Ultraspinning instability of anti-de Sitter black holes

TL;DR

The paper extends the ultrasound of ultraspinning instabilities from asymptotically flat to asymptotically AdS singly-spinning Myers-Perry black holes by numerically solving a modified Lichnerowicz eigenvalue problem for stationary axisymmetric perturbations.Thermodynamic zero-modes l=0 and l=1 delineate the ultraspinning surface and, beyond it, a ladder of non-thermodynamic zero-modes with l>=2 signals classical ultraspinning instabilities and potential bifurcations to pinched AdS black hole families.All identified ultraspinning instabilities occur in the superradiant regime (Omega_H ell > 1), implying intertwined dynamical behavior and possible inheritance by the new pinched branches, with implications for AdS/CFT phase structure and holography.The work provides a framework to map onset as a function of the cosmological constant, motivates time-dependent analyses to determine growth rates, and supports conjectured connections between MP-AdS branches and more complex AdS black hole configurations.

Abstract

Myers-Perry black holes with a single spin in d>5 have been shown to be unstable if rotating sufficiently rapidly. We extend the numerical analysis which allowed for that result to the asymptotically AdS case. We determine numerically the stationary perturbations that mark the onset of the instabilities for the modes that preserve the rotational symmetries of the background. The parameter space of solutions is thoroughly analysed, and the onset of the instabilities is obtained as a function of the cosmological constant. Each of these perturbations has been conjectured to represent a bifurcation point to a new phase of stationary AdS black holes, and this is consistent with our results.

Figures (5)

• Figure 1: Phase diagram of singly-spinning MP-AdS black holes in $d\geq 6$. We plot the entropy $S$ vs. the angular momentum $J$, at a fixed value of the mass $M$, in units of the AdS curvature radius $\ell$. The figure illustrates the conjecture of Ref. Caldarelli:2008pz. At sufficiently large spin the MP-AdS solution becomes unstable for axisymmetric perturbations (dashed line), and at the threshold of the instability a new branch of black holes with a central pinch appear ($I$). As the spin grows, new branches of black holes with further axisymmetric pinches ($II$, $III$, …) appear. We determine numerically the points where the new branches appear, but it is not yet known in which directions they run.
• Figure 2: Parameter space of singly-spinning Myers-Perry$-$AdS black holes in $d=6$: dimensionless rotation parameter $a/r_m$ as a function of the dimensionless mass-radius parameter $r_m/\ell$. Regular black holes exist only for $a<\ell$. The thermodynamic $l=0$ and $l=1$ zero-mode curves, described by \ref{['Hessian0modeL0']} and \ref{['Hessian0modeL1']}, are plotted. The thermodynamic $l=1$ zero-mode curve defines also the ultraspinning surface, above which (region $C$) the black holes might be ultraspinning unstable (later section \ref{['sec:results']} and Fig. \ref{['fig:spectrum']} confirm this is indeed the case). The superradiant curve $\Omega_H\ell=1$ is also plotted. Above this curve all black holes are superradiant unstable. For $d>6$ the plot is qualitatively very similar.
• Figure 3: Dimensionless negative modes $\lambda r_m^2$ of the singly-spinning MP-AdS black hole in $d=6$ as a function of the dimensionless rotation parameter $a/r_m$ for fixed $r_m/\ell$. In the left plot, we represent the spectrum for $r_m/\ell=1.0$, while the right plot describes the spectrum for $r_m/\ell=0.5$. The latter has several (supposedly infinite) branches of zero-modes (for which $\lambda=0$) and the corresponding negative eigenvalues are labeled by the integer $l$. For this particular black hole family with $r_m/\ell=0.5$, the values of $a/r_m$ at which the first few branches intersect the $\lambda=0$ axis are: $a/r_m{\bigl|}_{l=1}\simeq 1.10$, $a/r_m{\bigl|}_{l=2} \simeq 1.49$, $a/r_m{\bigl|}_{l=3} \simeq 1.69$, $a/r_m{\bigl|}_{l=4} \simeq 1.81$, and $a/r_m{\bigl|}_{l=5} \simeq 1.88$. As opposed to the thermodynamic $l=0,1$ zero-modes (curve on left plot, first two curves on right plot), the zero-modes with $l\geq 2$ describe the onset of ultraspinning instabilities of the black hole.
• Figure 4: Number of negative modes of the singly-spinning MP-AdS black hole in $d=6$(left) and $d=7$(right). The plots describe the dimensionless rotation parameter $a/r_m$ as a function of the dimensionless mass-radius $r_m/\ell$. As we move from the bottom to the top, the new colored/dotted areas represent regions where a new negative mode of the (modified) Lichnerowicz operator gets excited: blue (one negative mode), red (2 n.m.), purple (3 n.m.), green (4 n.m.), yellow (5 n.m.), pink (6 n.m.), … The interpretation of the several curves plotted is described in Fig. \ref{['fig:thermo6d']}. (Note that the faults in these figures, e.g. where we expected to find blue dots, correspond to parameter space points where our numerical code failed. They do not correspond to black holes with no negative modes.)
• Figure 5: Number of negative modes of the singly-spinning MP-AdS black hole in $d=6$. This figure has the data of Fig. \ref{['fig:spectrum']}-left, now plotting the dimensionless angular momentum $J/\ell^{d-2}$ as a function of the dimensionless mass $M/\ell^{d-3}$. Regular MP-AdS black holes exist for $|J|<M\ell$. We represent the location of the $l=0$ and the $l=1$ zero-modes, for reference.