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'''Hawking radiation''' is [[black-body radiation]] that is predicted to be released by [[black hole]]s, due to quantum effects near the [[event horizon]]. It is named after the theoretical physicist [[Stephen Hawking]], who provided a theoretical argument for its existence in 1974.<ref>{{cite web |url-status=dead |url=http://www.charlierose.com/guest/view/6294 |archive-url=https://web.archive.org/web/20130329060031/http://www.charlierose.com/guest/view/6294 |archive-date=March 29, 2013 |first=Charlie |last=Rose |title=A conversation with Dr. Stephen Hawking & Lucy Hawking |website=charlierose.com }}</ref>

Hawking radiation reduces the mass and rotational energy of black holes and is therefore also known as '''black hole evaporation'''. Because of this, black holes that do not gain mass through other means are expected to shrink and ultimately vanish. [[Micro black hole]]s are predicted to be larger emitters of radiation than more massive black holes and should thus shrink and dissipate faster.<ref>{{Cite book|url=https://books.google.com/?id=zGxxDgAAQBAJ&pg=PA70&dq=Hawking+radiation+reduces+mass+and+energy+of+black+holes+and+is+also+known+as+black+hole+evaporation#v=onepage&q=Hawking%20radiation%20reduces%20mass%20and%20energy%20of%20black%20holes%20and%20is%20also%20known%20as%20black%20hole%20evaporation&f=false|title=Recent Developments in Intelligent Nature-Inspired Computing|last=Srikanta|first=Patnaik|date=2017-03-09|publisher=IGI Global|isbn=9781522523239|language=en}}</ref>

In June 2008, [[NASA]] launched the [[Fermi Gamma-ray Space Telescope|Fermi space telescope]], which is searching for the terminal gamma-ray flashes expected from evaporating [[primordial black hole]]s. In the event that speculative [[large extra dimension]] theories are correct, [[CERN]]'s [[Large Hadron Collider]] may be able to create micro black holes and observe their evaporation. No such micro black hole has ever been observed at CERN.<ref>{{Cite journal | doi = 10.1103/PhysRevD.65.056010|arxiv=hep-ph/0106219| title = High energy colliders as black hole factories: The end of short distance physics| journal = Physical Review D| volume = 65| issue = 5|page=056010| year = 2002| last1 = Giddings | first1 = S. | last2 = Thomas | first2 = S. |bibcode=2002PhRvD..65e6010G}}</ref><ref>{{Cite journal | doi = 10.1103/PhysRevLett.87.161602| pmid = 11690198| title = Black Holes at the Large Hadron Collider| journal = Physical Review Letters| volume = 87| issue = 16| page = 161602| year = 2001| last1 = Dimopoulos | first1 = S. | last2 = Landsberg | first2 = G. | bibcode = 2001PhRvL..87p1602D|arxiv = hep-ph/0106295 }}</ref><ref name="courier">{{cite web|url=http://cerncourier.com/cws/article/cern/29199|title=The case for mini black holes|work=CERN courier|date= November 2004}}</ref><ref>{{cite news| url=http://www.timesonline.co.uk/tol/news/uk/science/article4715761.ece | work=The Times | location=London | title=Stephen Hawkings 50 bet on the world the universe and the God particle | first=Mark | last=Henderson | date=September 9, 2008 | accessdate=May 4, 2010}}</ref>

In September 2010, a signal that is closely related to black hole Hawking radiation (see [[Analog models of gravity|analog gravity]]) was claimed to have been observed in a laboratory experiment involving optical light pulses. However, the results remain unverified and debatable.<ref name="Milanoguys">{{cite journal|title=Hawking radiation from ultrashort laser pulse filaments |first1=F. |last1=Belgiorno |first2=S. L. |last2=Cacciatori |first3=M. |last3=Clerici |first4=V. |last4=Gorini |first5=G. |last5=Ortenzi |first6=L. |last6=Rizzi |first7=E. |last7=Rubino |first8=V. G. |last8=Sala |first9=D. |last9=Faccio |arxiv=1009.4634 |doi=10.1103/PhysRevLett.105.203901 |date=2010 |journal=Phys. Rev. Lett. |volume=105 |issue=20 |page=203901 |bibcode=2010PhRvL.105t3901B}}</ref><ref>{{cite news|title=Ultrafast Laser Pulse Makes Desktop Black Hole Glow|newspaper=[[Wired (magazine)|Wired]]|date=September 29, 2010|first=Lisa|last=Grossman|url=https://www.wired.com/wiredscience/2010/09/hawking-radiation-in-the-lab/|accessdate=April 30, 2012}}</ref> Other projects have been launched to look for this radiation within the framework of [[Analog models of gravity|analog gravity]].

== Overview ==

[[Black holes]] are sites of immense [[gravity|gravitational attraction]]. Classically, the gravitation generated by the [[gravitational singularity]] inside a black hole is so powerful that nothing, not even [[electromagnetic radiation]], can escape from the black hole. It is yet unknown how [[gravity]] can be incorporated into [[quantum mechanics]]. Nevertheless, far from the black hole, the gravitational effects can be weak enough for calculations to be reliably performed in the framework of [[quantum field theory in curved spacetime]]. Hawking showed that quantum effects allow [[black holes]] to emit exact [[black-body radiation]]. The [[electromagnetic radiation]] is produced as if emitted by a black body with a [[temperature]] [[Proportionality (mathematics)|inversely proportional]] to the [[mass]] of the black hole.

