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In [[physics]], the '''Bekenstein bound''' is an upper limit on the [[entropy]] ''S'', or [[information]] ''I'', that can be contained within a given finite region of space which has a finite amount of energy—or conversely, the maximum amount of information required to perfectly describe a given physical system down to the quantum level.<ref name="Bekenstein1981-1">Jacob D. Bekenstein, [http://www.phys.huji.ac.il/~bekenste/PRD23-287-1981.pdf "Universal upper bound on the entropy-to-energy ratio for bounded systems"], ''[[Physical Review|Physical Review D]]'', Vol. 23, No. 2, (January 15, 1981), pp. 287-298, {{doi|10.1103/PhysRevD.23.287}}, {{bibcode|1981PhRvD..23..287B}}. [http://www.webcitation.org/5pvt5c96N Mirror link].</ref> It implies that the information of a physical system, or the information necessary to perfectly describe that system, must be finite if the region of space and the energy is finite. In [[computer science]], this implies that there is a maximum information-processing rate ([[Bremermann's limit]]) for a physical system that has a finite size and energy, and that a [[Turing machine]] with finite physical dimensions and unbounded memory is not physically possible.


==Equations==
The universal form of the bound was originally found by [[Jacob Bekenstein]] as the [[inequality (mathematics)|inequality]]<ref name="Bekenstein1981-1"/><ref name="Bekenstein2005"/><ref name="Bekenstein2008"/>


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:<math>S \leq \frac{2 \pi k R E}{\hbar c}</math>
 
where ''S'' is the [[entropy]], ''k'' is [[Boltzmann's constant]], ''R'' is the [[radius]] of a [[sphere]] that can enclose the given system, ''E'' is the total [[mass–energy equivalence|mass–energy]] including any [[invariant mass|rest masses]], ''ħ'' is the [[Planck constant#Reduced Planck constant|reduced Planck constant]], and ''c'' is the [[speed of light]]. Note that while gravity plays a significant role in its enforcement, the expression for the bound does not contain [[Gravitational constant|Newton's Constant]]&nbsp;''G''.
 
In informational terms, the bound is given by
 
:<math>I \leq \frac{2 \pi R E}{\hbar c \ln 2}</math>
 
where ''I'' is the [[information]] expressed in number of [[bit]]s contained in the quantum states in the sphere. The [[Natural logarithm|ln]] 2 factor comes from defining the information as the [[logarithm]] to the [[radix|base]] [[Binary numeral system|2]] of the number of quantum states.<ref name="Tipler2005b">[[Frank J. Tipler]], [http://math.tulane.edu/~tipler/theoryofeverything.pdf "The structure of the world from pure numbers"], ''[[Reports on Progress in Physics]]'', Vol. 68, No. 4 (April 2005), pp. 897-964, {{doi|10.1088/0034-4885/68/4/R04}}, {{bibcode|2005RPPh...68..897T}}, p. 902. [http://www.webcitation.org/5nx3CxKm0 Mirror link]. Also released as [http://arxiv.org/abs/0704.3276 "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything"], {{arxiv|0704.3276}}, April 24, 2007, p. 8.</ref> Using [[mass energy equivalence]], the informational limit may be reformulated as
 
:<math>I \leq \frac{2 \pi c R m}{\hbar \ln 2} \approx 2.577\times 10^{43} m R</math>
 
where <math>m</math> is the mass of the system.
 
