A covariant entropy conjecture

Probably not a substantial claim: may not pass S3 (load-bearing).

Other prescriptions can be used for sufficiently small surfaces in a wide class of cosmological solutions.

“while other prescriptions [12,13,14,15,16] can be used for sufficiently small surfaces in a wide class of cosmological solutions.”

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

  The Conclusions (p. 11) propose a modified holographic bound for regions smaller than the Hubble volume, supporting the point about sufficiently small regions. But the proposal is specifically for the post-inflationary universe; the same section says a useful holographic bound cannot be deduced for closed universes. The claim’s ‘wide class of cosmological solutions’ is therefore broader than this paper establishes. (LLM, conf=0.85)

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

  The discussion on p. 2 proposes a Hubble entropy bound for fairly homogeneous cosmological situations, supporting the existence of an alternative prescription. It does not say that this prescription can be used for sufficiently small surfaces across a wide class of cosmological solutions. (LLM, conf=0.82)

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

  In “Cosmic Holography Tested Against FRW Unverse” (page unavailable), the paper discusses an apparent-horizon prescription and reports compatibility with open universes and restricted classes of flat universes. It does not establish that other prescriptions work for sufficiently small surfaces across a wide class of cosmological solutions; it also reports failures for closed universes and typical post-big-bang inflationary models. (LLM, conf=0.82)

  Experimental — accuracy not yet measured
• cites [15] 15. N. Kaloper and A. Linde: Cosmology vs. holography, hep-th/9904120. sourceNot checked — no full text of the source
• cites [16] 16. R. Brustein: The generalized second law of thermodynamics in cosmology, gr-qc/9904061. sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S3 (load-bearing).

The Fischler-Susskind bound applies to universes which are not closed or recollapsing.

“The Fischler-Susskind bound [9], for example, applies to universes which are not closed or recollapsing”

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

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The validity of the bound was checked for flat universes with 0 ≤ γ ≤ 1 in ref. [9] and for -1 < γ < 0 in ref. [15].

“The validity of the bound was checked for flat universes with 0 ≤ γ ≤ 1 in ref. [9] and for -1 < γ < 0 in ref. [15].”

• cites [9] 9. W. Fischler and L. Susskind: Holography and cosmology, hep-th/9806039. JHEP07(1999)004 sourceSupported
• cites [15] 15. N. Kaloper and A. Linde: Cosmology vs. holography, hep-th/9904120. sourceNot checked — no full text of the source

May combine two claims.

The FS bound cannot be applied in the normal region and would be violated.

“The FS bound cannot be applied here.It would continue to compare entropy and area of the particle horizon, and would be violated [9].”

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

  In “Non Flat Universe” (p. 6), Fischler and Susskind compare entropy with horizon area and say the holographic bound in a positively curved closed universe will be saturated and then threatened with violation. They do not say that the FS bound cannot be applied in a “normal region”; instead, they apply it to analyze that universe. Their discussion also leaves open whether new behavior prevents a violation or such universes are inconsistent. (LLM, conf=0.85)

  Experimental — accuracy not yet measured

May combine two claims.

The covariant entropy bound passes cosmological tests, and just passes them.

“We will see that it passes the test, and more significantly, it just passes it.”

May combine two claims.

In an attempt to violate the covariant entropy conjecture by collapsing a massive shell around a black hole, one cannot squeeze more entropy through L than a quarter of the area of B; otherwise L reaches the singularity before passing through the infalling shell entirely.

“In an attempt to violate the covariant entropy conjecture, we collapse a massive shell around the black hole.We find, however, that one cannot squeeze more entropy through L than a quarter of the area of B. Otherwise, L reaches the singularity before passing through the infalling shell entirely.”

May combine two claims.

The conjecture was tested on cosmological solutions and inside gravitationally collapsing regions, and under the most adverse assumptions the bound can be saturated but not exceeded.

“We tested the conjecture on cosmological solutions and inside gravitationally collapsing regions.We found, under the most adverse assumptions, that the bound can be saturated but not exceeded.”

May combine two claims.