Physical insight into the process may be gained by imagining that [[Elementary particle|particle]]–[[antiparticle]] radiation is emitted from just beyond the [[event horizon]]. This radiation does not come directly from the black hole itself, but rather is a result of [[virtual particle]]s being "boosted" by the black hole's gravitation into becoming real particles.{{cn|date=September 2019}} As the particle–antiparticle pair was produced by the black hole's gravitational energy, the escape of one of the particles lowers the mass of the black hole.<ref>{{cite book |last=Carroll |first=Bradley |first2=Dale |last2=Ostlie |page=673 |title=An Introduction to Modern Astrophysics |publisher=Addison Wesley |year=1996 |isbn=0-201-54730-9}}</ref>

An alternative view of the process is that [[vacuum fluctuation]]s cause a particle–antiparticle pair to appear close to the event horizon of a black hole. One of the pair falls into the black hole while the other escapes. In order to preserve total [[energy]], the particle that fell into the black hole must have had a [[negative energy]] (with respect to an observer far away from the black hole). This causes the black hole to lose mass, and, to an outside observer, it would appear that the black hole has just emitted a [[Elementary particle|particle]]. In another model, the process is a [[quantum tunnelling]] effect, whereby particle–antiparticle pairs will form from the vacuum, and one will tunnel outside the event horizon.{{cn|date=September 2019}}

An important difference between the black hole [[radiation]] as computed by Hawking and [[thermal radiation]] 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]], while the former fits the data better. Thus [[thermal radiation]] contains [[Information#As a property in physics|information]] about the body that emitted it,{{Citation needed|date=December 2017|reason=Describe what information about the emitter apart from its temperature is revealed}} while Hawking radiation seems to contain no such information, and depends only on the [[mass]], [[angular momentum]], and [[Charge (physics)|charge]] of the black hole (the [[no-hair theorem]]). This leads to the [[black hole information paradox]].

However, according to the conjectured [[String theory#Gauge-gravity duality|gauge-gravity duality]] (also known as the [[AdS/CFT correspondence]]), black holes in certain cases (and perhaps in general) are equivalent to solutions of [[quantum field theory]] at a non-zero [[temperature]]. 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 [[Hawking radiation#Trans-Planckian problem|below]]).

A black hole of one [[solar mass]] ({{Solar mass|link=y}}) has a temperature of only 60&nbsp;nanokelvins (60 [[Nano-|billionths]] of a [[kelvin]]); in fact, such a black hole would absorb far more [[cosmic microwave background radiation]] than it emits. A black hole of {{val|4.5|e=22|u=kg}} (about the mass of the [[Moon]], or about {{val|133|ul=um}} across) would be in equilibrium at 2.7&nbsp;K, absorbing as much radiation as it emits. Yet smaller [[primordial black hole]]s would emit more than they absorb and thereby lose mass.{{cn|date=September 2019}}

== 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]]s ought to create and emit particles.  When Hawking did the calculation, he found to his surprise that even non-rotating black holes produce radiation.<ref>{{cite book|last1=Hawking|first1=Stephen|authorlink=Stephen Hawking|title=[[A Brief History of Time]]|date=1988|publisher=Bantam Books|isbn=0-553-38016-8}}</ref>

== Trans-Planckian problem ==
The [[trans-Planckian problem]] is the issue that Hawking's original calculation includes [[quantum]] particles where the [[wavelength]] 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 [[Wikt:finite|finite]] [[frequency]], if traced back to the horizon, must have had an [[Infinity|infinite]] frequency, and therefore a trans-Planckian wavelength.

The [[Unruh effect]] and the Hawking effect both talk about field modes in the superficially stationary [[space-time]] 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 shift]]s the modes.

An outgoing Hawking-radiated [[photon]], 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 metric|Schwarzschild solution]], 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.<ref>{{Cite journal | doi = 10.1088/0034-4885/66/6/202|arxiv=gr-qc/0304042| title = Do black holes radiate?| journal = Reports on Progress in Physics| volume = 66| issue = 6| pages = 943–1008| year = 2003| last1 = Helfer | first1 = A. D. |bibcode=2003RPPh...66..943H}}</ref><ref>{{Cite journal | last1 = 't Hooft | first1 = G. | doi = 10.1016/0550-3213(85)90418-3 | title = On the quantum structure of a black hole | journal = Nuclear Physics B | volume = 256 | pages = 727–745 | year = 1985 | pmid =  | pmc = |bibcode = 1985NuPhB.256..727T }}</ref><ref>{{Cite journal | doi = 10.1103/PhysRevD.44.1731| title = Black-hole evaporation and ultrashort distances| journal = Physical Review D| volume = 44| issue = 6| pages = 1731–1739| year = 1991| last1 = Jacobson | first1 = T. | bibcode = 1991PhRvD..44.1731J}}</ref><ref>{{Cite journal | doi = 10.1103/PhysRevD.52.4559|arxiv=hep-th/9506121| title = Hawking radiation without trans-Planckian frequencies| journal = Physical Review D| volume = 52| issue = 8| pages = 4559–4568| year = 1995| last1 = Brout | first1 = R.| last2 = Massar | first2 = S.| last3 = Parentani | first3 = R.| last4 = Spindel | first4 = P. |bibcode=1995PhRvD..52.4559B}}</ref>

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]] 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.{{citation needed|date=August 2016}} The key point is that similar trans-Planckian problems occur when the modes occupied with Unruh radiation are traced back in time.<ref name="witt-blah">For an alternative derivation and more detailed discussion of Hawking radiation as a form of Unruh radiation see {{cite book|last=de Witt |first=Bryce |date=1980 |contribution=Quantum gravity: the new synthesis |page=696 |title=General Relativity: An Einstein Centenary Survey |editor1-first=S. |editor1-last=Hawking |editor2-first=W. |editor2-last=Israel |isbn=0-521-29928-4}}</ref> In the Unruh effect, the magnitude of the temperature can be calculated from ordinary [[Hermann Minkowski|Minkowski]] field theory, and is not controversial.

==Emission process==
Hawking radiation is required by [[Unruh effect|the Unruh effect]] 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.<ref name="witt-blah"/>

A [[Schwarzschild black hole]] has a metric:
:<math>\left(\operatorname{d}s\right)^2 = -\left(1-\tfrac{2M}{r}\right)\,\left(\operatorname{d}t\right)^2 + \frac{1}{\left(1-\frac{2M}{r}\right)} \,\left(\operatorname{d}r\right)^2 + r^2 \,\left(\operatorname{d}\Omega\right)^2\, </math>.