==Origins==
Bekenstein derived the bound from heuristic arguments involving [[black hole]]s. If a system exists that violates the bound, i.e. by having too much entropy, Bekenstein argued that it would be possible to violate the [[second law of thermodynamics]] by lowering it into a black hole. In 1995, [[Theodore Jacobson|Ted Jacobson]] demonstrated that the [[Einstein field equations]] (i.e., [[general relativity]]) can be derived by assuming that the Bekenstein bound and the [[laws of thermodynamics]] are true.<ref name="Jacobson1995">[[Theodore Jacobson|Ted Jacobson]], "Thermodynamics of Spacetime: The Einstein Equation of State", ''[[Physical Review Letters]]'', Vol. 75, Issue 7 (August 14, 1995), pp. 1260-1263, {{doi|10.1103/PhysRevLett.75.1260}}, {{bibcode|1995PhRvL..75.1260J}}. Also at {{arxiv|gr-qc/9504004}}, April 4, 1995. Also available [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.6675 here] and [http://www.math.ohio-state.edu/~gerlach/jacobson/jacobson.ps here]. Additionally available as [http://www.gravityresearchfoundation.org/pdf/awarded/1995/jacobson.pdf an entry] in the [[Gravity Research Foundation]]'s 1995 essay competition. [http://www.webcitation.org/5pw2xrwBb Mirror link].</ref><ref name="Smolin2002">[[Lee Smolin]], ''[[Three Roads to Quantum Gravity]]'' (New York, N.Y.: [[Basic Books]], 2002), pp. 173 and 175, ISBN 0-465-07836-2, {{LCCN|2007310371}}.</ref> However, while a number of arguments have been devised which show that some form of the bound must exist in order for the laws of thermodynamics and general relativity to be mutually consistent, the precise formulation of the bound has been a matter of debate.<ref name="Bekenstein2005">Jacob D. Bekenstein, "How Does the Entropy/Information Bound Work?", ''[[Foundations of Physics]]'', Vol. 35, No. 11 (November 2005), pp. 1805-1823, {{doi|10.1007/s10701-005-7350-7}}, {{bibcode|2005FoPh...35.1805B}}. Also at {{arxiv|quant-ph/0404042}}, April 7, 2004.</ref><ref name="Bekenstein2008">Jacob D. Bekenstein, [http://www.scholarpedia.org/wiki/index.php?title=Bekenstein_bound&oldid=50791 "Bekenstein bound"], ''[[Scholarpedia]]'', Vol. 3, No. 10 (October 31, 2008), p. 7374, {{doi|10.4249/scholarpedia.7374}}.</ref><ref name="Bousso1999-6">Raphael Bousso, "Holography in general space-times", ''[[Journal of High Energy Physics]]'', Vol. 1999, Issue 6 (June 1999), Art. No. 28, 24 pages, {{doi|10.1088/1126-6708/1999/06/028}}, {{bibcode|1999JHEP...06..028B}}. [http://www.webcitation.org/5pyuEsv6m Mirror link]. Also at {{arxiv|hep-th/9906022}}, June 3, 1999.</ref><ref name="Bousso1999-7">Raphael Bousso, "A covariant entropy conjecture", ''[[Journal of High Energy Physics]]'', Vol. 1999, Issue 7 (July 1999), Art. No. 4, 34 pages, {{doi|10.1088/1126-6708/1999/07/004}}, {{bibcode|1999JHEP...07..004B}}. [http://www.webcitation.org/5pyuO4nVn Mirror link]. Also at {{arxiv|hep-th/9905177}}, May 24, 1999.