In physics, **black hole thermodynamics**^{ [1] } 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 advent of the theory of quantum mechanics, 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.^{ [2] }

The second law of thermodynamics requires that black holes have entropy. 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.

Starting from theorems proved by Stephen Hawking, Jacob Bekenstein conjectured that the black hole entropy was proportional to the area of its event horizon divided by the Planck area. 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, Hawking showed that black holes emit thermal Hawking radiation ^{ [3] }^{ [4] } corresponding to a certain temperature (Hawking temperature).^{ [5] }^{ [6] } Using the thermodynamic relationship between energy, temperature and entropy, Hawking was able to confirm Bekenstein's conjecture and fix the constant of proportionality at :^{ [7] }^{ [8] }

where is the area of the event horizon, is the Boltzmann 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 (wherein the Bekenstein bound becomes an equality) was the main observation that led to the holographic principle.^{ [2] } 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.^{ [9] }

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^{ [10] } appeared to suggest that black holes could have only a single microstate. The situation changed in 1995 when Andrew Strominger and Cumrun Vafa calculated^{ [11] } the right Bekenstein–Hawking entropy of a supersymmetric black hole in string theory, using methods based on D-branes and string duality. Their calculation was followed by many similar computations of entropy of large classes of other extremal and near-extremal black holes, 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)^{ [nb 1] } 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.^{ [12] }^{ [13] } It is possible to derive, from the covariant formulation of full quantum theory (spinfoam) the correct relation between energy and area (1st law), the Unruh temperature and the distribution that yields Hawking entropy.^{ [14] } The calculation makes use of the notion of dynamical horizon 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 four **laws of black hole mechanics** are physical properties that black holes are believed to satisfy. The laws, analogous to the laws of thermodynamics, were discovered by Brandon Carter, Stephen Hawking, and James Bardeen.

The laws of black-hole mechanics are expressed in geometrized units.

The horizon has constant surface gravity for a stationary black hole.

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, is the surface gravity, is the horizon area, is the angular velocity, is the angular momentum, is the electrostatic potential and is the electric charge.

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.

It is not possible to form a black hole with vanishing surface gravity. That is, cannot be achieved.

The zeroth law is analogous to the zeroth law of thermodynamics, which states that the temperature is constant throughout a body in thermal equilibrium. It suggests that the surface gravity is analogous to temperature. *T* constant for thermal equilibrium for a normal system is analogous to constant over the horizon of a stationary black hole.

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. Analogously, the first law of thermodynamics is a statement of energy conservation, which contains on its right side the term .

The second law is the statement of Hawking's area theorem. Analogously, the second law of thermodynamics states that the change in entropy 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 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.^{ [15] } 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.^{ [16] }

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.^{ [15] }^{ [17] }

Extremal black holes ^{ [18] } have vanishing surface gravity. Stating that cannot go to zero is analogous to the third law of thermodynamics, 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.

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,^{ [10] } 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 (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

Gary Gibbons 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 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.^{ [19] }

There are also connections between black-hole entropy and fluid surface tension.^{ [20] }

- ↑ Carlip, S (2014). "Black Hole Thermodynamics".
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A **black hole** is a region of spacetime where gravity is so strong that nothing—no particles or even electromagnetic radiation such as light—can escape from it. The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of the region from which no escape is possible is called the event horizon. Although the event horizon has an enormous effect on the fate and circumstances of an object crossing it, it has no locally detectable features. In many ways, a black hole acts like an ideal black body, as it reflects no light. Moreover, quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with the same spectrum as a black body of a temperature inversely proportional to its mass. This temperature is on the order of billionths of a kelvin for black holes of stellar mass, making it essentially impossible to observe.

The **holographic principle** is a tenet of 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 light-like boundary like a gravitational horizon. First proposed by Gerard 't Hooft, it was given a precise string-theory interpretation by Leonard Susskind who combined his ideas with previous ones of 't Hooft and Charles Thorn. As pointed out by Raphael Bousso, Thorn observed in 1978 that string theory admits a lower-dimensional description in which gravity emerges from it in what would now be called a holographic way. The prime example of holography is the AdS/CFT correspondence.

A **gravitational singularity**, **spacetime singularity** or simply **singularity** is a location in spacetime where the gravitational field of a celestial body is predicted to become infinite by general relativity in a way that does not depend on the coordinate system. The quantities used to measure gravitational field strength are the scalar invariant curvatures of spacetime, which includes a measure of the density of matter. Since such quantities become infinite at the singularity, the laws of normal spacetime break down.

**T-symmetry** or **time reversal symmetry** is the theoretical symmetry of physical laws under the transformation of time reversal:

**Hawking radiation** is black-body radiation that is predicted to be released by black holes, due to quantum effects near the black hole event horizon. It is named after the theoretical physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974.

The **no-hair theorem** states that all black hole solutions of the Einstein–Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three *externally* observable classical parameters: mass, electric charge, and angular momentum. All other information about the matter which formed a black hole or is falling into it, "disappears" behind the black-hole event horizon and is therefore permanently inaccessible to external observers. Physicist John Archibald Wheeler expressed this idea with the phrase "black holes have no hair" which was the origin of the name. In a later interview, Wheeler said that Jacob Bekenstein coined this phrase.