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
:<math>r = 2\,M + \frac{\rho^2}{\ 8\,M\ }\,</math>.

The local metric to lowest order is
:<math>\left(\operatorname{d}s\right)^2 \;=\; - \left(\frac{\rho}{4\,M}\right)^2 \,\left(\operatorname{d}t\right)^2 + \left(\operatorname{d}\rho\right)^2 + \left(\operatorname{d}X_\perp\right)^2 \;=\; - \rho^2 \,\left(\operatorname{d}\tau\right)^2 + \left(\operatorname{d}\rho\right)^2 + \left(\operatorname{d}X_\perp\right)^2\,</math>,

which is [[Rindler coordinates|Rindler]] in terms of {{math|''τ'' {{=}} {{sfrac|''t''|4''M''}}}}. The metric describes a frame that is accelerating to keep from falling into the black hole. The local acceleration, {{math|''α'' {{=}} {{frac|1|''ρ''}}}}, diverges as {{math|''ρ'' → 0}}.

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
:<math>T \;=\; \frac{\alpha}{2\pi} \;=\; \frac{1}{\,2\pi\rho\,} \;=\;\frac{1}{\,4\pi\sqrt{2M\left(r-2M\right)\,}\,}\,</math>;

which is the [[Unruh effect]].

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:

:<math>T\left(r'\right) \;=\; \frac{1}{\,4\pi\sqrt{2M(r-2M)\,}\,} \sqrt\frac{1 - \frac{2M}{r}}{\,1 - \frac{2M}{r'}\,} \;=\; \frac{1}{4\pi\sqrt{2Mr\left(1 - \frac{2M}{r'}\right)}}\,</math>.

The inverse temperature redshifted to {{mvar|r′}} at infinity is

:<math>T(\infty) = \frac{1}{4 \pi \sqrt{2 M r\,}}</math>

and {{mvar|r}} is the near-horizon position, near {{math|2''M''}}, so this is really:

:<math>T(\infty) = \frac{1}{\,8 \pi M\,} </math>.

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

:<math>T_\text{H} = \frac{1}{\,8 \pi M\,} </math>.

This can be expressed in a cleaner way in terms of the [[surface gravity]] of the black hole; this is the parameter that determines the acceleration of a near-horizon observer. In [[natural units]] ({{math|1=''G'' = ''c'' = ''ħ'' = ''k''{{sub|B}} = 1}}), the temperature is

:<math>T_\text{H} = \frac{\kappa}{\,2 \pi\,}\,</math>,

where {{mvar|κ}} is the [[surface gravity]] of the horizon. So a black hole can only be in equilibrium with a gas of radiation at a finite temperature. Since radiation incident on the black hole is absorbed, the black hole must emit an equal amount to maintain [[detailed balance]]. The black hole acts as a [[blackbody radiation|perfect blackbody]] radiating at this temperature.

In [[SI units]], the radiation from a [[Schwarzschild metric|Schwarzschild]] black hole is [[blackbody radiation]] with temperature

:<math>T = \frac{\hbar\,c^3}{\,8\,\pi\,G\,k_\text{B}\,M\,} \;\approx\; 1.227 \times 10^{+23} \text{K·kg}\, \times \frac{1}{M} \;=\;  6.169 \times 10^{-8}\text{K}\, \times \frac{M_\odot}{M} \,</math>,

where {{mvar|ħ}} is the [[Planck's constant|reduced Planck constant]], {{mvar|c}} is the [[speed of light]], {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]], {{mvar|G}} is the [[gravitational constant]], {{math|{{solar mass}}}} is the [[solar mass]], and {{mvar|M}} is the [[mass]] of the black hole.

From the black hole temperature, it is straightforward to calculate the black hole entropy. The change in entropy when a quantity of heat {{mvar|dQ}} is added is:

:<math>\operatorname{d}S \;=\; \frac{\operatorname{d}Q}{T} \;=\; 8 \pi M\, \operatorname{d}Q\,</math>.

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

:<math>\operatorname{d}S \;=\; 8 \pi M\, \operatorname{d}M \;=\; \operatorname{d}\left(4\,\pi\,M^2\right)\, </math>.

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

:<math>S = \pi\,R^2 = \frac{\,A\,}{4} </math>.

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 [[holographic principle|somehow encoded]] onto a bounding surface.

In 2019, Biancalana, Robson and Villari from Heriot-Watt University in Edinburgh (UK), showed that Hawking's radiation temperature is a purely topological quantity that can be calculated very simply by computing the Euler characteristics of the black hole spacetime.<ref name="RobsonDi Mauro Villari2019">{{cite journal |last1=Robson |first1=Charles W. |last2=Di Mauro Villari |first2=Leone |last3=Biancalana |first3=Fabio |title=Topological nature of the Hawking temperature of black holes |journal=Physical Review D |volume=99 |issue=4 |year=2019 |issn=2470-0010 |doi=10.1103/PhysRevD.99.044042|arxiv=1810.09322 }}</ref>

==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 [[special relativity|Einstein's equation]] [[E = mc²|{{math|''E'' {{=}} ''mc''<sup>2</sup>}}]]).

===1976 Page numerical analysis===

In 1976 [[Don Page (physicist)|Don Page]] calculated the power produced, and the time to evaporation, for a nonrotating, non-charged [[Schwarzschild black hole]] of mass {{mvar|M}}.<ref>{{cite journal |last=Page |first=Don N. |authorlink=Don Page (physicist) |date=1976 |title=Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole |journal=Physical Review D |volume=13 |issue=2 |pages=198–206 |doi=10.1103/PhysRevD.13.198 |url= |accessdate= |bibcode = 1976PhRvD..13..198P }}</ref> 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]] goes down in a complicated, [[spin (physics)|spin]]-dependent manner as frequency decreases, especially when the wavelength becomes comparable to the size of the event horizon. Note that writing in 1976, Page erroneously postulates 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 [[Neutrino#Flavor,_mass,_and_their_mixing|3 flavors of neutrinos with nonzero masses]].

