An instability of higher-dimensional rotating black holes

TL;DR

This work provides the first demonstration of a linearized gravitational instability for an asymptotically flat vacuum black hole in higher dimensions, showing that nine-dimensional Myers-Perry black holes with equal angular momenta become unstable when spinning exceeds a critical value. By exploiting cohomogeneity-1 symmetry and scalar perturbations on CP^N, the authors reduce the problem to a tractable eigenvalue problem solved via spectral methods, revealing zero-modes that signal bifurcations to new black-hole branches with reduced rotational symmetry. They show that, in D=9, there exists a non-thermodynamic, time-dependent instability beyond a certain spin a2, as well as an ultraspinning-criterion-driven sequence of stationary modes, and they connect these instabilities to a large 70-parameter family of single-symmetry black holes. The paper also analyzes the Gregory-Laflamme instability of rotating black strings, demonstrating that rotation generally enhances string instability and that thermodynamic considerations explain the presence of negative modes and threshold modes. Together, these results illuminate the rich landscape of higher-dimensional black holes and imply a proliferation of new solutions with minimal symmetry, challenging uniqueness in higher dimensions.

Abstract

We present the first example of a linearized gravitational instability of an asymptotically flat vacuum black hole. We study perturbations of a Myers-Perry black hole with equal angular momenta in an odd number of dimensions. We find no evidence of any instability in five or seven dimensions, but in nine dimensions, for sufficiently rapid rotation, we find perturbations that grow exponentially in time. The onset of instability is associated with the appearance of time-independent perturbations which generically break all but one of the rotational symmetries. This is interpreted as evidence for the existence of a new 70-parameter family of black hole solutions with only a single rotational symmetry. We also present results for the Gregory-Laflamme instability of rotating black strings, demonstrating that rotation makes black strings more unstable.

Figures (7)

• Figure 1: Results in $D=5$, $\kappa=0$. We represent this mode for fixed values of $\Gamma \,r_M$ (first graph) and $a/ r_M$ (second graph). This unstable mode of the string corresponds to well-known Gregory-Laflamme mode.
• Figure 2: Results in $D=7$, $\kappa=0$. The graphs are entirely analogous to the ones in Fig. \ref{['fig:5dkzero']}.
• Figure 3: Results in $D=7$, $\kappa=1$. We represent this mode for fixed values of $\Gamma \,r_M$ (first graph) and $a/ r_M$ (second graph). It corresponds to a new Gregory-Laflamme instability of the rotating black string, appearing for the numerical value $(a_1/r_M)^\mathrm{num}=0.5949$. In the first graph, the vertical line to the left corresponds to the analytical prediction of \ref{['eqn:a1']}, $(a_1/r_M)=0.5946\,$, and the interrupted vertical line to the right corresponds to extremality. The second graph indicates clearly that the instability of the black brane does not extend to an instability of the black hole ($k = 0$).
• Figure 4: Results in $D=9$, $\kappa=1$. The graphs are entirely analogous to the ones in Fig. \ref{['fig:7d']}.
• Figure 5: Results in $D=9$, $\kappa=2$. We represent this mode for fixed values of $\Gamma \,r_M$ (first graph) and $a/ r_M$ (second graph). As opposed to the $\kappa=1$ case, the time-dependent mode extends all the way to $k=0$. There is not only a new Gregory-Laflamme instability of the black string, but also an instability of the black hole, appearing at $a=a_2$, where $a_2/r_M=0.6858 > a_1/r_M =0.6300\,$. In the first graph, the interrupted vertical line to the right corresponds to extremality. In the second graph, the curve shrinks to the origin as $a \rightarrow a_2$.