
In
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, the Bekenstein bound (named after
Jacob Bekenstein
Jacob David Bekenstein (; May 1, 1947 – August 16, 2015) was a Mexican-born American-Israeli theoretical physicist who made fundamental contributions to the foundation of black hole thermodynamics and to other aspects of the connections betwee ...
) is an upper limit on the
thermodynamic entropy
In classical thermodynamics, entropy () is a property of a thermodynamic system that expresses the direction or outcome of spontaneous changes in the system. The term was introduced by Rudolf Clausius in the mid-19th century to explain the relati ...
''S'', or
Shannon entropy
Shannon may refer to:
People
* Shannon (given name)
* Shannon (surname)
* Shannon (American singer), stage name of singer Brenda Shannon Greene (born 1958)
* Shannon (South Korean singer), British-South Korean singer and actress Shannon Arrum ...
''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.
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
where ''S'' is the
entropy
Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
, ''k'' is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
, ''R'' is the
radius
In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
of a
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
that can enclose the given system, ''E'' is the total
mass–energy including any
rest masses, ''ħ'' is the
reduced Planck constant
The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
, and ''c'' is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
. Note that while gravity plays a significant role in its enforcement, the expression for the bound does not contain the
gravitational constant
The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
''G'', and so, it ought to apply to
quantum field theory in curved spacetime
In theoretical physics, quantum field theory in curved spacetime (QFTCS) is an extension of quantum field theory from Minkowski spacetime to a general curved spacetime. This theory uses a semi-classical approach; it treats spacetime as a fixed ...
.
The
Bekenstein–Hawking boundary entropy of three-dimensional
black hole
A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
s exactly saturates the bound. The
Schwarzschild radius
The Schwarzschild radius is a parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius of a sphere in flat space that has the same surface area as that of the event horizon of a Schwarzschild black ho ...
is given by
and so the two-dimensional area of the black hole's event horizon is
and using the
Planck length
In particle physics and physical cosmology, Planck units are a system of units of measurement defined exclusively in terms of four universal physical constants: '' c'', '' G'', '' ħ'', and ''k''B (described further below). Expressing one of ...
the Bekenstein–Hawking entropy is
One interpretation of the bound makes use of the
microcanonical formula for entropy,
where
is the number of energy
eigenstate
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
s accessible to the system. This is equivalent to saying that the dimension of the
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
describing the system is
The bound is closely associated with
black hole thermodynamics
In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. As the study of the statistical mechanics of black-body radiation led to the deve ...
, the
holographic principle and the
covariant entropy bound of quantum gravity, and can be derived from a conjectured strong form of the latter.
Origins
Bekenstein derived the bound from heuristic arguments involving
black hole
A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
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
The second law of thermodynamics is a physical law based on Universal (metaphysics), universal empirical observation concerning heat and Energy transformation, energy interconversions. A simple statement of the law is that heat always flows spont ...
by lowering it into a black hole. In 1995,
Ted Jacobson demonstrated that the
Einstein field equations
In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...
(i.e.,
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
) can be derived by assuming that the Bekenstein bound and the
laws of thermodynamics
The laws of thermodynamics are a set of scientific laws which define a group of physical quantities, such as temperature, energy, and entropy, that characterize thermodynamic systems in thermodynamic equilibrium. The laws also use various param ...
are true.
Lee Smolin
Lee Smolin (; born June 6, 1955) is an American theoretical physicist, a faculty member at the Perimeter Institute for Theoretical Physics, an adjunct professor of physics at the University of Waterloo, and a member of the graduate faculty of th ...
, '' Three Roads to Quantum Gravity'' (New York, N.Y.: Basic Books
Basic Books is a book publisher founded in 1950 and located in New York City, now an imprint of Hachette Book Group. It publishes books in the fields of psychology, philosophy, economics, science, politics, sociology, current affairs, and his ...
, 2002), pp. 173 and 175, , . 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.
[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
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it, with more than 150 Nobel Pri ...
'', Vol. 289, No. 2 (August 2003), pp. 58-65
Mirror link
[. 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.]