For a mass much larger than 10<sup>17</sup>&nbsp;[[gram]]s, Page deduces that electron emission can be ignored, and that black holes of mass {{mvar|M}} in grams evaporate via massless electron and muon neutrinos, photons, and gravitons in a time {{mvar|τ}} of

:<math>\tau = 8.66 \times 10^{-27}\;\left[\frac{M}{\mathrm{g}}\right]^3\;\mathrm{s}\,.</math>

For a mass much smaller than 10<sup>17</sup>&nbsp;g, but much larger than {{val|5|e=14|u=g}}, the emission of [[ultrarelativistic limit|ultrarelativistic]] electrons and positrons will accelerate the evaporation, giving a lifetime of

:<math>\tau = 4.8 \times 10^{-27}\;\left[\frac{M}{\mathrm{g}}\right]^3\;\mathrm{s}\,.</math>

If black holes evaporate under Hawking radiation, a solar mass black hole will evaporate over 10<sup>64</sup> years.<ref name="Frautschi">Frautschi, S., 1982. Entropy in an expanding universe. Science, 217(4560), pp.593-599. See page 596: table 1 and section "black hole decay" and previous sentence on that page
{{quote|"Since we have assumed a maximum scale of gravitational binding – for instance, superclusters of galaxies – black hole formation eventually comes to an end in our model, with masses of up to 10<sup>14</sup>{{solar mass}} ... the timescale for black holes to radiate away all their energy ranges from to 10<sup>64</sup> years for black holes of one solar mass ..."}}</ref> A supermassive black hole with a mass of 10<sup>11</sup> (100 billion) {{Solar mass|}} will evaporate in around 2×10<sup>100</sup> years.<ref name=page>Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole, Don N. Page, ''Physical Review D'' '''13''' (1976), pp. 198–206.  {{doi|10.1103/PhysRevD.13.198}}. See in particular equation (27).</ref> Some monster black holes in the universe are predicted to continue to grow up to perhaps 10<sup>14</sup> {{solar mass}} during the collapse of superclusters of galaxies. Even these would evaporate over a timescale of up to 10<sup>106</sup> years<ref name="Frautschi"/>

===A crude analytic estimate===
{{Original research section|date=December 2015}}

The [[Power (physics)|power]] emitted by a black hole in the form of Hawking radiation can easily be estimated for the simplest case of a nonrotating, non-charged [[Schwarzschild black hole]] of mass {{mvar|M}}. Combining the formulas for the [[Schwarzschild radius]] of the black hole, the [[Stefan–Boltzmann law]] of blackbody radiation, the above formula for the temperature of the radiation, and the formula for the surface area of a [[sphere]] (the black hole's event horizon), several equations can be derived:

[[Stefan–Boltzmann constant]]:
:<math>\sigma = \frac{\pi^2 k_\mathrm{B}^4}{60 \hbar^3 c^2}</math>

[[Schwarzschild radius]]:
:<math>r_\mathrm{s} = \frac{2GM}{c^2}</math>

Acceleration due to gravity at the event horizon:
:<math>g = \frac{G M}{r_s^2} = \frac{c^4}{4 G M}</math>

Hawking radiation has a [[Blackbody radiation|blackbody]] (Planck) spectrum with a temperature {{mvar|T}} given by:
:<math>E = k_\mathrm{B} T = \frac{\hbar g}{2 \pi c} = \frac{\hbar}{2 \pi c} \left( \frac{c^4}{4 G M} \right) = \frac{\hbar c^3}{8 \pi G M}</math>

Hawking radiation temperature:
:{|cellpadding="2" style="border:2px solid #ccccff"
|<math>T_\mathrm{H} = \frac{\hbar c^3}{8 \pi G M k_\mathrm{B}} </math>
|}

For a one solar mass black hole, the peak Hawking radiation temperature is:
:<math>T_{\mathrm{H}} = \frac{\hbar c^3}{8 \pi G M_{\odot} k_\mathrm{B}} = 6.170 \times 10^{-8} \; \text{K} \,.</math>

The peak wavelength of this radiation is nearly 16 times the Schwarzschild radius of the black hole. Using [[Wien's displacement law|Wien's displacement constant]] {{math|''b'' {{=}} {{sfrac|''hc''|4.9651&thinsp;''k''<sub>B</sub>}}}} = {{val|2.8978|e=-3|u=m K}}:
:<math>\lambda_\mathrm{max} = \frac{b}{T_\mathrm{H}} = \frac{8\pi^2}{4.9651}\,r_\mathrm{s} = 15.902 \,r_\mathrm{s} </math>

Schwarzschild [[sphere]] surface area of [[Schwarzschild radius]] {{math|''r''<sub>s</sub>}}:
:<math>A_\mathrm{s} = 4 \pi r_\mathrm{s}^2 = 4 \pi \left( \frac{2 G M}{c^2} \right)^2 = \frac{16 \pi G^2 M^2}{c^4} </math>

[[Stefan–Boltzmann law|Stefan–Boltzmann]] power law:
:<math>P = A_\mathrm{s} j^\star = A_s \varepsilon \sigma T^4 </math>

For simplicity, assume a black hole is a perfect blackbody ({{math|''ε'' {{=}} 1}}).

Stefan–Boltzmann–Schwarzschild–Hawking black hole radiation power law derivation:
:<math>P = A_s \varepsilon \sigma T_\mathrm{H}^4 = \left( \frac{16 \pi G^2 M^2}{c^4} \right) \left( \frac{\pi^2 k_\mathrm{B}^4}{60 \hbar^3 c^2} \right) \left( \frac{\hbar c^3}{8 \pi G M k_\mathrm{B}} \right)^4 = \frac{\hbar c^6}{15360 \pi G^2 M^2} </math>

This yields the Bekenstein–Hawking luminosity of a black hole, under the assumption of pure photon emission (no other particles are emitted) and under the assumption that the horizon is the radiating surface:
:{|cellpadding="2" style="border:2px solid #ccccff"
|<math>P = \frac{\hbar c^6}{15360 \pi G^2 M^2} </math>
|}

where {{mvar|P}} is the luminosity, i.e., the radiated power, {{mvar|ħ}} is the [[reduced Planck constant]], {{mvar|c}} is the [[speed of light]],  {{mvar|G}} is the [[gravitational constant]] and {{mvar|M}} 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]].

