Superradiant and Trapping Instabilities in AdS

AdS black holes support distinct mechanisms for slow decay and growth. Rotation can supply negative boundary-time Killing-energy flux into the horizon; a timelike boundary can return a wave for repeated amplification; stable trapping produces long-lived approximate modes; and a Robin domain can admit a negative spatial eigenvalue. These statements have different proofs. In particular, an exponentially accurate approximate mode is not an exact quasinormal frequency, and reflection plus rotation alone does not prove exponential instability.

Evidence cutoff: 2026-08-12. Claims and publication status are assessed through this date.

Helpful background. Black-Hole Geometry and Mechanics supplies rotating AdS horizons; Partial Differential Equations and Green Functions supplies resonances and energy estimates; Formulations of General Relativity supplies physical perturbation sectors; Gravitational Actions and Boundary Terms supplies boundary flux; Causality and Global Structure supplies the AdS boundary; Black-Hole Perturbation Theory supplies superradiant flux conventions.

Rotational energy extraction in an AdS frame

Use $e^{-i\omega t+im\phi}$, where $t$ is normalized by the chosen nonrotating conformal-boundary frame. If the future-horizon generator is

$$\chi=\partial_t+\Omega_H\partial_\phi,$$

then a neutral bosonic mode has generator frequency

$$k_H=\omega-m\Omega_H.$$

For positive real $\omega$, the horizon energy flux is proportional to $\omega k_H$ in the standard positive normalization. The superradiant band is therefore

$$0 < \omega < m\Omega_H$$

for $m\Omega_H>0$. In Kerr–AdS, quoting $\Omega_H$ without saying whether it is measured relative to a rotating or nonrotating infinity changes this inequality. The boundary frame and azimuthal identification are physical input.

Amplification is a real-frequency scattering statement. A growing mode must also satisfy both complex-frequency endpoint conditions. For a massless scalar with fast falloff on small, slowly rotating Kerr–AdS$_4$ black holes, long-wavelength matching gives growth in the superradiant band Cardoso and Dias 2004, §III, pp. 3–5, Open PDF. The assumptions include $r_+\ll L$, $a\ll GM$, and $|\omega|GM\ll1$. This is distinct from the asymptotically flat mirror calculation.

When translating the source’s rotating-at-infinity coordinates, use $\omega=\omega_{\rm src}+ma/L^2$ and $\Omega_H=\Omega_{H,\rm src}+a/L^2$. Their difference $\omega-m\Omega_H$ is invariant; shifting only $\Omega_H$ would change the physical prediction. The calculation is not a theorem for arbitrary size, spin, field, or boundary domain.

The Hawking–Reall energy obstruction

Suppose a Killing field $K$ is timelike throughout the exterior, preserves the selected reflecting boundary domain, and agrees with the future generator on the horizon. For matter obeying the appropriate positive-energy condition, its $K$-energy on a regular spacelike slice is nonnegative. Zero outer flux and nonnegative inward horizon flux make this energy nonincreasing. If it controls the perturbation, a nonzero exponentially growing finite-energy solution is excluded. This is the Hawking–Reall energy argument Hawking and Reall 2000, §§2–4.

In the usual four-dimensional Kerr–AdS family, the simple condition is often expressed as $\Omega_HL\le1$ using the nonrotating boundary frame. Outside that regime the globally timelike generator is lost, opening the door to negative-energy states, but loss of an estimate is not construction of a growing mode. Conversely, the argument’s hypotheses matter: gravitational canonical energy, alternative Robin domains, charged fields, or matter violating the assumed positivity require their own analysis.

