On long time unitarity restoring processes in the presence of eternal black holes
Open accessAbstract
We discuss the various scales determining the temporal behaviour of correlation functions in the presence of eternal black holes. We point out the origins of the failure of the semiclassical gravity approximation to respect a unitarity‐based bound suggested by Maldacena. We find that the presence of a subleading (in the large N approximation involved) master field does restore the compliance with one bound but additional configurations are needed to explain the more detailed expected time dependence of the Poincare recurrences and their magnitude.
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TL;DR
The paper analyzes long-time correlation functions in eternal AdS black holes to test unitarity restoration beyond semiclassical gravity. It finds that including a subleading topology-changing master field restores the correct order of magnitude of the infinite-time average, but not the detailed Poincaré recurrence time and amplitude expected from the exact CFT.
Key findings
- In the semiclassical black-hole background, normal-mode frequencies form a continuous spectrum, giving \(L_{\rm bh}=0\), while pure AdS has a discrete spectrum and \(L_{\rm AdS}=O(1)\); this is the failure of semiclassical gravity to satisfy the unitarity-based bound.
- Coherently summing the black-hole Euclidean saddle \(X\) and the thermal AdS/gas saddle \(Y\) yields a nonvanishing recurrence index \(L\) with exponential suppression \(\exp(-2|I_X|)\sim \exp(-N^2)\), reproducing the expected scaling in order of magnitude.
- The full instanton time profile still disagrees: its resurgences have amplitude \(\exp(-N^2)\) and period \(t_H(Y)\sim N^0\), whereas the exact CFT is expected to have \(O(1)\) resurgences with period \(t_H\sim \exp(N^2)\); both have infinite-time average \(O(e^{-N^2})\).
- Large topological fluctuations become quantitatively important at \(t_c\sim \Delta I/\Gamma\sim N^2\), much shorter than the detailed unitarity restoration time \(t_H\sim \exp(N^2)\), so semiclassical singularity analysis is incomplete.
Methods
- Theoretical study using AdS/CFT, the large-\(N\) expansion, eternal AdS black holes, and long-time correlators \(G(t)\) / normalized \(L(t)\) to probe Poincaré recurrences and the Heisenberg time.
- Analyzes Euclidean saddle points: the black-hole manifold \(X\) with contractible thermal circle and the thermal AdS manifold \(Y\) with non-contractible \(S^1\); estimates instanton contributions and effective radial Schrödinger potentials for normal modes.
- No empirical data; results are analytic/order-of-magnitude estimates in the semiclassical and instanton approximations.
Limitations
- The paper states that the large-scale topology-changing fluctuations “begin to restore some of the fine structure required by unitarity, but fall short at the quantitative level”; the appropriate instantons are presumed to occur on microscopic scales and involve stringy dynamics.
- The paper states that its semiclassical analysis of the singularity is “bound to be incomplete” because detailed unitarity is restored only at \(t_H\sim \exp(N^2)\), while semiclassical probes act at \(t_c\sim N^2\).
- Evident from the text: the analysis is restricted to order-of-magnitude estimates and does not provide the microscopic CFT configurations needed for the exact recurrence profile.
Model-written summary. Check it against the paper before relying on it.
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Claims the paper makes
0 of 10 cited sources checked · 0 supported · 0 flagged
- Findingown data§ 3 Topological diversity and unitarity · p. 8
The time scale t_c for which large-scale instantons are quantitatively important on the graph of L(t) is t_c ∼ ∆I/Γ ∼ N^2.
“We can also find the time scale t c for which the large-scale instantons considered here are quantitatively important on the graph of L ( t ). This is shown in Fig. 5 and yields t c ∼ ∆ I/ Γ ∼ N 2 .”
Quote in the paperSubstantial claim- Rests on an assumption: The Bekenstein-Hawking entropy of large AdS black holes scales like that of N^2 conformal degrees of freedom at high energy.
- Findingnone given§ 2 Long-time details of thermal quasi-equilibrium · p. 6
In the perturbative approximation, L_bh = 0.
