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| [[File:BlackHole.jpg|thumb|Artist's depiction of a [[black hole]]]]
| | Hospitals and clinics the Clash of Clans hack tool; there is also hack tools with respect to other games. People young and old can check out all of the hacks and obtain many of those which they need. It is sure these people will have lost at fun once they own the hack tool saved.<br><br>If you have any inquiries relating to where and how to make use of [http://prometeu.net clash of clans hack no survey], you could contact us at the web site. Yet somehow Supercell, by allowing the actual illusion on the multiplayer game, taps into instinctual male drive to assist you to from the status hierarchy, and even though it''s unattainable to the top of your hierarchy if it's not necessary been [http://www.Dict.cc/englisch-deutsch/logging.html logging] in each day because the game turned out plus you invested fundamental money in extra builders, the drive for obtaining a small bit further obliges enough visitors to pay out a real income towards virtual 'gems'" that sport could be the top-grossing app within the Practical application Store.<br><br>At hand is a patch quest button that you will want to click after entering the desired values. when you check back directly on the game after 30 seconds to a minute, you will already gain the items. On that point is nothing wrong when it comes to making use of hacks. To hack is in fact the best way in enjoy clash of clans cheats. Make use of usually the Resources that you have, and take advantage pointing to this 2013 Clash of Clans download! Then why pay for coins or gems when you has the potential to get the needed foods with this tool! Hurry and get you are very own Clash created by Clans hack tool right now. The needed parts are just a quantity clicks away.<br><br>Discs are fun, nonetheless typically also be costly. The costs of gaming and consoles can cost more than many people would choose those to be, but this may often be eliminated.<br><br>Whether you are looking Conflict of Young couples Jewels Free, or your are just buying a Affect Conflict of Tribes, right now the smartest choice while on the internet, absolutely free and also only takes a couple of minutes to get all these products.<br><br>It's to select the most effective xbox game gaming console. In the beginning, you should think of your standard requirements like a video game player, as well as check out the specs made available from nearly every unit you are if you think about. Consider investigating on-line. Check consumer reviews to ascertain if more gamers have discovered difficulty with the unit. Ahead of buying a game process, you should know nearly everything you are able as a way to regarding it.<br><br>Which will master game play regarding shooter video games, guru your weapons. Do understand everything there is to learn about each and whatever weapon style in recreation. Each weapon excels in the certain ways, but occurs short in others. When you know that pluses and minuses involved with each weapon, you can use them to king advantage. |
| '''Hawking radiation''' is [[black body radiation]] that is predicted to be released by [[black hole]]s, due to [[quantum physics|quantum]] effects near the [[event horizon]].
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| It is named after the physicist [[Stephen Hawking]], who provided a theoretical argument for its existence in 1974,<ref>[http://www.charlierose.com/guest/view/6294 Charlie Rose: A conversation with Dr. Stephen Hawking & Lucy Hawking]</ref> and sometimes also after [[Jacob Bekenstein]], who predicted that black holes should have a finite, non-zero [[temperature]] and [[entropy]].<ref>{{cite news|last=Levi Julian|first=Hana|title='40 Years of Black Hole Thermodynamics' in Jerusalem|url=http://www.israelnationalnews.com/News/News.aspx/159585#.UErd_yJipNs|accessdate=8 September 2012|newspaper=[[Arutz Sheva]]|date=3 September 2012}}</ref>
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| Hawking's work followed his visit to [[Moscow]] in 1973 where the Soviet scientists [[Yakov Zeldovich]] and Alexei Starobinsky showed him that according to the quantum mechanical [[uncertainty principle]], [[rotating black hole]]s should create and emit particles.<ref>''A Brief History of Time'', Stephen Hawking, Bantam Books, 1988.</ref> Hawking radiation reduces the mass and the energy of the black hole and is therefore also known as ''black hole evaporation''. Because of this, black holes that lose more mass than they gain through other means are expected to shrink and ultimately vanish. [[Micro black hole]]s (MBHs) are predicted to be larger net emitters of radiation than larger black holes and should shrink and dissipate faster.