</ref><ref name="Bousso2000">Raphael Bousso, "The holographic principle for general backgrounds", ''[[Classical and Quantum Gravity]]'', Vol. 17, No. 5 (March 7, 2000), pp. 997-1005, {{doi|10.1088/0264-9381/17/5/309}}, {{bibcode|2000CQGra..17..997B}}. Also at {{arxiv|hep-th/9911002}}, November 2, 1999.</ref><ref name="Bekenstein2000">Jacob D. Bekenstein, "Holographic bound from second law of thermodynamics", ''[[Physics Letters|Physics Letters B]]'', Vol. 481, Issues 2-4 (May 25, 2000), pp. 339-345, {{doi|10.1016/S0370-2693(00)00450-0}}, {{bibcode|2000PhLB..481..339B}}. Also at {{arxiv|hep-th/0003058}}, March 8, 2000.</ref><ref name="Bousso2002">Raphael Bousso, [http://bib.tiera.ru/DVD-005/Bousso_R._The_holographic_principle_(2002)(en)(50s).pdf "The holographic principle"], ''[[Reviews of Modern Physics]]'', Vol. 74, No. 3 (July 2002), pp. 825-874, {{doi|10.1103/RevModPhys.74.825}}, {{bibcode|2002RvMP...74..825B}}. [http://www.webcitation.org/5pw1VZbGO Mirror link]. Also at {{arxiv|hep-th/0203101}}, March 12, 2002.</ref><ref name="Bekenstein2003">Jacob D. Bekenstein, [http://www.phys.huji.ac.il/~bekenste/Holographic_Univ.pdf "Information in the Holographic Universe: Theoretical results about black holes suggest that the universe could be like a gigantic hologram"], ''[[Scientific American]]'', Vol. 289, No. 2 (August 2003), pp. 58-65. [http://www.webcitation.org/5pvxM7hws Mirror link].</ref><ref name="BoussoEtAl2003">Raphael Bousso, Éanna É. Flanagan and [[Donald Marolf]], "Simple sufficient conditions for the generalized covariant entropy bound", Physical Review D, Vol. 68, Issue 6 (September 15, 2003), Art. No. 064001, 7 pages, {{doi|10.1103/PhysRevD.68.064001}}, {{bibcode|2003PhRvD..68f4001B}}. Also at {{arxiv|hep-th/0305149}}, May 19, 2003.</ref><ref name="Bekenstein2004">Jacob D. Bekenstein, "Black holes and information theory", ''[[Contemporary Physics]]'', Vol. 45, Issue 1 (January 2004), pp. 31-43, {{doi|10.1080/00107510310001632523}}, {{bibcode|2003ConPh..45...31B}}. Also at {{arxiv|quant-ph/0311049}}, November 9, 2003. Also at {{arxiv|quant-ph/0311049}}, November 9, 2003.</ref><ref name="Tipler2005">[[Frank J. Tipler]], [http://math.tulane.edu/~tipler/theoryofeverything.pdf "The structure of the world from pure numbers"], ''[[Reports on Progress in Physics]]'', Vol. 68, No. 4 (April 2005), pp. 897-964, {{doi|10.1088/0034-4885/68/4/R04}}, {{bibcode|2005RPPh...68..897T}}. [http://www.webcitation.org/5nx3CxKm0 Mirror link]. Also released as [http://arxiv.org/abs/0704.3276 "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything"], {{arxiv|0704.3276}}, April 24, 2007. Tipler gives a number of arguments for maintaining that Bekenstein's original formulation of the bound is the correct form. See in particular the paragraph beginning with "A few points ..." on p. 903 of the ''Rep. Prog. Phys.'' paper (or p. 9 of the ''arXiv'' version), and the discussions on the Bekenstein bound that follow throughout the paper.</ref>
 