“On the other hand, we have L bh = 0 in this approximation.”
Quote in the paperSubstantial claim - Findingreasoning§ 3 Topological diversity and unitarity · p. 6
The instanton approximation to L yields an exponential suppression factor of order exp(-N^2), reproducing the expected scaling at least in order of magnitude.
“Since I Y ∼ -N 0 and I X ∼ -N 2 , the exponential suppression factor is of order exp( -2 | I X | ) ∼ exp( -N 2 ), reproducing the expected scaling (9), at least in order of magnitude”
Quote in the paperSubstantial claim- Rests on an assumption: The Bekenstein-Hawking entropy of large AdS black holes scales like that of N^2 conformal degrees of freedom at high energy.
- Findingreasoning§ 2 Long-time details of thermal quasi-equilibrium · p. 5
In gravity perturbation theory in the AdS black hole background, L = 0.
“Indeed, one finds L = 0 in gravity perturbation theory in the AdS black hole background.”
Quote in the paperSubstantial claim - Findingreasoning§ 2 Long-time details of thermal quasi-equilibrium · p. 5
The recurrence index L scales as exp(-N^2) in the large N limit of the dual CFT.
“Since both β and Γ scale as N 0 in the largeN limit of the dual CFT, the 'recurrence index' L ∼ exp( -N 2 ) scales as a nonperturbative effect in the semiclassical approximation.”
Quote in the paperSubstantial claim- Rests on an assumption: The CFT lives in finite volume.
- Findingreasoning§ 4 Conclusions · p. 9
The characteristic time for large topological fluctuations to be important is t_c ∼ O(N^2) in the semiclassical approximation.
“We have also seen that the characteristic time for large topological fluctuations to be important is t c ∼ O ( N 2 ) in the semiclassical approximation.”
Quote in the paperSubstantial claim- Rests on an assumption: The Bekenstein-Hawking entropy of large AdS black holes scales like that of N^2 conformal degrees of freedom at high energy.
- Interpretationcitation§ 1 Introduction · p. 2
According to AdS/CFT, the formation and evaporation of small AdS black holes with Schwarzschild radius much less than the AdS radius should be described by a unitary process in terms of the CFT Hamiltonian.
“According to this definition, the formation and evaporation of small black holes with Schwarschild radius R S ≪ R , should be described by a unitary process in terms of the CFT Hamiltonian.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [2] [2] J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 hep-th/9711200. S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Phys. Lett. B428 (1998) 105 hep-th/9802109. E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253 hep-th/9802150. -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationreasoning§ 3 Topological diversity and unitarity · p. 6
The superposition of Euclidean saddle points, or master fields in the CFT language, corresponds to large-scale fluctuations in which the AdS black hole is converted into a graviton gas at the same temperature and vice versa.
“Physically, this superposition of Euclidean saddle points (or master fields, in the language of the CFT) corresponds to large-scale fluctuations in which the AdS black hole is converted into a graviton gas at the same temperature and viceversa.”
Quote in the paperSubstantial claim - Interpretationreasoning§ 1 Introduction · p. 3
The failure of G(t) to vanish as t tends to infinity can be used as a criterion for unitarity preservation beyond the semiclassical approximation.
“Hence, the failure of G ( t ) to vanish as t → ∞ can be used as a criterion for unitarity preservation beyond the semiclassical approximation.”
Quote in the paperSubstantial claim - Interpretationown data§ 3 Topological diversity and unitarity · p. 6
The apparent success of the instanton approximation for the infinite time average is somewhat coincidental when the full time profile of L(t) is considered.
“However, the apparent success of (14) turns out to be somewhat coincidental. If we consider the full time profile of L ( t ) rather than the infinite time average, we find”
Quote in the paperSubstantial claim - Premisecitation§ 1 Introduction · p. 3
Coupling an ordinary quantum mechanical system to random classical noise forces G(t) to decay exponentially for large t despite having a discrete energy spectrum.