The maximum width of a shell of mass M that can be completely penetrated by L before the light-sheet hits the singularity is w_max ≈ r0^2/(2M), which is inversely proportional to its mass and always less than r0.

“This requires w/2 ≤ ∆τ dead , whence w max ≈ r 2 0 /2 M .(6.6) The maximum width of the shell is thus inversely proportional to its mass, and is always less than r 0 .”

• AssumptionRests on an assumption: The dominant energy condition holds for matter crossing L.

May combine two claims.

In sec. 4, the covariant entropy conjecture was tested for anti-trapped and normal surfaces in cosmological space-times, and even the most non-adiabatic processes can only saturate, but not violate, the bound.

“In sec. 4 we tested the covariant entropy conjecture for anti-trapped and normal surfaces in cosmological space-times.We found that even the most non-adiabatic processes can only saturate, but not violate, the bound.”

May combine two claims.

At η = χ = 2π/3, the particle horizon covers two-thirds of the total entropy of the universe, and its area is much larger.

“At this time, the particle horizon covers two-thirds of the total entropy of the universe, which will be of order a max .Its area, however, is 4π(3/4) 3 a 2 max , which is much larger.”

May combine two claims. The quote only partly states this claim.

At least two of the four null directions orthogonal to B will have non-positive expansion and be allowed.

“Therefore we can be sure that at least two of the four null directions will be allowed.”

May combine two claims.

All physically reasonable forms of matter are believed to satisfy the dominant energy condition, so requiring it is not a significant restriction.

“It is believed that all physically reasonable forms of matter satisfy the dominant energy condition [18,19], so we are not imposing a significant restriction.”

• cites [18] 18. S.W. Hawking and G.F.R. Ellis: The large scale stucture of space-time. Cambridge University Press, Cambridge 1973. sourceNot checked — no full text of the source
• cites [19] 19. R.M. Wald: General Relativity. The University of Chicago Press, Chicago 1984.Not matched

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

May combine two claims.

The origin of the covariant entropy bound cannot be thermodynamic, but must be statistical.

“Because the conjecture is manifestly time reversal invariant, its origin cannot be thermodynamic, but must be statistical.”

• Also stated in: § 1.2 Outline · plain

  “Therefore the covariant bound must be linked to the statistical origin of entropy.”

May combine two claims.

When Bekenstein's conditions are not satisfied, his bounds can easily be violated, for example by a system undergoing gravitational collapse whose surface area becomes arbitrarily small while entropy cannot decrease.

“When the conditions set forth by Bekenstein are not satisfied, the bounds can easily be violated.The simplest example is a system undergoing gravitational collapse.Before it is destroyed on the black hole singularity, its surface area becomes arbitrarily small.Since the entropy cannot decrease, the bound is violated.”

May combine two claims.

If the expansion of one pair of null congruences vanishes on B, there will be three allowed directions; if both pairs vanish, all four null directions will be allowed.

“If the expansion of one pair of null congruences vanishes on B, there will be three allowed directions.If the expansion of both pairs vanishes on B, all four null directions will be allowed.”

May combine two claims.

In a highly disordered collapsing ball, the light-rays percolate and consume more affine time, so the system will not be swept out completely before the singularity is reached.

“Therefore the second system will not be swept out completely before the singularity is reached (see fig. 3).”

May combine two claims.

At a caustic, the expansion jumps from -∞ to +∞, so the light-ray must be stopped; caustics form the endpoints of the light-sheet.

“Thus, the expansion θ jumps from -∞ to +∞ at a caustic.Then the expansion is positive, and we must stop following the light-ray.This is why caustics form the endpoints of the light-sheet.”

May combine two claims.

The covariant entropy bound can be saturated, but not exceeded, in cosmological solutions and in the interior of black holes.

“We present evidence that the bound can be saturated, but not exceeded, in cosmological solutions and in the interior of black holes.”

• Also stated in: § 1.2 Outline · plain

  “We find again that the bound can be saturated but not exceeded.”

Classified as probably not a substantial claim.

In the covariant generalization, the quantitative formula S ≤ A/4 remains unchanged.

