A covariant entropy conjecture

In the paper

“Because of its symmetric treatment of the four light-like directions orthogonal to any surface, which marks its only significant difference from the Fischler-Susskind bound [9], our covariant prescription applies also to closed and recollapsing universes (sec.5.2).”

What the sources say

• Supported

  The citation identifies the Fischler–Susskind bound, not the citing authors’ claim about their own covariant prescription. In “Flat Universe” (p. 3), the source proposes an entropy bound using past-directed light rays; in “Non Flat Universe” (p. 6), it discusses difficulties for that bound in closed universes. The statement that the citing authors’ prescription applies to closed and recollapsing universes is their own conclusion, not a result attributed to this source. (LLM, conf=0.88)

  cites [9] 9. W. Fischler and L. Susskind: Holography and cosmology, hep-th/9806039. JHEP07(1999)004 source

In the paper

“But the possibility of using any spatial hypersurface is already excluded by the counterexamples given at the end of sec.1.1; this was first pointed out in ref. [9].”

What the sources say

• Supported

  In §2 (p. 2), Fischler and Susskind reject the proposal that the entropy in any spatial region is bounded by its boundary area: sufficiently large regions violate it. They instead formulate the bound using entropy passing through a light-like surface from the boundary (§2, p. 3). (LLM, conf=0.90)

  cites [9] 9. W. Fischler and L. Susskind: Holography and cosmology, hep-th/9806039. JHEP07(1999)004 source

In the paper

“Or consider a homogeneous spacelike hypersurface in a flat Friedmann-Robertson-Walker universe.The entropy of a sufficiently large spherical volume will exceed the boundary area [9].”

What the sources say

• Supported

  In “2. Flat Universe” (p. 2), the authors assume a flat Robertson–Walker cosmology with homogeneous, constant comoving entropy density. They state that entropy grows with a region’s volume while its boundary area grows more slowly, so for a sufficiently large region the entropy exceeds the area. (LLM, conf=0.94)

  cites [9] 9. W. Fischler and L. Susskind: Holography and cosmology, hep-th/9806039. JHEP07(1999)004 source

In the paper

“As an example, we apply a corollary derived in sec.5.1 to understand why the bound of ref. [14] cannot be applied to a flat universe with negative cosmological constant [15].”

What the sources say

• ?Could not verify

  The section “Cosmic Holography Tested Against FRW Unverse” discusses bounds for flat universes with specified equations of state and difficulties in closed or inflationary universes, but does not address a flat universe with a negative cosmological constant. Searches for that condition and for a negative cosmological constant yielded no passage establishing that the bound cannot be applied to it. (LLM, conf=0.74)

  cites [14] 14. D. Bak and S.-J. Rey: Cosmic holography, hep-th/9902173. sourceExperimental — accuracy not yet measured
• ?Not checked — no full text of the sourcecites [15] 15. N. Kaloper and A. Linde: Cosmology vs. holography, hep-th/9904120. source

In the paper

“In open universes the bound will be satisfied more comfortably than in flat ones [9].”

What the sources say

• Supported

  Section 3 (p. 6) says the holographic bound places no strong restriction on the equation of state in open universes at late times, and no lower limit on how slowly they may expand. Section 2 (p. 3) says the bound imposes a lower limit on the expansion rate in flat universes. This supports the comparison, though the paper notes that open universes behave like flat ones at early times. (LLM, conf=0.89)

  cites [9] 9. W. Fischler and L. Susskind: Holography and cosmology, hep-th/9806039. JHEP07(1999)004 source

In the paper

“Other proposals for cosmological entropy bounds [12,13,14,15,16] are based on the idea of defining a limited spatial region to which Bekenstein's bound can be JHEP07(1999)004 applied.”

What the sources say

• Supported

  The Introduction (p. 2) links the holographic principle to Bekenstein’s entropy bounds within a given volume. “The Story So Far” (p. 3) explains a cosmological proposal that limits the bound to regions smaller than the particle horizon, and “Flat Universe” (p. 6) discusses defining the region by the Hubble length. (LLM, conf=0.88)

  cites [12] 12. R. Easther and D.A. Lowe: Holography, cosmology and the second law of thermody- namics, hep-th/9902088. source
• Partly supported

  The discussion on p. 2 defines a cosmological entropy bound using limited Hubble-size regions, supporting that part of the claim. But it derives the bound using the Bekenstein–Hawking black-hole entropy formula and distinguishes it from Bekenstein’s general bound; it does not describe applying Bekenstein’s bound to each region. (LLM, conf=0.85)

  cites [13] 13. G. Veneziano: Pre-bangian origin of our entropy and time arrow, Phys. Lett. B 454 (1999) 22 [hep-th/9902126]. sourceExperimental — accuracy not yet measured
• Supported