“It is then shown in [7] that this random noise forces G ( t ) to decay exponentially for large t , despite having a discrete energy spectrum.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [7] [7] J.L.F. Barb´ on and E. Rabinovici, J. High Energy Phys. 0311 (2003) 047 hep-th/0308063. -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationreasoning§ 3 Topological diversity and unitarity · p. 6
Improving on the semiclassical prediction for L requires some sort of topology-change process.
“Therefore, it seems that improving on the semiclassical prediction for L requires some sort of topology-change process.”
Quote in the paperSubstantial claim - Interpretationreasoning§ 3 Topological diversity and unitarity · p. 6
The continuous spectrum in the spectral decomposition of G(t)_X is a consequence of the contractible topology of X, since the Hamiltonian foliation by τ = constant surfaces is singular at r = r0.
“The continuous spectrum arising in the spectral decomposition of G ( t ) X is a consequence of the contractible topology of X , since the Hamiltonian folliation by τ = constant surfaces is singular at r = r 0 .”
Quote in the paperSubstantial claim - Interpretationcitation§ 1 Introduction · p. 2
The AdS/CFT correspondence is not consistent with Hawking's semiclassical picture of black hole evaporation as a non-unitary process.
“The AdS/CFT correspondence [2] is not consistent with this picture.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [2] [2] J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 hep-th/9711200. S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Phys. Lett. B428 (1998) 105 hep-th/9802109. E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253 hep-th/9802150. -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationreasoning§ 2 Long-time details of thermal quasi-equilibrium · p. 5
The reason L = 0 is that the relevant eigenfrequencies, the normal modes of the black hole, form a continuous spectrum to all orders in the 1/N expansion.
“The reason is that the relevant eigenfrequencies ω (the so-called normal modes of the black hole) form a continuous spectrum to all orders in the 1 /N expansion.”
Quote in the paperSubstantial claim - Interpretationcitation§ 1 Introduction · p. 3
The long-time behaviour of correlators probes the strict quantum coherence of the bounded system.
“This shows that the long-time behaviour of correlators probes the strict quantum coherence of the bounded system.”
Quote in the paperSubstantial claim - Interpretationreasoning§ 1 Introduction · p. 2
Large AdS black holes with inverse Hawking temperature much less than the AdS radius describe the leading approximation to the thermodynamical functions of the canonical CFT state.
“Therefore, large AdS black holes with inverse Hawking temperature β ≪ R describe the leading approximation to the thermodynamical functions of the canonical CFT state”
Quote in the paperSubstantial claim- Rests on an assumption: Large AdS black holes with Schwarzschild radius much greater than the AdS radius are thermodynamically stable and can exist in equilibrium at fixed high temperatures.
- Rests on an assumption: The Bekenstein-Hawking entropy of large AdS black holes scales like that of N^2 conformal degrees of freedom at high energy.
- Interpretationreasoning§ 3 Topological diversity and unitarity · p. 6
Since Y has a discrete spectrum in perturbation theory, the net result for L should be non-vanishing in this approximation.
“Since Y has a discrete spectrum in perturbation theory, the net result for L should be non-vanishing in this approximation.”
Quote in the paperSubstantial claim - Interpretationreasoning§ 4 Conclusions · p. 9
Since t_c ≪ t_H, semiclassical analysis of the singularity is bound to be incomplete, as it misses whatever microphysics is responsible for detailed unitarity restoration in the quantum mechanical time evolution.
“Thus t c ≪ t H and we conclude that such semiclassical analysis of the singularity is bound to be incomplete, as it misses whatever microphysics is responsible for the detailed unitarity restoration in the quantum mechanical time evolution.”
Quote in the paperSubstantial claim - Interpretationreasoning§ 4 Conclusions · p. 9
Detailed unitarity is only restored on time scales of order t_H ∼ exp(N^2).
“However, we have seen that detailed unitarity is only restored on time scales of order t H ∼ exp( N 2 ).”
Quote in the paperSubstantial claim- Rests on an assumption: The CFT lives in finite volume.
