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{{Short description|Upper limit on entropy in physics}}
[[File:Bekenstein-Hawking entropy of a black hole.svg|thumb|According to the Bekenstein bound, the [[entropy]] of a [[black hole]] is proportional to the number of [[Planck units|Planck areas]] that it would take to cover the black hole's [[event horizon]]. ]]
In [[physics]], the '''Bekenstein bound''' (named after [[Jacob Bekenstein]]) is an upper limit on the [[Entropy (statistical thermodynamics)|thermodynamic entropy]] ''S'', or [[Entropy (Information theory)|Shannon entropy]] ''H'', that can be contained within a given finite region of space which has a finite amount of energy—or equivalently, the maximum amount of information that is required to perfectly describe a given physical system down to the quantum level.<ref name="Bekenstein1981-1">{{Cite journal |last1 = Bekenstein |first1 = Jacob D. |year = 1981 |title = Universal upper bound on the entropy-to-energy ratio for bounded systems |url=http://www.phys.huji.ac.il/~bekenste/PRD23-287-1981.pdf |journal = Physical Review D |volume = 23 |issue = 2 |pages = 287–298 |bibcode = 1981PhRvD..23..287B  |doi = 10.1103/PhysRevD.23.287|s2cid = 120643289 }}</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 are finite.

== Equations ==
The universal form of the bound was originally found by Jacob Bekenstein in 1981 as the [[inequality (mathematics)|inequality]]<ref name="Bekenstein1981-1"/><ref name="Bekenstein2005"/><ref name="Bekenstein2008"/>
<math display="block">S \leq \frac{2 \pi k R E}{\hbar c},</math>
where ''S'' is the [[entropy]], ''k'' is the [[Boltzmann 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 sometimes plays a significant role in its enforcement (e.g., in a black hole), the expression for the bound does not contain the [[gravitational constant]]&nbsp;''G'', so the bound ought to apply to [[quantum field theory in curved spacetime]].

The [[Black hole thermodynamics#Black hole entropy|Bekenstein–Hawking boundary entropy]] of three-dimensional [[black hole]]s exactly saturates the bound. The [[Schwarzschild radius]] is given by
<math display="block">r_{\rm s} = \frac{2 G M}{c^2},</math>
where ''M'' is the mass of the stellar-sized object
And so the two-dimensional area of the black hole's event horizon is
<math display="block">A = 4 \pi r_{\rm s}^2 = {16 \pi G^2 M^2}/{c^4},</math>
and using the [[Planck length]]
<math display="block">l_{\rm P}^2 = \hbar G/c^3,</math>
the Bekenstein–Hawking entropy is
<math display="block">S = \frac{kA}{4 \ l_{\rm P}^2} = \frac{4 \pi k G M^2}{\hbar c}.</math>

One interpretation of the bound makes use of the [[microcanonical ensemble|microcanonical]] formula for entropy,
<math display="block">S = k \log \Omega,</math>
where <math>\Omega</math> is the number of energy [[eigenstate]]s accessible to the system. This is equivalent to saying that the dimension of the [[Hilbert space]] describing the system is<ref name=":0" /><ref>{{cite arXiv|first=G. |last= 't Hooft |title=Dimensional reduction in quantum gravity |eprint=gr-qc/9310026 |date=1993-10-19 |author-link=Gerard 't Hooft }}</ref>
<math display="block">\dim \mathcal{H} = \exp \left(\frac{2\pi R E}{\hbar c}\right).</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.<ref name=":0" />

