Gravitational perturbations of higher dimensional rotating black holes: Tensor perturbations

Assessing the stability of higher-dimensional rotating black holes requires a study of linearized gravitational perturbations around such backgrounds. We study perturbations of Myers-Perry black holes with equal angular momenta in an odd number of dimensions (greater than five), allowing for a cosmological constant. We find a class of perturbations for which the equations of motion reduce to a single radial equation. In the asymptotically flat case, we find no evidence of any instability. In the asymptotically anti-de Sitter case, we demonstrate the existence of a superradiant instability that sets in precisely when the angular velocity of the black hole exceeds the speed of light from the point of view of the conformal boundary. We suggest that the end point of the instability may be a stationary, nonaxisymmetric black hole.

Figure 1

Plot of solution $\Psi (r)$ against $r/\ell$ for doubly transverse tensor perturbations with $N=2$, $l=6$, $m=8$, $r_{+}/\ell =0.7$, and ${\Omega }_H\ell =1.46242$.

Figure 2

Plot of ${\Omega }_H\ell$ against $r_{+}/\ell$ for doubly transverse tensor perturbations with $N=2$, $ϵ=1$. No black holes exist in the empty region bounded by the curve in the top right of the diagram. Curves ${\Omega }_H(r_{+},l,m)$ corresponding to existence of a normal mode with frequency $\omega =m{\Omega }_H$ are displayed for different values of $l$, $m$. From top to bottom, $(l,m)=(5,1)$, (4, 2), (3, 3), (2, 4), (3, 5), (4, 6), (5, 7), (6, 8).

Figure 3

Plots of critical value of ${\Omega }_H$ for doubly transverse tensor perturbations for $N=2$, $ϵ=1$, $(l,m)=(38,40)$ (top) and (78, 80). No black holes exist in the empty region bounded by the curve in the top right of the diagram. The fact that the ${\Omega }_H$ curves do not quite meet this curve is due to the limitations of our numerical method. Note the scale of the vertical axis.

Figure 4

Plots of $Z$ against $\omega /(m{\Omega }_H)$ for (from bottom to top) ${\Omega }_H/{\Omega }_{\max }=0.5$, 0.7, 0.9, 0.99, 0.999 with $ϵ=1$, $l=2$, and $m=4$. Note that $Z=1$ for $\omega =m{\Omega }_H$ but this point has been deleted from the topmost two curves to make the figure clearer.

Figure 5

Plot of $Z$ against $l=2,3,\dots$ for $m=m_{\max }=l+2$, $ϵ=1$, ${\Omega }_H/{\Omega }_{\max }=0.99$, and $\omega /(m{\Omega }_H)=0.99$.