bekenstein bound

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Bekenstein bound

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Inphysics, the Bekenstein bound is an upper limit on theentropyS, orinformationI, thatcan be contained within a given finite region of space which has a finite amount ofenergyor conversely, the maximum amount of information required to perfectly describea given physical system down to the quantum level.[1]It implies that the information of aphysical system, or the information necessary to perfectly describe that system, must befinite if the region of space and the energy is finite. In computer science, this implies thatthere is amaximum information-processing ratefor a physical system that has a finite sizeand energy, and that aTuring machinewith unbounded memory is not physically possible.

[edit] EquationsThe universal form of the bound was originally found byJacob Bekensteinas theinequality[1][2][3]

where Sis theentropy,kisBoltzmann's constant,R is theradiusof aspherethat canenclose the given system,Eis the totalmass-energyincluding anyrest masses, is thereduced Planck constant, and c is thespeed of light. Note that while gravity plays a

significant role in its enforcement, the expression for the bound does not containNewton'sConstantG.

In informational terms, the bound is given by

whereIis theinformationexpressed in number ofbitscontained in the quantum states inthe sphere. Theln2 factor comes from defining the information as thelogarithmto thebase2of the number of quantum states.[4]Theright-hand sideof the foregoingrelationisapproximately equal to 2.57690871043(mass in kilograms)(radius in meters).