==Examples==
===Black holes===
It happens that the [[Black_hole_thermodynamics#Black_hole_entropy|Bekenstein-Hawking Entropy]] of three-dimensional [[black hole]]s exactly saturates the bound
 
:<math>S =\frac{kA}{4}</math>
 
where ''A'' is the two-dimensional area of the black hole's event horizon in units of the [[Planck area]], <math>\hbar G/c^3</math>.
 
The bound is closely associated with [[black hole thermodynamics]], the [[holographic principle]] and the [[covariant entropy bound]] of quantum gravity, and can be derived from a conjectured strong form of the latter.
 
===Human brain===
An average [[human brain]] has a mass of 1.5&nbsp;kg and a volume of 1260&nbsp;cm³. If the brain is approximated by a sphere then [[Sphere#Volume_of_a_sphere|the radius will be]] 6.7 cm.
 
The informational Bekenstein bound will be <math>\approx 2.6 \times 10^{42}</math> bit and represents the maximum information needed to perfectly recreate an average human brain down to the quantum level. This means that the number <math>O=2^I</math> of [[quantum state|states]] of the human brain must be less than <math>\approx 10^{7.8 \times 10^{41}}</math>.
 
The existence of the Bekenstein bound implies that the storage capacity of human brain is finite, although potentially very large, if constrained only by ultimate physical limits. This makes [[mind uploading]] possible from the point of view of quantum mechanics, provided that [[physicalism]] is true.
 
==See also==
* [[Limits to computation]]
* [[Black hole thermodynamics#Black hole entropy|Black hole entropy]]
* [[Digital physics]]
* [[Entropy]]
 
==Further reading==
*J. D. Bekenstein, [http://www.phys.huji.ac.il/~barak_kol/Courses/Black-holes/reading-papers/Beken-Entropy.pdf "Black Holes and the Second Law"], ''[[Nuovo Cimento|Lettere al Nuovo Cimento]]'', Vol. 4, No 15 (August 12, 1972), pp. 737-740, {{doi|10.1007/BF02757029}}, {{bibcode|1972NCimL...4..737B}}. [http://www.webcitation.org/5pvpODjvK Mirror link].
*Jacob D. Bekenstein, [http://www.physics.princeton.edu/~mcdonald/examples/QM/bekenstein_prd_7_2333_73.pdf "Black Holes and Entropy"], ''[[Physical Review|Physical Review D]]'', Vol. 7, No. 8 (April 15, 1973), pp. 2333-2346, {{doi|10.1103/PhysRevD.7.2333}}, {{bibcode|1973PhRvD...7.2333B}}. [http://www.webcitation.org/5pvpyakNu Mirror link].
*Jacob D. Bekenstein, [http://www.phys.huji.ac.il/~bekenste/PRD9-3292-1974.pdf "Generalized second law of thermodynamics in black-hole physics"], ''[[Physical Review|Physical Review D]]'', Vol. 9, No. 12 (June 15, 1974), pp. 3292-3300, {{doi|10.1103/PhysRevD.9.3292}}, {{bibcode|1974PhRvD...9.3292B}}. [http://www.webcitation.org/5pvqDR9Rs Mirror link].
*Jacob D. Bekenstein, [http://www.physics.princeton.edu/~mcdonald/examples/QM/bekenstein_prd_12_3077_75.pdf "Statistical black-hole thermodynamics"], ''[[Physical Review|Physical Review D]]'', Vol. 12, No. 10 (November 15, 1975), pp. 3077-3085, {{doi|10.1103/PhysRevD.12.3077}}, {{bibcode|1975PhRvD..12.3077B}}. [http://www.webcitation.org/5pvqVyq9K Mirror link].
*Jacob D. Bekenstein, [http://www.phys.huji.ac.il/~bekenste/PT,33,24(1980).pdf "Black-hole thermodynamics"], ''[[Physics Today]]'', Vol. 33, Issue 1 (January 1980), pp. 24-31, {{doi|10.1063/1.2913906}}, {{bibcode|1980PhT....33a..24B}}. [http://www.webcitation.org/5pvqqPGuq Mirror link].
