Classical instability of Kerr-AdS black holes and the issue of final state

Oscar J. C. Dias; Shijun Yoshida; Vitor Cardoso

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Classical instability of Kerr-AdS black holes and the issue of final state

Vitor Cardoso, Oscar J. C. Dias, Shijun Yoshida

High Energy Physics - Theory · Submitted 22 Jul 2006 · 30 references

Pith's verdict: The spin-2 instability calculation is solid and worth citing; the endpoint is an explicitly conjectural add-on that the abstract oversells.

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arXiv IDhep-th/0607162 (version 2)

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CitationOscar J. C. Dias, Shijun Yoshida, Vitor Cardoso. "Classical instability of Kerr-AdS black holes and the issue of final state." arXiv:hep-th/0607162 (2006). https://pith.science/paper/IWJQBSE4

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The thing to know up front: the gravitational instability result is real, and it is the part that matters. Cardoso, Dias and Yoshida take the scalar superradiant instability of small Kerr-AdS they published earlier and redo it for spin-2 perturbations using the Chambers-Moss Teukolsky equation. The matched asymptotic calculation is standard — near-horizon hypergeometric, far-region pure-AdS solution, matching, expansion around the known AdS frequencies — and the conclusion that the growth rate is proportional to the superradiant factor (omega - m Omega_H) is exactly what you would expect. This upgrades 'test fields are unstable' to 'the metric is unstable,' which is the statement that matters for the spacetime itself. That part is clean and citable, and the higher-dimensional appendix is a reasonable sketch of how far it extends. The endpoint claim is where I would slow down. The geometric observation is genuinely nice: r+ = sqrt(a ell) is precisely where the boundary Einstein universe rotates at light speed, and that is a clean way to state the threshold. But the derivation of that threshold uses the large-l=m limit, while the paper elsewhere says the most unstable modes are l=m=2. The stress-test flags a factor of sqrt2 between the two thresholds. I do not think that lands as a flaw: the endpoint is where the last superradiant mode shuts off, and that is the large-m limit, not the fastest-growing mode. But the paper never says this, and a reader can easily be confused. That is an exposition gap worth fixing. The bigger soft spot is the adiabatic assumption. The hole is assumed to evolve quasi-statically through a sequence of Kerr-AdS solutions, with dA >= 0 forcing r+ up and a down. The growth rate scales like r+^{-2l-2} in the deep superradiant regime, so the instability can be fast, and whether the adiabatic approximation holds throughout is not established. The paper's condition (3.3) addresses perturbative validity, not adiabaticity. The authors are honest in Section III that the endpoint is an educated guess, but the abstract states it more confidently than the body supports, and 'slightly oblate' and 'co-exist with radiation' are explicitly expectations. One more thing to check: the sign in delta = -P1 P2 varpi versus 'instability whenever varpi is negative.' As written, varpi < 0 gives delta > 0, which decays under Psi proportional to e^{i omega t}. Probably a typo, but a referee should confirm. Who it is for: anyone working on AdS black hole stability, superradiance, or rotating AdS/CFT boundaries. The core result is worth citing; the endpoint is worth citing as a conjecture. Recommendation: send it out. The gravitational instability is a solid, citable result; the endpoint needs to be labeled as a conjecture, not a derived result.

Referee report

Major objections (3)