“Having decided to search for an entropy/area bound, our difficulty lies not in the quantitative formula, S ≤ A/4; this will remain unchanged.”

May combine two claims.

Local energy is not well-defined in general relativity, and global definitions of mass require the space-time to possess an infinity.

“Local energy is not well-defined in general relativity, and for global definitions of mass the space-time must possess an infinity [19,20].”

• cites [19] 19. R.M. Wald: General Relativity. The University of Chicago Press, Chicago 1984.Not matched

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

• cites [20] 20. C.W. Misner, K.S. Thorne and J.A. Wheeler eds.: Gravitation. Freeman, New York 1973. sourceNot checked — no full text of the source

Classified as probably not a substantial claim.

Bekenstein proposed a universal bound on the entropy S of any thermodynamic system of total energy M: S ≤ 2πRM.

“Bekenstein [1] has proposed the existence of a universal bound on the entropy S of any thermodynamic system of total energy M: S ≤ 2πRM .”

• cites [1] 1. J.D. Bekenstein: A universal upper bound on the entropy to energy ratio for bounded systems. Phys. Rev. D 23 (1981) 287.Not matched

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

The quote only partly states this claim.

In any four-dimensional space-time satisfying Einstein's equation with the dominant energy condition, for any connected two-dimensional spatial surface B of area A, and any hypersurface L bounded by B and generated by a null congruence orthogonal to B with non-positive expansion away from B, the total entropy S on L satisfies S ≤ A/4.

“We conjecture the following entropy bound to be valid in all space-times admitted by Einstein's equation: let A be the area of any two-dimensional surface.Let L be a hypersurface generated by surface-orthogonal null geodesics with non-positive expansion.Let S be the entropy on L. Then S ≤ A/4.”

• AssumptionThe authors state this as an assumption: The dominant energy condition holds for matter in the space-times considered.
• AssumptionThe authors state this as an assumption: The space-time is inextendible and contains no null or timelike (naked) singularities.

Probably not a substantial claim: may not pass S4 (based).

Area can always be covariantly defined as the proper area of a surface.

“Area, on the other hand, can always be covariantly defined as the proper area of a surface.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Thermodynamic entropy is never time reversal invariant, and neither is the generalized second law of thermodynamics.

“After all, thermodynamic entropy is never T-invariant, and neither is the generalized second law of thermodynamics, which underlies Bekenstein's bound.”

Probably not a substantial claim: may not pass S4 (based).

The future-directed light-sheet covers the same comoving volume as the past light-sheet.

“The future-directed light-sheet covers the same comoving volume as the past light-sheet.”

May combine two claims.

Some universes recollapse and necessarily contain trapped surfaces; trapped regions can occur in any case by gravitational collapse.

“Some universes, for example most closed universes, or a flat universe with negative cosmological constant [15], recollapse.Such universes necessarily contain trapped surfaces.But trapped regions can occur in any case by gravitational collapse.”

• cites [15] 15. N. Kaloper and A. Linde: Cosmology vs. holography, hep-th/9904120. sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Inflation ends on a spacelike hypersurface V at t = t_reheat, and at this time all the entropy in the universe is produced through reheating.

“Inflation ends on a spacelike hypersurface V at t = t reheat .At this time, all the entropy in the universe is produced through reheating.”

May combine two claims.

A backward light-sheet of area A specified at t = 2t_Pl will be truncated at t = t_Pl, sweep a volume of order A l_Pl, and the entropy bound will at most be saturated.

“A backward light-sheet of an area A specified at t = 2t Pl will be truncated at t = t Pl .It will sweep a volume of order Al Pl and the entropy bound will at most be saturated.”

• AssumptionRests on an assumption: At one Planck time after the singularity, there cannot have been more than one unit of entropy per Planck volume, up to a factor of order one.

May combine two claims.

For a Bekenstein system, the gravitational binding is not strong enough to form a black hole, so V is contained in the causal past of L.

“Since the gravitational binding of a Bekenstein system is not strong enough to form a black hole, V will be contained in the causal past of L.”

May combine two claims.

