Bekenstein bound

According to the Bekenstein bound, the entropy of a black hole is proportional to the number of 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 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 maximal 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 are finite. In computer science, this implies that there is a maximal 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 in 1981 as the inequality^[1][2][3]

${\displaystyle S\leq {\frac {2\pi kRE}{\hbar c}},}$

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 the gravitational constant G.

In informational terms, the relation between thermodynamic entropy S and Shannon entropy H is given by

${\displaystyle S=kH\ln 2,}$

whence

${\displaystyle H\leq {\frac {2\pi RE}{\hbar c\ln 2}},}$

where H is the Shannon entropy 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

${\displaystyle H\leq {\frac {2\pi cRM}{\hbar \ln 2}}\approx 2.5769082\times 10^{43}\ {\frac {\text{bit}}{{\text{kg}}\cdot {\text{m}}}}\cdot M\cdot R,}$

where ${\displaystyle M}$ is the mass (in kg), and ${\displaystyle R}$ is the radius (in meter) 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 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.^{[2][3][7][8][9][10][11][12][13][14][15]}

Proof in quantum field theory[]

A proof of the Bekenstein bound in the framework of quantum field theory was given in 2008 by Casini.^[16] 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 ${\displaystyle V}$, Casini defines the entropy on the left-hand side of the Bekenstein bound as

${\displaystyle S_{V}=S(\rho _{V})-S(\rho _{V}^{0})=-\mathrm {tr} (\rho _{V}\log \rho _{V})+\mathrm {tr} (\rho _{V}^{0}\log \rho _{V}^{0})}$

where ${\displaystyle S(\rho _{V})}$ is the Von Neumann entropy of the reduced 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 a Lorentz boost, and the natural analog of a boost in this situation is the 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,

${\displaystyle K_{V}=\mathrm {tr} (K\rho _{V})-\mathrm {tr} (K\rho _{V}^{0}).}$

With these definitions, the bound reads

${\displaystyle S_{V}\leq K_{V},}$

which can be rearranged to give

${\displaystyle \mathrm {tr} (\rho _{V}\log \rho _{V})-\mathrm {tr} (\rho _{V}\log \rho _{V}^{0})\geq 0.}$

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

Examples[]

Black holes[]

It happens that the Bekenstein–Hawking boundary entropy of three-dimensional black holes exactly saturates the bound

${\displaystyle r_{\rm {s}}={\frac {2GM}{c^{2}}},}$

${\displaystyle A=4\pi r_{\rm {s}}^{2}={\frac {16\pi G^{2}M^{2}}{c^{4}}},}$

${\displaystyle l_{\rm {P}}^{2}=\hbar G/c^{3},}$

${\displaystyle S={\frac {kA}{4\ l_{\rm {P}}^{2}}}={\frac {4\pi kGM^{2}}{\hbar c}},}$

where ${\displaystyle k}$ is Boltzmann's constant, A is the two-dimensional area of the black hole's event horizon and ${\displaystyle l_{\rm {P}}}$ is the Planck length.

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 about 2.6×10⁴² bits and represents the maximal information needed to perfectly recreate an average human brain down to the quantum level. This means that the number ${\displaystyle O=2^{I}}$ of states of the human brain must be less than ${\displaystyle \approx 10^{7.8\times 10^{41}}}$.

See also[]