Black hole instabilities and local Penrose inequalities

TL;DR

The paper develops a simple, gauge-invariant framework based on local Penrose inequalities to diagnose instabilities of higher-dimensional black holes via conformal perturbations of initial data. It demonstrates near-onset instability predictions for the Gregory-Laflamme instability of black strings, confirms ultraspinning instabilities of Myers-Perry black holes, and shows fat black rings are classically unstable under rotationally symmetric perturbations, while thin rings show no such instability in the studied regime. The method provides a practical alternative to full perturbation theory, yielding results that closely track known thresholds and offering a path to extend stability analyses to AdS and charged cases. Overall, the approach highlights a tight link between horizon-geometry inequalities and dynamical stability in higher-dimensional gravity.

Abstract

Various higher-dimensional black holes have been shown to be unstable by studying linearized gravitational perturbations. A simpler method for demonstrating instability is to find initial data that describes a small perturbation of the black hole and violates a Penrose inequality. An easy way to construct initial data is by conformal rescaling of the unperturbed black hole initial data. For a compactified black string, we construct initial data which violates the inequality almost exactly where the Gregory-Laflamme instability appears. We then use the method to confirm the existence of the "ultraspinning" instability of Myers-Perry black holes. Finally we study black rings. We show that "fat" black rings are unstable. We find no evidence of any rotationally symmetric instability of "thin" black rings.

Figures (10)

• Figure 1: ${\cal A}$ vs. $r_+/L$ for the $d=5$ black string. For other dimensions the plots look qualitatively similar. For large values of $r_+/L$, $\mathcal{A}$ is positive but it becomes negative at a certain critical value, signalling an instability. For any number of dimensions this critical value is always smaller from critical value for the onset of the GL instability. In $d=5$ we find $(r_+/L)_\textrm{crit}=0.8745$, which differs by less than $0.2$% from the GL critical value. The agreement gets better as the number of dimensions increases.
• Figure 2: The solutions $\dot{\psi}$ (left) and $\ddot{\psi}$ (right) for $a/r_M=2$ and $(c_1,c_2,c_3,c_4)=(3.51,3.12,1.50,0.239)$. The dotted line shows the position of the horizon of the unperturbed solution ($w=0$).
• Figure 3: The first order deviation of the apparent horizon for the solution of Fig. \ref{['fig:psis']}.
• Figure 4: The minimum value of $\bar{Q}$ is plotted against $a/r_M$. $\bar{Q}_\textrm{min}$ is negative for $a/r_M>1.933$.
• Figure 5: Adapted coordinates $(r_1,r_2)$ for $R=1$ and $\nu=0.6$. The thick curve $z/z_\textrm{max}=0$ corresponds to the event horizon of the background solution. The region inside the thick curve corresponds to the opposite side of the Einstein-Rosen bridge. We impose a Dirichlet boundary condition at $z/z_\textrm{max}=-0.5$ which is located behind the horizon $z/z_\textrm{max}=0$.