Bekenstein bound
Encyclopedia
In physics
, the Bekenstein bound is an upper limit on the entropy
S, or information
I, 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 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 is finite. In computer science
, this implies that there is a maximum information-processing rate
for a physical system that has a finite size and energy, and that a Turing machine
with unbounded memory is not physically possible.
as the inequality
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 bound is independent of Newton's Constant
G.
In informational terms, the bound is given by
where I is the information
expressed in number of bit
s 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. The right-hand side
of the foregoing relation
is approximately equal to 2.5769087×1043×(mass in kilograms)×(radius in meters).
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, 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. However, while a number of arguments have been 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 has been a matter of debate.
s exactly saturates the bound
where A is the two-dimensional area of the black hole's event horizon in units of the Planck area,
.
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.
has a mass of 1.5 kg
and a volume of 1260 cm3. The energy (E = mc2) will be 1.34813×1017 J
and if the brain is approximate to a sphere then the radius (V = 4πr3/3) will be 6.70030×10−2 m.
The Bekenstein bound (I ≤ ) will be 2.58991×1042 bit
and represent the maximum information
needed to perfectly recreate the average human brain down to the quantum level
(or anything of the same mass and volume). This implies that the number of different states (Ω=2I) of the human brain
is at most 107.79640×1041.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the Bekenstein bound is an upper limit on the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
S, or information
Information
Information in its most restricted technical sense is a message or collection of messages that consists of an ordered sequence of symbols, or it is the meaning that can be interpreted from such a message or collection of messages. Information can be recorded or transmitted. It can be recorded as...
I, 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 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 is finite. In computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, this implies that there is a maximum information-processing rate
Limits to computation
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 energy:...
for a physical system that has a finite size and energy, and that a Turing machine
Turing machine
A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...
with unbounded memory is not physically possible.
Equations
The universal form of the bound was originally found by Jacob BekensteinJacob Bekenstein
Jacob David Bekenstein is an Israeli theoretical physicist who has contributed to the foundation of black hole thermodynamics and to other aspects of the connections between information and gravitation.-Biography:...
as the inequality
where S is the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
, k is Boltzmann's constant, R is the radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
of a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
that can enclose the given system, E is the total mass-energy including any rest masses
Invariant mass
The invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...
, ħ is the reduced Planck constant, and c is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
. Note that while gravity plays a significant role in its enforcement, the bound is independent of Newton's Constant
Gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
G.
In informational terms, the bound is given by
where I is the information
Information
Information in its most restricted technical sense is a message or collection of messages that consists of an ordered sequence of symbols, or it is the meaning that can be interpreted from such a message or collection of messages. Information can be recorded or transmitted. It can be recorded as...
expressed in number of bit
Bit
A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...
s contained in the quantum states in the sphere. The ln
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
2 factor comes from defining the information as the logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
to the base
Radix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...
2
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
of the number of quantum states. The right-hand side
Sides of an equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric...
of the foregoing relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
is approximately equal to 2.5769087×1043×(mass in kilograms)×(radius in meters).
Origins
Bekenstein derived the bound from heuristic arguments involving black holeBlack hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
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
Second law of thermodynamics
The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and...
by lowering it into a black hole. In 1995, Ted Jacobson
Theodore Jacobson
Theodore A. "Ted" Jacobson is an American theoretical physicist. He is known for his work on the connection between gravity and thermodynamics. In particular, in 1995 Jacobson proved that the Einstein field equations describing relativistic gravity can be derived from thermodynamic...
demonstrated that the Einstein field equations
Einstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
(i.e., general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
) can be derived by assuming that the Bekenstein bound and the laws of thermodynamics
Laws of thermodynamics
The four laws of thermodynamics summarize its most important facts. They define fundamental physical quantities, such as temperature, energy, and entropy, in order to describe thermodynamic systems. They also describe the transfer of energy as heat and work in thermodynamic processes...
are true. However, while a number of arguments have been 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 has been a matter of debate.
Black holes
It happens that the Bekenstein-Hawking Entropy of three-dimensional black holeBlack hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
s exactly saturates the bound
where A is the two-dimensional area of the black hole's event horizon in units of the Planck area,
.The bound is closely associated with black hole thermodynamics
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...
, the holographic principle
Holographic principle
The holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon...
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 brainHuman brain
The human brain has the same general structure as the brains of other mammals, but is over three times larger than the brain of a typical mammal with an equivalent body size. Estimates for the number of neurons in the human brain range from 80 to 120 billion...
has a mass of 1.5 kg
Kilogram
The kilogram or kilogramme , also known as the kilo, is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype Kilogram , which is almost exactly equal to the mass of one liter of water...
and a volume of 1260 cm3. The energy (E = mc2) will be 1.34813×1017 J
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...
and if the brain is approximate to a sphere then the radius (V = 4πr3/3) will be 6.70030×10−2 m.
