Ultraspinning Limits and Classical Phase Structure

Rapidly rotating AdS black holes exhibit several phenomena that are often called “ultraspinning,” but they do not share one universal limit or endpoint. A standard Kerr–AdS family can approach a singular $a\to L$ boundary; a rescaled azimuth can instead produce a new noncompact finite-area horizon; and higher-dimensional spherical horizons can develop membrane-like perturbative zero modes before either limit. Thermodynamic branches, stationary lumpy solutions, and time-dependent instabilities supply different kinds of evidence. The status below is checked through 2026-08-12 and concerns classical bulk geometry, not a boundary-field-theory phase diagram.

Helpful background. Exact Solutions of Einstein’s Equations supplies Kerr–AdS and limiting metrics; Causality and Global Structure supplies event horizons and timelike-boundary evolution; The Einstein Equation supplies the vacuum and linearized equations; The Gibbons-Hawking-York Boundary Term supplies asymptotic charges and ensemble data; and Black-Hole Geometry and Mechanics supplies the stability and first-law distinctions used in comparing branches.

Three inequivalent ultraspinning regimes

For the standard globally identified Kerr–AdS family, $\Xi=1-a^2/L^2$ and regularity requires $\Xi\gt0$. Taking $a\to L$ with the usual azimuth and fixed metric parameter makes the area and charges diverge; scaling the parameter to keep charges finite instead drives the horizon toward a singular or vanishing limit. This is not a regular member of the original family Dias, Figueras, Monteiro, and Santos 2010, §2.1, equations (2.1)–(2.5), Open PDF, pp. 3–6.

The super-entropic limit first changes to an azimuth rotating at infinity, rescales that azimuth by $\Xi$, takes $a\to L$, and then imposes a new finite period. It changes the global spacetime: the rotation axis is excised and the horizon is a sphere with two punctures, noncompact but of finite area. It has no smooth asymptotically flat limit Hennigar, Kubizňák, Mann, and Musoke 2015, §§2.1–2.2, Open PDF, pp. 4–7; §5, pp. 24–25. This exact generating limit is not the endpoint of evolving ordinary Kerr–AdS data because its quotient and asymptotic frame have changed.

The membrane-like regime occurs for singly spinning spherical Myers–Perry–AdS black holes in $D\ge6$ when rotation is large compared with the horizon thickness. Here “ultraspinning” labels a region of ordinary parameter space in which stationary axisymmetric perturbations with harmonic number $\ell\ge2$ appear. It need not mean that $a=L$ has been taken.

The threshold plots of Dias and collaborators use a mass-radius parameter $r_m$, defined for their singly spinning $D$-dimensional family by

$$\Delta_r=(r^2+a^2)\left(1+\frac{r^2}{L^2}\right) -\frac{r_m^{D-3}}{r^{D-5}}.$$

Thus $r_m$ is neither the horizon radius nor the physical ADM mass. If $G_D$ is the $D$-dimensional Newton constant and $\Omega_{D-2}$ is the area of the unit $(D-2)$-sphere, then their physical mass is

$$M=\frac{\Omega_{D-2}r_m^{D-3}}{8\pi G_D\Xi^2} \left(1+\frac{D-4}{2}\Xi\right).$$

The horizon radius $r_+$ is instead the largest positive root of $\Delta_r=0$, equivalently

$$r_m^{D-3}=r_+^{D-5}(r_+^2+a^2) \left(1+\frac{r_+^2}{L^2}\right).$$

The ratio $r_m/L$ used in the worked diagnostic below therefore labels the metric family before a horizon root is found Dias, Figueras, Monteiro, and Santos 2010, §2.1, equations (2.1)–(2.4), Open PDF, pp. 3–5.

Ultraspinning branches and nonlinear evolutions

Each entry states, in order, the dimension and theory; asymptotics and regularity; data and boundary condition; observable and approximation; and evidential status.

• Spherical thresholds. $D=6,7$ vacuum Einstein gravity; global AdS and regular spherical horizons; stationary axisymmetric perturbations with Dirichlet AdS falloff; zero eigenvalues of the modified Lichnerowicz operator from two-dimensional spectral numerics. The $\ell\ge2$ modes occur only beyond the thermodynamic ultraspinning surface and are strong evidence for instability onsets and pinched branches, but the calculation did not include time dependence Dias, Figueras, Monteiro, and Santos 2010, §4, Open PDF, pp. 13–17.
• Helically symmetric lumps. $D=4$ vacuum Einstein gravity; global AdS with a regular spherical Killing horizon; reflecting boundary metric and a helical Killing field; fully nonlinear Einstein–DeTurck solutions with convergence and charge checks. Black resonators bifurcate from Kerr–AdS superradiant onsets and prove nonuniqueness, but all constructed examples have $\Omega_HL\gt1$ and remain unstable to higher azimuthal modes Dias, Santos, and Way 2015, “Numerical Approach,” “Results,” and “Outlook,” Open PDF, pp. 1–4.
• Competing ring topology. $D=5$ vacuum Einstein gravity; global AdS and regular $S^1\!\times S^2$ horizons; stationary reflecting-boundary data; nonlinear Einstein–DeTurck numerics and first-law charge reconstruction. Both thin and membrane-like ring branches exist, yet the computed rings have $\Omega_HL\gt1$ and are expected to be superradiantly unstable Figueras and Tunyasuvunakool 2015, §1, Open PDF, pp. 1–4; §§4–5, pp. 13–19.
• Rescaled finite-area family. $D\ge4$ Einstein or specified gauged-supergravity examples; asymptotically AdS with a punctured, quotient horizon; exact stationary solutions after changing the azimuthal identification; horizon area, charges, and extended first law computed analytically. Existence is exact, while dynamical formation from standard compact-horizon data is not established.

