REVIEW
Charged and rotating AdS black holes and their CFT duals
High Energy Physics - Theory · Submitted 16 Aug 1999 · 32 references
Pith's verdict: The stability no-go is correct and kills a published speculation; the CFT half is a clean computation whose abstract sentence about the 4/3 factor claims more than the body shows.
T0 · AI review
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Referee report
The paper
The paper claims
Black holes that curve anti-de Sitter space can rotate in two distinct ways, and the authors argue that neither kind can be spun down by superradiant scattering as long as the hole is the thermodynamically preferred phase.
The reading
What carries the argument
The load-bearing device is the corotating Killing vector and the energy functional built from it. In Kerr-AdS the authors use coordinates in which the boundary is manifestly an Einstein universe $R imes S^3$ with metric $-dT^2 + dTheta^2 + sin^2Theta, dPhi_1^2 + cos^2Theta, dPhi_2^2$, and in which the horizon angular velocities are $Omega_i = a_i(1+r_+^2)/(r_+^2+a_i^2)$. The vector $chi = partial_T + Omega_1 partial_{Phi_1} + Omega_2 partial_{Phi_2}$ is tangent to the horizon's null generators; if it is timelike everywhere outside the horizon, the flux $J^mu = T^{mu u}chi_ u$ is non-spacelike and the integrated energy on a slice is bounded below by zero and non-increasing, which forbids extraction. The same construction, applied to the ten-dimensional lift of Reissner-Nordstrom-AdS, gives a norm that is manifestly negative outside $r_+$. Supporting machinery includes the Euclidean action relative to an identified AdS background, the Jacobian determinant of the map from $(r_+, a_1, a_2)$ to $(T, Omega_1, Omega_2)$ versus $(M, J_1, J_2)$, and the decomposition of free-field partition functions into $SU(2) imes SU(2)$ representations of the Einstein universe isometry group, summed in closed form as geometric series in a winding number $n$.
Extended reading notes
Core claim
The central claim is that superradiant instability in asymptotically anti-de Sitter black holes is controlled by a single geometric fact: whether the corotating Killing vector $k = 0$ is timelike everywhere outside the horizon. For Kerr-AdS with both rotation parameters satisfying $a_i^2 < 1$, and for every Reissner-Nordstrom-AdS solution, it is; the authors verify the charged case explicitly by computing the norm of $k$ in the ten-dimensional metric and finding it negative for all $r > r_+$. Given the dominant energy condition and reflecting boundary conditions at infinity, the energy measured in the corotating frame on a spacelike slice is non-negative and non-increasing, so no net energy can cross the horizon outward. The escape hatch is closed too: the parameter range where the corotating vector does become spacelike requires $r_+ < 1$, hence positive Euclidean action, hence suppression relative to pure AdS in the grand canonical partition function. A separate Jacobian computation shows the Legendre transform from mass and angular momenta to temperature and angular velocities is non-singular precisely where the action is negative, so the thermodynamically preferred holes are also locally stable. On the field-theory side, the paper finds that a free CFT on an Einstein universe with a $U(1)$ chemical potential Bose-condenses at $|Phi| = 1$, exactly the value at which the internal sphere would rotate at the speed of light, while the supergravity dual shows only a first-order phase transition and no condensation; and that in the limit where the boundary universe rotates at the speed of light at finite temperature, the free partition function and the supergravity action diverge in the same way, as $(1-a_i^2)^{-1}$, with the familiar $3/4$ ratio between them recovered at high temperature and modified at finite temperature.
Load-bearing premise
The monotonicity argument assumes that no energy enters the spacetime through the anti-de Sitter boundary, a reflecting-boundary condition stated in a single clause rather than derived from the asymptotics.
Editorial extensions
If this is right
- Reissner-Nordstrom-AdS black holes, including the extremal non-supersymmetric ones, are classically stable against charged superradiance, correcting the speculation in the earlier literature that they might not be.
- Any classical instability of Kerr-AdS black holes is confined to the positive-action regime, so it cannot alter the phase structure of the dual strongly coupled CFT, which is governed by negative-action holes.
- In the grand canonical ensemble, local thermodynamic instability of these holes coincides with positive action, so the region where the Legendre transform degenerates is already excluded by the AdS-preferred phase.
- A free gauge theory on an Einstein universe with a $U(1)$ R-symmetry chemical potential condenses at $|Phi|=1$, and the absence of any corresponding transition at strong coupling is a qualitative, not merely numerical, disagreement between weak and strong coupling.