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| 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">Hawking radiation from ultrashort laser pulse filaments
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| Authors: F. Belgiorno, S.L. Cacciatori, M. Clerici, V. Gorini, G. Ortenzi, L. Rizzi, E. Rubino, V.G. Sala, D. Faccio http://arxiv.org/abs/1009.4634</ref><ref>{{cite web|title=Ultrafast Laser Pulse Makes Desktop Black Hole Glow|work=[[Wired (magazine)|Wired]]|date=29 September 2010|author=Lisa Grossman|url=http://www.wired.com/wiredscience/2010/09/hawking-radiation-in-the-lab/|accessdate=30 April 2012}}</ref> Other projects have been launched to look for this radiation within the framework of [[Analog models of gravity|analog gravity]]. In June 2008, [[NASA]] launched the [[Fermi Gamma-ray Space Telescope|Fermi space telescope]], which will search 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|CERN's]] [[Large Hadron Collider]] may be able to create micro black holes and observe their evaporation.<ref>S.B. Giddings and S.D. Thomas, "High-energy colliders as black hole factories: The End of short distance physics," [http://arXiv.org/abs/hep-ph/0106219 arXiv:hep-ph/0106219], [http://prola.aps.org/abstract/PRD/v65/i5/e056010 Phys. Rev. D65:056010 (2002)].</ref><ref>S. Dimopoulos and G.L. Landsberg, "Black holes at the LHC", [http://arxiv.org/abs/hep-ph/0106295 arXiv:hep-ph/0106295],
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| [http://prola.aps.org/abstract/PRL/v87/i16/e161602 Phys. Rev. Lett. 87:161602 (2001)]</ref><ref name="courier">{{cite web|url=http://cerncourier.com/cws/article/cern/29199|title=CERN courier - ''The case for mini black holes. Nov 2004''}}</ref><ref>American Institute of Physics Bulletin of Physics News, Number 558, September 26, 2001, by Phillip F. Schewe, Ben Stein, and James Riordon</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>
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| ==Overview==
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| {{general relativity}}
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| [[Black holes]] are sites of immense [[gravity|gravitational attraction]]. Classically, the gravitation is so powerful that nothing, not even [[electromagnetic radiation]] (including [[light]]), 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]], which is the average thermal radiation emitted by an idealized thermal source known as a black body. The [[electromagnetic radiation]] is as if it were emitted by a black body with a [[temperature]] that is [[proportionality (mathematics)|inversely proportional]] to the black hole's [[mass]].
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| 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.<ref name="kumar2012">{{cite doi|10.3968/j.ans.1715787020120502.1817}}</ref> As the particle-antiparticle pair was produced by the black hole's gravitational energy, the escape of one of the particles takes away some of the mass of the black hole.<ref>{{Citation
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| | last = Carroll
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| | first = Bradley
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| | coauthors = Dale Ostlie
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| | pages = 673
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| | title = An Introduction to Modern Astrophysics
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| | publisher = Addison Wesley
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| | year = 1996
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| | isbn = 0-201-54730-9 }}</ref>
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| A slightly more precise, but still much simplified, 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). By this process, the black hole loses 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.<ref name="kumar2012"/>
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| 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]] about the body that emitted it, 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]].
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| 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 no such loss exists in the [[quantum field theory]]), 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]]).
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| A black hole of one [[solar mass]] has a temperature of only 60 nanokelvin (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 4.5 × 10<sup>22</sup> kg (about the mass of the [[Moon]]) would be in equilibrium at 2.7 kelvin, absorbing as much radiation as it emits. Yet smaller [[primordial black hole]]s would emit more than they absorb, and thereby lose mass.<ref name="kumar2012"/>
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| == Trans-Planckian problem ==
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| The [[trans-Planckian problem]] is the observation that Hawking's original calculation requires talking about [[quantum]] particles in which the [[wavelength]] becomes shorter than the [[Planck length]] near the black hole's horizon. It is due to the peculiar behavior near a gravitational horizon 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 there and a trans-Planckian wavelength.