[Section III, Eqs. (3.4)-(3.5) and final paragraph] The endpoint r+ = √(aℓ) is derived in the large-l=m limit, but the paper also states in the final paragraph of Section III that 'the most unstable modes are those with l = m = 2.' For l = m = 2, n = 0, and c_s = 2, the superradiance condition ω < m Ω_H gives Ω_H > 2/ℓ, i.e., r+ < √(aℓ/2), not √(aℓ). The paper does not reconcile which mode determines the endpoint. If the fastest-growing mode (l = m = 2) dominates the evolution, the endpoint would be at r+ = √(aℓ/2); if the endpoint is set by the last mode to shut off (large m), it is at r+ = √(aℓ). The paper should state explicitly that the endpoint is set by the large-m limit and explain why the l = m = 2 threshold does not determine it. This is load-bearing for the abstract's claim (abstract:s4).
[Section III, Eq. (2.44)] The analytic growth rate δ = -P1 P2 ϖ appears to increase with l for l ≪ ℓ/r+, since P1 P2 scales like (l²/(4e² r+²))^l in the small-ϖ limit. This would make large-l modes more unstable than l = 2, contradicting the paper's statement that l = m = 2 is the most unstable. The paper should either justify the l = m = 2 claim (e.g., by numerical computation, or by noting where the near-region approximation breaks down) or correct it. This bears directly on the endpoint argument, which relies on the large-m limit.
[Section III, Eq. (3.3)] The paper assumes the perturbative regime is valid at all times, but for small r+ the growth rate δ ~ 1/r+⁴ (for l = 2) may be so large that the field amplitude becomes large quickly, invalidating the perturbative treatment. The paper should check whether condition (3.3) is actually satisfied for the relevant modes and parameters, or acknowledge explicitly that the endpoint argument is heuristic rather than a controlled evolution.

Minor objections (6)

[Eq. (2.4) and Eq. (3.6)] The horizon area appears as A = 4π(r+² + a²)Σ, but the standard Kerr-AdS result is A = 4π(r+² + a²)/Σ with Σ = 1 - a²/ℓ². The same factor appears in Eq. (3.6). Please check and correct if this is a typo; the sign structure of the dA ≥ 0 argument is unaffected.
[Appendix A] The appendix sketches the higher-dimensional matching but does not actually compute the instability timescale, despite the introduction stating 'we compute in Appendix A the instability timescale for higher dimensional geometries.' The matching is left implicit. Please complete the computation or soften the claim in the introduction.
[Section II.B] The boundary conditions for gravitational perturbations in AdS are adopted as Dirichlet, with the caveat that the results might hold for more general conditions. For spin-2 fields the boundary conditions are more subtle than for scalars; please justify the choice or cite a reference that establishes it for gravitational perturbations.
[Abstract] The abstract states 'we identify its endpoint state,' but Section III explicitly says 'It is in general impossible to predict the endpoint of an unstable system but... make some educated guesses.' Please align the abstract's framing with the body's acknowledged uncertainty.
[Eq. (2.16)] The expression (K_r² + 2iK_r Δ_r')² appears to have an extra square; the preceding line suggests the quantity is K_r² + 2iK_r Δ_r'. Please check.
[Section III, final paragraph] The statement that 'the most unstable modes are those with l = m = 2' is not supported by a reference or derivation in the paper. Please provide a justification or cite a source.

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The paper

The paper claims

Small rotating black holes in anti-de Sitter space are unstable even to gravitational waves, and the instability should run until the black hole's boundary rotates exactly at the speed of light.

The reading

A rotating black hole in anti-de Sitter space sits inside a cavity: the negative cosmological constant acts like a reflecting wall, so waves that the hole amplifies by superradiance come back and are amplified again. Earlier work showed this makes small Kerr-AdS holes unstable to scalar fields. This paper shows the same happens for gravitational perturbations, i.e. for the metric itself, and computes the growth rate. The authors then argue that the runaway stops only when the hole has shed enough spin and grown enough in radius that $r_+ = r_c^+ = sqrt{a,ell}$, the point at which the Einstein universe living on the AdS boundary rotates at exactly the speed of light. If correct, the metric of these black holes is genuinely unstable, not merely a scalar-field artefact, and the endpoint is a specific, identifiable geometry rather than an unknown mess.

What carries the argument

The superradiant black-hole-bomb mechanism inside a box: AdS reflects outgoing radiation back onto the hole, and a rotating horizon amplifies any wave satisfying $omega < mOmega_H$, so repeated round trips give exponential growth. The carrier of the quantitative result is the matched-asymptotics computation — a hypergeometric near-horizon solution matched to the exact Regge-Wheeler-in-AdS solution — which turns the instability question into the sign of $varpi$. The carrier of the endpoint claim is the identification of the superradiance threshold $r_+ = sqrt{a,ell}$ with the light-speed condition $Omega = 1/ell$ for the boundary Einstein universe.