[edit] Origins

Bekenstein derived the bound from heuristic arguments involvingblack holes. If a systemexists that violates the bound, i.e. by having too much entropy, Bekenstein argued that itwould be possible to violate thesecond law of thermodynamicsby lowering it into a black
http://en.wikipedia.org/w/index.php?title=Bekenstein_Bound&redirect=nohttp://en.wikipedia.org/w/index.php?title=Bekenstein_Bound&redirect=nohttp://en.wikipedia.org/w/index.php?title=Bekenstein_Bound&redirect=nohttp://en.wikipedia.org/wiki/Bekenstein_Bound#mw-headhttp://en.wikipedia.org/wiki/Bekenstein_Bound#mw-headhttp://en.wikipedia.org/wiki/Bekenstein_Bound#mw-headhttp://en.wikipedia.org/wiki/Bekenstein_Bound#p-searchhttp://en.wikipedia.org/wiki/Bekenstein_Bound#p-searchhttp://en.wikipedia.org/wiki/Bekenstein_Bound#p-searchhttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Entropyhttp://en.wikipedia.org/wiki/Entropyhttp://en.wikipedia.org/wiki/Entropyhttp://en.wikipedia.org/wiki/Informationhttp://en.wikipedia.org/wiki/Informationhttp://en.wikipedia.org/wiki/Informationhttp://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bekenstein1981-1-0http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bekenstein1981-1-0http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bekenstein1981-1-0http://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Limits_to_computationhttp://en.wikipedia.org/wiki/Limits_to_computationhttp://en.wikipedia.org/wiki/Limits_to_computationhttp://en.wikipedia.org/wiki/Turing_machinehttp://en.wikipedia.org/wiki/Turing_machinehttp://en.wikipedia.org/wiki/Turing_machinehttp://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=1http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=1http://en.wikipedia.org/wiki/Jacob_Bekensteinhttp://en.wikipedia.org/wiki/Jacob_Bekensteinhttp://en.wikipedia.org/wiki/Jacob_Bekensteinhttp://en.wikipedia.org/wiki/Inequality_(mathematics)http://en.wikipedia.org/wiki/Inequality_(mathematics)http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bekenstein2005-1http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bekenstein2005-1http://en.wikipedia.org/wiki/Entropyhttp://en.wikipedia.org/wiki/Entropyhttp://en.wikipedia.org/wiki/Entropyhttp://en.wikipedia.org/wiki/Boltzmann%27s_constanthttp://en.wikipedia.org/wiki/Boltzmann%27s_constanthttp://en.wikipedia.org/wiki/Boltzmann%27s_constanthttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Spherehttp://en.wikipedia.org/wiki/Spherehttp://en.wikipedia.org/wiki/Spherehttp://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalencehttp://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalencehttp://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalencehttp://en.wikipedia.org/wiki/Invariant_masshttp://en.wikipedia.org/wiki/Invariant_masshttp://en.wikipedia.org/wiki/Invariant_masshttp://en.wikipedia.org/wiki/Planck_constant#Reduced_Planck_constanthttp://en.wikipedia.org/wiki/Planck_constant#Reduced_Planck_constanthttp://en.wikipedia.org/wiki/Speed_of_lighthttp://en.wikipedia.org/wiki/Speed_of_lighthttp://en.wikipedia.org/wiki/Speed_of_lighthttp://en.wikipedia.org/wiki/Gravitational_constanthttp://en.wikipedia.org/wiki/Gravitational_constanthttp://en.wikipedia.org/wiki/Gravitational_constanthttp://en.wikipedia.org/wiki/Gravitational_constanthttp://en.wikipedia.org/wiki/Informationhttp://en.wikipedia.org/wiki/Informationhttp://en.wikipedia.org/wiki/Informationhttp://en.wikipedia.org/wiki/Bithttp://en.wikipedia.org/wiki/Bithttp://en.wikipedia.org/wiki/Bithttp://en.wikipedia.org/wiki/Natural_logarithmhttp://en.wikipedia.org/wiki/Natural_logarithmhttp://en.wikipedia.org/wiki/Natural_logarithmhttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Radixhttp://en.wikipedia.org/wiki/Radixhttp://en.wikipedia.org/wiki/Radixhttp://en.wikipedia.org/wiki/Binary_numeral_systemhttp://en.wikipedia.org/wiki/Binary_numeral_systemhttp://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Tipler2005b-3http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Tipler2005b-3http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Tipler2005b-3http://en.wikipedia.org/wiki/Sides_of_an_equationhttp://en.wikipedia.org/wiki/Sides_of_an_equationhttp://en.wikipedia.org/wiki/Sides_of_an_equationhttp://en.wikipedia.org/wiki/Binary_relationhttp://en.wikipedia.org/wiki/Binary_relationhttp://en.wikipedia.org/wiki/Binary_relationhttp://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=2http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=2http://en.wikipedia.org/wiki/Black_holehttp://en.wikipedia.org/wiki/Black_holehttp://en.wikipedia.org/wiki/Black_holehttp://en.wikipedia.org/wiki/Second_law_of_thermodynamicshttp://en.wikipedia.org/wiki/Second_law_of_thermodynamicshttp://en.wikipedia.org/wiki/Second_law_of_thermodynamicshttp://en.wikipedia.org/wiki/Second_law_of_thermodynamicshttp://en.wikipedia.org/wiki/Black_holehttp://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=2http://en.wikipedia.org/wiki/Binary_relationhttp://en.wikipedia.org/wiki/Sides_of_an_equationhttp://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Tipler2005b-3http://en.wikipedia.org/wiki/Binary_numeral_systemhttp://en.wikipedia.org/wiki/Radixhttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Natural_logarithmhttp://en.wikipedia.org/wiki/Bithttp://en.wikipedia.org/wiki/Informationhttp://en.wikipedia.org/wiki/Gravitational_constanthttp://en.wikipedia.org/wiki/Gravitational_constanthttp://en.wikipedia.org/wiki/Speed_of_lighthttp://en.wikipedia.org/wiki/Planck_constant#Reduced_Planck_constanthttp://en.wikipedia.org/wiki/Invariant_masshttp://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalencehttp://en.wikipedia.org/wiki/Spherehttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Boltzmann%27s_constanthttp://en.wikipedia.org/wiki/Entropyhttp://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bekenstein2005-1http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bekenstein2005-1http://en.wikipedia.org/wiki/Inequality_(mathematics)http://en.wikipedia.org/wiki/Inequality_(mathematics)http://en.wikipedia.org/wiki/Jacob_Bekensteinhttp://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=1http://en.wikipedia.org/wiki/Turing_machinehttp://en.wikipedia.org/wiki/Limits_to_computationhttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bekenstein1981-1-0http://en.wikipedia.org/wiki/Informationhttp://en.wikipedia.org/wiki/Entropyhttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Bekenstein_Bound#p-searchhttp://en.wikipedia.org/wiki/Bekenstein_Bound#mw-headhttp://en.wikipedia.org/w/index.php?title=Bekenstein_Bound&redirect=no


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hole. In 1995,Ted Jacobsondemonstrated that theEinstein field equations(i.e.,generalrelativity) can be derived by assuming that the Bekenstein bound and thelaws ofthermodynamicsare true.[5][6]However, while a number of arguments have been devisedwhich show that some form of the bound must exist in order for the laws ofthermodynamics 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]

[edit] Examples

[edit] Black holes

It happens that theBekenstein-Hawking Entropyof three-dimensionalblack holesexactlysaturates the bound

whereA is the two-dimensional area of the black hole's event horizon in units of thePlanck

area, .

