Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's

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 ...

because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical argument for its existence in 1974. Hawking radiation is a purely

kinematic Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fiel ...

effect that is generic to Lorentzian geometries containing event horizons or local apparent horizons. Hawking radiation reduces the mass and rotational energy of black holes and is therefore also theorized to cause black hole evaporation. Because of this, black holes that do not gain mass through other means are expected to shrink and ultimately vanish. For all except the smallest black holes, this would happen extremely slowly. The radiation temperature is inversely proportional to the black hole's mass, so

micro black hole Micro black holes, also called mini black holes or quantum mechanical black holes, are hypothetical tiny (<1 )

Overview

Black holes are astrophysical objects of interest primarily because of their compact size and immense

gravitational attraction In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...

. They were first predicted by Einstein's 1915 theory of

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 ...

, before astrophysical evidence began to mount half a century later. A black hole can form when enough

matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic part ...

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 compressed into a volume small enough that the escape velocity is greater than the speed of light. Nothing can travel that fast, so nothing within a distance, proportional to the mass of the black hole, can escape beyond that distance. The region beyond which not even light can escape is the

; an observer outside it cannot observe, become aware of, or be affected by events within the event horizon. The essence of a black hole is its event horizon, a theoretical demarcation between events and their causal relationships. Andrew Hamilton schwarzchild waterfall

Alternatively, using a set of infalling coordinates in general relativity, one can conceptualize the event horizon as the region beyond which space is infalling faster than the speed of light. (Although nothing can travel ''through'' space faster than light, space itself can infall at any speed.) Once matter is inside the event horizon, all of the matter inside falls inexorably into a

gravitational singularity A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravity is so intense that spacetime itself breaks down catastrophically. As such, a singularity is by definition no longer part of the regular sp ...

, a place of infinite curvature and zero size, leaving behind a warped spacetime devoid of any matter. A classical black hole is pure empty

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...

, and the simplest (nonrotating and uncharged) is characterized just by its mass and event horizon. Our current understandings of quantum physics can be used to investigate what may happen in the region around the event horizon. In 1974,

British British may refer to: Peoples, culture, and language * British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies. ** Britishness, the British identity and common culture * British English, ...

physicist Stephen Hawking used

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 ...

to show that in theory, the force of gravity at the event horizon was strong enough to cause thermal radiation to be emitted and energy to "leak" into the wider universe from a tiny distance around and outside the event horizon. In effect this energy acted as if the black hole itself was slowly

evaporating Evaporation is a type of vaporization that occurs on the surface of a liquid as it changes into the gas phase. High concentration of the evaporating substance in the surrounding gas significantly slows down evaporation, such as when humidi ...

(although it actually came from outside it). An important difference between the black hole radiation as computed by Hawking and

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 ...

emitted from a black body is that the latter is statistical in nature, and only its average satisfies what is known as

Planck's law of black-body radiation In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature , when there is no net flow of matter or energy between the body and its environment. At ...

, while the former fits the data better. Thus,

contains

information Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random ...

about the body that emitted it, while Hawking radiation seems to contain no such information, and depends only on the

mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...

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 ...

, and

charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...

of the black hole (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 ...

). This leads to the

black hole information paradox The black hole information paradox is a puzzle that appears when the predictions of quantum mechanics and general relativity are combined. The theory of general relativity predicts the existence of black holes that are regions of spacetime from wh ...

. However, according to the conjectured gauge-gravity duality (also known as 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 ...

), black holes in certain cases (and perhaps in general) are equivalent to solutions of quantum field theory at a non-zero

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 ...

. This means that no information loss is expected in black holes (since the theory permits no such loss) and the radiation emitted by a black hole is probably the usual thermal radiation. If this is correct, then Hawking's original calculation should be corrected, though it is not known how (see below). A black hole of one solar mass () has a temperature of only 60 nanokelvins (60 billionths of a

kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phy ...

); in fact, such a black hole would absorb far more

cosmic microwave background radiation In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all space ...

than it emits. A black hole of (about the mass of the

Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...

, or about across) would be in equilibrium at 2.7 K, absorbing as much radiation as it emits.

