REVIEW
AdS/CFT Duals of Topological Black Holes and the Entropy of Zero-Energy States
High Energy Physics - Theory · Submitted 4 Jun 1999 · 40 references
Pith's verdict: A useful, honest paper whose headline reading rests on an unproven protection argument the author himself flags.
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Referee report
The paper
The paper claims
Hyperbolic black holes in anti-de Sitter space suggest that strongly coupled Super-Yang-Mills theory carries states that add entropy without adding energy, including a zero-temperature ground state with large degeneracy.
The reading
What carries the argument
The machinery is a three-way comparison. On the gravity side, the paper uses the counterterm-subtraction prescription for the quasilocal stress tensor of asymptotically AdS solutions, which yields $⟨T_{μν}⟩$, the energy $E(β)$ and the entropy $S(β)$ for the hyperbolic black hole family as functions of the horizon radius, with the pure-AdS member sitting at $β=2πl$ and the extremal member at $β→∞$. On the field theory side, it uses the Plancherel measure for the wave operator on hyperbolic space to compute one-loop energy and entropy densities for fields of spin 0, $1/2$ and $1$ on $S^1 × H^3$, then specializes to the $N=4$ supermultiplet content. The comparison is made meaningful by the fact that the $β=2πl$ state is conformally flat and its energy is fixed by the central charge, so both regimes can be trusted there. A conformal transformation between the hyperboloid and Rindler space then reinterprets the whole comparison as a statement about accelerating observers in flat space.
Extended reading notes
Core claim
The central claim is that hyperbolic black holes expose a genuine strong-coupling surprise in $N=4$ Super-Yang-Mills on a hyperboloid. Two states carry the argument. First, the extremal hyperbolic black hole has a stress tensor that vanishes identically, so its energy density is zero, yet its Bekenstein-Hawking entropy is nonzero; the author identifies this with a highly degenerate ground state at infinite 't Hooft coupling, in contrast to the free-field result of zero entropy. Second, the solution that is locally isometric to pure $AdS_5$ (temperature $1/(2πl)$) has an energy that matches the one-loop field theory value exactly, but an entropy larger than the weak-coupling value by a factor of $3/2$. The author stresses that the pattern is the same in both cases: entropy is enhanced at strong coupling while the energy is not, so the strong-coupling theory appears to have more states contributing to entropy than to energy. He also shows that hyperbolic black holes are, in the flat-space description of the dual theory, thermal Rindler states, so the same statements become claims about the degeneracy of the Rindler vacuum and about entanglement entropy across a Rindler horizon in exact $AdS_5$.
Load-bearing premise
The claim that the $β=2πl$ entropy mismatch is a real strong-coupling effect rests on the assumption that all string $α'$ corrections can be written in a scheme built from the Weyl tensor, so that the conformally flat pure-AdS solution is uncorrected and its free energy is protected; the author calls this reasonable rather than established, and it conflicts with a background-subtraction calculation in which that same state is corrected.
Editorial extensions
If this is right
- At infinite 't Hooft coupling the ground state of the theory on a hyperboloid is highly degenerate even though its energy density vanishes, so the usual identification of zero energy with a unique vacuum fails in this sector.
- The entropy of the $β=2πl$ state is $3/2$ times the one-loop value while its energy and specific heat agree exactly, which points to states that contribute to entropy but not to the stress tensor.
- Because hyperbolic black holes are thermal Rindler states of the flat-space dual, the mismatch becomes a statement that the Minkowski vacuum, as seen by an accelerating observer, has more entanglement entropy in the strong-coupling description than free-field counting suggests.
- The ratio $S/E$ at strong coupling exceeds the weak-coupling ratio at every finite temperature, with equality only in the infinite-temperature limit.
- In the $AdS_4$ and $AdS_7$ cases the mismatch is sharper still: the state isometric to $AdS_4$ has finite temperature and nonzero entropy but zero energy density, and there are states of negative total energy.
- The author notes that the required hidden degrees of freedom resemble the 'precursor' states postulated elsewhere to preserve causality in AdS/CFT scattering, suggesting a common origin.