Extended reading notes

Core claim

Using the Chambers-Moss-Teukolsky radial equation for spin-2 perturbations of four-dimensional Kerr-AdS, the authors solve the near-horizon region with a hypergeometric ingoing wave and match it to the exact pure-AdS far-region solution with Dirichlet boundary conditions. The resulting eigenvalue equation, expanded around the pure-AdS frequency $omega = (2n + l + 2)/ell + i,delta$, gives $delta = -P_1 P_2,varpi$ with $P_1, P_2 > 0$ and $varpi propto (omega - mOmega_H)$. So every mode with $omega < mOmega_H$ — the superradiant regime — grows exponentially, and small Kerr-AdS black holes are unstable against gravitational perturbations, not only scalar ones. The authors then follow the instability: superradiance drains angular momentum ($a$ falls) while the area law forces $r_+$ up, so the critical radius $r_c^+ = sqrt{a,ell}$ falls; the evolution should halt when the two meet, at $r_+ = sqrt{a,ell}$. That same condition is shown, via the asymptotically-AdS coordinates in which the horizon angular velocity is $Omega = a(1 + r_+^2/ell^2)/(r_+^2 + a^2)$, to be exactly the condition $Omega = 1/ell$: the boundary Einstein universe rotates at light speed.

Load-bearing premise

The endpoint claim assumes the unstable hole evolves quasi-statically through a sequence of Kerr-AdS solutions, that the area law then forces $r_+$ up while $a$ falls, and that the evolution stops exactly at $r_+ = sqrt{a,ell}$ rather than overshooting; the authors themselves write in Section III that it is in general impossible to predict the endpoint of an unstable system and that this part is an educated guess.

Editorial extensions

If this is right

Kerr-AdS black holes with $r_+ < sqrt{a,ell}$ are dynamically unstable at the level of the metric, so their linear-mode spectrum and thermodynamic description are not the whole story.
The instability timescale is computable in closed form, so one can estimate how fast a small spinning AdS hole loses angular momentum.
The endpoint is a Kerr-AdS hole with $r_+ = sqrt{a,ell}$, whose conformal boundary is an Einstein universe rotating at the speed of light — a concrete target for a dual field theory description.
Because the endpoint is expected to be slightly oblate and to coexist with trapped radiation, there should exist non-Kerr stationary solutions of Einstein's equations with AdS asymptotics, at least to first order in the deformation.
The same instability is expected in higher dimensions; the appendix sketches the matched-asymptotics calculation for a sector of perturbations that exists only for $D geq 5$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the light-speed boundary condition is the true stopping point, it suggests a general criterion: AdS black holes settle where the boundary rotation saturates causality, which could be checked against other rotating AdS solutions with different asymptotics.
The paper's own caveat that the endpoint is an educated guess could be tested numerically: a time-domain evolution of a perturbed small Kerr-AdS hole would show whether the trajectory in the $(a, r_+)$ plane actually terminates on $r_+ = sqrt{a,ell}$ or crosses it.
The coexistence with outside radiation suggests the endpoint is not a vacuum solution but a hairy, radiation-dressed configuration; that would put it in the same family as other AdS black holes with scalar or gravitational hair, and its stability would itself need checking.
The quoted first-order oblate deformation is a concrete prediction that a perturbative construction could confirm or refute independently of the dynamical argument.

Original abstract

It is now established that small Kerr-Anti-de Sitter (Kerr-AdS) black holes are unstable against scalar perturbations, via superradiant amplification mechanism. We show that small Kerr-AdS black holes are also unstable against gravitational perturbations and we compute the features of this instability. We also describe with great detail the evolution of this instability. In particular, we identify its endpoint state. It corresponds to a Kerr-AdS black hole whose boundary is an Einstein universe rotating with the light velocity. This black hole is expected to be slightly oblate and to co-exist in equilibrium with a certain amount of outside radiation.