Substituting the numerical values of the physical constants in the formula for luminosity we obtain P= {{val|3.562|e=32|u=W kg^2}}/M^2. The power of the Hawking radiation from a [[solar mass]] ({{math|{{solar mass}}}}) black hole turns out to be minuscule:
:<math>P = \frac{\hbar c^6}{15360 \pi G^2 M_\odot^2} = 9.007 \times 10^{-29} \; \text{W} \,.</math>

It is indeed an extremely good approximation to call such an object 'black'. Under the assumption of an otherwise empty universe, so that no [[matter]], [[cosmic microwave background radiation]], or other radiation falls into the black hole, it is possible to calculate how long it would take for the black hole to dissipate:
:<math>K_\mathrm{ev} = \frac{\hbar c^6}{15360 \pi G^2} = 3.562 \times 10^{32} \; \text{W} \; \text{kg}^2 \,.</math>

Given that the power of the Hawking radiation is the rate of evaporation energy loss of the black hole:
:<math>P = - \frac{dE}{dt} = \frac{K_\mathrm{ev}}{M^2} \,.</math>

Since the total energy {{mvar|E}} of the black hole is related to its mass {{mvar|M}} by Einstein's mass–energy formula {{math|''E'' {{=}} ''Mc''<sup>2</sup>}}:

:<math>P = - \frac{dE}{dt} = - \left( \frac{d}{dt} \right) M c^2 = -c^2 \frac{dM}{dt} \,.</math>

We can then equate this to our above expression for the power:
:<math>-c^2 \frac{dM}{dt} = \frac{K_\mathrm{ev}}{M^2} \,.</math>

This differential equation is separable, and we can write:
:<math>M^2\, dM = - \frac{K_\mathrm{ev}}{c^2}\, dt \,.</math>

The black hole's mass is now a function {{math|''M''(''t'')}} of time {{mvar|t}}. Integrating over {{mvar|M}} from {{math|''M''<sub>0</sub>}} (the initial mass of the black hole) to zero (complete evaporation), and over {{mvar|t}} from zero to {{math|t<sub>ev</sub>}}:
:<math>\int_{M_0}^0 M^2\, dM = - \frac{K_\mathrm{ev}}{c^2} \int_0^{t_\mathrm{ev}}\, dt \,.</math>

The evaporation time of a black hole is proportional to the cube of its mass:
:<math>t_\mathrm{ev} = \frac{c^2 M_0^3}{3 K_\mathrm{ev}} = \left( \frac{c^2 M_0^3}{3} \right) \left( \frac{15360 \pi G^2}{\hbar c^6} \right) = \frac{5120 \pi G^2 M_0^3}{\hbar c^4} = 8.4109 \times 10^{-17}\; \left[\frac{M_0}{\mathrm{kg}}\right]^3\; \mathrm{s} \,.</math>

Therefore, the evaporation time of a black hole is also proportional to its volume (and the cube of its Schwarzschild radius):
:<math>r_s = \frac{2 G M_0}{c^2}</math>
:<math>V_0 = \frac{4 \pi r_s^3}{3} = \frac{32 \pi G^3 M_0^3}{3 c^6}</math>
:<math>t_{ev} = \frac{5120 \pi G^2 M_0^3}{\hbar c^4} = \frac{480 c^2 V_0}{\hbar G} = 6.129 \times 10^{63} \frac{V_0}{m^3} s</math>

The time that the black hole takes to dissipate is:
:{|cellpadding="2" style="border:2px solid #ccccff"
|<math>t_\mathrm{ev} = \frac{5120 \pi G^2 M_0^3}{\hbar c^4} = \frac{480 c^2 V_0}{\hbar G} </math>
|}

where {{math|''M''<sub>0</sub>}} and {{math|''V''<sub>0</sub>}} are the mass and (Schwarzschild) volume of the black hole.

The lower classical quantum limit for mass for this equation is equivalent to the [[Planck mass]], {{math|''m''<sub>P</sub>}}.

Hawking radiation evaporation time for a Planck mass quantum black hole:
:<math>t_\mathrm{ev} = \frac{5120 \pi G^2 m_\mathrm{P}^3}{\hbar c^4} = 5120 \pi t_\mathrm{P} = 5120 \pi \sqrt{\frac{\hbar G}{c^5}} = 8.671 \times 10^{-40} \; \text{s} </math>

:{|cellpadding="2" style="border:2px solid #ccccff"
|<math>t_\mathrm{ev} = 5120 \pi \sqrt{\frac{\hbar G}{c^5}} </math>
|}

where {{math|''t''<sub>P</sub>}} is the [[Planck time]].

For a black hole of one [[solar mass]] ({{math|{{solar mass}}}} = {{val|1.98892|e=30|u=kg}}), we get an evaporation time of {{val|2.098|e=67}}&nbsp;years—much longer than the current [[age of the universe]] at {{val|13.799|0.021|e=9}}&nbsp;years:<ref name="Planck 2015">{{cite journal |author=Planck Collaboration |date=2016 |title=Planck 2015 results: XIII. Cosmological parameters |url=https://www.research.manchester.ac.uk/portal/files/52915797/aa25830_15.pdf |format=PDF |journal=[[Astronomy & Astrophysics|Astron. Astrophys.]] |volume=594 |at=A13, p. 31, Table 4 |arxiv=1502.01589 |bibcode=2016A&A...594A..13P |doi=10.1051/0004-6361/201525830 |hdl=10261/140585 |via=The University of Manchester's Research Explorer |access-date=27 October 2019}}</ref>

:<math>t_\mathrm{ev} = \frac{5120 \pi G^2 M_\odot^3}{\hbar c^4} = 6.617 \times 10^{74} \; \text{s}\,. </math>

But for a black hole of {{val|e=11|u=kg}}, the evaporation time is 2.667&nbsp;billion years. This is why some astronomers are searching for signs of exploding [[primordial black holes]].