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| 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 which are regular across the horizon. This is necessarily so, since to stay outside a horizon requires acceleration which constantly [[Doppler shift]]s the modes.
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| An outgoing Hawking radiated [[photon]], if the mode is traced back in time, has a frequency which 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 only stays regular if the mode is extended back into the past region where no observer can go. That region doesn't seem to be observable and is physically suspect, so Hawking used a black hole solution without a past region which forms at a finite time in the past. In that case, the source of all the outgoing photons can be identified–it is a microscopic point right at the moment that the black hole first formed.
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| 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>[http://arxiv.org/abs/gr-qc/0304042 Adam D. Helfer: Do black holes radiate?]</ref><ref>G. 't Hooft, Nuclear Phys B256, 727 (1985)</ref><ref>T. Jacobson, Phys Review D 44 1731 (1991)</ref><ref name="autogenerated1">http://www.fys.ruu.nl/~wwwthe/lectures/itfuu-0196.ps (p.46)</ref><ref>R. Brout, S. Massar, R. Parentani, Ph. Spindel, ''Hawking radiation without trans-Planckian frequencies'', Phys. Rev. D 52, 4559 - 4568 (1995)</ref><ref>Helfer, Adam D. ''Trans-Planckian Modes, Back-Reaction, and the Hawking Process'', http://arxiv.org/abs/gr-qc/0008016</ref>
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| The trans-Planckian problem is nowadays mostly considered a mathematical artifact of horizon calculations.<ref name="autogenerated1" /><ref>[http://relativity.livingreviews.org/open?pubNo=lrr-2005-12&page=articlesu11.html Analog Gravity<!-- Bot generated title -->]</ref> The same effect occurs for regular matter falling onto a [[white hole]] solution. Matter which 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 which 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.
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| There exist alternative physical pictures which 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.<ref name="witt-blah">For an alternative derivation and more detailed discussion of Hawking radiation as a form of Unruh radiation see [[Bryce de Witt]]'s chapter ''Quantum gravity: the new synthesis'' pg c.696 in
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| ''General Relativity: An Einstein Centenary'' eds S Hawking and W 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.
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| ==Emission process==
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| Hawking radiation is required by 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"/>
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| A [[Schwarzschild black hole]] has a metric
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| :<math>
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| ds^2 = -\left(1-{2M\over r}\right)dt^2 + {1\over 1- 2M/r} dr^2 + r^2 d\Omega^2.</math>
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| The black hole is the background spacetime for a quantum field theory.
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| 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 = 2M + u^2/2M</math>. The local metric to lowest order is
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| :<math>
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| ds^2 = - {u^2\over 4M^2} dt^2 + 4 du^2 + dX_\perp^2 = - \rho^2 d\tau^2 + d\rho^2 + dX_\perp^2,</math>
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| which is [[Rindler coordinates|Rindler]] in terms of <math>\tau=t/4M</math> and <math>\rho=2u</math>. The metric describes a frame that is accelerating to keep from falling into the black hole. The local acceleration diverges as <math>u\rightarrow 0</math>.
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| 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 inverse temperature
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| :<math>\beta(u)=2\pi \rho = (4\pi) u = 4\pi \sqrt{2M(r-2M)};</math>
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| this is the [[Unruh effect]].
