A covariant entropy conjecture

In the paper

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

What the sources say

• Partly 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)

  cites [12] 12. R. Easther and D.A. Lowe: Holography, cosmology and the second law of thermody- namics, hep-th/9902088. sourceExperimental — accuracy not yet measured
• Partly 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)

  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
• Partly 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)

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

What the sources say

• Supported

  Section 3 (p. 6) says closed universes reach or threaten to violate the holographic bound, either while growing or during recollapse, and suggests they may be inconsistent with the holographic principle. It contrasts this with open universes, for which it continues to analyze the bound; Section 2 analyzes flat universes. (LLM, conf=0.86)

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

In the paper

“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].”

What the sources say

• Supported

  Section 2, “Flat Universe” (pp. 3–4), checks the holographic bound for flat cosmologies with an equation-of-state parameter γ. It concludes that violating the resulting bound would require sound to travel faster than light and gives a γ = 1 case that can saturate it. The sentence assigns the −1 < γ < 0 range to a different reference, so that range is not attributed to this paper. (LLM, conf=0.86)

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

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

What the sources say

• Partly 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)

  cites [9] 9. W. Fischler and L. Susskind: Holography and cosmology, hep-th/9806039. JHEP07(1999)004 sourceExperimental — accuracy not yet measured

In the paper

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

In the paper

“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.”

In the paper

“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.”

In the paper

“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 .”

In the paper

“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.”

In the paper

“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.”

In the paper

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

In the paper

“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.”

What the sources say

• ?Not checked — no full text of the sourcecites [18] 18. S.W. Hawking and G.F.R. Ellis: The large scale stucture of space-time. Cambridge University Press, Cambridge 1973. source
• ?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

“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.”

In the paper

“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.”

In the paper

“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.”

In the paper

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

In the paper

“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.”

In the paper

“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.”

In the paper

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

In the paper

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

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.
• ?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

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

What the sources say

• ?Not matched

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

  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.

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

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

In the paper

“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.”

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

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

In the paper

“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.”

In the paper

“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.”

In the paper

“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.”

In the paper

“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.”

In the paper

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

In the paper

“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.”

In the paper

“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.”

In the paper

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

In the paper

“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.”

In the paper

“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).”

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

“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.”

In the paper

“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).”

In the paper

“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.”

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

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

In the paper

“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.”

In the paper

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

In the paper

“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.”

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

“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.”

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

“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.”

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

“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).”

In the paper

“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.”

In the paper

“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.”

In the paper

“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.”

In the paper

“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.”

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

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