The trouble with de Sitter space
Open accessAbstract
In this paper we assume the de Sitter Space version of Black Hole Complementarity which states that a single causal patch of de Sitter space is described as an isolated finite temperature cavity bounded by a horizon which allows no loss of information. We discuss the how the symmetries of de Sitter space should be implemented. Then we prove a no go theorem for implementing the symmetries if the entropy is finite. Thus we must either give up the finiteness of the de Sitter entropy or the exact symmetry of the classical space. Each has interesting implications for the very long time behavior. We argue that the lifetime of a de Sitter phase can not exceed the Poincare recurrence time. This is supported by recent results of Kachru, Kallosh, Linde and Trivedi.
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- Premisecitation
The Gibbons-Hawking formula relates de Sitter entropy to horizon area as S = Horizon Area / 4G.
“The value of the entropy is connected to the size of the de Sitter space by the Gibbons Hawking [19] formula for de Sitter entropy: S = Horizon Area 4G .”
Quote in the paperSubstantial claim1 of 1 sources checked- cites [19] M.K. Parikh, 2003, Phys. Rev. source
- Findingreasoning
H cannot have a countable spectrum.
“This proves that H can not have a countable spectrum.and therefore that the E i cannot be discrete and the entropy cannot be finite.”
Quote in the paperSubstantial claim - Findingreasoning
The rate of thermal tunneling is always larger than the rate of Poincaré recurrences.
“Since the recurrence time is exp S 0 it is clear that the rate of thermal tunelling is always bigger than the rate of Poincaré recurrences.”
Quote in the paperSubstantial claim - Findingreasoning
Finite entropy is incompatible with the existence of the symmetry generators H, Ji, and Kj together with the requirement that H be Hermitian.
“Then we prove a no go theorem for implementing the symmetries if the entropy is finite.”
- Also stated in: § JHEP07(2003)056 · plain
“The finiteness of the entropy is incompatible with the existence of the symmetry generators (H, J i , K j ), and the requirement that H be a hermetian operator.”
- Also stated in: § 4. Conclusions · hedged
“Finiteness of entropy appears to be incompatible with de Sitter symmetry.”
Quote in the paperSubstantial claim - Also stated in: § JHEP07(2003)056 · plain
- Interpretationreasoning
Because the lifetime is shorter than the recurrence time, each energy level must develop a width gamma at least of order exp(-S).
“Since the lifetime is always shorter than the recurrence time, each energy level must develop a width γ at least of order exp -S.”
Quote in the paperSubstantial claim
- Premisenone given
The Minkowski vacuum coincides with the thermofield double state.
“However, once it is realized that the Minkowski vacuum coincides with the thermofield double |ψ , it becomes clear that the action of the translations can be represented in the product Hilbert space H tf = H 1 ⊗ H 2 .”
Quote in the paperSubstantial claim - Premisereasoning
The Rindler quadrant can be described by the Unruh thermal state with temperature 1/(2 pi).
“The Rindler quadrant may be described by the Unruh thermal state with temperature T rind = 1 2π = 1 β rind .”
Quote in the paperNot a substantial claim - Interpretationreasoning
In string theory, a metastable de Sitter phase cannot live for a recurrence time.
“Thus is appears that in string theory, a metastable de Sitter phase can not live for a recurrence time.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [27] L.V. Keldysh, 1964, Sov. Phys. JETP
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationreasoning
If the entropy is finite, then the spectrum of H1 minus H2 must be countable.
“This condition implies that the spectrum of (H 1 -H 2 ) must be countable.”
Quote in the paperSubstantial claim - Interpretationnone given
If spacetime has three or more non-compact directions, the horizon is infinite and the total entropy diverges.
“Obviously if the number of non-compact directions of spacetime is 3 or more, the horizon is infinite and the total entropy does diverge.”
Quote in the paperSubstantial claim - Interpretationcitation
The lifetime of a de Sitter phase cannot exceed the Poincaré recurrence time.
“We argue that the lifetime of a de Sitter phase can not exceed the Poincaré recurrence time.This is supported by recent results of Kachru, Kallosh, Linde and Trivedi.”
- Also stated in: § 5. Note added in revised version · plain
“We have argued on general grounds that de Sitter behavior can not persist for times of order the Poincaré recurrence time.”
