Black Hole Thermodynamics
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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 A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
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 obj ...
s. As the study of the
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
of
black-body radiation Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous spect ...
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 chemistry, ...
, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
, 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 transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
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'' observable ...
s. In 1973 Bekenstein suggested \frac\approx 0.276 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 arg ...
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 ther ...
relationship between energy, temperature and entropy, Hawking was able to confirm Bekenstein's conjecture and fix the constant of proportionality at 1/4: :S_\text = \frac, where A is the area of the event horizon, k_\text 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 \ell_\text = \sqrt 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 A. 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 co ...
(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 In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, 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 w ...
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 Tehran ...
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 In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
, using methods based on
D-branes In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polchi ...
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 anim ...
. Their calculation was followed by many similar computations of entropy of large classes of other
extremal In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" in ...
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 Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. It is an attem ...
(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 Loop quantum gravity (LQG) is a theory of quantum gravity, which aims to merge quantum mechanics and general relativity, incorporating matter of the Standard Model into the framework established for the pure quantum gravity case. It is an attem ...
. 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 hole A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...
s 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 the ...
, 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 t ...
, 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 A geometrized unit system, geometric unit system or geometrodynamic unit system is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum, ''c'', and the gravitational constant, ''G'', are set eq ...
.


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 :dE = \frac\,dA + \Omega\,dJ + \Phi\,dQ, where E 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 heat a ...
, \kappa 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 ...
, A is the horizon area, \Omega is the
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
, J 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 ...
, \Phi 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 amber ...
and Q 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 In theory of relativity, relativistic classical field theory, classical field theories of gravitation, particularly general relativity, an energy condition is a generalization of the statement "the energy density of a region of space cannot be ne ...
, a non-decreasing function of time: :\frac \geq 0. 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, \kappa = 0 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 in ...
. 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 measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
. ''T'' constant for thermal equilibrium for a normal system is analogous to \kappa constant over the horizon of a stationary black hole.


The first law

The left side, dE, 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 a ...
. 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 amoun ...
is a statement of
energy conservation Energy conservation is the effort to reduce wasteful energy consumption by using fewer energy services. This can be done by using energy more effectively (using less energy for continuous service) or changing one's behavior to use less service (f ...
, which contains on its right side the term T dS.


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 transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
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 In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
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 N nucleons, S_ - S > 0, where S_ is the entropy of a black hole and S 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 \kappa 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, \Delta S will reach zero at zero temperature, but S 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'' observable ...
, 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) is ...
(Hawking radiation) at a temperature :T_\text = \frac. From the first law of black hole mechanics, this determines the multiplicative constant of the Bekenstein–Hawking entropy, which is (in
geometrized units A geometrized unit system, geometric unit system or geometrodynamic unit system is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum, ''c'', and the gravitational constant, ''G'', are set eq ...
) :S_\text = \frac. 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 giving ...
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: : \Gamma=\int d^4x\, \sqrt\,\bigg(\frac+c_1(\mu)R^2 +c_2(\mu)R_R^ +c_3(\mu)R_R^\bigg)-\int d^4 x \sqrt\bigg alpha R\ln\left(\frac\right)R +\beta R_\ln\left(\frac\right)R^ + \gamma R_\ln\left(\frac\right)R^\bigg where \mu is an energy scale. The exact values of the coefficients c_1,c_2,c_3 are unknown, as they depend on the nature of the ultra-violet theory of quantum gravity. \ln\left(\Box/\mu^2\right) is an operator with the integral representation : \ln\left(\frac\right)=\int_0^ds\, \left(\frac-\frac\right). 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 r_h of the event horizon into account: : S_=-2\pi \int\limits_ d\Sigma\, \epsilon_\epsilon_\frac. Here, \mathcal is the Lagrangian density of the theory, d\Sigma=r^2\sin\theta d\theta d\phi, R_ is the Riemann tensor and \epsilon_ is an antisymmetric tensor normalised as \epsilon_\epsilon^=-2. 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 : S_=\frac+64\pi^2c_3+64\pi^2\gamma\Big ln\left(4G^2M^2\mu^2\right)+2\gamma_E-2\Big 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 Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to f ...
.


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 Scholarpedia



Black hole entropy on arxiv.org
{{Stephen Hawking Black holes Branches of thermodynamics