Dynamics and Stability of Black Rings

TL;DR

This work analyzes the dynamics and stability of neutral five-dimensional black rings, revealing a rich instability structure beyond spherical black holes. By combining off-shell radial perturbations, GL-instability analysis, and null-geodesic tests, it shows fat rings are radially unstable while thin rings can be radially stable, though most thin rings suffer GL-type instabilities, with a cusp marking a zero mode and a potential narrow stable window. The study also investigates fragmentation into multiple black holes, emission/absorption processes, and Euclidean thermodynamics, finding no regular real Euclidean section and a grand-canonical preference for Myers-Perry black holes. Collectively, these results suggest neutral black rings are dynamically unstable over a broad parameter space, though special parameter ranges and charged/dipole variants may yield stability or different endstates.

Abstract

We examine the dynamics of neutral black rings, and identify and analyze a selection of possible instabilities. We find the dominating forces of very thin black rings to be a Newtonian competition between a string-like tension and a centrifugal force. We study in detail the radial balance of forces in black rings, and find evidence that all fat black rings are unstable to radial perturbations, while thin black rings are radially stable. Most thin black rings, if not all of them, also likely suffer from Gregory-Laflamme instabilities. We also study simple models for stability against emission/absorption of massless particles. Our results point to the conclusion that most neutral black rings suffer from classical dynamical instabilities, but there may still exist a small range of parameters where thin black rings are stable. We also discuss the absence of regular real Euclidean sections of black rings, and thermodynamics in the grand-canonical ensemble.

Figures (12)

• Figure 1: Phase diagram of the five-dimensional black rings (solid) and MP black holes (dotted). We plot area ( i.e., entropy) vs. angular momentum $j^2$, both in reduced units for fixed mass.
• Figure 2: Temperature (left) and angular velocity (right) vs. angular momentum $j$ for black rings (solid) and MP black holes (dotted) of fixed mass. Both $T$ and $\Omega$ have been rendered dimensionless by multiplying by $\sqrt{GM}$.
• Figure 3: Black ring visuals. The azimuthal angle of the $S^2$ is suppressed. The plot shows the isometric embedding of the $S^2$ cross section (see fig. \ref{['fig:S2dist']}) with the size of the $S^1$ estimated as the inner radius of the horizon. All three rings shown have the same mass, and the black rings with $\nu=0.05$ and $\nu=0.95$ also have the same horizon area.
• Figure 4: Isometric embedding of the cross-section of the black ring two-sphere for fixed mass $G M =1$. The azimuthal angle is suppressed. The table shows values of the reduced angular momentum $j$ for each $\nu$. The black ring with minimal spin $j^2 = 27/32$ ($\nu=1/2$) is shown in gray. Solutions on the thin black ring branch ($\nu<1/2$) have nearly round two-spheres. The spheres flatten out on the fat black ring branch ($1/2 < \nu < 1$) as $j \to 1$. A black ring with $\nu=\nu_0$ has the same horizon area as one with $\nu=1-\nu_0$.
• Figure 5: Sketch of a diametrical cross section of the black ring horizon showing some definitions of radii characterizing the black ring horizon: the 'equator' radius $R_2^\circ$ of the $S^2$, the inner and outer $S^1$ radii, $R_1^\mathrm{inner}$ and $R_1^\mathrm{outer}$, and finally $R_1^\circ$ being the radius of the $S^1$ evaluated at the 'equator' of the $S^2$.