  In “Cosmological Apparent Horizon and Gravitational Entropy” (page not specified), the paper proposes the apparent horizon as the boundary of a limited region and bounds the particle entropy inside it by the horizon’s gravitational entropy. This supports citing it as an example of a cosmological entropy-bound proposal based on a defined spatial region. (LLM, conf=0.88)

  cites [14] 14. D. Bak and S.-J. Rey: Cosmic holography, hep-th/9902173. source
• ?Not checked — no full text of the sourcecites [15] 15. N. Kaloper and A. Linde: Cosmology vs. holography, hep-th/9904120. source
• ?Not checked — no full text of the sourcecites [16] 16. R. Brustein: The generalized second law of thermodynamics in cosmology, gr-qc/9904061. source

In the paper

“The entropy in the spatial interior of the apparent horizon [14] will not be bounded by its area in this region.”

What the sources say

• Partly supported

  In “Cosmic Holography Tested Against FRW Unverse” (page unavailable), the paper reports that the entropy–area bound can be violated in closed-universe models. But it also finds the bound satisfied in open universes and some flat-universe cases. The unqualified claim that entropy inside the apparent horizon will not be bounded by its area overstates those findings. (LLM, conf=0.86)

  cites [14] 14. D. Bak and S.-J. Rey: Cosmic holography, hep-th/9902173. sourceExperimental — accuracy not yet measured

In the paper

“This leaves no choice but to conclude that the number of degrees of freedom in nature is fundamentally limited, as proposed in refs.[10,11].”

What the sources say

• ?Not checked — no full text of the sourcecites [10] 10. G. 't Hooft: Dimensional reduction in quantum gravity, gr-qc/9310026. source
• Supported

  In “Counting Degrees of Freedom” (pp. 2–3), Susskind argues that a full description of nature requires fewer degrees of freedom than a three-dimensional Planck-scale lattice would imply. He describes a bound corresponding to a two-dimensional lattice at a region’s boundary. This supports the citing sentence’s attribution of the proposal, without treating it as an established fact. (LLM, conf=0.93)

  cites [11] 11. L. Susskind: The world as a hologram. J. Math. Phys. 36 (1995) 6377 [hep-th/9409089]. source

In the paper

“Using Friedmann's equation, H 2 = 8πρ 3 - k a 2 ,(4.6) one finds r AH = 3 8πρ , (4.7) where ρ is the energy density of matter.”

In the paper

“This is exactly a quarter of the area of the apparent horizon.”

What the sources say

• ?Not checked — no full text of the sourcecites [20] 20. C.W. Misner, K.S. Thorne and J.A. Wheeler eds.: Gravitation. Freeman, New York 1973. source

In the paper

“Thus the shell entropy cannot exceed πr 2 0 , which by eq. ( 6.1) is a quarter of the area bounding the light-sheet: S shell ≤ A 4 . (6.9)”

In the paper

“Therefore the shell has a maximum entropy of S shell = 2πwM .(6.8)”

In the paper

“What we must ensure, however, is that L actually penetrates the entire shell and samples all of its entropy.We will now show that this requirement keeps the shell entropy within the conjectured bound.”

In the paper

“We conclude that A/4 is the maximum amount of entropy one can transport through a future-directed outgoing light-sheet L bounded by a black hole apparent horizon of area A, using an exactly spherically symmetric shell of matter.”

In the paper

“Therefore we must use null hypersurfaces bounded by B.”

In the paper

“It thus places a fundamental limit on the number of degrees of freedom in nature.”

• Also stated in: § 1.2 Outline · plain

  “Thus the covariant bound implies a fundamental limit on the total number of independent degrees of freedom that are actually present in nature.”

In the paper

“For a system contained in a spherical volume, gravitational stability requires that M ≤ R/2.Thus eq.(1.1) implies S ≤ A 4 .”

In the paper

“If the expansion takes the negative value θ 0 at any point on a geodesic in the congruence, eq.(3.5) implies that the expansion will become negative infinite, θ → -∞, along that geodesic within affine time ∆λ ≤ 2/|θ 0 | [19].”

What the sources say

• ?Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  cites [19] 19. R.M. Wald: General Relativity. The University of Chicago Press, Chicago 1984.

In the paper

“It follows that θ cannot increase along any geodesic.”

In the paper

“This eliminates any hope of obtaining a completely general bound involving mass.”