The interior of the apparent horizon is the largest system for which Bekenstein's conditions can be considered to hold; for larger systems, inconsistencies arise.

“We will show below, purely from the point of view of Bekenstein's bound, that the interior of the apparent horizon is in fact the largest system for which Bekenstein's conditions can be considered to hold; for larger systems, inconsistencies arise.”

Probably not a substantial claim: may not pass S4 (based).

The covariant entropy conjecture is a universal law valid for all space-times satisfying Einstein's equations with the dominant energy condition holding for matter.

“Since we claim that the conjecture is a universal law valid for all space-times satisfying Einstein's equations (with the dominant energy condition holding for matter), it must be valid for cosmological solutions in particular.”

• AssumptionThe authors state this as an assumption: The dominant energy condition holds for matter.

Probably not a substantial claim: may not pass S4 (based).

An anti-trapped surface B contains two past-directed light-sheets.

“An anti-trapped surface B contains two past-directed light-sheets.”

May combine two claims.

The covariant entropy bound may be saturated on the apparent horizon, but it will not be violated, no matter how hard we try to produce entropy.

“The covariant entropy bound may be saturated on the apparent horizon, but it will not be violated, no matter how hard we try to produce entropy.”

May combine two claims.

If the evolution is non-adiabatic, the entropy bound predicts S/A ≤ 1/4, implying a limit on how rapidly the universe can produce entropy.

“If the evolution is non-adiabatic, the entropy bound nevertheless predicts that S/A ≤ 1/4, implying that there is a limit on how rapidly the universe can produce entropy.”

May combine two claims.

The exponent in the adiabatic entropy/area ratio is non-positive, so the entropy/area ratio does not increase.

“The exponent is non-positive, so the entropy/area ratio does not increase.”

May combine two claims.

In FRW, near χ = 0 (and near the opposite pole for closed universes), spherical surfaces are normal; larger spheres beyond the apparent horizon are anti-trapped.

“In the vicinity of χ = 0 (and for closed universes, also on the opposite pole, near χ = π), the spherical surfaces will be normal.The larger spheres beyond the apparent horizon(s) will be anti-trapped.”

May combine two claims.

The normal region contains a past- and a future-directed light-sheet, both ingoing.

“The normal region contains a past-and a future-directed light-sheet.Both of them are ingoing, i.e. they are directed towards the center of the region at χ = 0 (or χ = π for the normal region near the opposite pole in closed universes).”

May combine two claims.

Therefore the spacelike projection theorem does not apply to any but the smallest surfaces B ⊂ V; the covariant conjecture does not relate area to enclosed entropy for other spheres.

“Therefore the spacelike projection theorem does not apply to any but the smallest of the surfaces B ⊂ V .The other spheres may be exponentially large, but the covariant conjecture does not relate their area to the enclosed entropy.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The most enthropic system is a saturated Bekenstein system.

“The most enthropic system is a saturated Bekenstein system, so they should convert all the matter into such systems.”

May combine two claims.

The spacelike projection theorem is not T-invariant; it refers to past and future explicitly.

“The spacelike projection theorem is not T-invariant; it refers to past and future explicitly.”

May combine two claims.

The covariant entropy conjecture singles out the apparent horizon as a special surface; it marks the largest surface to which the spacelike projection theorem can possibly apply, and the region inside it is the largest region one can hope to treat as a Bekenstein system.

“Thus the covariant entropy conjecture singles out the apparent horizon as a special surface.It marks the largest surface to which the spacelike projection theorem can possibly apply, and hence the region inside it is the largest region one can hope to treat as a Bekenstein system.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

Trapped surfaces occur not only in collapsing universes, but arise generally during gravitational collapse and inside black holes.

“We now turn to trapped surfaces, which occur not only in collapsing universes, but arise generally during gravitational collapse and inside black holes.”

Classified as probably not a substantial claim.

In a collapsing region, the direction in which surfaces are getting smaller is the future.

“This makes sense because in a collapsing region, loosely speaking, the direction in which surfaces are getting smaller is the future.”

May combine two claims.