However, since the universe contains the [[cosmic microwave background radiation]], in order for the black hole to dissipate, it must have a temperature greater than that of the present-day blackbody radiation of the universe of 2.7&nbsp;K = {{val|2.3|e=-4|u=eV}}. This implies that {{mvar|M}} must be less than 0.8% of the mass of the Earth<ref>{{cite arxiv |last=Kapusta |first=Joseph |date=1999 |eprint=astro-ph/9911309 |title=The Last Eight Minutes of a Primordial Black Hole}}</ref> – approximately the mass of the Moon.

[[Cosmic microwave background radiation]] universe temperature:
:<math>T_\mathrm{u} = 2.725 \; \text{K} </math>

Hawking total black hole mass:
:<math>M_\mathrm{H} \leq \frac{\hbar c^3}{8 \pi G k_\mathrm{B} T_\mathrm{u}} \leq 4.503 \times 10^{22} \; \text{kg} </math>

:{|cellpadding="2" style="border:2px solid #ccccff"
|<math>M_\mathrm{H} \leq \frac{\hbar c^3}{8 \pi G k_\mathrm{B} T_\mathrm{u}} </math>
|}

:<math>\frac{M_\mathrm{H}}{M_\oplus} = 7.539 \times 10^{-3} = 0.754 \; \% </math>

where {{math|{{earth mass}}}} is the total Earth mass.

In common units,

:<math>P = 3.563\,45 \times 10^{32}\; \left[\frac{\mathrm{kg}}{M}\right]^2\; \mathrm{W} </math>

:<math>t_\mathrm{ev} = 8.410\,92 \times 10^{-17} \;\left[\frac{M_0}{\mathrm{kg}}\right]^3\; \mathrm{s} \quad \approx\ 2.665\,32 \times 10^{-24}\; \left[\frac{M_0}{\mathrm{kg}}\right]^3\; \mathrm{yr} </math>

:<math>M_0 = 2.282\,71 \times 10^5\; \left[\frac{t_\mathrm{ev}}{\mathrm{s}}\right]^\frac13\; \mathrm{kg} \quad \approx\ 7.2 \times 10^7\; \left[\frac{t_\mathrm{ev}}{\mathrm{yr}}\right]^\frac13\; \mathrm{kg} </math>

So, for instance, a 1-second-life black hole has a mass of {{val|2.28|e=5|u=kg}}, equivalent to an energy of {{val|2.05|e=22|ul=J}} that could be released by {{val|5|e=6}} [[TNT equivalent|megatons of TNT]]. The initial power is {{val|6.84|e=21|ul=W}}.

Black hole evaporation has several significant consequences:
* Black hole evaporation produces a more consistent view of [[black hole thermodynamics]] 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]]s. A complete description of this dissolution requires a model of [[quantum gravity]], however, as it occurs when the black hole approaches [[Planck mass]] and [[planck length|Planck radius]].
* The simplest models of black hole evaporation lead to the [[black hole information paradox]]. 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.

==Large extra dimensions==
The formulae 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 (~{{val|e=-8|u=kg}}), they result in impossible lifetimes below the Planck time (~{{val|e=-43|u=s}}). 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]]s (10 or 11), the values of Planck constants can be radically different, and the formulae 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)<ref>{{cite journal|last1=Cheung|first1=Kingman|title=Black Hole Production and Large Extra Dimensions|arxiv=hep-ph/0110163|doi=10.1103/PhysRevLett.88.221602|journal=Phys. Rev. Lett.|volume=88|issue=22|page=221602|date=2002|bibcode=2002PhRvL..88v1602C}}</ref> and equations 25 and 26 in Carr (2005).<ref>{{Cite journal|last1=Carr|first1=B. J.|title=Primordial Black Holes – Recent Developments|arxiv=astro-ph/0504034|date=2005|journal=22nd Texas Symposium at Stanford, 12–17 December 2004|pages=89|bibcode=2005tsra.conf...89C}}</ref>

:<math>\tau \sim \frac{1}{M_*} \left( \frac{M_\mathrm{BH}}{M_*} \right) ^\frac{n+3}{n+1}\,, </math>

where {{math|''M''<sub>∗</sub>}} is the low energy scale, which could be as low as a few TeV, and {{mvar|n}} 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" ~{{val|e=-26|u=s}}.

==In loop quantum gravity==
A detailed study of the quantum geometry of a black hole [[event horizon]] has been made using [[loop quantum gravity]].<ref>{{Cite journal | doi = 10.1103/PhysRevLett.80.904 | last1 = Ashtekar  | first1 = A. | last2 = Baez | first2 = J. | last3 = Corichi | first3 = A. | last4 = Krasnov | first4 = K. | title = Quantum Geometry and Black Hole Entropy | bibcode=1998PhRvL..80..904A | volume=80 | issue = 5  | journal=Phys. Rev. Lett. | pages=904–907|arxiv = gr-qc/9710007 | year = 1998  }}</ref> Loop-quantization reproduces the result for [[black hole entropy]] originally discovered by [[Jacob Bekenstein|Bekenstein]] and [[Stephen Hawking|Hawking]]. Further, it led to the computation of quantum gravity corrections to the entropy and radiation of black holes.

