Instability and new phases of higher-dimensional rotating black holes

It has been conjectured that higher-dimensional rotating black holes become unstable at a sufficiently large value of the rotation, and that new black holes with pinched horizons appear at the threshold of the instability. We search numerically and find the stationary axisymmetric perturbations of Myers-Perry black holes with a single spin that mark the onset of the instability and the appearance of the new black hole phases. We also find new ultraspinning Gregory-Laflamme instabilities of rotating black strings and branes.

Figure 1

Diagram of entropy vs spin, at fixed mass, for MP black holes in $d\ge 6$ illustrating the conjecture of 3 (see also 6): at sufficiently large spin the MP solution becomes unstable, and at the threshold of the instability a new branch of black holes with a central pinch appears ($A$). As the spin grows new of branches of black holes with further axisymmetric pinches ($B,C,\dots$) appear. We determine the points where the new branches appear, but it is not yet known in which directions they run. We also indicate that at the inflection point (0), where ${\partial }^{2}S/\partial J^2=0$, there is a stationary perturbation that should not correspond to an instability nor a new branch but rather to a zero-mode that moves the solution along the curve of MP black holes.

Figure 2

Negative eigenvalues for the MP black hole in $d=7$.

Figure 3

Embedding diagram at $(a/r_m{)}_{\mathrm{crit}}$ of the $d=7$ black hole horizon, unperturbed (solid), and with the first unstable harmonic perturbation ($\ell =2$, $k=0$) (dashed). The embedding Cartesian coordinates $Z$ and $X$ lie respectively along the rotation axis $\theta =0$ and the rotation plane $\theta =\pi /2$. We also show the logarithmic difference between the embeddings of the perturbed ($Z_{\ell =2}$) and unperturbed ($Z_0$) horizons. The spikes represent the points where the two embeddings intersect. The perturbation has two nodes, so the horizon squeezes around the rotation axis, then bulges out, and squeezes again at the equator, as in the conjectured shape $A$ in Fig. 1.

Figure 4

Like Fig. 3, for $\ell =3$: between the first two nodes of the perturbation the horizon has a pinch (shape $B$ in Fig. 1).

Figure 5

Like Fig. 3, for $\ell =4$: the four nodes deform the horizon into shape $C$ of Fig. 1.