- In the lightlike-rotation limit at finite temperature, the divergence structure of the free CFT partition function matches that of the supergravity action, giving a new regime in which the $3/4$ ratio can be tracked away from the high-temperature limit.
Reading between the lines
- The no-go argument is really a statement about energy, not about spectra: it rules out net energy flow across the horizon but leaves open the possibility of a mode that grows while carrying no net corotating energy, which is why the authors concede in the discussion that a full proof for small Kerr-AdS holes would require solving the wave equations.
- Because the obstruction is the existence of a global timelike Killing vector, the same test should apply to any asymptotically AdS hole with a boundary rotating slower than light, including charged rotating solutions in four and seven dimensions that the paper does not enumerate.
- The mismatch between free-field Bose condensation and the strong-coupling first-order transition suggests that the free theory is a poor guide wherever a chemical potential approaches a bosonic energy level, which is a caution for reading chemical-potential physics off weakly coupled dual descriptions generally.
- The matching divergence as $Omega_i o 1$ at finite temperature looks like a candidate for a sharper test of the correspondence than the $3/4$ ratio, since it is a statement about a whole function of temperature rather than a single number.
Original abstract
Black hole solutions that are asymptotic to $ AdS_5 \times S^5$ or $ AdS_4 \times S^7$ can rotate in two different ways. If the internal sphere rotates then one can obtain a Reissner-Nordstrom-AdS black hole. If the asymptotically AdS space rotates then one can obtain a Kerr-AdS hole. One might expect superradiant scattering to be possible in either of these cases. Superradiant modes reflected off the potential barrier outside the hole would be re-amplified at the horizon, and a classical instability would result. We point out that the existence of a Killing vector field timelike everywhere outside the horizon prevents this from occurring for black holes with negative action. Such black holes are also thermodynamically stable in the grand canonical ensemble. The CFT duals of these black holes correspond to a theory in an Einstein universe with a chemical potential and a theory in a rotating Einstein universe. We study these CFTs in the zero coupling limit. In the first case, Bose-Einstein condensation occurs on the boundary at a critical value of the chemical potential. However the supergravity calculation demonstrates that this is not to be expected at strong coupling. In the second case, we investigate the limit in which the angular velocity of the Einstein universe approaches the speed of light at finite temperature. This is a new limit in which to compare the CFT at strong and weak coupling. We find that the free CFT partition function and supergravity action have the same type of divergence but the usual factor of 4/3 is modified at finite temperature.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Black hole solutions that are asymptotic to $ AdS_5 \times S^5$ or $ AdS_4 \times S^7$ can rotate in two different ways. If the internal sphere rotates then one can obtain a Reissner-Nordstrom-AdS black hole. If the asymptotically AdS space rotates then one can obtain a Kerr-AdS hole. One might expect superradiant scattering to be possible in either of these cases. Superradiant modes reflected off the potential barrier outside the hole would be re-amplified at the horizon, and a classical instability would result. We point out that the existence of a Killing vector field timelike everywhere outside the horizon prevents this from occurring for black holes with negative action. Such black holes are also thermodynamically stable in the grand canonical ensemble. The CFT duals of these black holes correspond to a theory in an Einstein universe with a chemical potential and a theory in a rotating Einstein universe. We study these CFTs in the zero coupling limit. In the first case, Bose-Einstein condensation occurs on the boundary at a critical value of the chemical potential. However the supergravity calculation demonstrates that this is not to be expected at strong coupling. In the second case, we investigate the limit in which the angular velocity of the Einstein universe approaches the speed of light at finite temperature. This is a new limit in which to compare the CFT at strong and weak coupling. We find that the free CFT partition function and supergravity action have the same type of divergence but the usual factor of 4/3 is modified at finite temperature.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Black hole solutions that are asymptotic to $ AdS_5 \times S^5$ or $ AdS_4 \times S^7$ can rotate in two different ways. | background 1.00 | 0.09 | 0.02 | asserted 0.99 |
| s1 | If the internal sphere rotates then one can obtain a Reissner-Nordstrom-AdS black hole. | background 0.82 | 0.32 | 0.04 | hedged possibility 0.73 |
| s2 | If the asymptotically AdS space rotates then one can obtain a Kerr-AdS hole. | background 0.97 | 0.23 | 0.04 | hedged possibility 0.75 |
| s3 | One might expect superradiant scattering to be possible in either of these cases. | background 0.90 | 0.07 | 0.03 | speculative 0.96 |
