In
physics
Physics is the natural science that studies matter, its 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 which r ...
, black hole thermodynamics is the area of study that seeks to reconcile 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 paramet ...
with the existence of
black hole event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact ob ...
s. As the study of the
statistical mechanics of
black-body radiation led to the development of the theory of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of
quantum gravity, leading to the formulation of the
holographic principle
The holographic principle is an axiom in string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region — such as a ...
.
Overview
The
second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unles ...
requires that black holes have
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
. If black holes carried no entropy, it would be possible to violate the second law by throwing mass into the black hole. The increase of the entropy of the black hole more than compensates for the decrease of the entropy carried by the object that was swallowed.
In 1972,
Jacob Bekenstein
Jacob David Bekenstein ( he, יעקב בקנשטיין; May 1, 1947 – August 16, 2015) was an American and Israeli theoretical physicist who made fundamental contributions to the foundation of black hole thermodynamics and to other aspects of ...
conjectured that black holes should have an entropy, where by the same year, he proposed
no-hair theorem
The no-hair theorem states that all stationary black hole solutions of the Einstein–Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three independent ''externally'' observabl ...
s.
In 1973 Bekenstein suggested
as the constant of proportionality, asserting that if the constant was not exactly this, it must be very close to it. The next year, in 1974,
Stephen Hawking showed that black holes emit thermal
Hawking radiation
Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical a ...
corresponding to a certain temperature (Hawking temperature). Using the
thermodynamic
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
relationship between energy, temperature and entropy, Hawking was able to confirm Bekenstein's conjecture and fix the constant of proportionality at
:
:
where
is the area of the event horizon,
is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, and
is the
Planck length. This is often referred to as the Bekenstein–Hawking formula. The subscript BH either stands for "black hole" or "Bekenstein–Hawking". The black hole entropy is proportional to the area of its event horizon
. The fact that the black hole entropy is also the maximal entropy that can be obtained by the
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 c ...
(wherein the Bekenstein bound becomes an equality) was the main observation that led to the
holographic principle
The holographic principle is an axiom in string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region — such as a ...
.
This area relationship was generalized to arbitrary regions via the
Ryu–Takayanagi formula, which relates the entanglement entropy of a boundary conformal field theory to a specific surface in its dual gravitational theory.
[
]
Although Hawking's calculations gave further thermodynamic evidence for black hole entropy, until 1995 no one was able to make a controlled calculation of black hole entropy based on
statistical mechanics, which associates entropy with a large number of microstates. In fact, so called "
no-hair" theorems
appeared to suggest that black holes could have only a single microstate. The situation changed in 1995 when
Andrew Strominger
Andrew Eben Strominger (; born 1955) is an American theoretical physicist who is the director of Harvard's Center for the Fundamental Laws of Nature. He has made significant contributions to quantum gravity and string theory. These include his ...
and
Cumrun Vafa
Cumrun Vafa ( fa, کامران وفا ; born 1 August 1960) is an Iranian-American theoretical physicist and the Hollis Professor of Mathematics and Natural Philosophy at Harvard University.
Early life and education
Cumrun Vafa was born in Tehra ...
calculated the right Bekenstein–Hawking entropy of a
supersymmetric
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
black hole in
string theory, using methods based on
D-branes and
string duality
String or strings may refer to:
* String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects
Arts, entertainment, and media Films
* ''Strings'' (1991 film), a Canadian ani ...
. Their calculation was followed by many similar computations of entropy of large classes of other
extremal and
near-extremal black hole
In theoretical physics, a near-extremal black hole is a black hole which is not far from the minimal possible mass that can be compatible with the given charges and angular momentum. The calculations of the properties of near-extremal black hole ...
s, and the result always agreed with the Bekenstein–Hawking formula. However, for the
Schwarzschild black hole, viewed as the most far-from-extremal black hole, the relationship between micro- and macrostates has not been characterized. Efforts to develop an adequate answer within the framework of string theory continue.
In
loop quantum gravity (LQG)
[See ]List of loop quantum gravity researchers {{Short description, none
This is a list researchers in the physics field of loop quantum gravity who have Wikipedia articles.
* Abhay Ashtekar, Pennsylvania State University, United States
* John Baez, University of California, Riverside, United ...
. it is possible to associate a geometrical interpretation with the microstates: these are the quantum geometries of the horizon. LQG offers a geometric explanation of the finiteness of the entropy and of the proportionality of the area of the horizon.
It is possible to derive, from the covariant formulation of full quantum theory (
spinfoam
In physics, the topological structure of spinfoam or spin foam consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity. These structu ...
) the correct relation between energy and area (1st law), the
Unruh temperature
The Unruh effect (also known as the Fulling–Davies–Unruh effect) is a kinematic prediction of quantum field theory that an accelerating observer will observe a thermal bath, like blackbody radiation, whereas an inertial observer would observe ...
and the distribution that yields Hawking entropy.