Reading between the lines
- If states carrying entropy but no local energy really exist in the gauge theory, they should show up as an anomalous contribution to the free energy on $S^1 × H^3$ that is invisible in the stress tensor; a direct resummation or lattice-style check of the one-loop result beyond leading order would be the natural place to look.
- The $3/2$ factor is numerically the reciprocal of the familiar $3/4$ reduction seen for planar black holes, which suggests the two discrepancies may be two faces of one coupling-dependent reweighting of degrees of freedom rather than two separate puzzles.
- Because the argument for protection relies on the solution being conformally flat, any state with a nonzero Weyl tensor should show ordinary coupling dependence; comparing the size of the $β=2πl$ anomaly with the known $α'^3$ corrections at nearby temperatures would test whether the effect is a threshold jump or a smooth drift.
- The Rindler reformulation makes the claim in principle accessible to entanglement-entropy calculations in the field theory itself, independent of any black hole background, which is a cleaner setting in which to confirm or refute the enhancement.
Original abstract
The horizon of a static black hole in Anti-deSitter space can be spherical, planar, or hyperbolic. The microscopic dynamics of the first two classes of black holes have been extensively discussed recently within the context of the AdS/CFT correspondence. We argue that hyperbolic black holes introduce new and fruitful features in this respect, allowing for more detailed comparisons between the weak and strong coupling regimes. In particular, by focussing on the stress tensor and entropy of some particular states, we identify unexpected increases in the entropy of Super-Yang-Mills theory at strong coupling that are not accompanied by increases in the energy. We describe a highly degenerate state at zero temperature and zero energy density. We also find that the entanglement entropy across a Rindler horizon in exact AdS_5 is larger than might have been expected from the dual SYM theory. Besides, we show that hyperbolic black holes can be described as thermal Rindler states of the dual conformal field theory in flat space.
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
The horizon of a static black hole in Anti-deSitter space can be spherical, planar, or hyperbolic. The microscopic dynamics of the first two classes of black holes have been extensively discussed recently within the context of the AdS/CFT correspondence. We argue that hyperbolic black holes introduce new and fruitful features in this respect, allowing for more detailed comparisons between the weak and strong coupling regimes. In particular, by focussing on the stress tensor and entropy of some particular states, we identify unexpected increases in the entropy of Super-Yang-Mills theory at strong coupling that are not accompanied by increases in the energy. We describe a highly degenerate state at zero temperature and zero energy density. We also find that the entanglement entropy across a Rindler horizon in exact AdS_5 is larger than might have been expected from the dual SYM theory. Besides, we show that hyperbolic black holes can be described as thermal Rindler states of the dual conformal field theory in flat space.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The horizon of a static black hole in Anti-deSitter space can be spherical, planar, or hyperbolic. | background 1.00 | 0.06 | 0.02 | asserted 1.00 |
| s1 | The microscopic dynamics of the first two classes of black holes have been extensively discussed recently within the context of the AdS/CFT correspondence. | background 0.99 | 0.04 | 0.02 | asserted 1.00 |
| s2 | We argue that hyperbolic black holes introduce new and fruitful features in this respect, allowing for more detailed comparisons between the weak and strong coupling regimes. | contribution 0.75 | 0.27 | 0.27 | asserted 0.75 |
| s3 | In particular, by focussing on the stress tensor and entropy of some particular states, we identify unexpected increases in the entropy of Super-Yang-Mills theory at strong coupling that are not accompanied by increases in the energy. | result 0.98 | 0.95 | 0.51 | asserted 1.00 |
| s4 | We describe a highly degenerate state at zero temperature and zero energy density. | result 0.95 | 0.93 | 0.14 | asserted 1.00 |
| s5 | We also find that the entanglement entropy across a Rindler horizon in exact AdS_5 is larger than might have been expected from the dual SYM theory. | result 1.00 | 0.94 | 0.08 | asserted 0.93 |
| s6 | Besides, we show that hyperbolic black holes can be described as thermal Rindler states of the dual conformal field theory in flat space. | result 0.87 | 0.92 | 0.23 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
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Figures
Figures reproduced from arXiv:hep-th/9906040 by the authors.
References
How later papers used it
86 later papers cite it
1 later paper by other authors is listed below: 1 used its method or tool.