- May combine two claims. 8
- Classified as probably not a substantial claim. 4
May combine two claims.
Findingreasoning§ 2 Long-time details of thermal quasi-equilibrium · p. 6The thermal entropy of perturbative gravity excitations in vacuum AdS scales as N^0, so the perturbative Heisenberg time of AdS spacetime is O(1) in the large N limit, leading to L_AdS = O(1).
“Since the thermal entropy of perturbative gravity excitations in the vacuum AdS spacetime scales as S ( β ) AdS ∼ N 0 , we see that the perturbative Heisenberg time of the AdS spacetime is of O (1) in the largeN limit, leading to L AdS = O (1).”
Quote in the paperSubstantial claim- Rests on an assumption: The CFT lives in finite volume.
May combine two claims.
Findingreasoning§ 2 Long-time details of thermal quasi-equilibrium · p. 5The black-hole horizon has a non-degenerate zero g(r0)=0, which induces a universal scaling.
“The distinguishing feature of the black-hole horizon is a a non-degenerate zero, g ( r 0 ) = 0, which induces the universal scaling”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ 2 Long-time details of thermal quasi-equilibrium · p. 5The vacuum AdS manifold behaves like a finite cavity, with a discrete spectrum, while the AdS black hole has a continuous spectrum.
“The vacuum AdS manifold, corresponding to the choice g ( r ) = 1 + r 2 /R 2 in (10), behaves like a finite cavity, as expected.”
- Also stated in: § 2 Long-time details of thermal quasi-equilibrium · p. 6 · plain
“The spectrum is discrete in pure AdS, and continuous in the AdS black hole.”
Quote in the paperSubstantial claim- Also stated in: § 2 Long-time details of thermal quasi-equilibrium · p. 6 · plain
May combine two claims.
Interpretationown data§ 4 Conclusions · p. 8Large-scale topology-changing fluctuations proposed in [3] begin to restore some of the fine structure required by unitarity, but fall short at the quantitative level.
“We have seen that large scale topology-changing fluctuations proposed in [3] begin to restore some of the fine structure required by unitarity, but fall short at the quantitative level.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [3] [3] J. Maldacena, J. High Energy Phys. 0304 (2003) 021 hep-th/0106112. -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Classified as probably not a substantial claim.
Interpretationcitation§ 4 Conclusions · p. 8A systematic set of corrections could generate a stretched horizon of Planckian thickness.
“Roughly, one needs a systematic set of corrections that could generate a 'stretched horizon' of Planckian thickness [12].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [12] [12] K.S. Thorne, R.H Price and D.A. McDonald, 'Black Holes: The Membrane Paradigm'. Yale University Press 1986. L. Susskind, L. Thorlacius and J. Uglum, Phys. Rev. D48 (1993) 3743 hep-th/9306069. -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
May combine two claims.
Premisecitation§ 1 Introduction · p. 2Because the CFT lives in finite volume, its spectrum is discrete, and the correlator must show nontrivial long-time structure in the form of Poincaré recurrences.
“However, because the CFT lives in finite volume, the spectrum is actually discrete (c.f. Fig 1), and the correlator must show nontrivial long time structure in the form of Poincar´ e recurrences (see [4, 5]).”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [4] [4] L Susskind, Contribution to Stephen Hawking's 60th birthday celebration. hep-th/0204027. L. Dyson, J. Lindesay and L. Susskind, J. High Energy Phys. 0208 (2002) 045 hep-th/0202163. N. Goheer, M. Kleban and L. Susskind, J. High Energy Phys. 0307 (2003) 056 hep-th/0212209. -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites [5] [5] L. Dyson, M. Kleban and L. Susskind, J. High Energy Phys. 0210 (2002) 011 hep-th/0208013. -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Rests on an assumption: The CFT lives in finite volume.
May combine two claims.
Interpretationcitation§ 4 Conclusions · p. 8Semiclassical black holes faithfully reproduce coarse-grained inclusive properties such as entropy, but additional dynamical features of the horizon may be necessary to resolve finer details of the information loss problem.
