Dynamic instability analysis for bumblebee black holes: The odd parity

Spherical black hole (BH) solutions have been found in the bumblebee gravity where a vector field nonminimally couples to the Ricci tensor. We study dynamic (in)stability associated with the gravitational and vector perturbations of odd parity against these bumblebee BHs. Under the plane-wave approximation, we find that bumblebee BHs do not suffer ghost instability, but gradient instability and tachyonic instability exist when the bumblebee charge exceeds certain values. The existence of the instabilities also depends on the nonminimal coupling constant $\xi$ such that there is a minimal value $\xi \sim 4\pi G$ with $G$ the gravitational constant for the instabilities to happen. The theoretical consideration for bumblebee BH stability turns out to place stronger constraints on the parameter space than those from the recent observations of supermassive BH shadows by the Event Horizon Telescope Collaboration. It is also reminiscent of Penrose’s cosmic censorship conjecture since the charge of bumblebee BHs cannot be too large due to the dynamic instabilities. Specifically, for $\xi (\xi -16\pi G)>0$, we find that the charge of a bumblebee BH cannot be larger than its mass.

Figure 1

BH horizon $r_h$ versus BH charge $|Q|$ for different $\xi$.

Figure 2

$h-\xi b_t^2$ against $r$ for various $\xi$ around $0.5\kappa$. We verify that when $\xi \le 0.5\kappa$, $h-\xi b_t^2$ is a monotonically increasing function with respect to $r$.

Figure 3

The relation between the gradient (in)stability indicator, $1-\xi {\mu }_{\infty }^2$, and the charge of bumblebee BHs.

Figure 4

Constraints on the $\xi \text{−}|Q|$ plane from EHT observations and the gradient instability. The dark red region shows the parameter space where there are two branches of solutions but only one of them has gradient instability while the blue region shows that both two branches of BHs have gradient instability. Furthermore, the light red region shows the parameter space where there is only one solution and it is unstable, with the red solid line giving the boundary for the gradient instability to happen. The orange and green dot-dashed lines show the bounds given by the observations of the shadows of supermassive BHs in M87 and Milky Way, respectively [15]. The black lines give the boundary of the existence domain of BH solutions. The region enclosed by the black and blue lines indicates that there exist two branches of BHs for the same $|Q|$ and $\xi /\kappa$.