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

It is now established that small Kerr-Anti-de Sitter (Kerr-AdS) black holes are unstable against scalar perturbations, via superradiant amplification mechanism. We show that small Kerr-AdS black holes are also unstable against gravitational perturbations and we compute the features of this instability. We also describe with great detail the evolution of this instability. In particular, we identify its endpoint state. It corresponds to a Kerr-AdS black hole whose boundary is an Einstein universe rotating with the light velocity. This black hole is expected to be slightly oblate and to co-exist in equilibrium with a certain amount of outside radiation.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	It is now established that small Kerr-Anti-de Sitter (Kerr-AdS) black holes are unstable against scalar perturbations, via superradiant amplification mechanism.	background 1.00	0.05	0.08	asserted 1.00
s1	We show that small Kerr-AdS black holes are also unstable against gravitational perturbations and we compute the features of this instability.	contribution 0.58	0.96	0.38	asserted 1.00
s2	We also describe with great detail the evolution of this instability.	contribution 0.99	0.92	0.06	asserted 0.99
s3	In particular, we identify its endpoint state.	contribution 0.64	0.93	0.16	asserted 0.99
s4	It corresponds to a Kerr-AdS black hole whose boundary is an Einstein universe rotating with the light velocity.	result 1.00	0.83	0.52	asserted 1.00
s5	This black hole is expected to be slightly oblate and to co-exist in equilibrium with a certain amount of outside radiation.	result 0.67	0.39	0.17	hedged possibility 0.95

What kind of paper the classifier reports this is: theoretical result 0.69 · evidence from simulation or computation 0.83.

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References

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47 later papers cite it

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1 paper by the same authors is listed below: 1 compared its own results with it.

Compares with it 1 of 4 papers with this label · 1 by the same authors

By the same authors

Reported that its own numerical results for small black holes agree with this paper's analytic estimate derived assuming Mω_R ≪ 1.

Breit-Wigner resonances and the quasinormal modes of anti-de Sitter black holes (2009) (same authors) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

45 arXiv papers cite this in 67 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 42 sentences without a confident label

Uses its method or tool 1 paper

We do this by matching asymptotic expansions of the solution in an intermediate region, see e.g., [ref] for asymptotically flat spacetimes and [ref] for asymptotically AdS.

Quasinormal modes of Proca and Maxwell fields in $d$-dimensional Schwarzschild-AdS black holes · Introduction · 0.93

Compares with it 4 papers

We estimate the growth-timescale to be of order $4×10^6M$, which agrees with the estimate of Ref. [ref] and with frequency-domain calculations [ref] .

Black holes in anti-de Sitter: quasinormal modes, tails and tales of flat spacetime · Results · 0.98
Our results have been benchmarked using previous results in the literature, specifically for scalar field perturbations of Kerr-AdS BHs [ref] .

Holographic thermalization, quasinormal modes and superradiance in Kerr-AdS · Numerical procedures · 0.99
For small BHs our numerical results agree with the following analytic estimate, derived under the assumption that $Mω_R ≪1$ [ref] :

Breit-Wigner resonances and the quasinormal modes of anti-de Sitter black holes · Gravitational perturbations · 0.98

Cites it as support 1 paper

Finally, we have have indications – directly in the time-domain – that small, rotating BHs in AdS are unstable against a superradiant mechanism in the sense that small non-axisymmetric fluctuations grow [ref] .

Black holes in anti-de Sitter: quasinormal modes, tails and tales of flat spacetime · Conclusions · 0.98

Cites it as background 18 papers

This is usually known as a BH bomb or superradiant instability [ref] .

Black holes in anti-de Sitter: quasinormal modes, tails and tales of flat spacetime · Results · 0.84
Recent prosperity of AdS spacetimes follows the historic claiming of the AdS/CFT duality [ref] , and for rotating AdS black holes, rotational superradiant instability has been widely studied [ref] .

Energy extraction from AdS black holes via superradiance · Introduction · 0.98
Several different reflection mechanisms have been considered: for example, a “mirror” surrounding the hole imposed as a boundary condition [ref] , the mass of perturbation fields [ref] , nonminimal couplings [ref] , and the spacetime curvature itself for asymptotically AdS black holes [ref] .

Superradiant instability of the Kerr brane · Introduction · 0.95

Discussion

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