An elementary spectral check reaches the same boundary from the opposite side. For a massless scalar in global AdS$_4$ with the standard fast falloff,

$$\omega_{n\ell}L=2n+\ell+3, \qquad n\geq0, \qquad |m|\leq\ell.$$

In the Hawking–Reall regime,

$$\omega_{n\ell}-m\Omega_H \geq\frac{2n+\ell+3-m}{L}>0 \qquad (0\leq m\leq\ell, 0\leq\Omega_HL\leq1)$$

for every corotating seed mode. Counterrotating modes have $\omega_{n\ell}-m\Omega_H\geq\omega_{n\ell}\gt0$ separately. No member of this pure-AdS family is therefore in the superradiant band while $\Omega_HL\leq1$. Once $\Omega_HL>1$, the inequality no longer excludes a crossing, but it still does not supply an upper-half-plane eigenvalue: the black-hole deformation, boundary condition, and horizon flux must be followed on one continuous branch.

Stable trapping is slow decay, not growth

Reflection at timelike infinity and an angular-momentum barrier can keep waves far from the absorbing horizon. The reflected rays need not stay away from the boundary itself. Three assertions must be separated.

An upper decay bound. Fix a four-dimensional Kerr–AdS exterior with $|a|\lt L$ and $r_+^2\gt|a|L$, and a massless scalar with homogeneous fast falloff and sufficiently regular finite-energy data. In units of $L$, on the stationary horizon-penetrating slices $\Sigma_{t^*}$, define

$$\|\psi\|_{H^1_{\rm AdS}}^2 =\int_{\Sigma_{t^*}} \left[r^2|\partial_r\psi|^2+|\nabla_S\psi|^2+|\psi|^2\right] r^2\,dr\,d\Omega .$$

Here $\nabla_S$ uses the induced metric on the constant-$(t^*,r)$ two-surfaces, $d\Omega=\sin\theta\,d\theta\,d\phi$, and $\partial_r$ holds $t^*$ fixed. For the finite initial higher-order norm $\mathcal E_2$ defined in the AdS decay discussion, which controls one additional derivative, the theorem gives

$$\|\psi\|_{H^1_{\rm AdS}(\Sigma_{t^*})} \leq\frac{C\mathcal E_2^{1/2}}{\log t^*},\qquad t^*\geq2.$$

This is a spatial field-norm upper bound, not an asymptotic equality or a statement that each mode decays logarithmically Holzegel and Smulevici 2013, Theorem 1.1, p. 4, norms pp. 10–11, Open PDF. The corresponding nondegenerate energy bound is stated separately in that discussion.

An obstruction to faster uniform decay. A different theorem constructs axisymmetric, real-frequency approximate solutions $\psi_\ell=e^{-i\omega_\ell t}\varphi_\ell$ with $|\omega_\ell|L$ of order $\ell$, supported outside a trapping barrier. Their wave-equation residual is exponentially small in $\ell$ in every fixed weighted Sobolev norm. Duhamel’s formula then keeps a genuine solution close to the approximate one for exponentially long times. This rules out uniform decay faster than $(\log t^*)^{-2}$ for the semilocal energy with the specified two-derivative initial norm Holzegel and Smulevici 2014, Theorem 1.2 and Corollary 1.1, pp. 4–5, Open PDF. The obstruction concerns energy in $r\geq r_{\max}$, with $r_{\max}$ the barrier maximum, and does not exclude faster decay in other regions or after stronger frequency restrictions.

Exact resonances. Turning those approximate solutions into exact QNMs, or proving an asymptotic law for their imaginary parts, requires an additional resonance theorem. Neither cited result alone establishes $-\operatorname{Im}\omega_\ell\sim e^{-c\ell}$. Long-lived approximate solutions and a uniform-decay obstruction are already sufficient for this page’s comparison. They are not upper-half-plane eigenvalues.

Schwarzschild–AdS is a useful discriminating case. It has $\Omega_H=0$, so a positive-frequency neutral mode cannot satisfy the superradiant inequality, even though high-angular-momentum quasimodes can exhibit the stable-trapping obstruction. It therefore supplies a trapped but non-superradiant comparison: long lifetime is not evidence for positive $\operatorname{Im}\omega$.