Uses its method or tool 1 of 5 papers with this label
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Computed the energy and entropy of this state via the AdS/CFT method of this paper.
Holographic de Sitter Universe (2011) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
77 arXiv papers cite this in 141 sentences
Papers citing it · 98 sentences without a confident label
Builds on it 2 papers
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The analysis of [ref] was done only for uncharged hyperbolic black holes, but the main conclusions relevant for our purposes extend to the charged case.
Black funnels and droplets in thermal equilibrium · Holographic interpretation of black droplets and funnels · 0.90
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One can also study the string in the hyperbolic AdS black hole (which generalizes [ref] by replacing $-1+r^2/ℓ^2→-1+μ/r^d-1+r^2/ℓ^2$ [ref] ; for $μ≠0$ this describes the exterior of a true black hole with non-zero mass and a singularity at $r=0$), which is dual to a stationary quark in $ℝ×H^d-1$ at finite temperature.
Chaos and the reparametrization mode on the AdS$_2$ string · Discussion · 0.96
Uses its method or tool 4 papers
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We now compute the energy and entropy of this state via AdS/CFT method [ref] - [ref] .
Holographic de Sitter Universe · Holographic Static de Sitter · 1.00
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The $n$-th Rényi entropy is obtained by extending the periodicity to $τ∼τ+ 2πn$, which in the bulk corresponds to replacing $f(ζ)$ by the topological black hole [ref]
Nonlocal Nonstabilizerness from Holographic Schwinger Pair Production · Casini–Huerta–Myers (CHM) map and topological black holes · 0.89
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The states $|Ψ_β⟩$ for free conformal field theories can be obtained by first solving for the thermal state of the field in $ℝ×H_n-1$, which can be done with spectral methods, and then Weyl-transforming to Rindler space [ref] .
Tearing down spacetime with quantum disentanglement · Rindler entanglement for free conformal fields · 0.96
Uses its data or values 1 paper
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First of all note that the boundary stress tensor for the hyperbolic black hole hypbh has been previously determined in [ref] :
Holographic models of de Sitter QFTs · Higher-dimensional conformal field theories on de Sitter spacetime. · 0.89
Compares with it 8 papers
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Of course, this is the same factor that appears in the usual AdS/CFT computations of the CFT at finite temperature [ref] .
Strongly Coupled Radiation from Moving Mirrors and Holography in the Karch-Randall model · Bootstrap Domain Wall Spacetimes · 0.86
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Comparison with [ref], taking $α' = 0$ and $β' = -1$ and thus $α= β= 1$, gives us g^(n)_ab x^a x^b = - f(r) t^2 + r^2/f(r) , ϕ = - ln(H'r) = - ln(r/ℓ) , and the corresponding AdS metric obtained via the map [ref] reads [equation] Changing coordinates to $z = ℓ/(H r)$ and $t = τ/(Hℓ)$, we obtain [equation] with [equation] which is the metric for an AdS black hole with hyperbolic horizon geometry [ref] .
Mapping AdS to dS spaces and back · Black holes · 0.86
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The results agree with those found in [ref] in the counterterm formalism of Balasubramanian and Kraus [ref] , but we include here some of the computational steps as done in the counterterm formalism of de Haro, Skenderis and Solodukhin [ref] , since comparison with this computation will allow a concise treatment of the nonstatic foliation in Section [ref].
Isotropic AdS/CFT fireball · Static FRW as the conformal boundary · 0.91
Cites it as background 26 papers
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The thermodynamics of such black holes has long been know to have some odd features of which the interpretation has been for the most part unclear [ref] .
Holography of negative energy states · Introduction · 0.93
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Global $AdS_d+1$ space can be “Rindlerized” pretty much like flat spacetime resulting in bulk acceleration horizons [ref] .
No Holography for Eternal AdS Black Holes · Review of Rindler-AdS · 0.92
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In the AdS/CFT context these have been studied in [ref] and references therein.] – a quotient of AdS space obtained by an identification of global $AdS_5$ along a boost [ref] .
Real time response on dS_3: the Topological AdS Black Hole and the Bubble · Introduction · 0.97
Discussion
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