*Jacob D. Bekenstein, [http://www.phys.huji.ac.il/~bekenste/PRD23-287-1981.pdf "Universal upper bound on the entropy-to-energy ratio for bounded systems"], ''[[Physical Review|Physical Review D]]'', Vol. 23, No. 2, (January 15, 1981), pp. 287-298, {{doi|10.1103/PhysRevD.23.287}}, {{bibcode|1981PhRvD..23..287B}}. [http://www.webcitation.org/5pvt5c96N Mirror link].
*Jacob D. Bekenstein, [http://www.physics.princeton.edu/~mcdonald/examples/QM/bekenstein_prl_46_623_81.pdf "Energy Cost of Information Transfer"], ''[[Physical Review Letters]]'', Vol. 46, No. 10 (March 9, 1981), pp. 623-626, {{doi|10.1103/PhysRevLett.46.623}}, {{bibcode|1981PhRvL..46..623B}}. [http://www.webcitation.org/5pvuFpfRA Mirror link].
*Jacob D. Bekenstein, "Specific entropy and the sign of the energy", ''[[Physical Review|Physical Review D]]'', Vol. 26, No. 4 (August 15, 1982), pp. 950-953, {{doi|10.1103/PhysRevD.26.950}}, {{bibcode|1982PhRvD..26..950B}}.
*Jacob D. Bekenstein, [http://128.112.100.2/~kirkmcd/examples/QM/bekenstein_prd_30_1669_84.pdf "Entropy content and information flow in systems with limited energy"], ''[[Physical Review|Physical Review D]]'', Vol. 30, No. 8, (October 15, 1984), pp. 1669-1679, {{doi|10.1103/PhysRevD.30.1669}}, {{bibcode|1984PhRvD..30.1669B}}. [http://www.webcitation.org/5pvud0uay Mirror link].
*Jacob D. Bekenstein, [http://pm1.bu.edu/~tt/qcl/pdf/bekenstj19867766031b.pdf "Communication and energy"], ''[[Physical Review|Physical Review A]]'', Vol. 37, Issue 9 (May 1988), pp. 3437-3449, {{doi|10.1103/PhysRevA.37.3437}}, {{bibcode|1988PhRvA..37.3437B}}. [http://www.webcitation.org/5pvutswfg Mirror link].
*Marcelo Schiffer and Jacob D. Bekenstein, "Proof of the quantum bound on specific entropy for free fields", ''[[Physical Review|Physical Review D]]'', Vol. 39, Issue 4 (February 15, 1989), pp. 1109-1115, {{doi|10.1103/PhysRevD.39.1109}} PMID 9959747, {{bibcode|1989PhRvD..39.1109S}}.
*Jacob D. Bekenstein, "Is the Cosmological Singularity Thermodynamically Possible?", ''[[International Journal of Theoretical Physics]]'', Vol. 28, Issue 9 (September 1989), pp. 967-981, {{doi|10.1007/BF00670342}}, {{bibcode|1989IJTP...28..967B}}.
*Jacob D. Bekenstein, "Entropy bounds and black hole remnants", ''[[Physical Review|Physical Review D]]'', Vol. 49, Issue 4 (February 15, 1994), pp. 1912-1921, {{doi|10.1103/PhysRevD.49.1912}}, {{bibcode|1994PhRvD..49.1912B}}. Also at {{arxiv|gr-qc/9307035}}, July 25, 1993.
*Oleg B. Zaslavskii, "Generalized second law and the Bekenstein entropy bound in ''Gedankenexperiments'' with black holes", ''[[Classical and Quantum Gravity]]'', Vol. 13, No. 1 (January 1996), pp. L7-L11, {{doi|10.1088/0264-9381/13/1/002}}, {{bibcode|1996CQGra..13L...7Z}}. See also O. B. Zaslavskii, "Corrigendum to 'Generalized second law and the Bekenstein entropy bound in ''Gedankenexperiments'' with black holes'", ''[[Classical and Quantum Gravity]]'', Vol. 13, No. 9 (September 1996), p. 2607, {{doi|10.1088/0264-9381/13/9/024}}, {{bibcode|1996CQGra..13.2607Z}}.
*Jacob D. Bekenstein, "Non-Archimedean character of quantum buoyancy and the generalized second law of thermodynamics", ''[[Physical Review|Physical Review D]]'', Vol. 60, Issue 12 (December 15, 1999), Art. No. 124010, 9 pages, {{doi|10.1103/PhysRevD.60.124010}}, {{bibcode|1999PhRvD..60l4010B}}. Also at {{arxiv|gr-qc/9906058}}, June 16, 1999.
 
