Hawking radiation

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In astrophysics, Hawking radiation is thermal radiation emitted by black holes. It is named after British physicist Stephen Hawking who worked out the theoretical argument for its existence in 1975.

Black holes are sites of immense gravitational attraction into which surrounding matter is drawn by gravitational forces. It was originally thought that the gravitation was so powerful that nothing, not even radiation, could escape from the black hole, but Hawking theorized that (particle-antiparticle) radiation would be 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. This would sap some of the black hole's energy, and so when these particles escape the black hole would lose a small amount of mass.

The radiation from a black hole is blackbody radiation with temperature:

<math>T={\hbar\,c^3\over8\pi k\,G M}</math>

where <math>\hbar</math> 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.

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�נ10���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 holes would emit more than they absorb, and thereby lose mass.

Black hole evaporation

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. Wikipedia contains all the necessary ingredients: the formula 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). Combining all these formulae, the emitted power comes out as:

<math>P={\hbar\,c^6\over15360\,\pi\,G^2M^2}</math>

where <math>\hbar</math> is the reduced Planck constant, c is the speed of light, and G is the gravitational constant.

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 will take for the black hole to evaporate. The black hole's mass is now a function M(t) of time t. Taking the time derivative of Einstein's famous equation E�=�Mc�, we get

<math>P=-{dE\over d\,t}=-c^2{dM\over d\,t}</math>

leading to

<math>{dM\over d\,t}=-{\hbar\,c^4\over15360\,\pi\,G^2M^2}</math>

which, assuming then mass is M₀ at time t�=�0, has the solution

<math>M(t)=\sqrt[3]{M_0^3-{\hbar\,c^4\over5120\,\pi\,G^2}\,t}</math>

Finally, setting M(t_ev)�=�0 we get the time it takes the black hole to evaporate:

<math>t_{\operatorname{ev}}={5120\,\pi\,G^2M_0^{\,3}\over\hbar\,c^4}</math>

For a black hole of one solar mass, 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.

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In common units,

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

<math> t_{ev} = 8.407\,16 \times 10^{-17} \left[\frac{\mathrm{Mass}}{kg}\right]^3 sec

\ \ \approx\ 2.66 \times 10^{-24} \left[\frac{\mathrm{Mass}}{kg}\right]^3 year </math>

<math> M_0 = 2.282\,71 \times 10^5 \left[\frac{\mathrm{Time}}{sec}\right]^{1/3} kg \ \ \approx\ 7.2 \times 10^7 \left[\frac{\mathrm{Time}}{yr}\right]^{1/3} kg </math>

So a 1-second black hole has a mass of <math> 2.28 \times 10^5 kg \Rightarrow 2.05 \times 10^{22} J

=           5    \times 10^6 </math> Megatons.

The initial power is <math>4.31 \times 10^{21} W</math>

External links

• http://casa.colorado.edu/~ajsh/hawk.html

References

• Adapted from the Wikipedia article, "Hawking_radiation" http://en.wikipedia.org/wiki/Hawking_radiation, used under the GNU Free Documentation License