The bound is closely associated withblack hole thermodynamics, theholographic principleand thecovariant entropy boundof quantum gravity, and can be derived from a conjecturedstrong form of the latter.

[edit] Human brain

An averagehuman brainhas a mass of 1.5kgand a volume of 1260 cm3. The energy (E =mc2) will be 1.348131017Jand if the brain is approximate to a sphere then the radius (V =4r

3/3) will be 6.70030102 m.

The Bekenstein bound (I 2rEc ln 2) will be 2.589911042bitand represent the maximuminformationneeded to perfectly recreate the average human brain down to thequantumlevel(or anything of the same mass and volume). This implies that the number of differentstates(=2I) of thehuman brainis at most 107.796401041.

[edit] See also

Black hole entropy Digital physics Entropy

[edit] Further reading
http://en.wikipedia.org/wiki/Theodore_Jacobsonhttp://en.wikipedia.org/wiki/Theodore_Jacobsonhttp://en.wikipedia.org/wiki/Theodore_Jacobsonhttp://en.wikipedia.org/wiki/Einstein_field_equationshttp://en.wikipedia.org/wiki/Einstein_field_equationshttp://en.wikipedia.org/wiki/Einstein_field_equationshttp://en.wikipedia.org/wiki/General_relativityhttp://en.wikipedia.org/wiki/General_relativityhttp://en.wikipedia.org/wiki/General_relativityhttp://en.wikipedia.org/wiki/General_relativityhttp://en.wikipedia.org/wiki/Laws_of_thermodynamicshttp://en.wikipedia.org/wiki/Laws_of_thermodynamicshttp://en.wikipedia.org/wiki/Laws_of_thermodynamicshttp://en.wikipedia.org/wiki/Laws_of_thermodynamicshttp://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Jacobson1995-4http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Jacobson1995-4http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Jacobson1995-4http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bekenstein2005-1http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bekenstein2005-1http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bousso1999-6-6http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bousso2000-8http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bousso2002-10http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-BoussoEtAl2003-12http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Tipler2005-14http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Tipler2005-14http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=3http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=3http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=4http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=4http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=4http://en.wikipedia.org/wiki/Black_hole_thermodynamics#Black_hole_entropyhttp://en.wikipedia.org/wiki/Black_hole_thermodynamics#Black_hole_entropyhttp://en.wikipedia.org/wiki/Black_hole_thermodynamics#Black_hole_entropyhttp://en.wikipedia.org/wiki/Black_holehttp://en.wikipedia.org/wiki/Black_holehttp://en.wikipedia.org/wiki/Black_holehttp://en.wikipedia.org/wiki/Planck_areahttp://en.wikipedia.org/wiki/Planck_areahttp://en.wikipedia.org/wiki/Planck_areahttp://en.wikipedia.org/wiki/Planck_areahttp://en.wikipedia.org/wiki/Black_hole_thermodynamicshttp://en.wikipedia.org/wiki/Black_hole_thermodynamicshttp://en.wikipedia.org/wiki/Black_hole_thermodynamicshttp://en.wikipedia.org/wiki/Holographic_principlehttp://en.wikipedia.org/wiki/Holographic_principlehttp://en.wikipedia.org/wiki/Holographic_principlehttp://en.wikipedia.org/wiki/Covariant_entropy_boundhttp://en.wikipedia.org/wiki/Covariant_entropy_boundhttp://en.wikipedia.org/wiki/Covariant_entropy_boundhttp://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=5http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=5http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=5http://en.wikipedia.org/wiki/Human_brainhttp://en.wikipedia.org/wiki/Human_brainhttp://en.wikipedia.org/wiki/Human_brainhttp://en.wikipedia.org/wiki/Kilogramhttp://en.wikipedia.org/wiki/Kilogramhttp://en.wikipedia.org/wiki/Kilogramhttp://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalencehttp://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalencehttp://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalencehttp://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalencehttp://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalencehttp://en.wikipedia.org/wiki/Joulehttp://en.wikipedia.org/wiki/Joulehttp://en.wikipedia.org/wiki/Joulehttp://en.wikipedia.org/wiki/Sphere#Volume_of_a_spherehttp://en.wikipedia.org/wiki/Sphere#Volume_of_a_spherehttp://en.wikipedia.org/wiki/Sphere#Volume_of_a_spherehttp://en.wikipedia.org/wiki/Sphere#Volume_of_a_spherehttp://en.wikipedia.org/wiki/Sphere#Volume_of_a_spherehttp://en.wikipedia.org/wiki/Sphere#Volume_of_a_spherehttp://en.wikipedia.org/wiki/Bithttp://en.wikipedia.org/wiki/Bithttp://en.wikipedia.org/wiki/Bithttp://en.wikipedia.org/wiki/Informationhttp://en.wikipedia.org/wiki/Informationhttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_statehttp://en.wikipedia.org/wiki/Quantum_statehttp://en.wikipedia.org/wiki/Human_brainhttp://en.wikipedia.org/wiki/Human_brainhttp://en.wikipedia.org/wiki/Human_brainhttp://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=6http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=6http://en.wikipedia.org/wiki/Black_hole_thermodynamics#Black_hole_entropyhttp://en.wikipedia.org/wiki/Black_hole_thermodynamics#Black_hole_entropyhttp://en.wikipedia.org/wiki/Digital_physicshttp://en.wikipedia.org/wiki/Digital_physicshttp://en.wikipedia.org/wiki/Entropyhttp://en.wikipedia.org/wiki/Entropyhttp://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=7http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=7http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=7http://en.wikipedia.org/wiki/Entropyhttp://en.wikipedia.org/wiki/Digital_physicshttp://en.wikipedia.org/wiki/Black_hole_thermodynamics#Black_hole_entropyhttp://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=6http://en.wikipedia.org/wiki/Human_brainhttp://en.wikipedia.org/wiki/Quantum_statehttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Informationhttp://en.wikipedia.org/wiki/Bithttp://en.wikipedia.org/wiki/Sphere#Volume_of_a_spherehttp://en.wikipedia.org/wiki/Sphere#Volume_of_a_spherehttp://en.wikipedia.org/wiki/Joulehttp://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalencehttp://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalencehttp://en.wikipedia.org/wiki/Kilogramhttp://en.wikipedia.org/wiki/Human_brainhttp://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=5http://en.wikipedia.org/wiki/Covariant_entropy_boundhttp://en.wikipedia.org/wiki/Holographic_principlehttp://en.wikipedia.org/wiki/Black_hole_thermodynamicshttp://en.wikipedia.org/wiki/Planck_areahttp://en.wikipedia.org/wiki/Planck_areahttp://en.wikipedia.org/wiki/Black_holehttp://en.wikipedia.org/wiki/Black_hole_thermodynamics#Black_hole_entropyhttp://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=4http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=3http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Tipler2005-14http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-BoussoEtAl2003-12http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-BoussoEtAl2003-12http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bousso2002-10http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bousso2002-10http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bousso2000-8http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bousso2000-8http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bousso1999-6-6http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bousso1999-6-6http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bekenstein2005-1http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Bekenstein2005-1http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Jacobson1995-4http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_note-Jacobson1995-4http://en.wikipedia.org/wiki/Laws_of_thermodynamicshttp://en.wikipedia.org/wiki/Laws_of_thermodynamicshttp://en.wikipedia.org/wiki/General_relativityhttp://en.wikipedia.org/wiki/General_relativityhttp://en.wikipedia.org/wiki/Einstein_field_equationshttp://en.wikipedia.org/wiki/Theodore_Jacobson