Based on the fluctuations of the horizon area, a quantum black hole exhibits deviations from the Hawking spectrum that would be observable were [[X-ray]]s from Hawking radiation of evaporating [[primordial black holes]] to be observed.<ref>{{Cite journal | doi = 10.1016/j.nuclphysb.2007.01.009 | last1 = Ansari  | first1 = M. H.  | title = Spectroscopy of a canonically quantized horizon | bibcode=2007NuPhB.783..179A | volume=783 | issue = 3  | journal=Nucl. Phys. B | pages=179–212|arxiv = hep-th/0607081 | year = 2007  }}</ref> The quantum effects are centered at a set of discrete and unblended frequencies highly pronounced on top of Hawking radiation spectrum.<ref>{{Cite journal | doi = 10.1016/j.nuclphysb.2007.11.038  | last1 = Ansari  | first1 = M. H.  | title = Generic degeneracy and entropy in loop quantum gravity | bibcode=2008NuPhB.795..635A | volume=795 | issue = 3  | journal=Nucl. Phys. B | pages=635–644|arxiv = gr-qc/0603121 | year = 2008  }}</ref>

==Experimental observation==
Under experimentally achievable conditions for gravitational systems this effect is too small to be observed directly. However, in September 2010 an experimental set-up created a laboratory "white hole event horizon" that the experimenters claimed was shown to radiate an optical analog to Hawking radiation,<ref>{{cite news|url=http://www.technologyreview.com/blog/arxiv/25805/ |title=First Observation of Hawking Radiation |website=MIT Technology Review |date=September 27, 2010 |publisher=MIT}}</ref> although its status as a genuine confirmation remains in doubt.<ref>{{cite news|last=Matson |first=John|title=Artificial event horizon emits laboratory analog to theoretical black hole radiation|journal=Scientific American|date=October 1, 2010|url=http://www.scientificamerican.com/article.cfm?id=hawking-radiation}}</ref> Some scientists predict that Hawking radiation could be studied by analogy using [[sonic black hole]]s, in which [[phonon|sound perturbations]] are analogous to light in a gravitational black hole and the flow of an approximately [[perfect fluid]] is analogous to gravity.<ref>{{cite journal | last1 = Barceló | first1 = C. | last2 = Liberati | first2 = S. | last3 = Visser | first3 = M. | date = 2003 | title = Towards the observation of Hawking radiation in Bose–Einstein condensates| arxiv=gr-qc/0110036 | url = | journal = Int. J. Mod. Phys. A | volume = 18 | issue = 21| pages = 3735–3745 | doi=10.1142/s0217751x0301615x|bibcode = 2003IJMPA..18.3735B }}</ref><ref>{{cite journal|last=Steinhauer |first=J. |title=Observation of quantum Hawking radiation and its entanglement in an analogue black hole |journal=Nature Physics |date=2016 |url=http://www.nature.com/articles/nphys3863.epdf |volume=12 |issue=10 |pages=959–965 |doi=10.1038/nphys3863|arxiv=1510.00621 |bibcode=2016NatPh..12..959S }}</ref>

==See also==
{{Portal|Astronomy|Physics}}
* [[Analog models of gravity]]
* [[Black hole starship]]
* [[Gibbons–Hawking effect]]
* [[Blandford–Znajek process]] and [[Penrose process]], other extractions of black-hole energy
* [[Thorne–Hawking–Preskill bet]]
* [[Unruh effect]]

