Inphysics, theBekenstein bound (named afterJacob Bekenstein) is an upper limit on thethermodynamic entropyS, orShannon entropyH, 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.^[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 are finite.

The universal form of the bound was originally found by Jacob Bekenstein in 1981 as theinequality^[1][2][3]$${\displaystyle S\le \frac{2\pi kRE}{\hslash c},}$$whereS is theentropy,k is theBoltzmann constant,R is theradius of asphere that can enclose the given system,E is the totalmass–energy including anyrest masses,ħ is thereduced Planck constant, andc is thespeed 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 thegravitational constant G, so the bound ought to apply toquantum field theory in curved spacetime.

TheBekenstein–Hawking boundary entropy of three-dimensionalblack holes exactly saturates the bound. TheSchwarzschild radius is given by$${\displaystyle r_{\mathrm{s}}=\frac{2GM}{c^2},}$$whereM is the mass of the stellar-sized objectAnd so the two-dimensional area of the black hole's event horizon is$${\displaystyle A=4\pi r_{\mathrm{s}}^2=16\pi G^2{M}^2/c^4,}$$and using thePlanck length$${\displaystyle l_{\mathrm{P}}^2=\hslash G/c^3,}$$the Bekenstein–Hawking entropy is$${\displaystyle S=\frac{kA}{4l_{\mathrm{P}}^2}=\frac{4\pi kG{M}^2}{\hslash c}.}$$

One interpretation of the bound makes use of themicrocanonical formula for entropy,$${\displaystyle S=k\log \mathrm{\Omega },}$$where${\displaystyle \mathrm{\Omega }}$ is the number of energyeigenstates accessible to the system. This is equivalent to saying that the dimension of theHilbert space describing the system is^[4][5]$${\displaystyle \dim \mathcal{H}=\exp \left(\frac{2\pi RE}{\hslash c}\right).}$$

The bound is closely associated withblack hole thermodynamics, theholographic principle and thecovariant entropy bound of quantum gravity, and can be derived from a conjectured strong form of the latter.^[4]

Bekenstein derived the bound from heuristic arguments involvingblack 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 thesecond law of thermodynamics by lowering it into a black hole. In 1995,Ted Jacobson demonstrated that theEinstein field equations (i.e.,general relativity) can be derived by assuming that the Bekenstein bound and thelaws of thermodynamics are true.^[6][7] 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.^{[2][3][8][9][10][11][12][13][14][15][16]}

The following is a heuristic derivation that shows${\displaystyle S\le KkRE/\hslash c}$ for some constant⁠${\displaystyle K}$⁠. Showing that${\displaystyle K=2\pi }$ requires a more technical analysis.

Suppose we have a black hole of mass⁠${\displaystyle M}$⁠, then theSchwarzschild radius of the black hole is⁠${\displaystyle R_{\text{bh}}\sim GM/c^2}$⁠, and the Bekenstein–Hawking entropy of the black hole is⁠${\displaystyle \sim \frac{k{c}^3{R}_{\text{bh}}^2}{\hslash G}\sim kG{M}^2/\hslash c}$⁠.

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

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

A proof of the Bekenstein bound in the framework ofquantum field theory was given in 2008 by Casini.^[17] 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 fromultraviolet divergences. 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 thevacuum state. For example, given a spatial region⁠${\displaystyle V}$⁠, Casini defines the entropy on the left-hand side of the Bekenstein bound as$${\displaystyle S_V=S({\rho }_V)-S({\rho }_V^0)=-\mathrm{t}\mathrm{r}({\rho }_V\log {\rho }_V)+\mathrm{t}\mathrm{r}({\rho }_V^0\log {\rho }_V^0)}$$where${\displaystyle S({\rho }_V)}$ is theVon Neumann entropy of thereduced density matrix${\displaystyle {\rho }_V}$ associated with${\displaystyle V}$ in the excited state⁠${\displaystyle \rho }$⁠, and${\displaystyle S({\rho }_V^0)}$ is the corresponding Von Neumann entropy for the vacuum state⁠${\displaystyle {\rho }^0}$⁠.

On the right-hand side of the Bekenstein bound, a difficult point is to give a rigorous interpretation of the quantity⁠${\displaystyle 2\pi RE}$⁠, where${\displaystyle R}$ is a characteristic length scale of the system and${\displaystyle E}$ is a characteristic energy. This product has the same units as the generator of aLorentz boost, and the natural analog of a boost in this situation is themodular Hamiltonian of the vacuum state⁠${\displaystyle K=-\log {\rho }_V^0}$⁠. 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,$${\displaystyle K_V=\mathrm{t}\mathrm{r}(K{\rho }_V)-\mathrm{t}\mathrm{r}(K{\rho }_V^0).}$$

With these definitions, the bound reads$${\displaystyle S_V\le K_V,}$$which can be rearranged to give$${\displaystyle \mathrm{t}\mathrm{r}({\rho }_V\log {\rho }_V)-\mathrm{t}\mathrm{r}({\rho }_V\log {\rho }_V^0)\ge 0.}$$

This is simply the statement of positivity ofquantum relative entropy, which proves the Bekenstein bound.

However, the modular Hamiltonian can only be interpreted as a weighted form of energy forconformal field theories, and when${\displaystyle V}$ is a sphere.

This construction allows us to make sense of theCasimir effect^[4] where the localized energy density islower than that of the vacuum, i.e. anegative 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.

• Margolus–Levitin theorem
• Landauer's principle
• Bremermann's limit
• Kolmogorov complexity
• Beyond black holes
• Digital physics
• Limits of computation
• Chandrasekhar limit

• Jacob D. Bekenstein,"Bekenstein-Hawking entropy",Scholarpedia, Vol. 3, No. 10 (2008), p. 7375,doi:10.4249/scholarpedia.7375.
• Jacob D. Bekenstein's website atthe Racah Institute of Physics,Hebrew University of Jerusalem, which contains a number of articles on the Bekenstein bound.
• O'Dowd, Matt (September 12, 2018)."How Much Information is in the Universe?".PBS Space Time.Archived from the original on 2021-12-12 – viaYouTube.

• Limits of computation
• Thermodynamic entropy
• Quantum information science
• Black holes

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