The calculation makes use of the notion of
dynamical horizon
In theoretical physics, a dynamical horizon (DH) is a local description (i.e. independent of the global structure of Space–time) of evolving black-hole horizons. In the literature there exist two different mathematical formulations of DHs—the 2 ...
and is done for non-extremal black holes. There seems to be also discussed the calculation of Bekenstein–Hawking entropy from the point of view of
loop quantum gravity. The current accepted microstate ensemble for black holes is the microcanonical ensemble. The partition function for black holes results in a negative heat capacity. In canonical ensembles, there is limitation for a positive heat capacity, whereas microcanonical ensembles can exist at a negative heat capacity.
The laws of black hole mechanics
The four laws of black hole mechanics are physical properties that
black holes are believed to satisfy. The laws, analogous to the laws of
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...
, were discovered by
Jacob Bekenstein
Jacob David Bekenstein ( he, יעקב בקנשטיין; May 1, 1947 – August 16, 2015) was an American and Israeli theoretical physicist who made fundamental contributions to the foundation of black hole thermodynamics and to other aspects of ...
,
Brandon Carter
Brandon Carter, (born 1942) is an Australian theoretical physicist, best known for his work on the properties of black holes and for being the first to name and employ the anthropic principle in its contemporary form. He is a researcher at th ...
, and
James Bardeen. Further considerations were made by
Stephen Hawking.
Statement of the laws
The laws of black hole mechanics are expressed in
geometrized units.
The zeroth law
The horizon has constant
surface gravity
The surface gravity, ''g'', of an astronomical object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as the acceleration due to gravity experien ...
for a stationary black hole.
The first law
For perturbations of stationary black holes, the change of energy is related to change of area, angular momentum, and electric charge by
:
where
is the
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
,
is the
surface gravity
The surface gravity, ''g'', of an astronomical object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as the acceleration due to gravity experien ...
,
is the horizon area,
is the
angular velocity,
is the
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
,
is the
electrostatic potential
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for ambe ...
and
is the
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
.
The second law
The horizon area is, assuming the
weak energy condition, a non-decreasing function of time:
:
This "law" was superseded by Hawking's discovery that black holes radiate, which causes both the black hole's mass and the area of its horizon to decrease over time.
The third law
It is not possible to form a black hole with vanishing surface gravity. That is,
cannot be achieved.
Discussion of the laws
The zeroth law
The zeroth law is analogous to the
zeroth law of thermodynamics
The zeroth law of thermodynamics is one of the four principal laws of thermodynamics. It provides an independent definition of temperature without reference to entropy, which is defined in the second law. The law was established by Ralph H. Fowl ...
, which states that the temperature is constant throughout a body in
thermal equilibrium
Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be i ...
. It suggests that the surface gravity is analogous to
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer.
Thermometers are calibrated in various Conversion of units of temperature, temp ...
. ''T'' constant for thermal equilibrium for a normal system is analogous to
constant over the horizon of a stationary black hole.
The first law
The left side,
, is the change in energy (proportional to mass). Although the first term does not have an immediately obvious physical interpretation, the second and third terms on the right side represent changes in energy due to rotation and
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
. Analogously, the
first law of thermodynamics
The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amou ...
is a statement of
energy conservation, which contains on its right side the term
.
The second law
The second law is the statement of Hawking's area theorem. Analogously, the
second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unles ...
states that the change in
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
in an isolated system will be greater than or equal to 0 for a spontaneous process, suggesting a link between entropy and the area of a black hole horizon. However, this version violates the second law of thermodynamics by matter losing (its) entropy as it falls in, giving a decrease in entropy. However, generalizing the second law as the sum of black hole entropy and outside entropy, shows that the second law of thermodynamics is not violated in a system including the universe beyond the horizon.
The generalized second law of thermodynamics (GSL) was needed to present the second law of thermodynamics as valid. This is because the second law of thermodynamics, as a result of the disappearance of
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
near the exterior of black holes, is not useful. The GSL allows for the application of the law because now the measurement of interior, common entropy is possible. The validity of the GSL can be established by studying an example, such as looking at a system having entropy that falls into a bigger, non-moving black hole, and establishing upper and lower entropy bounds for the increase in the black hole entropy and entropy of the system, respectively.
One should also note that the GSL will hold for theories of gravity such as
Einstein gravity,
Lovelock gravity, or Braneworld gravity, because the conditions to use GSL for these can be met.
However, on the topic of black hole formation, the question becomes whether or not the generalized second law of thermodynamics will be valid, and if it is, it will have been proved valid for all situations. Because a black hole formation is not stationary, but instead moving, proving that the GSL holds is difficult. Proving the GSL is generally valid would require using
quantum-statistical mechanics, because the GSL is both a
quantum and
statistical law. This discipline does not exist so the GSL can be assumed to be useful in general, as well as for prediction. For example, one can use the GSL to predict that, for a cold, non-rotating assembly of
nucleons,
, where
is the entropy of a black hole and
is the sum of the ordinary entropy.
The third law
Extremal black hole
In theoretical physics, an extremal black hole is a black hole with the minimum possible mass that is compatible with its electric charge, charge and angular momentum.