Discovery

Hawking's discovery followed a visit to Moscow in 1973, where the Soviet scientists Yakov Zel'dovich and Alexei Starobinsky convinced him that

rotating black hole A rotating black hole is a black hole that possesses angular momentum. In particular, it rotates about one of its axes of symmetry. All celestial objects – planets, stars (Sun), galaxies, black holes – spin. Types of black holes Ther ...

s ought to create and emit particles, while Russian physicist

Vladimir Gribov Vladimir Naumovich Gribov (Russian Влади́мир Нау́мович Гри́бов; March 25, 1930, LeningradAugust 13, 1997, Budapest) was a prominent Russian theoretical physicist, who worked on high-energy physics, quantum field theory an ...

believed that even a non-rotating black hole should emit radiation. When Hawking did the calculation, he found to his surprise that it was true. 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 the black holes should have an entropy, where by the same year, he proposed

s. Bekenstein's discovery and results are commended by Stephen Hawking, which also led him to think about radiation due to this formalism. According to the physicist Dmitri Diakonov, there was an argument between Zeldovich and

at the Zeldovich Moscow 1972–1973 seminar. Zeldovich believed that only a rotating black hole could emit radiation, while Gribov believed that even a non-rotating black hole emits radiation due to the laws of quantum mechanics. This account is confirmed by Gribov's obituary in the

Physics-Uspekhi ''Physics-Uspekhi'' is a peer-reviewed scientific journal. It is an English translation of the Russian journal of physics, ''Uspekhi Fizicheskikh Nauk'' (russian: Успехи физических наук, ''Advances in Physical Sciences'') which ...

Vitaly Ginzburg Vitaly Lazarevich Ginzburg, ForMemRS (russian: Вита́лий Ла́заревич Ги́нзбург, link=no; 4 October 1916 – 8 November 2009) was a Russian physicist who was honored with the Nobel Prize in Physics in 2003, together wit ...

and others.

Emission process

Hawking radiation is required by the

Unruh effect 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 equivalence principle applied to black-hole horizons. Close to the event horizon of a black hole, a local observer must accelerate to keep from falling in. An accelerating observer sees a thermal bath of particles that pop out of the local acceleration horizon, turn around, and free-fall back in. The condition of local thermal equilibrium implies that the consistent extension of this local thermal bath has a finite temperature at infinity, which implies that some of these particles emitted by the horizon are not reabsorbed and become outgoing Hawking radiation.For an accessible discussion of the Unruh effect and Hawking radiation, see: A Schwarzschild black hole has a metric :

(\mathrms)^2 = -\left(1 - \frac\right)\,(\mathrmt)^2 + \frac \,(\mathrmr)^2 + r^2\,(\mathrm\Omega)^2.

The black hole is the background spacetime for a quantum field theory. The field theory is defined by a local path integral, so if the boundary conditions at the horizon are determined, the state of the field outside will be specified. To find the appropriate boundary conditions, consider a stationary observer just outside the horizon at position :

r = 2M + \frac.

The local metric to lowest order is :

(\mathrms)^2 = -\left(\frac\right)^2 \,(\mathrmt)^2 + (\mathrm\rho)^2 + (\mathrmX_\perp)^2 = -\rho^2 \,(\mathrm\tau)^2 + (\mathrm\rho)^2 + (\mathrmX_\perp)^2,

which is Rindler in terms of . The metric describes a frame that is accelerating to keep from falling into the black hole. The local acceleration, , diverges as . The horizon is not a special boundary, and objects can fall in. So the local observer should feel accelerated in ordinary Minkowski space by the principle of equivalence. The near-horizon observer must see the field excited at a local temperature :

T = \frac = \frac = \frac,

which is the

. The gravitational redshift is given by the square root of the time component of the metric. So for the field theory state to consistently extend, there must be a thermal background everywhere with the local temperature redshift-matched to the near horizon temperature: :

T(r') = \frac \sqrt\frac = \frac.

The inverse temperature redshifted to at infinity is :

T(\infty) = \frac,

and is the near-horizon position, near , so this is really :

T(\infty) = \frac.

Thus a field theory defined on a black-hole background is in a thermal state whose temperature at infinity is :

T_\text = \frac.

From the black-hole temperature, it is straightforward to calculate the black-hole entropy . The change in entropy when a quantity of heat is added is :

\mathrmS = \frac = 8 \pi M \,\mathrmQ.

The heat energy that enters serves to increase the total mass, so :

\mathrmS = 8 \pi M \,\mathrmM = \mathrm(4 \pi M^2).

The radius of a black hole is twice its mass in Planck units, so the entropy of a black hole is proportional to its surface area: :

S = \pi R^2 = \frac.

Assuming that a small black hole has zero entropy, the integration constant is zero. Forming a black hole is the most efficient way to compress mass into a region, and this entropy is also a bound on the information content of any sphere in space time. The form of the result strongly suggests that the physical description of a gravitating theory can be somehow encoded onto a bounding surface.

Black hole evaporation

When particles escape, the black hole loses a small amount of its energy and therefore some of its mass (mass and energy are related by Einstein's equation ). Consequently, an evaporating black hole will have a finite lifespan. By

dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as mi ...