“While semiclassical black holes do faithfully reproduce 'coarse grained' inclusive properties of the system such as the entropy (c.f. [11]), additional dynamical features of the horizon may be necessary to resolve finer details of the information loss problem.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [11] [11] G.W. Gibbons and S.W. Hawking, Phys. Rev. D15 (1977) 2752. -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Classified as probably not a substantial claim.
Premisecitation§ 3 Topological diversity and unitarity · p. 6The proposal of [3] is to sum coherently the contribution of X and Y instead of evaluating the semiclassical correlators on X.
“The proposal of [3] is precisely that: instead of evaluating the semiclassical correlators on X , one should sum coherently the contribution of X and Y .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [3] [3] J. Maldacena, J. High Energy Phys. 0304 (2003) 021 hep-th/0106112. -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
May combine two claims.
Interpretationreasoning§ 3 Topological diversity and unitarity · p. 8The very long time behaviour of L(t)_inst is very different from the expected exact CFT behaviour, although the infinite time average comes out right in order of magnitude.
“Hence, the very long time behaviour as shown in Fig. 6 is very different from the expected one, although the infinite time average comes out right in order of magnitude:”
Quote in the paperSubstantial claim- Rests on an assumption: The Bekenstein-Hawking entropy of large AdS black holes scales like that of N^2 conformal degrees of freedom at high energy.
Classified as probably not a substantial claim.
Premisereasoning§ 2 Long-time details of thermal quasi-equilibrium · p. 3Poincaré recurrences occur in general bounded systems.
“Poincar´ e recurrences occur in general bounded systems.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Premisecitation§ 4 Conclusions · p. 8The brick-wall model replaces the horizon by a reflecting boundary condition at Planck distance ǫ ∼ l_P from the horizon, defining a mutilated X_ǫ manifold of cylindrical topology, leading to a discrete spectrum of the right spacing in order of magnitude.
“The crudest model of such stretched horizon is the brick-wall model of 't Hooft [10]. In this phenomenological description one replaces the horizon by a reflecting boundary condition at Planck distance ǫ ∼ ℓ P from the horizon. This defines a 'mutilated' X ǫ manifold, of cylindrical topology, leading to a discrete spectrum of the right spacing in order of magnitude.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [10] [10] G. 't Hooft, Nucl. Phys. B256 (1985) 727. -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
The paper relies on this without support.
Large AdS black holes with Schwarzschild radius much greater than the AdS radius are thermodynamically stable and can exist in equilibrium at fixed high temperatures.
“For large AdS black holes with Schwarschild radius R S ≫ R one may attempt to rise to the challenge, since they are thermodynamically stable and can exist in equilibrium at fixed (high) temperatures 1 /β ≫ 1 /R .”
The paper relies on this without support.
The Bekenstein-Hawking entropy of large AdS black holes scales like that of N^2 conformal degrees of freedom at high energy.
“Indeed, the corresponding Bekenstein-Hawking entropy scales like that of N 2 conformal degrees of freedom at high energy,”
The paper relies on this without support.
In the semiclassical approximation, correlators of the form <A(t)A(0)> decay as exp(-Γ t) with Γ ∼ β^{-1}.
“In the semiclassical approximation, one expects these correlators to decay as exp( -Γ t ) with Γ ∼ β -1 .”
The paper relies on this without support.
The CFT lives in finite volume.
“However, because the CFT lives in finite volume, the spectrum is actually discrete”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
By combining the detailed decay with the very late RMT behavior, one may estimate the dip time in a CFT2 dual to the AdS3 black hole as exponentially large in the central charge c.
“By combining the detailed decay with the very late RMT behavior, one may estimate the 'dip' time in a CFT 2 dual to the AdS 3 black hole as [11]”
From Scrambling the spectral form factor: Unitarity constraints and exact results (2017) · cites this paper as [11]
Including the one-loop fluctuation determinant around the saddle gives a prefactor leading to algebraic decay at late times.