In the paper

“By requiring the consistency of Bekenstein's conditions, we show that the apparent horizon is indeed the largest surface whose interior one can hope to treat as as a Bekenstein system.”

In the paper

“Our conclusion will be that the expansion becomes positive only at caustics.”

In the paper

“For systems with limited self-gravity it reduces to Bekenstein's bound.”

• Also stated in: § 1.2 Outline · plain

  “Through this theorem, we recover Bekenstein's bound as a special case.”

In the paper

“The conjecture is manifestly time reversal invariant.We regard this as its most significant property.”

• Also stated in: § Additional information · plain

  “In the final paragraphs of secs.3.1 and 3.4 we discussed the most important property of the conjecture: it is manifestly time reversal invariant.”

In the paper

“The past-and future-directed ingoing families are allowed.Both encounter caustics when they reach the center of the sphere.Thus they each sweep the interior of the sphere exactly once.”

In the paper

“The dominant energy condition, which we are assuming, implies that the null convergence condition will hold.”

In the paper

“In order to create the maximum amount of entropy, however, it is best to put the matter into a few large saturated Bekenstein systems, rather than many small ones.”

In the paper

“It follows that the apparent horizon is the largest sphere whose interior may be treated as a Bekenstein system.”

In the paper

“For a spherical Bekenstein system of a given radius and mass, the entropy/mass bound is always at least as tight as the entropy/area bound.”

In the paper

“If outgoing light-sheets existed even as χ → 0, the area would become arbitrarily small while the entropy remained finite.”

In the paper

“Thus the calculation holds independently of the equation of state.”

In the paper

“The size and total entropy of the reheating hypersurface V is thus irrelevant.”

In the paper

“The considerations above imply that this system will also saturate the entropy/mass bound.”

In the paper

“This is one of the reasons why the Fischler-Susskind proposal does not apply to closed universes (see sec.5.2).”

In the paper

“Therefore the covariant entropy bound will be satisfied on it if the evolution is adiabatic.”

In the paper

“For larger systems, Bekenstein's assumptions would not be self-consistent.”

In the paper

“Spacelike projection theorem.Let A be the area of a closed surface B possessing a future-directed light-sheet L with no boundary other than B. Let the spatial region V be contained in the intersection of the causal past of L with any spacelike hypersurface containing B. Let S be the entropy on V .Then S ≤ A/4.”

In the paper

“The truncation has the desirable effect that the volume swept by the light-sheets grows not like A 3/2 , but roughly like the area.”

In the paper

“Since the bound is satisfied at the Planck time, it will remain satisfied later.”

In the paper

“The above considerations have shown, however, that light-sheets percolate rather slowly through highly enthropic systems, because the geodesics follow a kind of random walk.”

In the paper

“the ratio of entropy to area behaves as [15] S A ≤ σt γ-1 γ+1 (4.8) for any past-directed light-sheet of areas specified at a later time t.”

What the sources say

• ?Not checked — no full text of the sourcecites [15] 15. N. Kaloper and A. Linde: Cosmology vs. holography, hep-th/9904120. source

In the paper

“Cosmological corollary.Let V be a spatial region within the apparent horizon of an observer.If the future-directed ingoing light-sheet L of the apparent horizon has no other boundaries, and if V is entirely contained in the causal past of L, then the entropy on V cannot exceed a quarter of the area of the apparent horizon.”

In the paper

“We address this problem in some generality in sec.6, where we argue that the conjecture holds even in such regions.”

In the paper

“The property r = 2M(r) is special to the apparent horizon.”

In the paper

“Because inflation cannot produce more than one unit of entropy per Planck volume, the bound will be satisfied by the same arguments that were given above for universes with a big bang.”

In the paper

“In general, however, the light-sheets prescribed by the covariant entropy bound provide the only consistent way of comparing entropy and area.”

In the paper

“It is this property which forces us to conclude that the origin of the covariant bound is not thermodynamic, but statistical (sec.7).”

In the paper

“Yet, it holds independently of the microscopic properties of matter.”

In the paper

“We pointed out in sec.3.1 that the covariant entropy bound is T-invariant.”

In the paper

“Both before and after reheating, all spheres will be anti-trapped except in a small neighbourhood of r = 0 (or small neighbourhoods of the poles of the S 3 spacelike slice in a closed universe), of the size of the apparent horizon.”

In the paper

“If the region V is contained in the causal past of the light-sheet L, the dominant energy condition implies that all matter in the region V must eventually pass through the light-sheet L.”