The following is a heuristic derivation that shows
for some constant . Showing that
requires a more technical analysis.
Suppose we have a black hole of mass , then the
Schwarzschild radius
The Schwarzschild radius is a parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius of a sphere in flat space that has the same surface area as that of the event horizon of a Schwarzschild black ho ...
of the black hole is , and the Bekenstein–Hawking entropy of the black hole is .
Now take a box of energy , entropy , and side length . If we throw the box into the black hole, the mass of the black hole goes up to , and the entropy goes up by . Since entropy does not decrease, .
In order for the box to fit inside the black hole, . If the two are comparable, , then we have derived the BH bound: .
Proof in quantum field theory
A proof of the Bekenstein bound in the framework of
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
was given in 2008 by Casini.
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
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. However, the quantum vacuum is not a simple ...
. For example, given a spatial region , Casini defines the entropy on the left-hand side of the Bekenstein bound as
where
is the
Von Neumann entropy
In physics, the von Neumann entropy, named after John von Neumann, is a measure of the statistical uncertainty within a description of a quantum system. It extends the concept of Gibbs entropy from classical statistical mechanics to quantum statis ...
of the
reduced density matrix
Quantum entanglement is the phenomenon where the quantum state of each particle in a group cannot be described independently of the state of the others, even when the particles are separated by a large distance. The topic of quantum entangleme ...
associated with
in the excited state , and
is the corresponding Von Neumann entropy for the vacuum state .
On the right-hand side of the Bekenstein bound, a difficult point is to give a rigorous interpretation of the quantity , where
is a characteristic length scale of the system and
is a characteristic energy. This product has the same units as the generator of a
Lorentz boost
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation ...
, and the natural analog of a boost in this situation is the
modular Hamiltonian of the vacuum state . 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,
With these definitions, the bound reads
which can be rearranged to give
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
is a sphere.
This construction allows us to make sense of the
Casimir effect
In quantum field theory, the Casimir effect (or Casimir force) is a physical force (physics), force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of a field (physics), field. The term Casim ...
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
Hawking radiation is black-body radiation released outside a black hole's event horizon due to quantum effects according to a model developed by Stephen Hawking in 1974.
The radiation was not predicted by previous models which assumed that onc ...
can be explained by dumping localized negative energy into a black hole.
See also
*
Margolus–Levitin theorem
*
Landauer's principle
Landauer's principle is a physical principle pertaining to a lower theoretical limit of energy consumption of computation. It holds that an irreversible change in information stored in a computer, such as merging two computational paths, dissipa ...
*
Bremermann's limit
*
Kolmogorov complexity
In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that prod ...
*
Beyond black holes
*
Digital physics
Digital physics is a speculative idea suggesting that the universe can be conceived of as a vast, digital computation device, or as the output of a deterministic or probabilistic computer program. The hypothesis that the universe is a digital com ...
*
Limits of computation
The limits of computation are governed by a number of different factors. In particular, there are several physical and practical limits to the amount of computation or data storage that can be performed with a given amount of mass, volume, or ener ...
*
Chandrasekhar limit
The Chandrasekhar limit () is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about (). The limit was named after Subrahmanyan Chandrasekhar.
White dwarfs resist gravitational collapse pr ...
References
External links
* Jacob D. Bekenstein
"Bekenstein-Hawking entropy" ''
Scholarpedia
''Scholarpedia'' is an English-language wiki-based online encyclopedia with features commonly associated with Open access (publishing), open-access online academic journals, which aims to have quality content in science and medicine.
''Scholarpe ...
'', Vol. 3, No. 10 (2008), p. 7375, .
Jacob D. Bekenstein's websiteat
the Racah Institute of Physics,
Hebrew University of Jerusalem
The Hebrew University of Jerusalem (HUJI; ) is an Israeli public university, public research university based in Jerusalem. Co-founded by Albert Einstein and Chaim Weizmann in July 1918, the public university officially opened on 1 April 1925. ...
, which contains a number of articles on the Bekenstein bound.
*
{{DEFAULTSORT:Bekenstein Bound
Limits of computation
Thermodynamic entropy
Quantum information science
Black holes