==References==
{{Reflist|30em}}

==Further reading==
{{columns list|colwidth=30em|style=font-size:90%|
* {{cite journal |last=Hawking |first=S. W. |authorlink= Stephen Hawking|date=1974 |title=Black hole explosions? |journal=Nature |volume=248 |issue=5443 |pages=30–31 |doi=10.1038/248030a0 |url= |accessdate= |bibcode = 1974Natur.248...30H }} → Hawking's first article on the topic
* {{cite journal |last=Page |first=Don N. |authorlink= Don Page (physicist)|date=1976 |title=Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole |journal=Phys. Rev. D |volume=13 |issue=2 |pages=198–206 |doi=10.1103/PhysRevD.13.198 |url= |accessdate= |bibcode = 1976PhRvD..13..198P }} → first detailed studies of the evaporation mechanism
* {{cite journal |last=Carr |first=B. J. |last2=Hawking |first2=S. W. |author2-link=Stephen Hawking |date=1974 |title=Black holes in the early universe |journal=[[Monthly Notices of the Royal Astronomical Society|Mon. Not. R. Astron. Soc.]] |volume=168 |issue=2 |pages=399–415 |bibcode=1974MNRAS.168..399C |url= |accessdate= |doi=10.1093/mnras/168.2.399|arxiv=1209.2243 }} → links between primordial black holes and the early universe
* {{cite journal |last=Barrau |first=A. |display-authors=etal |date=2002 |title=Antiprotons from primordial black holes |journal=Astron. Astrophys. |volume=388 |issue= 2|pages=676–687 |doi=10.1051/0004-6361:20020313 |url= |accessdate= |arxiv = astro-ph/0112486 |bibcode = 2002A&A...388..676B }}
* {{cite journal |last=Barrau |first=A. |display-authors=etal |date=2003 |title=Antideuterons as a probe of primordial black holes |journal=Astron. Astrophys. |volume=398 |issue= 2|pages=403–410 |doi=10.1051/0004-6361:20021588 |url= |accessdate= |arxiv = astro-ph/0207395 |bibcode = 2003A&A...398..403B }}
* {{cite journal |last=Barrau |first=A. |authorlink= |last2=Féron |first2=C. |last3=Grain |first3=J. |date=2005 |title=Astrophysical Production of Microscopic Black Holes in a Low–Planck-Scale World |journal=J. Am. Astronom. Soc. |volume=630 |issue= 2|pages=1015–1019 |doi=10.1086/432033 |url= |accessdate= |arxiv = astro-ph/0505436 |bibcode = 2005ApJ...630.1015B }} → experimental searches for primordial black holes thanks to the emitted antimatter
* {{cite arXiv |last=Barrau |first=A. |authorlink= |last2=Boudoul |first2=G. |title=Some aspects of primordial black hole physics |version= |pages= |url= |eprint=astro-ph/0212225 |accessdate= |year=2002 }} → cosmology with primordial black holes
* {{cite journal |last=Barrau |first=A. |authorlink= |last2=Grain |first2=J. |last3=Alexeyev |first3=S. O. |date=2004 |title=Gauss–Bonnet black holes at the LHC: beyond the dimensionality of space |journal=Phys. Lett. B |volume=584 |issue=1–2 |pages=114–122 |doi=10.1016/j.physletb.2004.01.019 |url= |accessdate= |arxiv = hep-ph/0311238 |bibcode = 2004PhLB..584..114B }} → searches for new physics (quantum gravity) with primordial black holes
* {{cite journal |last=Kanti |first=Panagiota |authorlink= |date=2004 |title=Black Holes in Theories with Large Extra Dimensions: a Review |journal=Int. J. Mod. Phys. A |volume=19 |issue=29 |pages=4899–4951 |doi=10.1142/S0217751X04018324 |url= |accessdate= |arxiv = hep-ph/0402168 |bibcode = 2004IJMPA..19.4899K }} → evaporating black holes and extra-dimensions
* {{cite journal|first1=Daisuke |last1=Ida |first2=Kin'ya |last2=Oda |first3=Seong-chan |last3=Park |arxiv=hep-th/0212108 |journal=Phys. Rev. D |volume=67 |issue=6 |date=2003 |page=064025 |title=Rotating black holes at future colliders: Greybody factors for brane fields |doi=10.1103/PhysRevD.67.064025|bibcode=2003PhRvD..67f4025I }}
* {{cite journal|first1=Daisuke |last1=Ida |first2=Kin'ya |last2=Oda |first3=Seong-chan |last3=Park |arxiv=hep-th/0503052 |journal=Phys. Rev. D |volume=71 |issue=12 |date=2005 |page=124039 |title=Rotating black holes at future colliders. II. Anisotropic scalar field emission |doi=10.1103/PhysRevD.71.124039|bibcode=2005PhRvD..71l4039I }}
* {{cite journal|first1=Daisuke |last1=Ida |first2=Kin'ya |last2=Oda |first3=Seong-chan |last3=Park |arxiv=hep-th/0602188 |journal=Phys. Rev. D |volume=73 |issue=12 |date=2006 |page=124022 |title=Rotating Black Holes at Future Colliders. III. Determination of Black Hole Evolution |doi=10.1103/PhysRevD.73.124022|bibcode=2006PhRvD..73l4022I }} → determination of black hole's life and extra dimensions
* {{cite journal|first1=Nistor |last1=Nicolaevici |journal=J. Phys. A: Math. Gen. |volume=36 |issue=27 |date=2003 |pages=7667–7677 |doi=10.1088/0305-4470/36/27/317 |title=Blackbody spectrum from accelerated mirrors with asymptotically inertial trajectories|bibcode=2003JPhA...36.7667N }} → consistent derivation of the Hawking radiation in the Fulling–Davies mirror model.
* {{cite news|authorlink=Lee Smolin|first=L. |last=Smolin |archive-url=https://web.archive.org/web/20080910170107/http://www.physicstoday.org/vol-59/iss-11/pdf/vol59no11p44_48.pdf |url=http://www.physicstoday.org/vol-59/iss-11/pdf/vol59no11p44_48.pdf |archive-date=September 10, 2008 |title=Quantum gravity faces reality |date=November 2006 |newspaper=Physics Today}} consists of the recent developments and predictions of [[loop quantum gravity]] about gravity in small scales including the deviation from Hawking radiation effect by [[Mohammad H Ansari|Ansari]].
* {{cite journal|authorlink=Mohammad H Ansari|last=Ansari |first=Mohammad H. |arxiv=hep-th/0607081 |title=Spectroscopy of a canonically quantized horizon |journal=Nucl. Phys. B |volume=783 |issue=3 |pages=179–212 |date=2007 |doi=10.1016/j.nuclphysb.2007.01.009 |bibcode=2007NuPhB.783..179A}} → studies the deviation of a loop quantized black hole from Hawking radiation. A novel observable quantum effect of black hole quantization is introduced.
* {{cite book|first1=Stuart L. |last1=Shapiro |first2=Saul A. |last2=Teukolsky |date=1983 |title=Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects |page=366 |publisher=Wiley-Interscience |isbn=978-0-471-87316-7}} → Hawking radiation evaporation formula derivation.
* {{Cite journal| last = Leonhardt | first = Ulf | authorlink = | last2= Maia |first2=Clovis |last3= Schuetzhold |first3=Ralf | date = 2010 | title = Focus on Classical and Quantum Analogs for Gravitational Phenomena and Related Effects | journal = [[New Journal of Physics|New J. Phys.]] | volume = 14 | issue = 10| pages =105032 | doi = 10.1088/1367-2630/14/10/105032| url = http://iopscience.iop.org/1367-2630/focus/Focus%20on%20Classical%20and%20Quantum%20Analogues%20for%20Gravitational%20Phenomena%20and%20Related%20Effects | accessdate = | bibcode = 2012NJPh...14j5032L }}
}}

==External links==
* [http://xaonon.dyndns.org/hawking/ Hawking radiation calculator tool]
* [https://cerncourier.com/the-case-for-mini-black-holes/ The case for mini black holes] A. Barrau & J. Grain explain how the Hawking radiation could be detected at colliders
* [http://www.ejtp.com/articles/ejtpv13i35p91.pdf Black Holes – Any''body'' out there?]
* [http://casa.colorado.edu/~ajsh/hawk.html University of Colorado at Boulder]
* [http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=448732 Hawking radiation on arxiv.org]
* [http://news.slashdot.org/story/10/09/27/1256236/Hawking-Radiation-Claimed-Created-In-a-Lab Hawking radiation observed in laboratory?]

{{quantum gravity}}
{{Stephen Hawking}}

{{DEFAULTSORT:Hawking Radiation}}
[[Category:Black holes]]
[[Category:Quantum field theory]]
[[Category:Stephen Hawking]]
[[Category:Astronomical hypotheses]]
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