Ultraspinning instability of rotating black holes

Rapidly rotating Myers-Perry black holes in $d\ge 6$ dimensions were conjectured to be unstable by Emparan and Myers. In a previous publication, we found numerically the onset of the axisymmetric ultraspinning instability in the singly spinning Myers-Perry black hole in $d=7$, 8, 9. This threshold also signals a bifurcation to new branches of axisymmetric solutions with pinched horizons that are conjectured to connect to the black ring, black Saturn and other families in the phase diagram of stationary solutions. We firmly establish that this instability is also present in $d=6$ and in $d=10$, 11. The boundary conditions of the perturbations are discussed in detail for the first time, and we prove that they preserve the angular velocity and temperature of the original Myers-Perry black hole. This property is fundamental to establishing a thermodynamic necessary condition for the existence of this instability in general rotating backgrounds. We also prove a previous claim that the ultraspinning modes cannot be pure gauge modes. Finally we find new ultraspinning Gregory-Laflamme instabilities of rotating black strings and branes that appear exactly at the critical rotation predicted by the aforementioned thermodynamic criterium. The latter is a refinement of the Gubser-Mitra conjecture.

Figure 1

Phase diagram of entropy vs angular momentum, at fixed mass, for MP black holes in $d\ge 6$ illustrating the conjecture of 45 (see also 40): 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 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 correspond to neither an instability nor a new branch but rather to a zero-mode that moves the solution along the curve of MP black holes, as discussed in Sec. 2b.

Figure 2

Negative modes of the singly spinning MP black hole in $d=5$ (left), $d=6$ ( center), and $d=7$ (right).

Figure 3

Functions $\delta {\mu }_0(x,y)$ and $\delta {\mu }_1(x,y)$ for the $\ell =1$ mode [Figs. 3a, 3c] and $\ell =2$ mode [Figs. 3b, 3d], respectively. The number of zeros at $y=0$ (the horizon) coincides with the integer $\ell$.

Figure 4

Functions $\delta \chi (x,y)$ and $\delta \omega (x,y)$ for the $\ell =1$ mode [Figs. 4a, 4c] and $\ell =2$ mode [Figs. 4b, 4d], respectively.

Figure 5

Comparison between our $\Delta (r)h_{rr}$ for $a/r_m=4.38$ and $k_c^2{r}_m^2=0.114$ in $d=7$ (dots) and the fitting function $\alpha J_0(\kappa \sigma )$ (dashed line). The solid (red) line corresponds to the interpolating polynomial to our data and should only serve to guide the eye. To obtain the plots, the minimum value of $x$ was taken to be $x_{\min }=0.69804$, since the comparison should be relevant near the rotation axis at $x=1$.

Figure 6

Embedding diagrams [Figs. 6a, 6b, 6c] at $(a/r_m{)}_{\mathrm{crit}}$ of the $d=7$ black hole horizon, unperturbed (light blue), and perturbed (dark red) with the $\ell =2$, 3, 4 harmonics (in this order). The embedding Cartesian coordinates $Z$ and $X$ lie along the rotation axis $\theta =0$ and the rotation plane $\theta =\pi /2$, respectively. We also show the logarithmic difference between the embeddings of the perturbed ($Z_{\ell =2,3,4}$) and unperturbed ($Z_0$) horizons. The number of spikes corresponds to the number of crossings between the two embeddings. Each $+$ sign indicates that the perturbation bulges out relative to the background, and the $-$ signs indicate the opposite situation. Figures 6d, 6e, 6f provide a three-dimensional illustration of the spatial sections of the horizon, with the ($d-4$)-sphere suppressed at every point, with the same color code.