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| The gravitational redshift is 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:
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| :<math>\beta(r') = 4\pi \sqrt{2M(r-2M)} \sqrt{1- 2M/r' \over 1- 2M/r}.</math>
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| The inverse temperature redshifted to r' at infinity is
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| :<math>\beta(\infty) = (4\pi)\sqrt{2Mr} \;</math>
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| and <math>r</math> is the near-horizon position, near <math>2M</math>, so this is really
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| :<math>\beta = 8 \pi M.</math>
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| So a field theory defined on a black hole background is in a thermal state whose temperature at infinity is
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| :<math>T_H = {1 \over 8 \pi M}.</math>
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| This can be expressed more cleanly 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> G = c = \hbar = k_\text{B} = 1</math>), the temperature is
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| :<math>T_H = \frac{\kappa}{2 \pi},</math>
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| where <math>\kappa</math> 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.
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| In [[SI units]], the radiation from a [[Schwarzschild metric|Schwarzschild]] black hole is [[black-body radiation]] with temperature
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| :<math>T = {\hbar \, c^3 \over 8 \pi G M k_\text{B}} \;\quad \left(\approx {1.227 \times 10^{23}\; \text{kg} \over M}\; \text{K} \right),</math>
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| where <math>\hbar</math> is the [[Planck's constant|reduced Planck constant]], ''c'' is the [[speed of light]], ''k''<sub>B</sub> is the [[Boltzmann constant]], ''G'' is the [[gravitational constant]], and ''M'' is the [[mass]] of the black hole.
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| From the black hole temperature, it is straightforward to calculate the black hole entropy. The change in entropy when a quantity of heat ''dQ'' is added is
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| :<math>dS = {dQ\over T} = 8\pi M dQ.</math>
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| The heat energy that enters serves to increase the total mass, so
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| :<math>dS = 8 \pi M dM = d(4 \pi M^2).</math>.
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| 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:
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| :<math>S = \pi R^2 = {A \over 4}.</math>
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| 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 [[holography|somehow encoded]] onto a bounding surface.
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| ==Black hole evaporation==
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| 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]]'').
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| 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 <math>M</math>. Combining the formulas for the [[Schwarzschild radius]] of the black hole, the [[Stefan–Boltzmann law]] of black-body 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), equation derivation:
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| [[Stefan–Boltzmann constant]]:
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| :<math>\sigma = \frac{\pi^2 k_B^4}{60 \hbar^3 c^2} \;</math>
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| [[Schwarzschild radius]]:
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| :<math>r_s = \frac{2GM}{c^2} \;</math>
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| Black hole [[surface gravity]] at the horizon:
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| :<math>g = \frac{G M}{r_s^2} = \frac{c^4}{4 G M} \;</math>
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| Hawking radiation has a [[Black-body radiation|black-body]] (Planck) spectrum with a temperature T given by:
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| :<math>E = k_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>
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| Hawking radiation temperature:
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| :{|cellpadding="2" style="border:2px solid #ccccff"
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| |<math>T_H = \frac{\hbar c^3}{8 \pi G M k_B} \;</math>
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| |}
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| Schwarzschild [[sphere]] surface area of [[Schwarzschild radius]] <math>r_s</math>:
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| :<math>A_s = 4 \pi r_s^2 = 4 \pi \left( \frac{2 G M}{c^2} \right)^2 = \frac{16 \pi G^2 M^2}{c^4} \;</math>
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| [[Stefan–Boltzmann law|Stefan–Boltzmann]] power law:
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| :<math>P = A_s j^{\star} = A_s \epsilon \sigma T^{4} \;</math>
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| A black hole is a perfect black-body:
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| :<math>\epsilon = 1 \;</math>
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| Stefan–Boltzmann–Schwarzschild–Hawking black hole radiation power law derivation:
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| :<math>P = A_s \epsilon \sigma T_H^4 = \left( \frac{16 \pi G^2 M^2}{c^4} \right) \left( \frac{\pi^2 k_B^4}{60 \hbar^3 c^2} \right) \left( \frac{\hbar c^3}{8 \pi G M k_B} \right)^4 = \frac{\hbar c^6}{15360 \pi G^2 M^2} \;</math>
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| Stefan–Boltzmann-Schwarzschild-Hawking power law:
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| :{|cellpadding="2" style="border:2px solid #ccccff"
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| |<math>P = \frac{\hbar c^6}{15360 \pi G^2 M^2} \;</math>
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| |}
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| Where <math>P</math> is the energy outflow, ''<math>\hbar</math>'' is the [[reduced Planck constant]], <math>c</math> is the [[speed of light]], and <math>G</math> is the [[gravitational constant]]. It is worth mentioning that the above formula has not yet been derived in the framework of [[semiclassical gravity]].
