. However, since $L$ appears in the metric and its effect cannot be erased by redefining the rotation parameter $\tilde{a}=\sqrt{1+L}a$, it indeed affects the bound state spectrum and the superradiance. We calculate the bound state spectrum via the continued-fraction method and show the influence of $L$ on the maximum binding energy and the damping rate. The superradiant instability could occur since the superradiance condition and the bound state condition could be both satisfied. Compared with Kerr black hole, the nature of the superradiant instability of this black hole depends non-monotonously not only on the rotation parameter of the black hole $\tilde{a}$ and the product of the black hole mass $M$ and the field mass $\mu$, but also on the Lorentz breaking parameter $L$. Through the Monte Carlo method, we find that for $l=m=1$ state the most unstable mode occurs at $L=-0.79637$, $\tilde{a}/M=0.99884$ and $M\mu=0.43920$, with the maximum growth rate of the field $\omega_{I}M=1.676\times10^{-6}$, which is about 10 times of that in Kerr black hole.