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J. D. Bekenstein,"Black Holes and the Second Law",Lettere al Nuovo Cimento,Vol. 4, No 15 (August 12, 1972), pp. 737-740,doi:10.1007/BF02757029,Bibcode:1972NCimL...4..737B.Mirror link.

Jacob D. Bekenstein,"Black Holes and Entropy",Physical Review D, Vol. 7, No. 8(April 15, 1973), pp. 2333-2346,doi:10.1103/PhysRevD.7.2333,Bibcode:

1973PhRvD...7.2333B.Mirror link. Jacob D. Bekenstein,"Generalized second law of thermodynamics in black-holephysics",Physical Review D, Vol. 9, No. 12 (June 15, 1974), pp. 3292-3300,doi:10.1103/PhysRevD.9.3292,Bibcode:1974PhRvD...9.3292B.Mirror link.

Jacob D. Bekenstein,"Statistical black-hole thermodynamics",Physical Review D,Vol. 12, No. 10 (November 15, 1975), pp. 3077-3085,doi:10.1103/PhysRevD.12.3077,Bibcode:1975PhRvD..12.3077B.Mirror link.

Jacob D. Bekenstein,"Black-hole thermodynamics",Physics Today, Vol. 33, Issue1 (January 1980), pp. 24-31,doi:10.1063/1.2913906,Bibcode:1980PhT....33a..24B.Mirror link.

Jacob D. Bekenstein,"Universal upper bound on the entropy-to-energy ratio forbounded systems",Physical Review D, Vol. 23, No. 2, (January 15, 1981), pp. 287-298,doi:10.1103/PhysRevD.23.287,Bibcode:1981PhRvD..23..287B.Mirror link.

Jacob D. Bekenstein,"Energy Cost of Information Transfer",Physical ReviewLetters, Vol. 46, No. 10 (March 9, 1981), pp. 623-626,doi:10.1103/PhysRevLett.46.623,Bibcode:1981PhRvL..46..623B.Mirror link.

Jacob D. Bekenstein, "Specific entropy and the sign of the energy",Physical ReviewD, Vol. 26, No. 4 (August 15, 1982), pp. 950-953,doi:10.1103/PhysRevD.26.950,Bibcode:1982PhRvD..26..950B.

Jacob D. Bekenstein,"Entropy content and information flow in systems with limitedenergy",Physical Review D, Vol. 30, No. 8, (October 15, 1984), pp. 1669-1679,doi:10.1103/PhysRevD.30.1669,Bibcode:1984PhRvD..30.1669B.Mirror link.

Jacob D. Bekenstein,"Communication and energy",Physical Review A, Vol. 37,Issue 9 (May 1988), pp. 3437-3449,doi:10.1103/PhysRevA.37.3437,Bibcode:1988PhRvA..37.3437B.Mirror link.

Marcelo Schiffer and Jacob D. Bekenstein, "Proof of the quantum bound on specificentropy for free fields",Physical Review D, Vol. 39, Issue 4 (February 15, 1989),pp. 1109-1115,doi:10.1103/PhysRevD.39.1109PMID 9959747,Bibcode:1989PhRvD..39.1109S.

Jacob D. Bekenstein, "Is the Cosmological Singularity ThermodynamicallyPossible?",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 ReviewD, Vol. 49, Issue 4 (February 15, 1994), pp. 1912-1921,doi:10.1103/PhysRevD.49.1912,Bibcode:1994PhRvD..49.1912B. Also atarXiv:gr-qc/9307035, July 25, 1993.

Oleg B. Zaslavskii, "Generalized second law and the Bekenstein entropy bound inGedankenexperiments 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 'Generalizedsecond law and the Bekenstein entropy bound in Gedankenexperiments with black
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4/5

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 thegeneralized second law of thermodynamics",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 atarXiv:gr-qc/9906058, June 16, 1999.

[edit] References

1. âbJacob D. Bekenstein,"Universal upper bound on the entropy-to-energy ratio forbounded systems",Physical Review D, Vol. 23, No. 2, (January 15, 1981), pp. 287-298,doi:10.1103/PhysRevD.23.287,Bibcode:1981PhRvD..23..287B.Mirror link.

2. âbJacob 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 atarXiv:quant-ph/0404042, April 7, 2004.

3. âbJacob D. Bekenstein,"Bekenstein bound",Scholarpedia, Vol. 3, No. 10 (October 31,2008), p. 7374,doi:10.4249/scholarpedia.7374.

4. ^Frank J. Tipler,"The structure of the world from pure numbers",Reports on Progress inPhysics, Vol. 68, No. 4 (April 2005), pp. 897-964,doi:10.1088/0034-4885/68/4/R04,Bibcode:2005RPPh...68..897T, p. 902.Mirror link. Also released as"Feynman-WeinbergQuantum Gravity and the Extended Standard Model as a Theory of Everything",arXiv:0704.3276, April 24, 2007, p. 8.

5. ^Ted Jacobson, "Thermodynamics of Spacetime: The Einstein Equation of State",PhysicalReview Letters, Vol. 75, Issue 7 (August 14, 1995), pp. 1260-1263,doi:10.1103/PhysRevLett.75.1260,Bibcode:1995PhRvL..75.1260J. Also atarXiv:gr-qc/9504004, April 4, 1995. Also availablehereandhere. Additionally available asan entryin theGravity 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,LCCN2007-310371.

7. ^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.Mirror link. Also atarXiv:hep-th/9906022, 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,doi:10.1088/1126-6708/1999/07/004,Bibcode:1999JHEP...07..004B.Mirror link. Also atarXiv:hep-th/9905177, May 24, 1999.