In the paper

“Then the second law of thermodynamics implies that the entropy on V , S V , cannot exceed the entropy on L, S L .”

In the paper

“In semi-classical gravity, a black hole, viewed from the outside, is an example of a saturated Bekenstein system”

In the paper

“While null hypersurfaces are certainly required in general, it turns out that there is a wide class of situations in which an entropy bound on spacelike hypersurfaces can be inferred from the covariant entropy conjecture.”

In the paper

“If we find a spherical thermodynamic system which saturates the entropy/area bound but does not saturate the entropy/mass bound, we must conclude that it was not a Bekenstein system in the first place.”

In the paper

“Therefore it is in fact quite plausible, if counter-intuitive, that the covariant entropy bound holds even during the gravitational collapse of a system initially saturating Bekenstein's bound.”

In the paper

“Therefore the conditions of the theorem are satisfied, and the entropy of the system must be less than A/4.We have recovered Bekenstein's bound.”

In the paper

“In particular, it can be used to show that area bounds entropy on spacelike sections of anti-de Sitter space.”

In the paper

“It follows that the entropy of the spatial region V cannot exceed a quarter of its boundary area: S V ≤ A/4.”

In the paper

“Since they end on the black hole singularity within finite affine time, they sample a smaller portion of a highly enthropic system than they would for a more regular system.”

In the paper

“On the other hand, the physical mechanisms responsible for its validity do not appear to be T-invariant, and of course the very concept of thermodynamic entropy has a built-in arrow of time.”

In the paper

“While we have formally JHEP07(1999)004 assumed the positivity of energy as a condition for the covariant entropy bound, we have argued in sec. 2 that the validity of the bound may well extend to space-times with a negative cosmological constant.”

In the paper

“Therefore the total entropy of the universe will not be larger than S max ∼ a(η Pl ) 3 ∼ a max .”

In the paper

“In order to introduce a sizable amount of entropy, we have to violate the conditions under which the solution is valid: homogeneity and zero temperature.”

In the paper

“Any surfaces met by the particle horizon in this region (such as B 1 ) possess a past-directed light-sheet that coincides with the particle horizon.”

In the paper

“Thus the entropy within the cosmological horizon cannot exceed 3π/Λ.”

In the paper

“The Tinvariance of the covariant entropy conjecture can be understood only if the bound is interpreted as a bound on the number of degrees of freedom of the matter systems present on the light-sheets.”

In the paper

“According to the covariant bound, the entropy contained on it has nothing to do with the surface that bounds it.”

In the paper

“The generalized second law of thermodynamics guarantees that the entropy conjecture will hold on this light-sheet, since A 4 = S bh ≥ S in .(6.2)”

In the paper

“By choosing B 4 to be very close to the future singularity, its area can be made arbitrarily small.”

In the paper

“But the cosmological corollary (sec.5.1) states that one can use the interior of the apparent horizon for an entropy/area bound only if the future-directed ingoing lightsheet is complete.”

In the paper

“Thus the particle horizon will initially be outside the apparent horizon, in an anti-trapped region.”

In the paper

“The limitations of the FS proposal can be understood in terms of this selection rule.”

In the paper

“This evidence suggests that the covariant entropy conjecture may be a universal law of physics.”

In the paper

“When the boundary of the system is a suffiently small proper time away from the future singularity, the light-sheet will not intersect the whole system, because it will be truncated by the singularity (or a surface where the Planck density is reached), much like the past light-sheet of spheres larger than the particle horizon in a flat FRW universe.”

In the paper

“As we pointed out in sec.1.1, Bekenstein's bound cannot be applied to regions containing a negative energy component, such as a negative cosmological constant.”

In the paper

“A future-directed ingoing light-sheet, starting at the surface B at a sufficiently early time but inside the event horizon, can easily traverse the ball before the singularity (or the Planck density, at R ∼ M 1/3 ) is reached.”

In the paper

“By demanding general coordinate invariance and constructing a selection rule, we arrived at a bound on the entropy present on null hypersurfaces in arbitrary space-times.”

In the paper

“Therefore the second law of thermodynamics, which underlies Bekenstein's bound, cannot be responsible for the covariant bound, and one is forced to contemplate the possibility of a different origin.”

In the paper

“By preparing the shell far outside the black hole and letting it collapse, we can thus transport an arbitrary amount of entropy to r 0 .”

In the paper

“The number of independent degrees of freedom on any light-sheet of a surface B cannot exceed a quarter of the area of B.”

In the paper

“Its generality, together with its Tinvariance, force us to the conclusion that a holographic principle underlies the description of nature.”