After t1 < t_turnaround, the future-directed light-sheet of the apparent horizon is truncated by the future singularity, so the conditions for the spacelike projection theorem and its corollary are no longer met.

“After the time t 1 , where t 1 < t turnaround , the future-directed light-sheet of the apparent horizon will have another boundary, namely on the future singularity of the space-time.Thus the conditions for the spacelike projection theorem, and the JHEP07(1999)004 corollary it implies, are no longer met.”

May combine two claims.

Light-rays not ending on caustics follow an irregular path similar to random walk and may not proceed far into the object before the singularity is reached, so it may be impossible for a light-sheet to penetrate a collapsing, highly entropic system far enough to sample excessive entropy.

“Any light-rays that do not end on caustics will follow an irregular path through the object similar to random walk.Since they waste affine time on covering angular directions, they may not proceed far into the object before the singularity is reached.Thus it may well be impossible for a light-sheet to penetrate through a collapsing, highly entropic system far enough to sample excessive entropy.”

May combine two claims.

Inhomogeneities that break spherical symmetry deflect light-sheet rays and cause angular caustics, at which the light-sheet ends because expansion becomes positive.

“Inhomogeneities that break spherical symmetry will deflect the rays in the light-sheet and cause dents in their spherical cross-sections.Such dents will develop into “angular” caustics.At any caustic, the light-sheet ends, because the expansion becomes positive (sec.3.2).”

Probably not a substantial claim: may not pass S3 (load-bearing).

The Bak-Rey bound is exceeded in flat, adiabatic FRW universes with an arbitrarily small but non-vanishing negative cosmological constant.

“But Kaloper and Linde [15] have shown that this bound is exceeded in flat, adiabatic FRW universes with an arbitrarily small but non-vanishing negative cosmological constant.”

• cites [15] 15. N. Kaloper and A. Linde: Cosmology vs. holography, hep-th/9904120. sourceNot checked — no full text of the source

May combine two claims.

The FS hypersurface will often be one of the light-sheets selected by the covariant rule, but not always.

“The FS hypersurface will often be one of them, but not always.”

May combine two claims.

This inhomogeneity makes it more difficult for a light-sheet to penetrate the system completely, unless it originates near the beginning of the collapse when the area is still large.

“This makes it more difficult for a light-sheet to penetrate the system completely, unless it originates near the beginning of the collapse, when the area is still large.”

Probably not a substantial claim: may not pass S2 (checkable).

The corollary defines the range of validity of Bekenstein's entropy/area bound in cosmological solutions.

“Thus, the role of the corollary is to define the range of validity of Bekenstein's entropy/area bound in cosmological solutions.”

May combine two claims.

The Oppenheimer-Snyder solution describes a dust ball at zero temperature and exactly homogeneous, so it contains zero entropy.

“Strictly, the Oppenheimer-Snyder solution describes a dust ball at zero temperature.Since it also must be exactly homogeneous, it contains not even the usual positional entropy equal to the particle number.Thus the entropy is zero.”

• cites [20] 20. C.W. Misner, K.S. Thorne and J.A. Wheeler eds.: Gravitation. Freeman, New York 1973. sourceNot checked — no full text of the source

May combine two claims.

As the turnaround time is approached in that universe, the apparent horizon moves out to spatial infinity, and the enclosed volume grows without bound; because entropy density is constant and area grows more slowly than volume, the entropy eventually exceeds the Bak-Rey bound.

“As the turnaround time is approached, the apparent horizon moves out to spatial infinity [15], and the enclosed volume grows without bound.The entropy density is constant, and the area grows more slowly than the volume.Thus the entropy eventually exceeds the Bak-Rey bound.”

• cites [15] 15. N. Kaloper and A. Linde: Cosmology vs. holography, hep-th/9904120. sourceNot checked — no full text of the source

May combine two claims.

Light-rays emitted from the surface of a star inside its gravitational radius can traverse the interior entirely, but this is not a violation of the covariant entropy bound because a collapsing dust star does not have much entropy.

“Light-rays emitted from the surface of the star when it is inside its gravitational radius can traverse the interior entirely.This is not a violation of the covariant entropy bound because a collapsing dust star does not have much entropy.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

Bekenstein has shown that the entropy of a thermodynamic system with limited self-gravity is bounded by its area.