9. ^Raphael Bousso, "The holographic principle for general backgrounds",Classical andQuantum Gravity, Vol. 17, No. 5 (March 7, 2000), pp. 997-1005,doi:10.1088/0264-9381/17/5/309,Bibcode:2000CQGra..17..997B. Also atarXiv:hep-th/9911002, November2, 1999.

10.^Jacob D. Bekenstein, "Holographic bound from second law of thermodynamics",PhysicsLetters 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 atarXiv:hep-th/0003058, March8, 2000.

11.^Raphael Bousso,"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.Mirror link. Also atarXiv:hep-th/0203101, March 12, 2002.

12.^Jacob D. Bekenstein,"Information in the Holographic Universe: Theoretical results aboutblack holes suggest that the universe could be like a gigantic hologram",ScientificAmerican, Vol. 289, No. 2 (August 2003), pp. 58-65.Mirror link.
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er/theoryofeverything.pdfhttp://en.wikipedia.org/wiki/Frank_J._Tiplerhttp://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-Tipler2005b_3-0http://dx.doi.org/10.4249%2Fscholarpedia.7374http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Scholarpediahttp://www.scholarpedia.org/wiki/index.php?title=Bekenstein_bound&oldid=50791http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-Bekenstein2008_2-1http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-Bekenstein2008_2-0http://arxiv.org/abs/quant-ph/0404042http://arxiv.org/abs/quant-ph/0404042http://en.wikipedia.org/wiki/ArXivhttp://adsabs.harvard.edu/abs/2005FoPh...35.1805Bhttp://en.wikipedia.org/wiki/Bibcodehttp://dx.doi.org/10.1007%2Fs10701-005-7350-7http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Foundations_of_Physicshttp://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-Bekenstein2005_1-1http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-Bekenstein2005_1-0http://www.webcitation.org/5pvt5c96Nhttp://adsabs.harvard.edu/abs/1981PhRvD..23..287Bhttp://en.wikipedia.org/wiki/Bibcodehttp://dx.doi.org/10.1103%2FPhysRevD.23.287http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Physical_Reviewhttp://www.phys.huji.ac.il/~bekenste/PRD23-287-1981.pdfhttp://www.phys.huji.ac.il/~bekenste/PRD23-287-1981.pdfhttp://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-Bekenstein1981-1_0-1http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-Bekenstein1981-1_0-0http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=8http://arxiv.org/abs/gr-qc/9906058http://en.wikipedia.org/wiki/ArXivhttp://adsabs.harvard.edu/abs/1999PhRvD..60l4010Bhttp://en.wikipedia.org/wiki/Bibcodehttp://dx.doi.org/10.1103%2FPhysRevD.60.124010http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Physical_Reviewhttp://adsabs.harvard.edu/abs/1996CQGra..13.2607Zhttp://en.wikipedia.org/wiki/Bibcodehttp://dx.doi.org/10.1088%2F0264-9381%2F13%2F9%2F024http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Classical_and_Quantum_Gravity


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13.^Raphael Bousso, anna . Flanagan andDonald Marolf, "Simple sufficient conditions forthe generalized covariant entropy bound", Physical Review D, Vol. 68, Issue 6 (September15, 2003), Art. No. 064001, 7 pages,doi:10.1103/PhysRevD.68.064001,Bibcode:2003PhRvD..68f4001B. Also atarXiv:hep-th/0305149, May 19, 2003.

14.^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 atarXiv:quant-ph/0311049, November 9, 2003. Also atarXiv:quant-ph/0311049, November 9, 2003.

15.^Frank J. Tipler,"The structure of the world from pure numbers",Reports on Progress inPhysics, Vol. 68, No. 4 (April 2005), pp. 897-964,doi:10.1088/0034-4885/68/4/R04,Bibcode:2005RPPh...68..897T.Mirror link. Also released as"Feynman-WeinbergQuantum 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 thatBekenstein's original formulation of the bound is the correct form. See in particular theparagraph beginning with "A few points ..." on p. 903 of theRep. Prog. Phys. paper (or p. 9of the arXiv version), and the discussions on the Bekenstein bound that follow throughoutthe paper.

[edit] External links

Jacob D. Bekenstein,"Bekenstein bound",Scholarpedia, Vol. 3, No. 10 (2008), p.7374,doi:10.4249/scholarpedia.7374.

Jacob D. Bekenstein,"Bekenstein-Hawking entropy",Scholarpedia, Vol. 3, No. 10(2008), p. 7375,doi:10.4249/scholarpedia.7375.