==References==
{{reflist}}
 
==External links==
* Jacob D. Bekenstein, [http://www.scholarpedia.org/article/Bekenstein_bound "Bekenstein bound"], ''[[Scholarpedia]]'', Vol. 3, No. 10 (2008), p.&nbsp;7374, {{doi|10.4249/scholarpedia.7374}}.
* Jacob D. Bekenstein, [http://www.scholarpedia.org/article/Bekenstein-Hawking_entropy "Bekenstein-Hawking entropy"], ''[[Scholarpedia]]'', Vol. 3, No. 10 (2008), p.&nbsp;7375, {{doi|10.4249/scholarpedia.7375}}.
* [http://www.phys.huji.ac.il/~bekenste/ Jacob D. Bekenstein's website] at [[the Racah Institute of Physics]], [[Hebrew University of Jerusalem]], which contains a number of articles on the Bekenstein bound.
 
{{DEFAULTSORT:Bekenstein Bound}}
[[Category:Thermodynamic entropy]]
[[Category:Quantum information science]]

Revision as of 07:44, 28 November 2013

In physics, the Bekenstein bound is an upper limit on the entropy S, or information I, that can be contained within a given finite region of space which has a finite amount of energy—or conversely, the maximum amount of information required to perfectly describe a given physical system down to the quantum level.[1] It implies that the information of a physical system, or the information necessary to perfectly describe that system, must be finite if the region of space and the energy is finite. In computer science, this implies that there is a maximum information-processing rate (Bremermann's limit) for a physical system that has a finite size and energy, and that a Turing machine with finite physical dimensions and unbounded memory is not physically possible.

Equations

The universal form of the bound was originally found by Jacob Bekenstein as the inequality[1][2][3]

S≤2πkREℏc

where S is the entropy, k is Boltzmann's constant, R is the radius of a sphere that can enclose the given system, E is the total mass–energy including any rest masses, ħ is the reduced Planck constant, and c is the speed of light. Note that while gravity plays a significant role in its enforcement, the expression for the bound does not contain Newton's Constant G.

In informational terms, the bound is given by

I≤2πREℏcln⁡2

where I is the information expressed in number of bits contained in the quantum states in the sphere. The ln 2 factor comes from defining the information as the logarithm to the base 2 of the number of quantum states.[4] Using mass energy equivalence, the informational limit may be reformulated as

I≤2πcRmℏln⁡2≈2.577×1043mR

where m is the mass of the system.

Origins

Bekenstein derived the bound from heuristic arguments involving black holes. If a system exists that violates the bound, i.e. by having too much entropy, Bekenstein argued that it would be possible to violate the second law of thermodynamics by lowering it into a black hole. In 1995, Ted Jacobson demonstrated that the Einstein field equations (i.e., general relativity) can be derived by assuming that the Bekenstein bound and the laws of thermodynamics are true.[5][6] However, while a number of arguments have been devised which show that some form of the bound must exist in order for the laws of thermodynamics and general relativity to be mutually consistent, the precise formulation of the bound has been a matter of debate.[2][3][7][8][9][10][11][12][13][14][15]

Examples

Black holes

It happens that the Bekenstein-Hawking Entropy of three-dimensional black holes exactly saturates the bound

S=kA4

where A is the two-dimensional area of the black hole's event horizon in units of the Planck area, ℏG/c3.

The bound is closely associated with black hole thermodynamics, the holographic principle and the covariant entropy bound of quantum gravity, and can be derived from a conjectured strong form of the latter.

Human brain

An average human brain has a mass of 1.5 kg and a volume of 1260 cm³. If the brain is approximated by a sphere then the radius will be 6.7 cm.

The informational Bekenstein bound will be ≈2.6×1042 bit and represents the maximum information needed to perfectly recreate an average human brain down to the quantum level. This means that the number O=2I of states of the human brain must be less than ≈107.8×1041.

The existence of the Bekenstein bound implies that the storage capacity of human brain is finite, although potentially very large, if constrained only by ultimate physical limits. This makes mind uploading possible from the point of view of quantum mechanics, provided that physicalism is true.