**Jacob David Bekenstein** was a Mexican-born Israeli-American theoretical physicist who made fundamental contributions to the foundation of black hole thermodynamics and to other aspects of the connections between information and gravitation.

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 spaces (AdS) which are used in theories of quantum gravity, formulated in terms of string theory or M-theory. On the other side of the correspondence are conformal field theories (CFT) which are quantum field theories, including theories similar to the Yang–Mills theories that describe elementary particles.

The **Immirzi parameter** is a numerical coefficient appearing in loop quantum gravity (LQG), a nonperturbative theory of quantum gravity. The Immirzi parameter measures the size of the quantum of area in Planck units. As a result, its value is currently fixed by matching the semiclassical black hole entropy, as calculated by Stephen Hawking, and the counting of microstates in loop quantum gravity.

In theoretical physics, an **extremal black hole** is a black hole with the minimal possible mass that can be compatible with a given charge and angular momentum. In other words, this is the smallest possible black hole that can exist while rotating at a given fixed constant speed.

The **Unruh effect** is the prediction that an accelerating observer will observe blackbody radiation where an inertial observer would observe none. In other words, the background appears to be warm from an accelerating reference frame; in layman's terms, a thermometer waved around in empty space, subtracting any other contribution to its temperature, will record a non-zero temperature. For a uniformly accelerating observer, the ground state of an inertial observer is seen as in thermodynamic equilibrium with a non-zero temperature.

**Micro black holes**, also called **quantum mechanical black holes** or **mini black holes**, are hypothetical tiny black holes, for which quantum mechanical effects play an important role. The concept that black holes may exist that are smaller than stellar mass was introduced in 1971 by Stephen Hawking.

The **black hole information paradox** is a puzzle resulting from the combination of quantum mechanics and general relativity. Calculations suggest that physical information could permanently disappear in a black hole, allowing many physical states to devolve into the same state. This is controversial because it violates a core precept of modern physics—that *in principle* the value of a wave function of a physical system at one point in time should determine its value at any other time. A fundamental postulate of the Copenhagen interpretation of quantum mechanics is that complete information about a system is encoded in its wave function up to when the wave function collapses. The evolution of the wave function is determined by a unitary operator, and unitarity implies that information is conserved in the quantum sense.

**Induced gravity** is an idea in quantum gravity that space-time curvature and its dynamics emerge as a mean field approximation of underlying microscopic degrees of freedom, similar to the fluid mechanics approximation of Bose–Einstein condensates. The concept was originally proposed by Andrei Sakharov in 1967.

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 maximal 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 maximal information-processing rate for a physical system that has a finite size and energy, and that a Turing machine with finite physical dimensions and unbounded memory is not physically possible.

The mathematical expressions for thermodynamic entropy in the statistical thermodynamics formulation established by Ludwig Boltzmann and J. Willard Gibbs in the 1870s are similar to the information entropy by Claude Shannon and Ralph Hartley, developed in the 1940s.

**Ruppeiner geometry** is thermodynamic geometry using the language of Riemannian geometry to study thermodynamics. George Ruppeiner proposed it in 1979. He claimed that thermodynamic systems can be represented by Riemannian geometry, and that statistical properties can be derived from the model.

**Entropic gravity**, also known as **emergent gravity**, is a theory in modern physics that describes gravity as an *entropic force*—a force with macro-scale homogeneity but which is subject to quantum-level disorder—and not a fundamental interaction. The theory, based on string theory, black hole physics, and quantum information theory, describes gravity as an *emergent* phenomenon that springs from the quantum entanglement of small bits of spacetime information. As such, entropic gravity is said to abide by the second law of thermodynamics under which the entropy of a physical system tends to increase over time.

The **asymptotic safety** approach to quantum gravity provides a nonperturbative notion of renormalization in order to find a consistent and predictive quantum field theory of the gravitational interaction and spacetime geometry. It is based upon a nontrivial fixed point of the corresponding renormalization group (RG) flow such that the running coupling constants approach this fixed point in the ultraviolet (UV) limit. This suffices to avoid divergences in physical observables. Moreover, it has predictive power: Generically an arbitrary starting configuration of coupling constants given at some RG scale does not run into the fixed point for increasing scale, but a subset of configurations might have the desired UV properties. For this reason it is possible that — assuming a particular set of couplings has been measured in an experiment — the requirement of asymptotic safety fixes all remaining couplings in such a way that the UV fixed point is approached.

The **Ryu–Takayanagi conjecture** is a conjecture within holography that posits a quantitative relationship between the entanglement entropy of a conformal field theory and the geometry of an associated anti-de Sitter spacetime. The formula characterizes "holographic screens" in the bulk; that is, it specifies which regions of the bulk geometry are "responsible to particular information in the dual CFT". The conjecture is named after Shinsei Ryu and Tadashi Takayanagi, who jointly published the result in 2006. As a result, the authors were awarded the 2015 New Horizons in Physics Prize for "fundamental ideas about entropy in quantum field theory and quantum gravity". The formula was generalized to a covariant form in 2007.

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