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In physics, Hawking radiation (also known as Bekenstein-Hawking radiation) is a thermal radiation with a black body spectrum predicted to be emitted by black holes due to "quantum effects". It is named after the British physicist Stephen Hawking who provided the theoretical argument for its existence in 1974, and sometimes also after the Israeli physicist Jacob Bekenstein who predicted that black holes should have a finite, non-zero temperature and entropy.

Because Hawking radiation allows black holes to lose mass, black holes which lose more matter than they gain through other means are expected to evaporate, shrink, and ultimately vanish. Smaller 'micro' black holes are currently predicted by theory to be larger net emitters of radiation than larger black holes, and to shrink and evaporate faster.

Hawking's analysis became the first convincing insight into quantum gravity. However, the existence of Hawking radiation has never been observed, nor are there currently viable experimental tests which would allow it to be observed. Hence there is still some theoretical dispute over whether Hawking radiation actually exists.^[1]

Contents 1 Overview 1.1 An example 1.2 Problems with the theory 2 Emission process 3 Black hole evaporation 4 Hawking radiation and large extra dimensions 5 Deviation from Hawking radiation in loop quantum gravity 6 See also 7 References 8 Notes 9 External links

Black holes are sites of immense gravitational attraction into which surrounding matter is drawn by gravitational forces. Classically, the gravitation is so powerful that nothing, not even radiation or light, can escape from the black hole. It is yet unknown how gravity can be incorporated into quantum mechanics, but nevertheless far from the black hole the gravitational effects can be weak enough that calculations can 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 radiation is as if it is emitted by a black body of temperature which is related (inverse proportional) to the black hole's mass.

Physical insight on the process may be gained by imagining that 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 particles being "boosted" by the black hole's gravitation into becoming real particles.

A more precise, but still much simplified view of the process is that vacuum fluctuations 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 whilst the other escapes. In order to preserve total energy, the particle which 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 particle.

An important difference between the black hole radiation as computed by Hawking and a 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 satisfies this law exactly. 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 of the black hole. This leads to the Black hole information paradox.

However, according to the conjectured 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 a usual thermal radiation. If this is correct, then Hawking's original computation should be corrected, though it is not known how (see below).

A black hole of one solar mass has a temperature of only 60 nanokelvin; in fact, such a black hole would absorb far more cosmic microwave background radiation than it emits. A black hole of 4.5 × 10²² kg (about the mass of the Moon) would be in equilibrium at 2.7 kelvins, absorbing as much radiation as it emits. Yet smaller primordial black holes would emit more than they absorb, and thereby lose mass.

The trans-Planckian problem may raise doubts on the physical validity of Hawking's result. Hawking's original derivation employed field modes of arbitrarily high frequency near the black hole horizon, although these do not appear in the final result. In particular, he used modes of frequency higher than the inverse Planck time, and at these scales the physical laws are unknown.

A number of alternative approaches to the Hawking radiation have appeared in order to try to overcome or address this problem, such as modelling Hawking radiation as a form of Unruh radiation^[2]. In this approach the exterior gravitational field around the black hole is the source of the radiation, rather than the event horizon. The same results are obtained with either approach.

The Hawking Radiation shows that the laws of black hole thermodynamics have a complete physical meaning.

A black hole emits thermal radiation at a temperature

,

in natural units with G, c, and k equal to 1, and where κ is the surface gravity of the horizon.

In particular, the radiation from a Schwarzschild black hole is black-body radiation with temperature:

where is the reduced Planck constant, c is the speed of light, k is the Boltzmann constant, G is the gravitational constant, and M is the mass of the black hole.

When particles escape, the black hole loses a small amount of its energy and therefore of its mass (recall that mass and energy are related by Einstein's famous equation E = mc²).

The 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 M. Combining the formulae 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) we get:

where P is the energy outflow, is the reduced Planck constant, c is the speed of light, and G is the gravitational constant. It is worth mentioning that the above formula has not yet been derived in the framework of semiclassical gravity.

The power in the Hawking radiation from a solar mass black hole turns out to be a minuscule 10⁻²⁸ watts. 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 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 evaporate. The black hole's mass is now a function M(t) of time t. The time that the black hole takes to evaporate is:

For a black hole of one solar mass (about 2 × 10³⁰ kg), we get an evaporation time of 10⁶⁷ years—much longer than the current age of the universe. But for a black hole of 10¹¹ kg, the evaporation time is about 3 billion years. This is why some astronomers are searching for signs of exploding primordial black holes.

In common units,

So, for instance, a 1 second-lived black hole has a mass of 2.28 × 10⁵ kg, equivalent to an energy of 2.05 × 10²² J that could be released by 5 × 10⁶ megatons of TNT. The initial power is 6.84 × 10²¹ 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 rays. 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 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 evaporates, as under these models the Hawking radiation is random (containing no 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.

Formulas 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 ( ~10⁻⁵ g ), they result in unphysical lifetimes below Planck time ( ~10⁻⁴³ s ). This is normally seen as an indication that Planck mass is the lower limit on the mass of a black hole.

In the model with large extra dimensions, 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

where M _* 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" ~10⁻²⁶ s.

A detailed study of the quantum geometry of a black hole horizon has been made using Loop quantum gravity. Loop-quantization reproduces the result for black hole entropy originally discovered by Bekenstein and 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-rays from Hawking radiation of evaporating primordial black holes to be observed. The deviation is such that the Hawking radiation is expected to be centered at a set of discrete and unblended energies.

• Black hole thermodynamics
• Black hole information paradox
• Meduso-anthropic principle
• Trans-Planckian problem
• Professor Stephen Hawking

• S.W. Hawking, Nature 248 (1974) 30 : Hawking's first article on the topic
• D. Page, Phys. Rev. D13 (1976) 198 : first detailed studies of the evaporation mechanism
• B.J. Carr & S.W. Hawking, Mon. Not. Roy. Astron. Soc 168 (1974) 399 : links between primordial black holes and the early universe
• A. Barrau et al., Astron. Astrophys. 388 (2002) 676 , Astron. Astrophys. 398 (2003) 403 , Astrophys. J. 630 (2005) 1015 : experimental searches for primordial black holes thanks to the emitted antimatter
• A. Barrau & G. Boudoul, Review talk given at the International Conference on Theoretical Physics TH2002 : cosmology with primordial black holes
• A. Barrau & J. Grain, [1], Phys. Lett. B 584 (2004) 114 : searches for new physics (quantum gravity) with primordial black holes
• P. Kanti, [2], Int. J. Mod. Phys. A19 (2004) 4899 : evaporating black holes and extra-dimensions
• D. Ida, K.-y. Oda & S.C.Park, Phys. Rev. D67 (2003) 064025,Phys. Rev. D71 (2005) 124039,[3]: determination of black hole's life and extra-dimensions
• N. Nicolaevici, J. Phys. A: Math. Gen. 36 (2003) 7667-7677 [4]: consistent derivation of the Hawking radiation in the Fulling-Davies mirror model.
• L. Smolin, 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 Spectroscopy of a canonically quantized horizon.

• http://casa.colorado.edu/~ajsh/hawk.htm
• The case for mini black holes A. Barrau & J. grain explain how the Hawking radiation could be detected at colliders
• Hawking radiation calculator tool