The comparison forbids two tempting inferences. A stationary zero mode is not by itself a nonlinear solution branch, and a solution with larger area at fixed charges is not thereby a dynamical attractor.

Thermodynamic branches do not determine dynamics

In the fixed-$J$ extended phase space, the Gibbs function of singly spinning Kerr–AdS has dimension-dependent branches. A numerical equation-of-state analysis finds ordinary small/large transitions in $D=4,5$ and a narrow large–small–large reentrant sequence for $D\ge6$ Altamirano, Kubizňák, and Mann 2013, equations (7)–(14), Open PDF, pp. 2–4. This is an ensemble-specific comparison at varied $P$; it neither proves a Lorentzian instability nor selects its endpoint.

A 2026 analysis reports positive heat-capacity regions for some super-entropic branches, with the result depending on dimension, charge, branch, and whether $C_P$ or $C_V$ is held fixed Di 2026, §III, Open PDF, pp. 9–14. This is useful contrary evidence against the blanket phrase “all ultraspinning black holes are thermodynamically unstable.” It is a published response-function analysis, not an independent perturbative or nonlinear stability calculation. Thermodynamic and classical dynamical stability must therefore be reported separately.

What nonlinear evolution currently supports

For $D=4$ Kerr–AdS with a fixed reflecting boundary metric, full $3+1$ Einstein evolutions show a two-stage superradiant growth. For the mass and spin evolved and two different seed perturbations, the late-time geometry became multi-oscillating and close to a hairy black resonator; the hair depended on the seed while the horizon area approached nearly the same plateau Chesler 2022, §§II–III, Open PDF, pp. 3–10. This is genuine nonlinear time-domain evidence, not merely a zero mode.

It is not a settled endpoint theorem. The simulations cover a limited data class, mass and spin, resolution, and duration; very slow modes were near numerical fidelity, and unprobed channels could remain unstable. The paper explicitly leaves longer, higher-resolution evolutions open Chesler 2022, §IV, Open PDF, pp. 10–11. For the $D\ge6$ axisymmetric AdS ultraspinning thresholds, the cited evidence contains no comparably converged finite-dimensional nonlinear evolution. Pinched spherical branches and their possible connections to rings or multi-horizon objects therefore remain conjectural in that setting.

Which nonlinear endpoint is selected?

The unresolved question is not whether new exact or numerical stationary geometries exist; several do. It is which, if any, is selected by generic smooth initial data at fixed dimension, theory, AdS boundary condition, and conserved charges. A universal endpoint cannot be inferred across $D=4$ superradiance, $D=5$ rings, $D\ge6$ axisymmetric ultraspinning modes, large-$D$ approximations, and quotient-changing super-entropic limits.

A stronger conclusion requires a branch construction or nonlinear evolution with fixed dimension, asymptotics, charges, regularity class, initial data, and boundary condition, together with quantitative convergence tests. This discussion covers results through 2026-08-12.

Worked transfer diagnostic

For the $D=6$ family with $r_m/L=0.5$ studied by Dias and collaborators, the reported stationary thresholds are $a/r_m\simeq1.10$ for the thermodynamic $\ell=1$ mode, $1.49$ for $\ell=2$, and $1.69$ for $\ell=3$. Classify a solution at $a/r_m=1.60$, and state what cannot be concluded.

Solution

The inequalities are

$$1.10\lt1.49\lt1.60\lt1.69.$$

The solution lies beyond the thermodynamic ultraspinning surface and beyond the first nonthermodynamic $\ell=2$ threshold, but before the $\ell=3$ threshold. Thus the stationary spectrum contains the $\ell=2$ negative-mode continuation interpreted as the first axisymmetric ultraspinning sector. One cannot conclude which nonlinear branch is reached, that a ring forms, or even extract a growth rate: the calculation found stationary thresholds, imposed a particular AdS boundary condition, and did not evolve time-dependent data.

Continue to Higher-Dimensional Black Holes and Black Objects to study topology, nonuniqueness, and stability across dimensions without assuming AdS asymptotics.

References

• Altamirano, N., Kubizňák, D., and Mann, R. B. “Reentrant Phase Transitions in Rotating Anti-de Sitter Black Holes.” Physical Review D 88 (2013): 101502(R). doi:10.1103/PhysRevD.88.101502. Open PDF.
• Chesler, P. M. “Hairy Black Resonators and the AdS4 Superradiant Instability.” Physical Review D 105 (2022): 024026. doi:10.1103/PhysRevD.105.024026. Open PDF.
• Di, Zhenbo. “Thermodynamics and Stability of Ultraspinning Black Holes.” Classical and Quantum Gravity 43 (2026): 135034. doi:10.1088/1361-6382/ae83fe. Open PDF.
• Dias, O. J. C., Figueras, P., Monteiro, R., and Santos, J. E. “Ultraspinning Instability of Anti-de Sitter Black Holes.” Journal of High Energy Physics 2010, no. 12 (2010): 067. doi:10.1007/JHEP12(2010)067. Open PDF.
• Dias, O. J. C., Santos, J. E., and Way, B. “Black Holes with a Single Killing Vector Field: Black Resonators.” Journal of High Energy Physics 2015, no. 12 (2015): 171. doi:10.1007/JHEP12(2015)171. Open PDF.
• Figueras, P., and Tunyasuvunakool, S. “Black Rings in Global Anti-de Sitter Space.” Journal of High Energy Physics 2015, no. 3 (2015): 149. doi:10.1007/JHEP03(2015)149. Open PDF.
• Hennigar, R. A., Kubizňák, D., Mann, R. B., and Musoke, N. “Ultraspinning Limits and Super-entropic Black Holes.” Journal of High Energy Physics 2015, no. 6 (2015): 096. doi:10.1007/JHEP06(2015)096. Open PDF.