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| The power in the Hawking radiation from a [[solar mass]] (<math>M_{\odot}</math>) black hole turns out to be a minuscule 9 × 10<sup>−29</sup> watts. It is indeed an extremely good approximation to call such an object 'black'.
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| :<math>P = \frac{\hbar c^6}{15360 \pi G^2 M_{\odot}^2} = 9.004 \times 10^{-29} \; \text{W} \;</math>
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| Under the assumption of an otherwise empty universe, so that no [[matter]] or [[cosmic microwave background radiation]] falls into the black hole, it is possible to calculate how long it would take for the black hole to dissipate:
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| :<math>K_{\operatorname{ev}} = \frac{\hbar c^6}{15360 \pi G^2} = 3.562 \times 10^{32} \; \text{W} \cdot \text{kg}^2 \;</math>
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| Given that the power of the Hawking radiation is the rate of evaporation energy loss of the black hole:
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| :<math>P = - \frac{dE}{dt} = \frac{K_{\operatorname{ev}}}{M^2} \;</math>
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| Since the total energy E of the black hole is related to its mass M by Einstein's mass-energy formula:
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| :<math>E = Mc^2 \;</math>
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| :<math>P = - \frac{dE}{dt} = - \left( \frac{d}{dt} \right) M c^2 = -c^2 \frac{dM}{dt} \;</math>
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| We can then equate this to our above expression for the power:
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| :<math>-c^2 \frac{dM}{dt} = \frac{K_{\operatorname{ev}}}{M^2} \;</math>
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| This differential equation is separable, and we can write:
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| :<math>M^2 dM = - \frac{K_{\operatorname{ev}}}{c^2} dt \;</math>
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| The black hole's mass is now a function ''M''(''t'') of time ''t''. Integrating over M from <math>M_0</math> (the initial mass of the black hole) to zero (complete evaporation), and over t from zero to <math>t_{\operatorname{ev}} \;</math>:
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| :<math>\int_{M_0}^0 M^2 dM = - \frac{K_{\operatorname{ev}}}{c^2} \int_0^{t_{\operatorname{ev}}} dt \;</math>
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| The evaporation time of a black hole is proportional to the cube of its mass:
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| :<math>t_{\operatorname{ev}} = \frac{c^2 M_0^3}{3 K_{\operatorname{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} = 8.410 \times 10^{-17} \left[\frac{M_0}{\mathrm{kg}}\right]^3 \mathrm{s} \;</math>
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| The time that the black hole takes to dissipate is:
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| :{|cellpadding="2" style="border:2px solid #ccccff"
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| |<math>t_{\operatorname{ev}} = \frac{5120 \pi G^2 M_0^{3}}{\hbar c^4} \;</math>
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| |}
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| Where <math>M_0</math> is the mass of the black hole.
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| The lower classical quantum limit for mass for this equation is equivalent to the [[Planck mass]], <math>m_P</math>.
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| Planck mass quantum black hole Hawking radiation evaporation time:
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| :<math>t_{\operatorname{ev}} = \frac{5120 \pi G^2 m_P^3}{\hbar c^4} = 5120 \pi t_P = 5120 \pi \sqrt{\frac{\hbar G}{c^5}} = 8.671 \times 10^{-40} \; \text{s} \;</math>
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| :{|cellpadding="2" style="border:2px solid #ccccff"
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| |<math>t_{\operatorname{ev}} = 5120 \pi \sqrt{\frac{\hbar G}{c^5}} \;</math>
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| |}
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| Where <math>t_P</math> is the [[Planck time]].
