Hawking radiation

From Example Problems
Jump to navigation Jump to search

In physics, Hawking radiation is thermal radiation thought to be emitted by black holes due to quantum effects. It is named after British physicist Stephen Hawking who worked out the theoretical argument for its existence in 1974. Hawking's discovery became the first convincing insight in quantum gravity. However, the existence of Hawking radiation remains controversial.

Overview

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, can escape from the black hole. However, by doing a calculation in the framework of quantum field theory in curved spacetimes, Hawking showed quantum effects allow black holes to emit radiation in a thermal spectrum.

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.

Example

A black hole of one solar mass has a temperature of only 60 nanokelvins; 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.

Problems with the theory

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. Some of these are in connection with the Unruh effect.

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

Emission process

A black hole emits thermal radiation at a temperature

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_H = \frac{\kappa}{2 \pi}} ,

in natural units with G, c, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar} and k equal to 1, and where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa} is the surface gravity of the horizon.

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T={\hbar\,c^3\over8\pi G M k}}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar} 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.

Black hole evaporation

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P={\hbar\,c^6\over15360\,\pi\,G^2M^2}}

where P is the energy outflow, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar} ; 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{\operatorname{ev}}={5120\,\pi\,G^2M_0^{\,3}\over\hbar\,c^4}}

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.

In common units,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = 3.563\,45 \times 10^{32} \left[\frac{\mathrm{kg}}{M}\right]^2 \mathrm{W}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_\mathrm{ev} = 8.407\,16 \times 10^{-17} \left[\frac{M_0}{\mathrm{kg}}\right]^3 \mathrm{s} \ \ \approx\ 2.66 \times 10^{-24} \left[\frac{M_0}{\mathrm{kg}}\right]^3 \mathrm{yr} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_0 = 2.282\,71 \times 10^5 \left[\frac{t_\mathrm{ev}}{\mathrm{s}}\right]^{1/3} \mathrm{kg} \ \ \approx\ 7.2 \times 10^7 \left[\frac{t_\mathrm{ev}}{\mathrm{yr}}\right]^{1/3} \mathrm{kg} }

So, for instance, a 1 second-lived black hole has a mass of 2.28 × 105 kg = 2.05 × 1022 J = 5 × 106 megatons of TNT. The initial power is 6.84 × 1021 W.

The black hole evaporation leads to the so-called black hole information paradox.

References


See also

also:

External links

ca:Radiació de Hawking de:Hawking-Strahlung es:Radiación de Hawking fr:Évaporation des trous noirs he:קרינת הוקינג lv:Hokinga radiācija nl:Hawkingstraling ru:Излучение Хокинга fi:Hawkingin säteily sv:Hawkingstrålning