Quote in the paperSubstantial claim0 of 2 sources checked- cites Kachru, Kallosh, Linde and Trivedi
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites [27] L.V. Keldysh, 1964, Sov. Phys. JETP
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Also stated in: § 5. Note added in revised version · plain
- Premisecitation
Kachru, Kallosh, Linde and Trivedi computed the tunneling rate over a barrier and found it always less than the recurrence time.
“Kachru, Kallosh, Linde and Trevedi compute the rate for tunelling over such a barrier and find that it is always less than the recurrence time.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [27] L.V. Keldysh, 1964, Sov. Phys. JETP
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationreasoning
A discrete energy spectrum introduces a new time scale of order 1/delta, where delta is the typical level spacing.
“The discreteness of the energy spectrum introduces a new time scale of order 1/δ, where δ is of order the typical spacing of levels.”
Quote in the paperSubstantial claim - Interpretationcitation
In typical situations of the type studied in [27], the width of each level is much greater than the level spacing.
“In a typical situation of the type studied in [27], the width of each level is vastly greater than the level spacing, which is reminiscent of a Schwarzschild black hole in flat space.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [27] L.V. Keldysh, 1964, Sov. Phys. JETP
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Premisecitation
In string theory, compactification to 1+1 dimensions cannot produce a theory with translation invariance.
“However, at least in the case of string theory it is known that compactification to 1 + 1 dimensions can not yield a theory with translation invariance [23].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [23] P. Martin, 1959, Phys. Rev. source
- Interpretationreasoning
Finiteness of entropy does not imply that the Hilbert space of states is finite dimensional; entropy can be finite even when the Hilbert space is infinite dimensional.
“This may be so, but it does not follow from the finiteness of the entropy.The Entropy is only equal to the dimensionality of the space of states when the temperature is infinite.Entropy can certainly be finite even though the Hilbert space of states is infinite dimensional.”
Quote in the paperSubstantial claim - Premisenone given
The de Sitter horizon is locally identical to the horizon of a Schwarzschild black hole.
“The horizon of de Sitter space is locally identical to that of a Schwarzschild black hole.”
Quote in the paperNot a substantial claim - Interpretationreasoning
If the thermal entropy of one causal patch is finite, then the eigenvalues of H1 must be discrete, with only finitely many below any specified energy.
“The consequences of that for the spectrum of H 1 are very clear: the eigenvalues E i must be discrete.More precisely, the number of eigenvalues with energy less than any specified value must be finite.”
Quote in the paperSubstantial claim - Interpretationreasoning
If de Sitter entropy is finite, then the symmetry among different causal patches must be broken, with Hamiltonians and energy spectra differing at least slightly.
“If the entropy is finite then the symmetry of different causal patches must be broken, the hamiltonians and energy spectra differing at least slightly.”
Quote in the paperSubstantial claim
- May combine two claims. 6
- Classified as probably not a substantial claim. 1
May combine two claims.
PremisecitationFor a system with entropy S, the level spacing is of order exp(-S), and the Poincaré recurrence time is of order exp(S).
“For a system with entropy S the level spacing is of order exp (-S) and the time scale is the Poincaré recurrence [15,16,17] time t p ∼ exp S.”
Quote in the paperSubstantial claim2 of 3 sources checked- cites [15] R. Bousso, 2000, J. High Energy Phys. source
The N-bound discussion (section 5, eq. 5.3) bounds observable entropy, but does not relate entropy to energy-level spacing or a Poincaré recurrence time. Searches of our copy for “Poincaré recurrence,” “recurrence,” and “level spacing” found no passage addressing either relation. (LLM, conf=0.72)
Experimental — accuracy not yet measured - cites [16] L. Dyson, 2002, J. High Energy Phys. source
- cites [17] L. Susskind
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
May combine two claims.
FindingreasoningApplying the same argument to Rindler space proves that the Rindler Hamiltonian has an all-real spectrum, so Rindler entropy must be infinite.
“The same argument can be applied to Rindler space.Using 1.14 and replacing the operator L by P 1 + P 0 the spectrum H rind is also proved to be all real numbers.Thus the entropy of Rindler space must be infinite.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningViolations of de Sitter symmetry should become important at times of order the Poincaré recurrence time, and recurrences should not respect the symmetry.
“Thus the violations of de Sitter symmetry should become important for JHEP07(2003)056 times of order t p and recurrences should not respect the symmetry.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningFor geometries with horizons, the complete symmetry acts on the thermofield double Hilbert space rather than on the Hilbert space accessible to an observer on one side, and the symmetry operations mix the degrees of freedom of the two copies.