Jacob D. Bekenstein's websiteatthe Racah Institute of Physics,Hebrew Universityof Jerusalem, which contains a number of articles on the Bekenstein bound.
http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-BoussoEtAl2003_12-0http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-BoussoEtAl2003_12-0http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-BoussoEtAl2003_12-0http://en.wikipedia.org/wiki/Donald_Marolfhttp://en.wikipedia.org/wiki/Donald_Marolfhttp://en.wikipedia.org/wiki/Donald_Marolfhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1103%2FPhysRevD.68.064001http://dx.doi.org/10.1103%2FPhysRevD.68.064001http://dx.doi.org/10.1103%2FPhysRevD.68.064001http://en.wikipedia.org/wiki/Bibcodehttp://en.wikipedia.org/wiki/Bibcodehttp://en.wikipedia.org/wiki/Bibcodehttp://adsabs.harvard.edu/abs/2003PhRvD..68f4001Bhttp://adsabs.harvard.edu/abs/2003PhRvD..68f4001Bhttp://en.wikipedia.org/wiki/ArXivhttp://en.wikipedia.org/wiki/ArXivhttp://arxiv.org/abs/hep-th/0305149http://arxiv.org/abs/hep-th/0305149http://arxiv.org/abs/hep-th/0305149http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-Bekenstein2004_13-0http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-Bekenstein2004_13-0http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-Bekenstein2004_13-0http://en.wikipedia.org/wiki/Contemporary_Physicshttp://en.wikipedia.org/wiki/Contemporary_Physicshttp://en.wikipedia.org/wiki/Contemporary_Physicshttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1080%2F00107510310001632523http://dx.doi.org/10.1080%2F00107510310001632523http://dx.doi.org/10.1080%2F00107510310001632523http://en.wikipedia.org/wiki/Bibcodehttp://en.wikipedia.org/wiki/Bibcodehttp://en.wikipedia.org/wiki/Bibcodehttp://adsabs.harvard.edu/abs/2003ConPh..45...31Bhttp://adsabs.harvard.edu/abs/2003ConPh..45...31Bhttp://en.wikipedia.org/wiki/ArXivhttp://en.wikipedia.org/wiki/ArXivhttp://arxiv.org/abs/quant-ph/0311049http://arxiv.org/abs/quant-ph/0311049http://arxiv.org/abs/quant-ph/0311049http://en.wikipedia.org/wiki/ArXivhttp://arxiv.org/abs/quant-ph/0311049http://arxiv.org/abs/quant-ph/0311049http://arxiv.org/abs/quant-ph/0311049http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-Tipler2005_14-0http://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-Tipler2005_14-0http://en.wikipedia.org/wiki/Frank_J._Tiplerhttp://en.wikipedia.org/wiki/Frank_J._Tiplerhttp://en.wikipedia.org/wiki/Frank_J._Tiplerhttp://math.tulane.edu/~tipler/theoryofeverything.pdfhttp://math.tulane.edu/~tipler/theoryofeverything.pdfhttp://math.tulane.edu/~tipler/theoryofeverything.pdfhttp://en.wikipedia.org/wiki/Reports_on_Progress_in_Physicshttp://en.wikipedia.org/wiki/Reports_on_Progress_in_Physicshttp://en.wikipedia.org/wiki/Reports_on_Progress_in_Physicshttp://en.wikipedia.org/wiki/Reports_on_Progress_in_Physicshttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1088%2F0034-4885%2F68%2F4%2FR04http://dx.doi.org/10.1088%2F0034-4885%2F68%2F4%2FR04http://dx.doi.org/10.1088%2F0034-4885%2F68%2F4%2FR04http://en.wikipedia.org/wiki/Bibcodehttp://en.wikipedia.org/wiki/Bibcodehttp://adsabs.harvard.edu/abs/2005RPPh...68..897Thttp://adsabs.harvard.edu/abs/2005RPPh...68..897Thttp://adsabs.harvard.edu/abs/2005RPPh...68..897Thttp://www.webcitation.org/5nx3CxKm0http://www.webcitation.org/5nx3CxKm0http://www.webcitation.org/5nx3CxKm0http://arxiv.org/abs/0704.3276http://arxiv.org/abs/0704.3276http://arxiv.org/abs/0704.3276http://arxiv.org/abs/0704.3276http://en.wikipedia.org/wiki/ArXivhttp://arxiv.org/abs/0704.3276http://arxiv.org/abs/0704.3276http://arxiv.org