“Bekenstein has shown that the entropy of a thermodynamic system with limited self-gravity is bounded by its area.”

May combine two claims.

The corollary follows from, but is not equivalent to, the covariant entropy bound.

“It follows from, but is not equivalent to, the covariant JHEP07(1999)004 entropy bound.”

May combine two claims.

Trapped surfaces have light-sheets on both sides, but only future-directed ones.

“Trapped surfaces, for example, have light-sheets on both sides, but only future-directed ones.”

May combine two claims.

The covariant bound remains valid in the trapped region because it applies only to future-directed null hypersurfaces, which are soon truncated by the singularity.

“The covariant bound remains valid, because it applies only to the future-directed null hypersurfaces, which are soon truncated by the singularity.”

May combine two claims.

The FS hypersurface would go to the past and pick up nearly half of the total entropy, so the FS bound cannot be used in the trapped region.

“The FS hypersurface would go to the past and would pick up nearly half of the total entropy, so the FS bound cannot be used in this region.”

May combine two claims.

For η = χ > 2π/3, the particle horizon ends on normal spheres and has entered a normal region surrounding the North pole bounded by a different apparent horizon.

“For η = χ > 2π/3 the particle horizon ends on normal, rather than anti-trapped spheres.It has entered a normal region surrounding the North pole and bounded by a different apparent horizon, η = 2(πχ).”

May combine two claims.

Because the future-directed light-sheet is truncated by the future singularity and the past-directed light-sheets by the past singularity, the comoving volume swept out by any of these light-sheets grows only like the area as one moves further along the apparent horizon; thus there is no contradiction with the covariant bound.

“Because the future-directed light-sheet is truncated by the future singularity, and the past-directed light-sheets are truncated by the past singularity, the comoving volume swept out by any of these light-sheet grows only like the area as one moves further along the apparent horizon.Thus there is no contradiction with the covariant bound.”

Classified as probably not a substantial claim. May combine two claims.

The covariant conjecture associates at least two hypersurfaces with any surface in any space-time, and bounds the entropy on those hypersurfaces.

“the corollary is far less general than the covariant conjecture, which associates at least two hypersurfaces with any surface in any space-time, and bounds the entropy on those hypersurfaces.”

May combine two claims.

The covariant entropy conjecture states that the entropy on the future-directed ingoing light-sheet, as well as the two past-directed light-sheets of the apparent horizon, will each be less than A/4.

“The covariant entropy conjecture states that the entropy on the future-directed ingoing light-sheet, as well as the two past-directed light-sheets of the apparent horizon will each be less than A/4.”

May combine two claims.

L has zero expansion and defines the black hole horizon as long as no additional matter falls in; when L encounters matter, its expansion becomes negative and it collapses to r=0 within a finite affine time.

“L has zero expansion and defines the black hole horizon as long as no additional matter falls in.When L does encounter matter, the expansion becomes negative and L collapses to r = 0 within a finite affine time.”

May combine two claims.

Highly enthropic systems undergoing gravitational collapse are very irregular internally and contain strong small scale density perturbations, making the collapse inhomogeneous with some regions reaching the singularity after a shorter proper time than others.

“By definition, highly enthropic systems undergoing gravitational collapse are very irregular internally and contain strong small scale density perturbations.This will make the collapse inhomogeneous, with some regions reaching the singularity after a shorter proper time than other regions.”

May combine two claims. The quote only partly states this claim.

Therefore the FS bound and the covariant bound should both be satisfied for those surfaces.

“Therefore the FS bound and the covariant bound should both be satisfied.”

Probably not a substantial claim: may not pass S2 (checkable).

This result is strong evidence in favor of the entropy bound proposed.

“We take this as strong evidence in favor of the entropy bound we propose.”

May combine two claims.

The covariant entropy bound applies even to surfaces in collapsing regions, or on cosmological scales.

“The covariant entropy bound, on the other hand, applies even to surfaces in collapsing regions, or on cosmological scales.”