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| For a black hole of one [[solar mass]] (<math>M_{\odot}</math> = 1.98892 × 10<sup>30</sup> kg), we get an evaporation time of 2.098 × 10<sup>67</sup> years—much longer than the current [[big bang|age of the universe]] at 13.798 ± 0.037 x 10<sup>9 </sup>years.<ref name='planck_cosmological_parameters'>{{cite journal | arxiv=1303.5076 | title=Planck 2013 results. XVI. Cosmological parameters | author=Planck collaboration | journal=Submitted to Astronomy & Astrophysics | year=2013|bibcode = 2013arXiv1303.5076P }}</ref>
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| :<math>t_{\operatorname{ev}} = \frac{5120 \pi G^2 M_{\odot}^3}{\hbar c^4} = 6.617 \times 10^{74} \; \text{s} \;</math>
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| But for a black hole of 10<sup>11</sup> kg, the evaporation time is 2.667 billion years. This is why some astronomers are searching for signs of exploding [[primordial black holes]].
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| 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 black-body radiation of the universe of 2.7 K = 2.3 × 10<sup>−4</sup> eV. This implies that <math>M</math> must be less than 0.8 of the mass of the [[Earth]]<ref>[http://arxiv.org/abs/astro-ph/9911309 [astro-ph/9911309] The Last Eight Minutes of a Primordial Black Hole<!-- Bot generated title -->]</ref> - approximately the mass of the [[Moon]].
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| [[Cosmic microwave background radiation]] universe temperature:
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| :<math>T_u = 2.725 \; \text{K} \;</math>
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| Hawking total black hole mass:
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| :<math>M_H \leq \frac{\hbar c^3}{8 \pi G k_B T_u} \leq 4.503 \times 10^{22} \; \text{kg} \;</math>
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| :{|cellpadding="2" style="border:2px solid #ccccff"
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| |<math>M_H \leq \frac{\hbar c^3}{8 \pi G k_B T_u} \;</math>
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| |}
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| :<math>\frac{M_H}{M_{\oplus}} = 7.539 \times 10^{-3} = 0.754 \; \% \;</math>
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| Where, <math>M_{\oplus}</math> is the total [[Earth]] mass.
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| In common units, | |
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| :<math>P = 3.563 \, 45 \times 10^{32} \left[\frac{\mathrm{kg}}{M}\right]^2 \mathrm{W} \;</math>
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| :<math>t_\mathrm{ev} = 8.407 \, 16 \times 10^{-17} \left[\frac{M_0}{\mathrm{kg}}\right]^3 \mathrm{s}
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| \ \ \approx\ 2.66 \times 10^{-24} \left[\frac{M_0}{\mathrm{kg}}\right]^3 \mathrm{yr} \;</math>
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| :<math>M_0 = 2.282 \, 71 \times 10^5 \left[\frac{t_\mathrm{ev}}{\mathrm{s}}\right]^{1/3} \mathrm{kg}
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| \ \ \approx\ 7.2 \times 10^7 \left[\frac{t_\mathrm{ev}}{\mathrm{yr}}\right]^{1/3} \mathrm{kg} \;</math>
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| So, for instance, a 1-second-lived black hole has a mass of 2.28 × 10<sup>5</sup> kg, equivalent to an energy of 2.05 × 10<sup>22</sup> J that could be released by 5 × 10<sup>6</sup> [[TNT equivalent|megatons of TNT]].
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| The initial power is 6.84 × 10<sup>21</sup> W.
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| Black hole evaporation has several significant consequences:
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| * 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.
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| * 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]].
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| * 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.