These are analytic sign and limiting-case counterchecks, not a newly computed Kerr–AdS continuation. No mode is numerically tracked across the Hawking–Reall boundary on this page, and no convergence claim for such a track is made.

A round-trip criterion

Let a wave packet return to the horizon after one cavity cycle with complex outer reflection coefficient $\mathcal R_B$, horizon scattering coefficient $\mathcal R_H$, and phase $e^{i\Phi}$. A mode condition has the schematic form

$$\mathcal R_B(\omega)\mathcal R_H(\omega)e^{i\Phi(\omega)}=1.$$

At a complex root the complete left side equals one, including its modulus. One must not replace this equation by a modulus-greater-than-one test at that root. For real-frequency scattering, a cycle can have net amplitude gain $|\mathcal R_B\mathcal R_H|\gt1$. If a nearby phase-matched frequency has positive group delay and slowly varying gain, that excess gives a growing-envelope estimate; reflection-phase dispersion and amplitude slopes must be included when they matter. The cavity derivation shows the approximation and the full complex-root linearization. An absorbing outer boundary can remove the excess gain, and a perfectly reflecting one can still miss the phase condition or the superradiant band.

AdS stability results through 2026-08-12

Mechanism	Controlled evidence	What it establishes	What it does not establish
Small Kerr–AdS superradiance	Matched asymptotics and numerical modes for declared fields and reflecting falloffs	Growing linear modes in specific bands	Instability of every size, spin, field, or boundary domain
Hawking–Reall regime	Positive conserved energy under a globally timelike generator and compatible reflecting boundary	Excludes the corresponding negative-energy growth mechanism under its hypotheses	Stability with arbitrary Robin data or a full nonlinear endpoint
Stable trapping	Real-frequency quasimodes and a lower obstruction complement a separate logarithmic upper bound	Long-lived approximate solutions; no faster uniform decay in the specified norm hierarchy	An exact QNM-width asymptotic or an upper-half-plane pole
Boundary-driven scalar growth	Charged/rotating backgrounds and some Robin domains can admit a negative mode or nonlinear transition; recent examples remain model-specific	Boundary choice can change stability, not merely frequency values	A universal claim about standard gravitational AdS boundary conditions
Nonlinear evolution	Black resonators, hairy branches, turbulence, and collapse provide candidate saturation channels in selected models	Growing linear modes can reorganize the endpoint landscape	A unique generic endpoint or proof that all small perturbations settle

A boundary-energy example

Ficek and Maliborski study a neutral, spherically symmetric conformal scalar with defocusing cubic nonlinearity on a fixed charged Reissner–Nordström–AdS background. In $L=1$ units, $\Box\phi+2\phi-\phi^3=0$. Charge belongs to the background, not to the scalar. With advanced time $v$, $y=1/r$, $\Phi=\phi/y$ and fixed real Robin angle $\beta_B$, their boundary condition is

$$\left[(-\partial_v\Phi+\partial_y\Phi)\cos\beta_B -\Phi\sin\beta_B\right]_{y=0}=0.$$

To specify the energy, write $V(y)=1-2g y+e^2y^2+y^{-2}$, where $g=GM$ and $e$ are the geometric mass and charge of this fixed background, and let $y_H=1/r_H$. Their bulk functional is

$$\begin{aligned} E_{\rm bulk}&=\int_0^{y_H}\mathcal E(v,y)\,dy,\\ \mathcal E&=\frac{y^2V}{2}(\partial_y\Phi)^2 -\frac12\left(yV'+\frac{2}{y^2}\right)\Phi^2+\frac14\Phi^4, \end{aligned}$$

with $V'=dV/dy$. The quadratic potential term is equivalently $(g y-e^2y^2)\Phi^2$, so its apparently singular terms cancel at $y=0$. Integrating the field equation by parts gives

$$\frac{dE_{\rm bulk}}{dv} =-(\partial_v\Phi)^2_{y=y_H} -\frac{\tan\beta_B}{2}\partial_v[\Phi(v,0)^2].$$