, the life span of a black hole can be shown to scale as the cube of its initial mass, and Hawking estimated that any black hole formed in the early universe with a mass of less than approximately 10¹⁵ g would have evaporated completely by the present day. In 1976, Don Page refined this estimate by calculating the power produced, and the time to evaporation, for a non-rotating, non-charged Schwarzschild black hole of mass . The time for the event horizon or entropy of a black hole to halve is known as the Page time. The calculations are complicated by the fact that a black hole, being of finite size, is not a perfect black body; the

absorption cross section Absorption cross section is a measure for the probability of an absorption process. More generally, the term cross section is used in physics to quantify the probability of a certain particle-particle interaction, e.g., scattering, electromagne ...

goes down in a complicated, spin-dependent manner as frequency decreases, especially when the wavelength becomes comparable to the size of the event horizon. Page concluded that primordial black holes could only survive to the present day if their initial mass were roughly or larger. Writing in 1976, Page using the understanding of neutrinos at the time erroneously worked on the assumption that neutrinos have no mass and that only two neutrino flavors exist, and therefore his results of black hole lifetimes do not match the modern results which take into account 3 flavors of neutrinos with nonzero masses. A 2008 calculation using the particle content of the Standard Model and the

WMAP The Wilkinson Microwave Anisotropy Probe (WMAP), originally known as the Microwave Anisotropy Probe (MAP and Explorer 80), was a NASA spacecraft operating from 2001 to 2010 which measured temperature differences across the sky in the cosmic mic ...

figure for the age of the universe yielded a mass bound of . If black holes evaporate under Hawking radiation, a solar mass black hole will evaporate over 10⁶⁴ years which is vastly longer than the age of the universe.See page 596: table 1 and the "black hole decay" section and previous sentence on that page in A supermassive black hole with a mass of 10¹¹ (100 billion) will evaporate in around . See in particular equation (27). Some monster black holes in the universe are predicted to continue to grow up to perhaps 10¹⁴ during the collapse of superclusters of galaxies. Even these would evaporate over a timescale of up to 10¹⁰⁶ years. The

power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...

emitted by a black hole in the form of Hawking radiation can be estimated for the simplest case of a nonrotating, non-charged Schwarzschild black hole of mass . Combining the formulas for the

Schwarzschild radius The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteris ...

of the black hole, the

Stefan–Boltzmann law The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths ...

of blackbody radiation, the above formula for the temperature of the radiation, and the formula for the surface area of a

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...

(the black hole's event horizon), several equations can be derived. The Hawking radiation temperature is:Hawking Radiation Calculator
/ref> :

T_\mathrm = \frac

The Bekenstein–Hawking luminosity of a black hole, under the assumption of pure photon emission (i.e. that no other particles are emitted) and under the assumption that the horizon is the radiating surface is: :

P = \frac

where is the luminosity, i.e., the radiated power, is the

reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...

, is the

speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...

, is the gravitational constant and is the mass of the black hole. It is worth mentioning that the above formula has not yet been derived in the framework of

semiclassical gravity Semiclassical gravity is the approximation to the theory of quantum gravity in which one treats matter fields as being quantum and the gravitational field as being classical. In semiclassical gravity, matter is represented by quantum matter fie ...

. The time that the black hole takes to dissipate is: :

t_\mathrm = \frac = \frac \approx 2.1\times10^\,\text \ \left(\frac\right)^3,

where and are the mass and (Schwarzschild) volume of the black hole. A black hole of one solar mass ( = ) takes more than to evaporate—much longer than the current

age of the universe In physical cosmology, the age of the universe is the time elapsed since the Big Bang. Astronomers have derived two different measurements of the age of the universe: a measurement based on direct observations of an early state of the universe, ...

at . But for a black hole of , the evaporation time is . This is why some astronomers are searching for signs of exploding primordial black holes. However, since the universe contains the

, in order for the black hole to dissipate, the black hole must have a temperature greater than that of the present-day blackbody radiation of the universe of 2.7 K. A study suggests that must be less than 0.8% of the mass of the

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...

– approximately the mass of the Moon. Black hole evaporation has several significant consequences: * Black hole evaporation produces a more consistent view of

black hole thermodynamics In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. As the study of the statistical mechanics of black-body radiation led to the developm ...

by showing how black holes interact thermally with the rest of the universe. * Unlike most objects, a black hole's temperature increases as it radiates away mass. The rate of temperature increase is exponential, with the most likely endpoint being the dissolution of the black hole in a violent burst of

gamma ray A gamma ray, also known as gamma radiation (symbol γ or \gamma), is a penetrating form of electromagnetic radiation arising from the radioactive decay of atomic nuclei. It consists of the shortest wavelength electromagnetic waves, typically ...

s. A complete description of this dissolution requires a model of quantum gravity, however, as it occurs when the black hole's mass approaches 1 Planck mass, when its radius will also approach two Planck lengths. * The simplest models of black hole evaporation lead to the

. The information content of a black hole appears to be lost when it dissipates, as under these models the Hawking radiation is random (it has no relation to the original information). A number of solutions to this problem have been proposed, including suggestions that Hawking radiation is perturbed to contain the missing information, that the Hawking evaporation leaves some form of remnant particle containing the missing information, and that information is allowed to be lost under these conditions.