“Considering in addition the one-loop fluctuation determinant around the saddle [11, 56] gives the prefactor”
From Scrambling the spectral form factor: Unitarity constraints and exact results (2017) · cites this paper as [11]
In holography the thermofield double state is dual to the eternal black hole, and the decay properties of S(β,t) can be interpreted as a diagnostic of information loss and recovery.
“In the latter context the thermofield double state is dual to the eternal black hole [5] and the decay properties of S ( β, t ) can be interpreted as a diagnostic of information loss and recovery [8, 11].”
From Scrambling the spectral form factor: Unitarity constraints and exact results (2017) · cites this paper as [11]
A better estimate of the decay of the spectral form factor in CFT2 is obtained by including the images under SL(2,Z) of the vacuum character, which results in a 1/t^3 power law.
“A better estimate of the decay of the spectral form factor in CFT 2 is obtained by including the images under SL (2 , Z ) of the vacuum character, which results in a 1 /t 3 power law [11].”
From Scrambling the spectral form factor: Unitarity constraints and exact results (2017) · cites this paper as [11]
The full analytical result matches earlier numerical studies and is remarkably similar at all times to the spectral form factor in various holographic models, such as the SYK model and the BTZ black hole.
“The full analytical result not only matches earlier numerical studies [8], but is also remarkably similar (at all times) to the spectral form factor in various holographic models, such as the SYK model [8] as well as the BTZ black hole [11].”
From Scrambling the spectral form factor: Unitarity constraints and exact results (2017) · cites this paper as [11]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Temporal bumps are assigned an inverse-time width
MinorThe paper uses exp(−Γt), making Γ an inverse-time decay rate. A bump's width on the time axis must have units of time, so it cannot be Γ as stated.
“The time average can be estimated by noticing that the graph of L ( t ) features positive 'bumps' of height ∆ L and width Γ, separated a time t H , so that”
“The instanton approximation to the correlator L ( t ) inst features the expected exponential decay exp( -Γ t ) induced by the contribution of the X -manifold”
Cites
The large $N$ limit of superconformal field theories and supergravity
Maldacena, J. (1998) · Advances in Theoretical and Mathematical Physics · doi:10.4310/atmp.1998.v2.n2.a1 · Cited by 14,524 · see citing works
Anti de Sitter space and holography
Witten, E. (1998) · Advances in Theoretical and Mathematical Physics · doi:10.4310/atmp.1998.v2.n2.a2 · Cited by 13,070 · see citing works
Gauge theory correlators from non-critical string theory
Gubser, S. S., Klebanov, I. R., & Polyakov, A. M. (1998) · Physics Letters B · doi:10.1016/s0370-2693(98)00377-3 · Cited by 9,307 · see citing works
Action integrals and partition functions in quantum gravity
Gibbons, G. W., & Hawking, S. W. (1977) · Physical Review D · doi:10.1103/physrevd.15.2752 · Cited by 3,449 · see citing works
Breakdown of predictability in gravitational collapse
Hawking, S. W. (1976) · Physical Review D · doi:10.1103/physrevd.14.2460 · Cited by 2,554 · see citing works
Eternal black holes in anti-de Sitter
Maldacena, J. (2003) · Journal of High Energy Physics · doi:10.1088/1126-6708/2003/04/021 · Cited by 1,555 · see citing works
On the quantum structure of a black hole
't Hooft, G. (1985) · Nuclear Physics B · doi:10.1016/0550-3213(85)90418-3 · Cited by 1,311 · see citing works
The stretched horizon and black hole complementarity
Susskind, L., Thorlacius, L., & Uglum, J. (1993) · Physical Review D · doi:10.1103/physrevd.48.3743 · Cited by 1,081 · see citing works
Disturbing Implications of a Cosmological Constant