The concept of an extremal black hole is theoretical and none have thusfar bee ...
s
have vanishing surface gravity. Stating that
cannot go to zero is analogous to the
third law of thermodynamics
The third law of thermodynamics states, regarding the properties of closed systems in thermodynamic equilibrium: This constant value cannot depend on any other parameters characterizing the closed system, such as pressure or applied magnetic fiel ...
, which states that the entropy of a system at absolute zero is a well defined constant. This is because a system at zero temperature exists in its ground state. Furthermore,
will reach zero at zero temperature, but
itself will also reach zero, at least for perfect crystalline substances. No experimentally verified violations of the laws of thermodynamics are known yet.
Interpretation of the laws
The four laws of black hole mechanics suggest that one should identify the surface gravity of a black hole with temperature and the area of the event horizon with entropy, at least up to some multiplicative constants. If one only considers black holes classically, then they have zero temperature and, by the
no-hair theorem
The no-hair theorem states that all stationary black hole solutions of the Einstein–Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three independent ''externally'' observabl ...
,
zero entropy, and the laws of black hole mechanics remain an analogy. However, when
quantum-mechanical effects are taken into account, one finds that black holes emit
thermal radiation
Thermal radiation is electromagnetic radiation generated by the thermal motion of particles in matter. Thermal radiation is generated when heat from the movement of charges in the material (electrons and protons in common forms of matter) i ...
(Hawking radiation) at a temperature
:
From the first law of black hole mechanics, this determines the multiplicative constant of the Bekenstein–Hawking entropy, which is (in
geometrized units)
:
which is the entropy of the black hole in Einstein's
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
.
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 treats spacetime as a fixed, classical background, while givi ...
can be utilized to calculate the entropy for a black hole in any covariant theory for gravity, known as the Wald entropy.
Quantum gravitational corrections to the entropy
The Hawking formula for the entropy receives corrections as soon as quantum effects are taken into account. Any UV finite theory of quantum gravity should reduce at low energy to General Relativity. Works pioneered by Barvinsky and Vilkovisky suggest as a starting point up to second order in curvature the following action, consisting of local and non-local terms:
:
where
is an energy scale. The exact values of the coefficients
are unknown, as they depend on the nature of the ultra-violet theory of quantum gravity.
is an operator with the integral representation
:
The new additional terms in the action modify the classical Einstein equations of motion. This implies that a given classical metric receives quantum corrections, which in turn shift the classical position of the event horizon. When computing the Wald entropy, one then takes the shifted position
of the event horizon into account:
:
Here,
is the Lagrangian density of the theory,
,
is the Riemann tensor and
is an antisymmetric tensor normalised as
This method was applied in 2021 by Calmet et al. for Schwarzschild black holes. The Schwarzschild metric does not receive quantum corrections at second order in curvature and the entropy is
:
A generalisation for charged (Reissner-Nordström) black holes was subsequently carried out by Campos Delgado.
Critique
While black hole thermodynamics (BHT) has been regarded as one of the deepest clues to a quantum theory of gravity, there remained some philosophical criticisms that it “is often based on a kind of caricature of thermodynamics” and "it’s unclear what the systems in BHT are supposed to be", leading to the conclusion -- "the analogy is not nearly as good as is commonly supposed".
These criticisms triggered a fellow skeptic to reexamine "the case for regarding black holes as thermodynamic systems", with particular attention paid to "the central role of Hawking radiation in permitting black holes to be in thermal contact with one another" and "the interpretation of Hawking radiation close to the black hole as a gravitationally bound thermal atmosphere", ending with the opposite conclusion -- "stationary black holes are not ''analogous'' to thermodynamic systems: they ''are'' thermodynamic systems, in the fullest sense."
Beyond black holes
Gary Gibbons
Gary William Gibbons (born 1 July 1946)
is a British theoretical physicist.
Education
Gibbons was born in Coulsdon, Surrey. He was educated at Purley County Grammar School and the University of Cambridge, where in 1969 he became a researc ...
and Hawking have shown that black hole thermodynamics is more general than black holes—that
cosmological event horizons also have an entropy and temperature.
More fundamentally,
't Hooft and
Susskind used the laws of black hole thermodynamics to argue for a general
holographic principle
The holographic principle is an axiom in string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region — such as a ...
of nature, which asserts that consistent theories of gravity and quantum mechanics must be lower-dimensional. Though not yet fully understood in general, the holographic principle is central to theories like the
AdS/CFT correspondence
In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter s ...
.
There are also connections between black hole entropy and fluid
surface tension.
See also
*
Joseph Polchinski
Joseph Gerard Polchinski Jr. (; May 16, 1954 – February 2, 2018) was an American theoretical physicist and string theorist.
Biography
Polchinski was born in White Plains, New York, the elder of two children to Joseph Gerard Polchinski Sr. (1929 ...
*
Robert Wald
The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' ( non, Hróðr) "fame, glory, honou ...
Notes
Citations
Bibliography
*
*
*
*
*
*
*
*
External links
Bekenstein-Hawking entropy on ScholarpediaBlack hole entropy on arxiv.org
{{Stephen Hawking
Black holes
Branches of thermodynamics