For $\cos\beta_B\ne0$, therefore, $E_{\rm total}=E_{\rm bulk}+\tfrac12\tan\beta_B\,\Phi(v,0)^2$ loses energy only through the horizon. The second term is exchange with a boundary energy, not necessarily work done by an external pump. Dirichlet data $\Phi(v,0)=0$ must be handled separately, without inserting an infinite tangent. This distinction between bulk and total energy explains how a Robin choice can affect stability even in a formally conservative boundary prescription Ficek and Maliborski 2025, §§2–4, Eqs. (3)–(7), unnumbered PDF pp. 1–3. We use this specified public preprint edition, not a claim about private refereeing. Its scalar phase diagram is not a vacuum-gravity phase diagram.

Distinguishing mode growth, trapping, and collapse

One position uses “AdS instability” broadly for any failure of uniform rapid decay, including stable trapping and weakly turbulent transfer. A narrower spectral position reserves linear instability for a mode with $\operatorname{Im}\omega>0$. Both usages occur, so a comparison must identify the effect: mode growth, logarithmic decay obstruction, nonlinear drift, collapse, or endpoint bifurcation. A generic nonlinear endpoint for rotating AdS black holes remains unresolved; no universal conclusion follows from the cited results.

Evidence against a proposed growing mode includes disappearance under increased resolution, violation of the horizon or boundary flux relation, a globally positive conserved energy under the actual domain, or migration below the real axis when unwanted falloff contamination is removed. A numerical claim of growth requires a converged upper-half-plane eigenvalue, an independent method, correct energy signs, and preferably time-domain growth at the same rate.

A rigorous stability or instability theorem, an independently converged new mode family, a nonlinear endpoint calculation with controlled boundary flux, or a correction to an onset curve could change this picture. The selected boundary condition must remain fixed during the comparison.

Exercises

Gain is not enough

A coherent fixed-delay packet model returns every time $T$, with constant horizon amplitude gain $|\mathcal R_H|=1.03$ and outer retention $|\mathcal R_B|=0.95$. Find its sampled cycle multiplier and amplitude rate. Repeat for retention $0.99$. What is still needed for a true frequency-dependent eigenmode?

Solution

The first multiplier is $1.03\times0.95=0.9785<1$, so the amplitude decays at rate

$$\gamma=T^{-1}\log(0.9785)<0.$$

With retention $0.99$, the multiplier is $1.0197\gt1$ and this fixed-delay model has $\gamma=T^{-1}\log(1.0197)\gt0$. Those are amplitude rates; quadratic energy rates are twice as large. A frequency-dependent mode requires phase closure and the complete analytically continued return equation, with both endpoint conditions satisfied. The sampled multiplier alone does not establish such a mode.

References

• Cardoso, V., and Dias, Ó. J. C. (2004). “Small Kerr–anti-de Sitter black holes are unstable.” Physical Review D 70, 084011. DOI. Open PDF.
• Ficek, F., and Maliborski, M. (2025). “Instability mechanisms on asymptotically AdS black hole backgrounds.” arXiv:2508.20483v1, submitted 2025-08-28. Open PDF.
• Hawking, S. W., and Reall, H. S. (2000). “Charged and rotating AdS black holes and their CFT duals.” Physical Review D 61, 024014. DOI.
• Holzegel, G., and Smulevici, J. (2013). “Decay properties of Klein–Gordon fields on Kerr–AdS spacetimes.” Communications on Pure and Applied Mathematics 66, 1751–1802. DOI. Open PDF.
• Holzegel, G., and Smulevici, J. (2014). “Quasimodes and a lower bound on the uniform energy decay rate for Kerr–AdS spacetimes.” Analysis & PDE 7, 1057–1090. DOI. Open PDF.