The Bekenstein bound (I ≤ ) will be 2.58991×1042 bit
Bit
A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...
and represent the maximum information
Information
Information in its most restricted technical sense is a message or collection of messages that consists of an ordered sequence of symbols, or it is the meaning that can be interpreted from such a message or collection of messages. Information can be recorded or transmitted. It can be recorded as...
needed to perfectly recreate the average human brain down to the quantum level
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
(or anything of the same mass and volume). This implies that the number of different states (Ω=2I) of the human brain
Human brain
The human brain has the same general structure as the brains of other mammals, but is over three times larger than the brain of a typical mammal with an equivalent body size. Estimates for the number of neurons in the human brain range from 80 to 120 billion...
is at most 107.79640×1041.
See also
- Black hole entropy
- Digital physicsDigital physicsIn physics and cosmology, digital physics is a collection of theoretical perspectives based on the premise that the universe is, at heart, describable by information, and is therefore computable...
- EntropyEntropyEntropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
Further reading
- J. D. Bekenstein, "Black Holes and the Second Law", Lettere al Nuovo CimentoNuovo CimentoNuovo Cimento is a series of peer-reviewed scientific journals of physics. The series was first established in 1855 in Pisa, Italy, when Carlo Matteucci and Raffaele Piria started publishing Il Nuovo Cimento as the continuation of Il Cimento, which they founded in 1844...
, Vol. 4, No 15 (August 12, 1972), pp. 737-740, , . Mirror link. - Jacob D. Bekenstein, "Black Holes and Entropy", Physical Review DPhysical ReviewPhysical Review is an American scientific journal founded in 1893 by Edward Nichols. It publishes original research and scientific and literature reviews on all aspects of physics. It is published by the American Physical Society. The journal is in its third series, and is split in several...
, Vol. 7, No. 8 (April 15, 1973), pp. 2333-2346, , . Mirror link. - Jacob D. Bekenstein, "Generalized second law of thermodynamics in black-hole physics", Physical Review DPhysical ReviewPhysical Review is an American scientific journal founded in 1893 by Edward Nichols. It publishes original research and scientific and literature reviews on all aspects of physics. It is published by the American Physical Society. The journal is in its third series, and is split in several...
, Vol. 9, No. 12 (June 15, 1974), pp. 3292-3300, , . Mirror link. - Jacob D. Bekenstein, "Statistical black-hole thermodynamics", Physical Review DPhysical ReviewPhysical Review is an American scientific journal founded in 1893 by Edward Nichols. It publishes original research and scientific and literature reviews on all aspects of physics. It is published by the American Physical Society. The journal is in its third series, and is split in several...
, Vol. 12, No. 10 (November 15, 1975), pp. 3077-3085, , . Mirror link. - Jacob D. Bekenstein, "Black-hole thermodynamics", Physics TodayPhysics TodayPhysics Today, created in 1948, is the membership journal of the American Institute of Physics. It is provided to 130,000 members of twelve physics societies, including the American Physical Society...
, Vol. 33, Issue 1 (January 1980), pp. 24-31, , . Mirror link. - Jacob D. Bekenstein, "Universal upper bound on the entropy-to-energy ratio for bounded systems", Physical Review DPhysical ReviewPhysical Review is an American scientific journal founded in 1893 by Edward Nichols. It publishes original research and scientific and literature reviews on all aspects of physics. It is published by the American Physical Society. The journal is in its third series, and is split in several...
, Vol. 23, No. 2, (January 15, 1981), pp. 287-298, , . Mirror link. - Jacob D. Bekenstein, "Energy Cost of Information Transfer", Physical Review LettersPhysical Review LettersPhysical Review Letters , established in 1958, is a peer reviewed, scientific journal that is published 52 times per year by the American Physical Society...
, Vol. 46, No. 10 (March 9, 1981), pp. 623-626, , . Mirror link. - Jacob D. Bekenstein, "Specific entropy and the sign of the energy", Physical Review DPhysical ReviewPhysical Review is an American scientific journal founded in 1893 by Edward Nichols. It publishes original research and scientific and literature reviews on all aspects of physics. It is published by the American Physical Society. The journal is in its third series, and is split in several...