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| ==Large extra dimensions==
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| Formulae from the previous section are only applicable if laws of gravity are approximately valid all the way down to the Planck scale. In particular, for black holes with masses below Planck mass (~10<sup>−5</sup> g), they result in unphysical lifetimes below Planck time (~10<sup>−43</sup> s). This is normally seen as an indication that Planck mass is the lower limit on the mass of a black hole.
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| In the model with [[large extra dimension]]s, values of Planck constants can be radically different, and formulas for Hawking radiation have to be modified as well. In particular, the lifetime of a micro black hole (with radius below the scale of extra dimensions) is given by
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| :<math>\tau \sim {1 \over M_*} \Bigl( {M_{BH} \over M_*} \Bigr) ^{(n+3)/(n+1)} </math>
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| where <math>M_*</math> is the low energy scale (which could be as low as a few TeV), and ''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 "new Planck time" ~10<sup>−26</sup> s.
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| ==Experimental observation of Hawking radiation==
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| Under experimentally achievable conditions for gravitational systems this effect is too small to be observed directly. In September 2010, however, an experimental set-up created a laboratory "white hole event horizon" that the experimenters claimed was shown to radiate Hawking radiation,<ref>[http://www.technologyreview.com/blog/arxiv/25805/ "First Observation of Hawking Radiation"] from the ''[[Technology Review]]''</ref> although its status as a genuine confirmation remains in doubt.<ref>{{cite journal|author=Matson, John|title=Artificial event horizon emits laboratory analog to theoretical black hole radiation|journal=Sci. Am|date=Oct 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>C. Barceló, S. Liberati and M. Visser, “Towards the observation of Hawking radiation in Bose–Einstein condensates,” [http://arxiv.org/abs/gr-qc/0110036 arXiv:gr-qc/0110036] Int. J. Mod. Phys. A 18, 3735 (2003) .</ref>
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| ==See also==
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| *[[Black hole starship]]
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| *[[Thorne–Hawking–Preskill bet]]
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| *[[Gibbons-Hawking effect]]
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| ==Notes==
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| {{Reflist|30em}}
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| ==Further reading==
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| *{{cite journal |last=Hawking |first=S. W. |authorlink= |coauthors= |year=1974 |month= |title=Black hole explosions? |journal=Nature |volume=248 |issue=5443 |page=30 |doi=10.1038/248030a0 |url= |accessdate= |bibcode = 1974Natur.248...30H }} → Hawking's first article on the topic
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| *{{cite journal |last=Page |first=Don N. |authorlink= |coauthors= |year=1976 |month= |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 }} → first detailed studies of the evaporation mechanism
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| *{{cite journal |last=Carr |first=B. J. |authorlink= |coauthors=Hawking, S. W. |year=1974 |month= |title=Black holes in the early universe |journal=Monthly Notices of the Royal Astronomical Society |volume=168 |issue=2 |pages=399–415 |bibcode=1974MNRAS.168..399C |url= |accessdate= |quote= }} → links between primordial black holes and the early universe
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| *{{cite journal |last=Barrau |first=A. |authorlink= |coauthors=''et al.'' |year=2002 |month= |title=Antiprotons from primordial black holes |journal=Astronomy & Astrophysics |volume=388 |issue= 2|pages=676–687 |doi=10.1051/0004-6361:20020313 |url= |accessdate= |arxiv = astro-ph/0112486 |bibcode = 2002A&A...388..676B }}
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| *{{cite journal |last=Barrau |first=A. |authorlink= |coauthors=''et al.'' |year=2003 |month= |title=Antideuterons as a probe of primordial black holes |journal=Astronomy & Astrophysics |volume=398 |issue= 2|pages=403–410 |doi=10.1051/0004-6361:20021588 |url= |accessdate= |arxiv = astro-ph/0207395 |bibcode = 2003A&A...398..403B }}
