Hawking radiation is radiation released outside a black hole's event horizon due to quantum effects according to a model developed by Stephen Hawking in 1974. This black-body radiation was not predicted by previous models which assumed that once electromagnetic radiation is inside the event horizon, it cannot escape. Hawking radiation is predicted to be extremely faint and is many orders of magnitude below the current best telescopes' detecting ability.
Hawking radiation would reduce the mass and rotational energy of black holes and consequently cause black hole evaporation. Because of this, black holes that do not gain mass through other means are expected to shrink and ultimately vanish. For all except the smallest black holes, this happens extremely slowly. The radiation temperature, called Hawking temperature, is inversely proportional to the black hole's mass, so micro black holes are predicted to be larger emitters of radiation than larger black holes and should dissipate faster per their mass. Consequently, if small black holes exist, as permitted by the hypothesis of primordial black holes, they will lose mass more rapidly as they shrink, leading to a final cataclysm of high energy radiation alone. Such radiation bursts have not yet been detected.
Background
Modern black holes were first predicted using Albert Einstein's general theory of relativity. Evidence of the astrophysical objects termed black holes began to mount half a century later, and these objects are of current interest primarily because of their compact size and immense gravitational attraction. Early research into black holes was done by individuals such as Karl Schwarzschild and John Wheeler, who modeled black holes as having zero entropy.
A black hole can form when enough matter or energy is compressed into a volume small enough that the escape velocity is greater than the speed of light. Because nothing can travel that fast, nothing within a certain distance, proportional to the mass of the black hole, can escape beyond that distance. The region beyond which not even light can escape is the event horizon: an observer outside it cannot observe, become aware of, or be affected by events within the event horizon.
Alternatively, using a set of infalling coordinates in general relativity, one can conceptualize the event horizon as the region beyond which space is falling faster than the speed of light. (Although nothing can travel space faster than light, space itself can fall at any speed.) Once matter is inside the event horizon, all of the matter inside falls inevitably into a gravitational singularity, a place of infinite curvature and zero size, leaving behind a warped spacetime devoid of any matter; a classical black hole is pure empty spacetime, and the simplest (non-rotating and uncharged) is characterized just by its mass and event horizon.
Discovery
In 1971, Soviet physicsists Yakov Zeldovich and Alexei Starobinsky proposed that rotating black holes ought to create and emit particles, reasoning by analogy with electromagnetic spinning metal spheres. The process is now called Zeldovich amplification. In 1972, Jacob Bekenstein developed a theory and reported that the black holes should have an entropy proportional to their surface area. Initially Stephen Hawking argued against Bekenstein's theory, viewing black holes as a simple object with no entropy. After meeting Zeldovich in Moscow in 1973, Hawking put these two ideas together using his mixture of quantum field theory and general relativity.
In his 1974 paper Hawking showed that in theory, a black hole radiates particles as if it were a blackbody. Particles escaping effectively drain energy from the black hole. In Hawking's prediction, black holes emit small amounts of thermal radiation at temperature
where is the reduced Planck constant, is the speed of light, is the gravitational constant, is the mass of the black hole and is the Boltzmann constant. By applying quantum field theory to black holes, Hawking determined that a black hole should continuously emit thermal blackbody radiation. This effect has become known as . This theory was supported by previous work by Jacob Bekenstein, who theorized that black holes should have a finite entropy proportional to their surface area, and therefore should also have a temperature. Due to Bekenstein's contribution to black hole entropy, it is also known as . Since Hawking's 1974 publication, he and others have mathematically verified the result through different approaches, such as by path integrals.
Hawking radiation derives from vacuum fluctuations. A vacuum fluctuation in the electromagnetic field can result in a photon outside of the black hole horizon paired with one on the inside. The horizon allows one to escape in each direction.
Emission process Hawking radiation is dependent on 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.For an accessible discussion of the Unruh effect and Hawking radiation, see:
A Schwarzschild black hole has a metric
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
The local metric to lowest order is
which is the Rindler coordinates in terms of . The metric describes a frame that is accelerating to keep from falling into the black hole. The local acceleration, , diverges as .
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
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:
The inverse temperature redshifted to at infinity is
and is the near-horizon position, near , so this is really
Thus a field theory defined on a black-hole background is in a thermal state whose temperature at infinity is
For a black hole mass equal to the Earth, this temperature would be . The wavelength of a typical quantum of Hawking radiation is similar to the size 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 is added is
The heat energy that enters serves to increase the total mass, so
So the entropy of a black hole is proportional to its surface area:
where, since the radius of the black hole is twice its mass, we have the area being given by
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 somehow encoded (by the holographic principle) onto a bounding surface.
Greybody factors Greybody factors are functions of frequency and angular momentum that characterize the deviation of the emission-spectrum of a black hole from a pure black-body spectrum. As a result of quantum effects, an isolated black hole emits radiation that, at the black-hole horizon, matches the radiation from a perfect black body. The rate at which a black hole emits particles with energy between and and with angular momentum quantum numbers is given by
where is the Boltzmann constant and is the Hawking temperature of the black hole. The constant in the denominator is for Bosons and for Fermions. The factors are called the greybody factors of the black hole. For a charged black hole, these factors may also depend on the charge of the emitted particles.
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 Einstein's equation of mass–energy equivalence, ). Consequently, an evaporating black hole will have a finite lifespan. By dimensional analysis, the life span of a black hole can be shown to scale as the cube of its initial mass.The time that the black hole takes to dissipate is:
where and are the mass and volume of the (Schwarzschild) black hole, the Planck mass and Planck time. A black hole of one solar mass () takes more than to evaporate—much longer than the current age of the universe at . The Hawking radiation temperature is:
Larger mass black holes have lower Hawking radiation temperatures. For the smallest predicted stellar black hole, about 3 solar masses, this temperature is . Since the universe contains the cosmic microwave background radiation at , no stellar black holes can evaporate: they are colder than outer space.
The Bekenstein–Hawking luminosity of a black hole, under the assumption of pure photon emission (i.e. that no other particles are emitted) and under the assumption that the horizon is the radiating surface is:
where is the luminosity or radiated power, is the reduced Planck constant, is the speed of light in a vacuum, is the gravitational constant, and is the mass of the black hole.
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's mass approaches 1 Planck mass, its radius will also approach two Planck lengths. 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.
Evaporation of primordial black holes
The relationship between mass and temperature for Hawking radiation then implies the mass would need to be less than 0.8% of the mass of the Earth. This in turn means any black hole that could dissipate cannot be one created by stellar collapse. Only primordial black holes might be created with this little mass.
Hawking estimated that any black hole formed in the early universe with a mass of less than approximately would have evaporated completely by the present day.