| s4 | Superradiant modes reflected off the potential barrier outside the hole would be re-amplified at the horizon, and a classical instability would result. | background 0.99 | 0.08 | 0.04 | asserted 0.67 |
| s5 | We point out that the existence of a Killing vector field timelike everywhere outside the horizon prevents this from occurring for black holes with negative action. | result 0.90 | 0.64 | 0.43 | asserted 1.00 |
| s6 | Such black holes are also thermodynamically stable in the grand canonical ensemble. | result 0.82 | 0.32 | 0.10 | asserted 1.00 |
| s7 | The CFT duals of these black holes correspond to a theory in an Einstein universe with a chemical potential and a theory in a rotating Einstein universe. | contribution 0.47 | 0.38 | 0.10 | asserted 1.00 |
| s8 | We study these CFTs in the zero coupling limit. | method 0.52 | 0.90 | 0.03 | asserted 0.97 |
| s9 | In the first case, Bose-Einstein condensation occurs on the boundary at a critical value of the chemical potential. | result 0.91 | 0.26 | 0.08 | asserted 1.00 |
| s10 | However the supergravity calculation demonstrates that this is not to be expected at strong coupling. | result 0.85 | 0.60 | 0.13 | asserted 1.00 |
| s11 | In the second case, we investigate the limit in which the angular velocity of the Einstein universe approaches the speed of light at finite temperature. | contribution 0.73 | 0.68 | 0.04 | asserted 0.84 |
| s12 | This is a new limit in which to compare the CFT at strong and weak coupling. | contribution 0.70 | 0.47 | 0.05 | asserted 0.99 |
| s13 | We find that the free CFT partition function and supergravity action have the same type of divergence but the usual factor of 4/3 is modified at finite temperature. | result 1.00 | 0.96 | 0.28 | asserted 1.00 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:hep-th/9908109 by the authors.
References
How the literature uses it
100 arXiv papers cite this in 161 sentences
Papers citing it · 118 sentences without a confident label
Compares with it 3 papers
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This decreasing critical temperature is consistent with the previous studies in rotating black holes in holography [ref] and rotating pure gluons [ref] .
Complex Langevin Method on Rotating Matrix Quantum Mechanics at Thermal Equilibrium · Summary of this work · 0.83
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Interestingly, we see that our rotating fluid has a direct analogy to the energy-momentum tensor of the extremal Myers-Perry black hole [ref] .
A New Paradigm for Topological or Rotational Non-Abelian Gauge Fields from Einstein-Skyrme Holography · Energy-momentum tensor · 0.99
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This is in close parallel to the instability of the over-spinning Kerr-AdS black holes [ref] .
The Yang-Mills duals of small AdS black holes · Conclusion · 1.00
Cites it as support 5 papers
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In fact, it has been shown that, for cetain ranges of charges or chemical potentials, (near-)extremal charged and/or rotating black holes with a macroscopic horizon are the dominant saddles in the zero temperature limit [ref] .[In certain cases, it has been only shown that they are locally stable due to difficulties in finding competing saddles in a given ensemble.
Bulk Cluster Decomposition in AdS/CFT and A No-Go Theorem for Correlators in Microstates of Extremal Black Holes · Introduction and summary · 0.97
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There are simple general arguments by Hawking and Reall [ref] that show that there is no super-radiant instability for matter that satisfies the dominant energy condition when the black hole is large and rotating slowly enough.
Black Hole Vacua and Rotation · Preliminaries: Motivation and Setup · 0.96
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Obviously, for Ω>1/L the Einstein Universe on the boundary rotates faster than the speed of light, and the corresponding CFT seems ill defined.[It was shown in [ref] , that a weakly coupled CFT does not admit a well-defined partition function when $ΩL>1$.
Holographic CFT Phase Transitions and Criticality for Rotating AdS Black Holes · Extended boundary thermodynamics · 0.98
Cites it as background 29 papers
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This has already been considered in [ref] , and the AdS/CFT correspondence about the Kerr-AdS black hole was investigated.
Phase Transitions of Charged Kerr-AdS Black Holes from Large-N Gauge Theories · Introduction and summary · 0.82
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Instability can be understood in terms of superradiance [ref] .[For charged scalar fields in non-rotating charged black hole spacetimes, superradiance is an amplified scattering of the scalar field around the black hole if its frequency $ω̃$ satisfies $ω̃ <qμ$ [ref] .
Hairy black holes in AdS with Robin boundary conditions · Instability of RNAdS · 0.85
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A similar process occurs for Reissner-Nordström (RN) BHs [ref] , with the charge playing the role of the angular momentum.
Nonlinear Evolution and Final Fate of Charged Anti-de Sitter Black Hole Superradiant Instability · Introduction · 0.84
Discussion
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