, Vol. 26, No. 4 (August 15, 1982), pp. 950-953, , . - Jacob D. Bekenstein, "Entropy content and information flow in systems with limited energy", Physical Review DPhysical ReviewPhysical Review is an American scientific journal founded in 1893 by Edward Nichols. It publishes original research and scientific and literature reviews on all aspects of physics. It is published by the American Physical Society. The journal is in its third series, and is split in several...
, Vol. 30, No. 8, (October 15, 1984), pp. 1669-1679, , . Mirror link. - Jacob D. Bekenstein, "Communication and energy", Physical Review APhysical ReviewPhysical Review is an American scientific journal founded in 1893 by Edward Nichols. It publishes original research and scientific and literature reviews on all aspects of physics. It is published by the American Physical Society. The journal is in its third series, and is split in several...
, Vol. 37, Issue 9 (May 1988), pp. 3437-3449, , . Mirror link. - Marcelo Schiffer and Jacob D. Bekenstein, "Proof of the quantum bound on specific entropy for free fields", Physical Review DPhysical ReviewPhysical Review is an American scientific journal founded in 1893 by Edward Nichols. It publishes original research and scientific and literature reviews on all aspects of physics. It is published by the American Physical Society. The journal is in its third series, and is split in several...
, Vol. 39, Issue 4 (February 15, 1989), pp. 1109-1115, PMID 9959747, . - Jacob D. Bekenstein, "Is the Cosmological Singularity Thermodynamically Possible?", International Journal of Theoretical PhysicsInternational Journal of Theoretical PhysicsThe International Journal of Theoretical Physics is a peer-reviewed scientific journal of physics published by Springer Science+Business Media since 1968. According to the Journal Citation Reports, the journal has a 2010 impact factor of 0.670 and publishes both original research and review articles...
, Vol. 28, Issue 9 (September 1989), pp. 967-981, , . - Jacob D. Bekenstein, "Entropy bounds and black hole remnants", Physical Review DPhysical ReviewPhysical Review is an American scientific journal founded in 1893 by Edward Nichols. It publishes original research and scientific and literature reviews on all aspects of physics. It is published by the American Physical Society. The journal is in its third series, and is split in several...
, Vol. 49, Issue 4 (February 15, 1994), pp. 1912-1921, , . Also at , July 25, 1993. - Oleg B. Zaslavskii, "Generalized second law and the Bekenstein entropy bound in Gedankenexperiments with black holes", Classical and Quantum GravityClassical and Quantum GravityClassical and Quantum Gravity is a peer-reviewed journal that covers all aspects of gravitational physics and the theory of spacetime.Its scope includes*Classical general relativity*Applications of relativity*Experimental gravitation...
, Vol. 13, No. 1 (January 1996), pp. L7-L11, , . See also O. B. Zaslavskii, "Corrigendum to 'Generalized second law and the Bekenstein entropy bound in Gedankenexperiments with black holes'", Classical and Quantum GravityClassical and Quantum GravityClassical and Quantum Gravity is a peer-reviewed journal that covers all aspects of gravitational physics and the theory of spacetime.Its scope includes*Classical general relativity*Applications of relativity*Experimental gravitation...
, Vol. 13, No. 9 (September 1996), p. 2607, , . - Jacob D. Bekenstein, "Non-Archimedean character of quantum buoyancy and the generalized second law of thermodynamics", Physical Review DPhysical ReviewPhysical Review is an American scientific journal founded in 1893 by Edward Nichols. It publishes original research and scientific and literature reviews on all aspects of physics. It is published by the American Physical Society. The journal is in its third series, and is split in several...
, Vol. 60, Issue 12 (December 15, 1999), Art. No. 124010, 9 pages, , . Also at , June 16, 1999.
External links
- Jacob D. Bekenstein, "Bekenstein bound", ScholarpediaScholarpediaScholarpedia is an English-language online wiki-based encyclopedia that uses the same MediaWiki software as Wikipedia, but has features more commonly associated with open-access online academic journals....
, Vol. 3, No. 10 (2008), p. 7374, . - Jacob D. Bekenstein, "Bekenstein-Hawking entropy", ScholarpediaScholarpediaScholarpedia is an English-language online wiki-based encyclopedia that uses the same MediaWiki software as Wikipedia, but has features more commonly associated with open-access online academic journals....
, Vol. 3, No. 10 (2008), p. 7375, . - Jacob D. Bekenstein's website at the Racah Institute of Physics, Hebrew University of JerusalemHebrew University of JerusalemThe Hebrew University of Jerusalem ; ; abbreviated HUJI) is Israel's second-oldest university, after the Technion – Israel Institute of Technology. The Hebrew University has three campuses in Jerusalem and one in Rehovot. The world's largest Jewish studies library is located on its Edmond J...
, which contains a number of articles on the Bekenstein bound.