== 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">{{cite journal |doi=10.1103/PhysRevLett.75.1260 |pmid=10060248 |bibcode=1995PhRvL..75.1260J |arxiv=gr-qc/9504004 |citeseerx=10.1.1.54.6675 |url=http://www.gravityresearchfoundation.org/pdf/awarded/1995/jacobson.pdf |title=Thermodynamics of Spacetime: The Einstein Equation of State |journal=Physical Review Letters |volume=75 |issue=7 |pages=1260–1263 |year=1995 |last1=Jacobson |first1=Ted |s2cid=13223728 |access-date=2010-05-23 |archive-date=2011-10-01 |archive-url=https://web.archive.org/web/20111001230201/http://www.gravityresearchfoundation.org/pdf/awarded/1995/jacobson.pdf }}</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 were 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 was a matter of debate until Horacio Casini's work in 2008.<ref name="Bekenstein2005">{{Cite journal | doi=10.1007/s10701-005-7350-7| title=How does the Entropy/Information Bound Work?| journal=Foundations of Physics| volume=35| issue=11| pages=1805–1823| year=2005| last1=Bekenstein| first1=Jacob D.| bibcode=2005FoPh...35.1805B| arxiv=quant-ph/0404042| s2cid=118942877}}</ref><ref name="Bekenstein2008">{{Cite journal | doi=10.4249/scholarpedia.7374| title=Bekenstein bound| journal=Scholarpedia| volume=3| issue=10| page=7374| year=2008| last1=Bekenstein| first1=Jacob| bibcode=2008SchpJ...3.7374B| doi-access=free}}</ref><ref name="Bousso1999-6">{{Cite journal |doi = 10.1088/1126-6708/1999/06/028|title = Holography in general space-times|journal = Journal of High Energy Physics|volume = 1999|issue = 6|page = 028|year = 1999|last1 = Bousso|first1 = Raphael|bibcode = 1999JHEP...06..028B|arxiv = hep-th/9906022|s2cid = 119518763}}</ref><ref name="Bousso1999-7">{{Cite journal |doi = 10.1088/1126-6708/1999/07/004|title = A covariant entropy conjecture|journal = Journal of High Energy Physics|volume = 1999|issue = 7|page = 004|year = 1999|last1 = Bousso|first1 = Raphael|bibcode = 1999JHEP...07..004B|arxiv = hep-th/9905177|s2cid = 9545752}}</ref><ref name="Bousso2000">{{Cite journal |doi = 10.1088/0264-9381/17/5/309|title = The holographic principle for general backgrounds|journal = Classical and Quantum Gravity|volume = 17|issue = 5|pages = 997–1005|year = 2000|last1 = Bousso|first1 = Raphael|bibcode = 2000CQGra..17..997B|arxiv = hep-th/9911002|s2cid = 14741276}}</ref><ref name="Bekenstein2000">{{Cite journal |doi = 10.1016/S0370-2693(00)00450-0|title = Holographic bound from second law of thermodynamics|journal = Physics Letters B|volume = 481|issue = 2–4|pages = 339–345|year = 2000|last1 = Bekenstein|first1 = Jacob D.|bibcode = 2000PhLB..481..339B|arxiv = hep-th/0003058|s2cid = 119427264}}</ref><ref name="Bousso2002">{{cite journal|doi=10.1103/RevModPhys.74.825|url=http://bib.tiera.ru/DVD-005/Bousso_R._The_holographic_principle_(2002)(en)(50s).pdf|title=The holographic principle|journal=Reviews of Modern Physics|volume=74|issue=3|pages=825–874|year=2002|last1=Bousso|first1=Raphael|bibcode=2002RvMP...74..825B|arxiv=hep-th/0203101|s2cid=55096624|access-date=2010-05-23|archive-date=2011-08-12|archive-url=https://web.archive.org/web/20110812102152/http://bib.tiera.ru/DVD-005/Bousso_R._The_holographic_principle_(2002)(en)(50s).pdf}}</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. [https://web.archive.org/web/20100703084644/http://www.phys.huji.ac.il/~bekenste/Holographic_Univ.pdf Mirror link].</ref><ref name="BoussoEtAl2003">{{Cite journal |doi = 10.1103/PhysRevD.68.064001|title = Simple sufficient conditions for the generalized covariant entropy bound|journal = Physical Review D|volume = 68|issue = 6|article-number = 064001|year = 2003|last1 = Bousso|first1 = Raphael|last2 = Flanagan|first2 = Éanna É.|last3 = Marolf|first3 = Donald|bibcode = 2003PhRvD..68f4001B|arxiv = hep-th/0305149|s2cid = 119049155}}</ref><ref name="Bekenstein2004">{{Cite journal | doi=10.1080/00107510310001632523| title=Black holes and information theory| journal=Contemporary Physics| volume=45| issue=1| pages=31–43| year=2004| last1=Bekenstein| first1=Jacob D.| bibcode=2004ConPh..45...31B| arxiv=quant-ph/0311049| s2cid=118970250}}</ref><ref name="Tipler2005">{{cite journal |doi=10.1088/0034-4885/68/4/R04 |url=http://math.tulane.edu/~tipler/theoryofeverything.pdf|title=The structure of the world from pure numbers|journal=Reports on Progress in Physics|volume=68|issue=4|pages=897–964|year=2005|last1=Tipler|first1=F. J.|bibcode=2005RPPh...68..897T|arxiv=0704.3276|s2cid=119620977 }}. 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>

The following is a heuristic derivation that shows <math>S \leq K{kRE}/{\hbar c} </math> for some constant {{tmath|1= K }}. Showing that <math>K = 2\pi</math> requires a more technical analysis.

Suppose we have a black hole of mass {{tmath|1= M }}, then the [[Schwarzschild radius]] of the black hole is {{tmath|1= R_\text{bh} \sim {GM}/{c^2} }}, and the Bekenstein–Hawking entropy of the black hole is {{tmath|1= \sim \frac{kc^3 R_\text{bh}^2}{\hbar G} \sim {kGM^2}/{\hbar c} }}.

Now take a box of energy {{tmath|1= E }}, entropy {{tmath|1= S }}, and side length {{tmath|1= R }}. If we throw the box into the black hole, the mass of the black hole goes up to {{tmath|1= M+{E}/{c^2} }}, and the entropy goes up by {{tmath|1= {kGME}/{\hbar c^3} }}. Since entropy does not decrease, {{tmath|1= {kGME}/{\hbar c^3}\gtrsim S }}.