See also

Further reading

References

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External links

  1. ↑ 1.0 1.1 Jacob D. Bekenstein, "Universal upper bound on the entropy-to-energy ratio for bounded systems", Physical Review D, Vol. 23, No. 2, (January 15, 1981), pp. 287-298, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Mirror link.
  2. ↑ 2.0 2.1 Jacob D. Bekenstein, "How Does the Entropy/Information Bound Work?", Foundations of Physics, Vol. 35, No. 11 (November 2005), pp. 1805-1823, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Also at Template:Arxiv, April 7, 2004.
  3. ↑ 3.0 3.1 Jacob D. Bekenstein, "Bekenstein bound", Scholarpedia, Vol. 3, No. 10 (October 31, 2008), p. 7374, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park..
  4. ↑ Frank J. Tipler, "The structure of the world from pure numbers", Reports on Progress in Physics, Vol. 68, No. 4 (April 2005), pp. 897-964, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode, p. 902. Mirror link. Also released as "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything", Template:Arxiv, April 24, 2007, p. 8.
  5. ↑ Ted Jacobson, "Thermodynamics of Spacetime: The Einstein Equation of State", Physical Review Letters, Vol. 75, Issue 7 (August 14, 1995), pp. 1260-1263, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Also at Template:Arxiv, April 4, 1995. Also available here and here. Additionally available as an entry in the Gravity Research Foundation's 1995 essay competition. Mirror link.
  6. ↑ Lee Smolin, Three Roads to Quantum Gravity (New York, N.Y.: Basic Books, 2002), pp. 173 and 175, ISBN 0-465-07836-2, Template:LCCN.
  7. ↑ Raphael Bousso, "Holography in general space-times", Journal of High Energy Physics, Vol. 1999, Issue 6 (June 1999), Art. No. 28, 24 pages, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Mirror link. Also at Template:Arxiv, June 3, 1999.
  8. ↑ Raphael Bousso, "A covariant entropy conjecture", Journal of High Energy Physics, Vol. 1999, Issue 7 (July 1999), Art. No. 4, 34 pages, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Mirror link. Also at Template:Arxiv, May 24, 1999.
  9. ↑ Raphael Bousso, "The holographic principle for general backgrounds", Classical and Quantum Gravity, Vol. 17, No. 5 (March 7, 2000), pp. 997-1005, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Also at Template:Arxiv, November 2, 1999.
  10. ↑ Jacob D. Bekenstein, "Holographic bound from second law of thermodynamics", Physics Letters B, Vol. 481, Issues 2-4 (May 25, 2000), pp. 339-345, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Also at Template:Arxiv, March 8, 2000.
  11. ↑ Raphael Bousso, "The holographic principle", Reviews of Modern Physics, Vol. 74, No. 3 (July 2002), pp. 825-874, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Mirror link. Also at Template:Arxiv, March 12, 2002.
  12. ↑ Jacob D. Bekenstein, "Information in the Holographic Universe: Theoretical results about black holes suggest that the universe could be like a gigantic hologram", Scientific American, Vol. 289, No. 2 (August 2003), pp. 58-65. Mirror link.
  13. ↑ Raphael Bousso, Éanna É. Flanagan and Donald Marolf, "Simple sufficient conditions for the generalized covariant entropy bound", Physical Review D, Vol. 68, Issue 6 (September 15, 2003), Art. No. 064001, 7 pages, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Also at Template:Arxiv, May 19, 2003.
  14. ↑ Jacob D. Bekenstein, "Black holes and information theory", Contemporary Physics, Vol. 45, Issue 1 (January 2004), pp. 31-43, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Also at Template:Arxiv, November 9, 2003. Also at Template:Arxiv, November 9, 2003.
  15. ↑ Frank J. Tipler, "The structure of the world from pure numbers", Reports on Progress in Physics, Vol. 68, No. 4 (April 2005), pp. 897-964, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., Template:Bibcode. Mirror link. Also released as "Feynman-Weinberg Quantum Gravity and the Extended Standard Model as a Theory of Everything", Template:Arxiv, April 24, 2007. Tipler gives a number of arguments for maintaining that Bekenstein's original formulation of the bound is the correct form. See in particular the paragraph beginning with "A few points ..." on p. 903 of the Rep. Prog. Phys. paper (or p. 9 of the arXiv version), and the discussions on the Bekenstein bound that follow throughout the paper.
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