Ultraspinning instability of rotating black holes

TL;DR

The paper establishes the onset of axisymmetric ultraspinning instabilities for singly-spinning Myers-Perry black holes in dimensions $d\ge6$, providing a detailed numerical study of the Lichnerowicz perturbation problem under TT gauge while enforcing boundary conditions that preserve the background's $\Omega_H$ and $T_H$. It reveals a thermodynamic zero-mode at the membrane threshold and an infinite ladder of higher-$\ell$ zero-modes signaling true ultraspinning instabilities, with critical rotations that increase with dimension. In $d=5$ only a single non-thermodynamic negative mode exists, while for $6\le d\le 11$ the $\ell\ge2$ modes indicate bifurcations to new pinched black hole branches and possible connections to black rings and Saturns. These findings reinforce the thermodynamic refinement of the Gubser-Mitra conjecture and illuminate the phase structure of higher-dimensional rotating black holes, with implications for blackfolds and holography.

Abstract

Rapidly rotating Myers-Perry black holes in d>5 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 signals also 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 establish 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.

Figures (6)

• Figure 1: Phase diagram of entropy vs. angular momentum, at fixed mass, for MP black holes in $d\geq 6$ illustrating the conjecture of Emparan:2003sy (see also Emparan:2007wm): 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 appear ($A$). 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 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, as discussed in subsection \ref{['subsec:ultrathermo']}.
• Figure 2: Negative modes of the singly-spinning MP black hole in $d=5$(left), $d=6$(centre), and $d=7$(right).
• Figure 3: Functions $\delta\mu_0(x,y)$ and $\delta\mu_1(x,y)$ for the $\ell=1$ (Figs. \ref{['subfig:mu0l1']} and \ref{['subfig:mu1l1']}) and $\ell=2$ (Figs. \ref{['subfig:mu0l2']} and \ref{['subfig:mu1l2']}) modes 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 $\ell=1$ (Figs. \ref{['subfig:chil1']} and \ref{['subfig:omegal1']}) and $\ell=2$ (Figs. \ref{['subfig:chil2']} and \ref{['subfig:omegal2']}) modes respectively.
• Figure 5: Comparison between our $\Delta(r)\,h_{rr}$ for $a/r_m=4.38$ and $k_c^2r_m^2=0.114$ in $d=7$ (red dots) and the fitting function $\alpha\,J_0(\kappa\,\sigma)$ (dashed line). The red line corresponds 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 $x_\textrm{min}=0.69804$, since the comparison should be relevant near the rotation axis at $x=1$.