In order for the box to fit inside the black hole, {{tmath|1= R \lesssim {GM}/{c^2} }}. If the two are comparable, {{tmath|1= R \sim {GM}/{c^2} }}, then we have derived the BH bound: {{tmath|1= S \lesssim {kRE}/{\hbar c} }}.

== Proof in quantum field theory ==

A proof of the Bekenstein bound in the framework of [[quantum field theory]] was given in 2008 by Casini.<ref>{{cite journal |last1=Casini |first1=Horacio |title=Relative entropy and the Bekenstein bound |journal= Classical and Quantum Gravity|date=2008 |volume=25 |issue=20 |article-number=205021 |doi=10.1088/0264-9381/25/20/205021 |arxiv=0804.2182 |bibcode=2008CQGra..25t5021C |s2cid=14456556 }}</ref> One of the crucial insights of the proof was to find a proper interpretation of the quantities appearing on both sides of the bound.

Naive definitions of entropy and energy density in Quantum Field Theory suffer from [[ultraviolet divergence]]s. In the case of the Bekenstein bound, ultraviolet divergences can be avoided by taking differences between quantities computed in an excited state and the same quantities computed in the [[Quantum vacuum state|vacuum state]]. For example, given a spatial region {{tmath|1= V }}, Casini defines the entropy on the left-hand side of the Bekenstein bound as 
<math display="block">S_V = S(\rho_V) - S(\rho^0_V) = - \mathrm{tr}(\rho_V \log \rho_V) + \mathrm{tr}(\rho_V^0 \log \rho_V^0)</math>
where <math>S(\rho_V)</math> is the [[Von Neumann entropy]] of the [[Density matrix#Systems and subsystems|reduced density matrix]] <math>\rho_V</math> associated with <math>V</math> in the excited state {{tmath|1= \rho }}, and <math>S(\rho^0_V)</math> is the corresponding Von Neumann entropy for the vacuum state {{tmath|1= \rho^0 }}.

On the right-hand side of the Bekenstein bound, a difficult point is to give a rigorous interpretation of the quantity {{tmath|1= 2\pi R E }}, where <math>R</math> is a characteristic length scale of the system and <math>E</math> is a characteristic energy. This product has the same units as the generator of a [[Lorentz transformation|Lorentz boost]], and the natural analog of a boost in this situation is the [[modular Hamiltonian]] of the vacuum state {{tmath|1= K=-\log \rho_V^0 }}. Casini defines the right-hand side of the Bekenstein bound as the difference between the expectation value of the modular Hamiltonian in the excited state and the vacuum state,
<math display="block"> K_V = \mathrm{tr}(K \rho_V) - \mathrm{tr}(K \rho^0_V) .</math>

With these definitions, the bound reads
<math display="block"> S_V \leq K_V ,</math>
which can be rearranged to give
<math display="block">\mathrm{tr}(\rho_V \log \rho_V) - \mathrm{tr}(\rho_V \log \rho_V^0) \geq 0 .</math>

This is simply the statement of positivity of [[quantum relative entropy]], which proves the Bekenstein bound.

However, the modular Hamiltonian can only be interpreted as a weighted form of energy for [[Conformal field theory|conformal field theories]], and when <math>V</math> is a sphere.

This construction allows us to make sense of the [[Casimir effect]]<ref name=":0">{{Cite journal |last=Bousso |first=Raphael |date=2004-02-12 |title=Bound states and the Bekenstein bound |journal=Journal of High Energy Physics |volume=2004 |issue=2 |page=025 |doi=10.1088/1126-6708/2004/02/025 |arxiv=hep-th/0310148 |bibcode=2004JHEP...02..025B |s2cid=17662307 |issn=1029-8479}}</ref> where the localized energy density is ''lower'' than that of the vacuum, i.e. a ''negative'' localized energy. The localized entropy of the vacuum is nonzero, and so, the Casimir effect is possible for states with a lower localized entropy than that of the vacuum. [[Hawking radiation]] can be explained by dumping localized negative energy into a black hole.

== See also ==

* [[Margolus–Levitin theorem]]
* [[Landauer's principle]]
* [[Bremermann's limit]]
* [[Kolmogorov complexity]]
* [[Black hole thermodynamics#Beyond black holes|Beyond black holes]]
* [[Digital physics]]
* [[Limits of computation]]
* [[Chandrasekhar limit]]

== References ==
{{reflist}}

== External links ==
* 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|doi-access=free}}.
* [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.
* {{cite web |first=Matt |last=O'Dowd |author-link=Matt O'Dowd (astrophysicist) |title=How Much Information is in the Universe? |date=September 12, 2018 |work=[[PBS Space Time]] |url=https://www.youtube.com/watch?v=XxVlGAFX7vA&list=PLsPUh22kYmNCHVpiXDJyAcRJ8gluQtOJR&index=5  |archive-url=https://ghostarchive.org/varchive/youtube/20211212/XxVlGAFX7vA| archive-date=2021-12-12 |url-status=live|via=[[YouTube]] }}{{cbignore}}

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