Dyson, L., Kleban, M., & Susskind, L. (2002) · Journal of High Energy Physics · doi:10.1088/1126-6708/2002/10/011 · Cited by 338 · see citing works
Difficulties for the evolution of pure states into mixed states
Banks, T., Susskind, L., & Peskin, M. E. (1984) · Nuclear Physics B · doi:10.1016/0550-3213(84)90184-6 · Cited by 308 · see citing works
The trouble with de Sitter space
Goheer, N., Kleban, M., & Susskind, L. (2003) · Journal of High Energy Physics · doi:10.1088/1126-6708/2003/07/056 · Cited by 281 · see citing works
Inside the horizon with AdS/CFT
Kraus, P., Ooguri, H., & Shenker, S. H. (2003) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.67.124022 · Cited by 247 · see citing works
Is There Really a de Sitter/CFT Duality
Dyson, L., Lindesay, J., & Susskind, L. (2002) · Journal of High Energy Physics · doi:10.1088/1126-6708/2002/08/045 · Cited by 165 · see citing works
Very long time scales and black hole thermal equilibrium
Barbón, J. L. F., & Rabinovici, E. (2003) · Journal of High Energy Physics · doi:10.1088/1126-6708/2003/11/047 · Cited by 113 · see citing works
Self-citationRelaxation in conformal field theory, Hawking-Page transition, and quasinormal or normal modes
Birmingham, D., Sachs, I., & Solodukhin, S. N. (2003) · Physical Review D · doi:10.1103/physrevd.67.104026 · Cited by 90 · see citing works
Twenty Years of Debate with StephenPreprint
Susskind, L. (2002) · arXiv (Cornell University) · doi:10.48550/arxiv.hep-th/0204027 · Cited by 27 · see citing works
Flagged first, then by citations
Cited by
Sorted by citations
Gauge-Gravity Duality and the Black Hole Interior
Marolf, D., & Polchinski, J. (2013) · Physical Review Letters · doi:10.1103/physrevlett.111.171301 · Cited by 260 · see citing works
Lessons from the Information Paradox
Raju, S. (2020) · arXiv · doi:10.48550/arxiv.2012.05770 · Cited by 234
Scrambling the spectral form factor: Unitarity constraints and exact results
del Campo, A., Molina-Vilaplana, J., & Sonner, J. (2017) · Physical Review D · doi:10.1103/physrevd.95.126008 · Cited by 133 · see citing works
Local Operators in the Eternal Black Hole
Papadodimas, K., & Raju, S. (2015) · Physical Review Letters · doi:10.1103/physrevlett.115.211601 · Cited by 133 · see citing works
On information loss in AdS3/CFT2
Fitzpatrick, A. L., Kaplan, J., Li, D., & Wang, J. (2016) · Journal of High Energy Physics · doi:10.1007/jhep05(2016)109 · Cited by 129 · see citing works
Eigenstate Thermalization and Disorder Averaging in Gravity
Pollack, J., Rozali, M., Sully, J., & Wakeham, D. (2020) · Physical Review Letters · doi:10.1103/physrevlett.125.021601 · Cited by 116 · see citing works
- Authors
- Barbón, J. L. F., & Rabinovici, E.
- Affiliations
- European Organization for Nuclear Research (CH)
- ORCID
- Barbón, J.L.F.
- Venue
- Fortschritte der Physik
- Publisher
- Wiley
- Published
- 2004-05-13
- Volume
- 52
- Issue
- 6-7
- Pages
- 642-649
- ISSN
- 0015-8208 (ISSN-L)
- 1521-3978
- OpenAlex
- W2763881955
- Semantic Scholar
- b57e9d39e970271e61f5901af20912f6ef5da6ae
- Cited by
- 49
- References
- 16
- Citations by year
- 2023: 1 · 2022: 2 · 2021: 4 · 2020: 1 · 2019: 5 · 2018: 3 · 2017: 2 · 2016: 2 · 2015: 2 · 2013: 2 · 2012: 1
- FWCI
- 3.11
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Fortschritte der Physik (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion, cc-by-nc-sa)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- eternal black holes, unitarity, correlation functions, semiclassical gravity, large-N approximation, master field, Poincaré recurrence, AdS/CFT correspondence
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-06.
Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-06.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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