/abs/0704.3276http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=9http://www.scholarpedia.org/article/Bekenstein_boundhttp://www.scholarpedia.org/article/Bekenstein_boundhttp://www.scholarpedia.org/article/Bekenstein_boundhttp://en.wikipedia.org/wiki/Scholarpediahttp://en.wikipedia.org/wiki/Scholarpediahttp://en.wikipedia.org/wiki/Scholarpediahttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.4249%2Fscholarpedia.7374http://dx.doi.org/10.4249%2Fscholarpedia.7374http://dx.doi.org/10.4249%2Fscholarpedia.7374http://www.scholarpedia.org/article/Bekenstein-Hawking_entropyhttp://www.scholarpedia.org/article/Bekenstein-Hawking_entropyhttp://www.scholarpedia.org/article/Bekenstein-Hawking_entropyhttp://en.wikipedia.org/wiki/Scholarpediahttp://en.wikipedia.org/wiki/Scholarpediahttp://en.wikipedia.org/wiki/Scholarpediahttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.4249%2Fscholarpedia.7375http://dx.doi.org/10.4249%2Fscholarpedia.7375http://dx.doi.org/10.4249%2Fscholarpedia.7375http://www.phys.huji.ac.il/~bekenste/http://www.phys.huji.ac.il/~bekenste/http://en.wikipedia.org/wiki/The_Racah_Institute_of_Physicshttp://en.wikipedia.org/wiki/The_Racah_Institute_of_Physicshttp://en.wikipedia.org/wiki/The_Racah_Institute_of_Physicshttp://en.wikipedia.org/wiki/Hebrew_University_of_Jerusalemhttp://en.wikipedia.org/wiki/Hebrew_University_of_Jerusalemhttp://en.wikipedia.org/wiki/Hebrew_University_of_Jerusalemhttp://en.wikipedia.org/wiki/Hebrew_University_of_Jerusalemhttp://en.wikipedia.org/wiki/Hebrew_University_of_Jerusalemhttp://en.wikipedia.org/wiki/Hebrew_University_of_Jerusalemhttp://en.wikipedia.org/wiki/The_Racah_Institute_of_Physicshttp://www.phys.huji.ac.il/~bekenste/http://dx.doi.org/10.4249%2Fscholarpedia.7375http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Scholarpediahttp://www.scholarpedia.org/article/Bekenstein-Hawking_entropyhttp://dx.doi.org/10.4249%2Fscholarpedia.7374http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Scholarpediahttp://www.scholarpedia.org/article/Bekenstein_boundhttp://en.wikipedia.org/w/index.php?title=Bekenstein_bound&action=edit&section=9http://arxiv.org/abs/0704.3276http://en.wikipedia.org/wiki/ArXivhttp://arxiv.org/abs/0704.3276http://arxiv.org/abs/0704.3276http://www.webcitation.org/5nx3CxKm0http://adsabs.harvard.edu/abs/2005RPPh...68..897Thttp://en.wikipedia.org/wiki/Bibcodehttp://dx.doi.org/10.1088%2F0034-4885%2F68%2F4%2FR04http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Reports_on_Progress_in_Physicshttp://en.wikipedia.org/wiki/Reports_on_Progress_in_Physicshttp://math.tulane.edu/~tipler/theoryofeverything.pdfhttp://en.wikipedia.org/wiki/Frank_J._Tiplerhttp://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-Tipler2005_14-0http://arxiv.org/abs/quant-ph/0311049http://en.wikipedia.org/wiki/ArXivhttp://arxiv.org/abs/quant-ph/0311049http://en.wikipedia.org/wiki/ArXivhttp://adsabs.harvard.edu/abs/2003ConPh..45...31Bhttp://en.wikipedia.org/wiki/Bibcodehttp://dx.doi.org/10.1080%2F00107510310001632523http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Contemporary_Physicshttp://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-Bekenstein2004_13-0http://arxiv.org/abs/hep-th/0305149http://en.wikipedia.org/wiki/ArXivhttp://adsabs.harvard.edu/abs/2003PhRvD..68f4001Bhttp://en.wikipedia.org/wiki/Bibcodehttp://dx.doi.org/10.1103%2FPhysRevD.68.064001http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Donald_Marolfhttp://en.wikipedia.org/wiki/Bekenstein_Bound#cite_ref-BoussoEtAl2003_12-0

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