Bekenstein bound
In physics, the Bekenstein bound (named after Jacob Bekenstein) is an upper limit on the thermodynamic entropy S, or Shannon entropy H, 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 that is required to perfectly describe a given physical system down to the quantum level.<ref name="Bekenstein1981-1">Template:Cite journal</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.
EquationsEdit
The universal form of the bound was originally found by Jacob Bekenstein in 1981 as the 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 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 the gravitational constant G, and so, it ought to apply to quantum field theory in curved spacetime.
The Bekenstein–Hawking boundary entropy of three-dimensional black holes exactly saturates the bound. The Schwarzschild radius is given by <math display="block">r_{\rm s} = \frac{2 G M}{c^2},</math> 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 formula for entropy, <math display="block">S = k \log \Omega,</math> where <math>\Omega</math> is the number of energy eigenstates accessible to the system. This is equivalent to saying that the dimension of the Hilbert space describing the system is<ref name=":0" /><ref>Template:Cite arXiv</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" />
OriginsEdit
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.<ref name="Jacobson1995">Template:Cite journal</ref><ref name="Smolin2002">Lee Smolin, Three Roads to Quantum Gravity (New York, N.Y.: Basic Books, 2002), pp. 173 and 175, Template:ISBN, Template:LCCN.</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 Casini's work in 2008.<ref name="Bekenstein2005">Template:Cite journal</ref><ref name="Bekenstein2008">Template:Cite journal</ref><ref name="Bousso1999-6">Template:Cite journal</ref><ref name="Bousso1999-7">Template:Cite journal</ref><ref name="Bousso2000">Template:Cite journal</ref><ref name="Bekenstein2000">Template:Cite journal</ref><ref name="Bousso2002">Template:Cite journal</ref><ref name="Bekenstein2003">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.</ref><ref name="BoussoEtAl2003">Template:Cite journal</ref><ref name="Bekenstein2004">Template:Cite journal</ref><ref name="Tipler2005">Template:Cite journal. 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 Template:Tmath. Showing that <math>K = 2\pi</math> requires a more technical analysis.
Suppose we have a black hole of mass Template:Tmath, then the Schwarzschild radius of the black hole is Template:Tmath, and the Bekenstein–Hawking entropy of the black hole is Template:Tmath.
Now take a box of energy Template:Tmath, entropy Template:Tmath, and side length Template:Tmath. If we throw the box into the black hole, the mass of the black hole goes up to Template:Tmath, and the entropy goes up by Template:Tmath. Since entropy does not decrease, Template:Tmath.
In order for the box to fit inside the black hole, Template:Tmath. If the two are comparable, Template:Tmath, then we have derived the BH bound: Template:Tmath.
Proof in quantum field theoryEdit
A proof of the Bekenstein bound in the framework of quantum field theory was given in 2008 by Casini.<ref>Template:Cite journal</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 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 the vacuum state. For example, given a spatial region Template:Tmath, 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 reduced density matrix <math>\rho_V</math> associated with <math>V</math> in the excited state Template:Tmath, and <math>S(\rho^0_V)</math> is the corresponding Von Neumann entropy for the vacuum state Template:Tmath.
On the right-hand side of the Bekenstein bound, a difficult point is to give a rigorous interpretation of the quantity Template:Tmath, 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 boost, and the natural analog of a boost in this situation is the modular Hamiltonian of the vacuum state Template:Tmath. 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 theories, and when <math>V</math> is a sphere.
This construction allows us to make sense of the Casimir effect<ref name=":0">Template:Cite journal</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 alsoEdit
- Margolus–Levitin theorem
- Landauer's principle
- Bremermann's limit
- Kolmogorov complexity
- Beyond black holes
- Digital physics
- Limits of computation
- Chandrasekhar limit
ReferencesEdit
External linksEdit
- Jacob D. Bekenstein, "Bekenstein-Hawking entropy", Scholarpedia, Vol. 3, No. 10 (2008), p. 7375, {{#invoke:doi|main}}.
- 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.
- {{#invoke:citation/CS1|citation
|CitationClass=web }}Template:Cbignore