“The lesson of this section is that the full symmetry of geometries with horizons acts on the thermofield double Hilbert space, and not on the space of available to an observer one one side of the horizon.Moreover the symmetry transformations mix the degrees of freedom of the two copies.”
Quote in the paperSubstantial claimMay combine two claims.
PremisereasoningThe de Sitter group belongs to the coordinate-transformation gauge group, so physical states should be invariant under its generators in the thermofield double.
“The de Sitter group is part of the group of coordinate transformations, which in general relativity is the gauge group.For this reason physical states should be invariant under the action of any of its generators.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningEither de Sitter entropy is not finite, or the exact symmetry of classical de Sitter space is not exact.
“Thus we must either give up the finiteness of de Sitter spaceentropy or the exact symmetry of the classical space.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe eternal AdS black hole should be described by two copies of the boundary conformal field theory, which are the thermofield double.
“Maldacena [20] has noted that the eternal black hole geometry has two boundaries, one in quadrant I of the Penrose diagram and one in quadrant III.Maldacena argues that the eternal black hole should be described by two copies of the usual boundary conformal field theory, and that the two copies are nothing but the thermofield double.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [20] G.W. Gibbons, 1977, Phys. Rev. source
The authors state this as an assumption.
The de Sitter version of black hole complementarity holds: a single causal patch of de Sitter space is an isolated finite-temperature cavity bounded by a horizon that allows no information loss.
“In this paper we assume the de Sitter space version of black hole Complementarity which states that a single causal patch of de Sitter space is described as an isolated finite temperature cavity bounded by a horizon which allows no loss of information.”
The authors state this as an assumption.
The thermal entropy of one causal patch is finite.
“There is one more condition that we must satisfy in order to have a quantum dual of de Sitter space; the thermal entropy of one causal patch is finite.”
“The only thing we will assume is that the entropy is finite.”
The authors state this as an assumption.
A holographic dual description of the de Sitter causal patch exists.
“Unlike the AdS case we do not know the details of the holographic dual that describes the causal patch, but we will assume that such a description exists.”
The authors state this as an assumption.
The spectrum of H1 is discrete.
“We are assuming only the weaker condition that the spectrum is 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?
The Euclidean approach to gravitational thermodynamics has been shown to fit within the AdS/CFT correspondence.
“Even at a nonperturbative level, the euclidean approach to gravitational thermodynamics has been shown to integrate nicely within the AdS/CFT correspondence [7].”
From Very long time scales and black hole thermal equilibrium (2003) · cites this paper as [7]
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.
Thermofield correlator omits its normalization
MinorThe state in equation 1.4 contains a factor of 1/√Z, so its expectation value must contain 1/Z. The expression in equation 1.8 has no such factor.
“ψ|A 1 (0)B 2 (t)|ψ = ij e -( β 2 E i + β 2 E j ) e i(E i -E j )t A ij B † ji . (1.8)”
“|ψ = 1 Z 1 2 i e -1 2 βE i |E i , E i ,(1.4)”
The cited KKLT reference has a different number in the list
MinorThe text identifies Kachru, Kallosh, Linde and Trivedi as reference [27], but the numbered list assigns their paper to 33 and assigns 27 to Keldysh.
“work by Kachru, Kallosh, Linde and Trivedi [27]”
“33. S. Kachru, R. Kallosh, A. Linde and S.P. Trivedi, De Sitter vacua in string theory, hep-th/0301240.”
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TASI/PITP/ISS LECTURES ON MODULI AND MICROPHYSICS
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- Authors
- Goheer, N., Kleban, M., & Susskind, L.
- Affiliations
- University of Cape Town (ZA)
- Stanford University (US)
- Korea Institute for Advanced Study (KR)
- Venue
- Journal of High Energy Physics
- Published
- 2003-07-24
- Volume
- 2003
- Issue
- 07
- Pages
- 056-056
- ISSN
- 1029-8479 (ISSN-L)
- OpenAlex
- W2027376174
- Cited by
- 281
- References
- 36
- Citations by year
- 2026: 6 · 2025: 9 · 2024: 15 · 2023: 21 · 2022: 13 · 2021: 15 · 2020: 7 · 2019: 12 · 2018: 7 · 2017: 5 · 2016: 11 · 2015: 7 · 2014: 14 · 2013: 3 · 2012: 8
- FWCI
- 13.96
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Journal of High Energy Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Keywords
- de Sitter space, black hole complementarity, causal patch, no-go theorem, de Sitter entropy, Poincaré recurrence, information retention
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