Problems and extensions

Trans-Planckian problem

The trans-Planckian problem is the issue that Hawking's original calculation includes quantum particles where the

wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...

becomes shorter than the Planck length near the black hole's horizon. This is due to the peculiar behavior there, where time stops as measured from far away. A particle emitted from a black hole with a

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...

frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

, if traced back to the horizon, must have had an

infinite Infinite may refer to: Mathematics * Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...

frequency, and therefore a trans-Planckian wavelength. The

and the Hawking effect both talk about field modes in the superficially stationary

that change frequency relative to other coordinates that are regular across the horizon. This is necessarily so, since to stay outside a horizon requires acceleration that constantly Doppler shifts the modes. An outgoing

photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...

of Hawking radiation, if the mode is traced back in time, has a frequency that diverges from that which it has at great distance, as it gets closer to the horizon, which requires the wavelength of the photon to "scrunch up" infinitely at the horizon of the black hole. In a maximally extended external

Schwarzschild solution In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assump ...

, that photon's frequency stays regular only if the mode is extended back into the past region where no observer can go. That region seems to be unobservable and is physically suspect, so Hawking used a black hole solution without a past region that forms at a finite time in the past. In that case, the source of all the outgoing photons can be identified: a microscopic point right at the moment that the black hole first formed. The quantum fluctuations at that tiny point, in Hawking's original calculation, contain all the outgoing radiation. The modes that eventually contain the outgoing radiation at long times are redshifted by such a huge amount by their long sojourn next to the event horizon that they start off as modes with a wavelength much shorter than the Planck length. Since the laws of physics at such short distances are unknown, some find Hawking's original calculation unconvincing. The trans-Planckian problem is nowadays mostly considered a mathematical artifact of horizon calculations. The same effect occurs for regular matter falling onto a

white hole In general relativity, a white hole is a hypothetical region of spacetime and singularity that cannot be entered from the outside, although energy-matter, light and information can escape from it. In this sense, it is the reverse of a black ho ...

solution. Matter that falls on the white hole accumulates on it, but has no future region into which it can go. Tracing the future of this matter, it is compressed onto the final singular endpoint of the white hole evolution, into a trans-Planckian region. The reason for these types of divergences is that modes that end at the horizon from the point of view of outside coordinates are singular in frequency there. The only way to determine what happens classically is to extend in some other coordinates that cross the horizon. There exist alternative physical pictures that give the Hawking radiation in which the trans-Planckian problem is addressed. The key point is that similar trans-Planckian problems occur when the modes occupied with Unruh radiation are traced back in time.For an alternative derivation and more detailed discussion of Hawking radiation as a form of Unruh radiation, see: In the Unruh effect, the magnitude of the temperature can be calculated from ordinary Minkowski field theory, and is not controversial.

Large extra dimensions

The formulas from the previous section are applicable only if the laws of gravity are approximately valid all the way down to the Planck scale. In particular, for black holes with masses below the Planck mass (~), they result in impossible lifetimes below the Planck time (~). This is normally seen as an indication that the Planck mass is the lower limit on the mass of a black hole. In a model with

large extra dimension In particle physics and string theory (M-theory), the ADD model, also known as the model with large extra dimensions (LED), is a model framework that attempts to solve the hierarchy problem. (''Why is the force of gravity so weak compared to the el ...

s (10 or 11), the values of Planck constants can be radically different, and the formulas for Hawking radiation have to be modified as well. In particular, the lifetime of a micro black hole with a radius below the scale of the extra dimensions is given by equation 9 in Cheung (2002) and equations 25 and 26 in Carr (2005). :

\tau \sim \frac \left( \frac \right)^\frac,

where is the low-energy scale, which could be as low as a few TeV, and is the number of large extra dimensions. This formula is now consistent with black holes as light as a few TeV, with lifetimes on the order of the "new Planck time" ~.

In loop quantum gravity

A detailed study of the quantum geometry of a black hole

has been made using loop quantum gravity.001.08833Black Hole evaporation: A Perspective from Loop Quantum Gravity"> Loop-quantization does not reproduce the result for

black hole entropy In physics, black hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of black hole event horizons. As the study of the statistical mechanics of black-body radiation led to the development ...

originally discovered by

Bekenstein Bekenstein is a surname. Notable people with the surname include: * Jacob Bekenstein (1947–2015), Mexican-born Israeli-American physicist * Joshua Bekenstein, American businessman {{Short pages monitor