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| *{{cite journal |last=Barrau |first=A. |authorlink= |coauthors=Féron, C.; Grain, J. |year=2005 |month= |title=Astrophysical Production of Microscopic Black Holes in a Low–Planck-Scale World |journal=American Astronomical Society |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
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| *{{cite arXiv |last=Barrau |first=A. |author= |authorlink= |coauthors=Boudoul, G. |title=Some aspects of primordial black hole physics |version= |pages= |date= |url= |eprint=astro-ph/0212225 |accessdate= }} → cosmology with primordial black holes
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| *{{cite journal |last=Barrau |first=A. |authorlink= |coauthors=Grain, J.; Alexeyev, S. O. |year=2004 |month= |title=Gauss–Bonnet black holes at the LHC: beyond the dimensionality of space |journal=Physics Letters 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
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| *{{cite journal |last=Kanti |first=Panagiota |authorlink= |coauthors= |year=2004 |month= |title=Black Holes in Theories with Large Extra Dimensions: a Review |journal=International Journal of Modern Physics 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
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| *D. Ida, K.-y. Oda & S.C.Park, [http://arxiv.org/abs/hep-th/0212108, Phys. Rev. D67 (2003) 064025],[http://arxiv.org/abs/hep-th/0503052, Phys. Rev. D71 (2005) 124039],[http://arxiv.org/abs/hep-th/0602188]: determination of black hole's life and extra-dimensions
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| *N. Nicolaevici, J. Phys. A: Math. Gen. 36 (2003) 7667-7677 [http://www.iop.org/EJ/abstract/0305-4470/36/27/317/]: consistent derivation of the Hawking radiation in the Fulling-Davies mirror model.
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| *L. Smolin, [http://www.physicstoday.org/vol-59/iss-11/pdf/vol59no11p44_48.pdf Quantum gravity faces reality], consists of the recent developments and predictions of [[loop quantum gravity]] about gravity in small scales including the deviation from Hawking radiation effect by Ansari [http://arxiv.org/hep-th/0607081 Spectroscopy of a canonically quantized horizon].
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| *M. Ansari, [http://xxx.lanl.gov/abs/hep-th/0607081 Area, ladder symmetry, degeneracy and fluctuations of a horizon] studies the deviation of a loop quantized black hole from Hawking radiation. A novel observable quantum effect of black hole quantization is introduced.
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| *Stuart L. Shapiro, Saul A. Teukolsky (1983), Black holes, white dwarfs, and neutron stars: The physics of compact objects. p. 366 Wiley-Interscience, Hawking radiation evaporation formula derivation.
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| *{{Cite journal| last = Leonhardt | first = Ulf | authorlink = | coauthors= Maia, Clovis; Schuetzhold, Ralf| year = 2010 | month = | title = Focus on Classical and Quantum Analogs for Gravitational Phenomena and Related Effects | journal = [[New Journal of Physics]] | volume = | issue = | pages = | doi = | url = http://iopscience.iop.org/1367-2630/focus/Focus%20on%20Classical%20and%20Quantum%20Analogues%20for%20Gravitational%20Phenomena%20and%20Related%20Effects | accessdate = | quote =}}
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| ==External links==
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| *[http://xaonon.dyndns.org/hawking/ Hawking radiation calculator tool]
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| *[http://www.cerncourier.com/main/article/44/9/22 The case for mini black holes] A. Barrau & J. Grain explain how the Hawking radiation could be detected at colliders
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| *[http://casa.colorado.edu/~ajsh/hawk.html University of Colorado at Boulder]
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| *[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=448732 Hawking radiation on arxiv.org]
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| *[http://news.slashdot.org/story/10/09/27/1256236/Hawking-Radiation-Claimed-Created-In-a-Lab Hawking radiation observed in laboratory?]
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| {{quantum gravity}}
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| {{Stephen Hawking}}
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| {{DEFAULTSORT:Hawking Radiation}}
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| [[Category:Black holes]]
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| [[Category:Quantum field theory]]
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| [[Category:Stephen Hawking]]
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| [[Category:Astronomical hypotheses]]
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