Hawking radiation

Black holes are astrophysical objects of interest primarily because of their compact size and immense gravitational attraction. They were first predicted by Einstein's 1915 theory of general relativity, before astrophysical evidence began to mount half a century later.

A black hole can form when enough matter and/or energy is compressed into a volume small enough that the escape velocity is greater than the speed of light. Nothing can travel that fast, so nothing within a 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. The essence of a black hole is its event horizon, a theoretical demarcation between events and their causal relationships.[5]^: 25–36

Picture of space infalling into a Schwarzschild black hole at the Newtonian escape speed. Outside/inside the horizon (red), the infalling speed is less/greater than the speed of light. At the event horizon, the infalling speed equals the speed of light.[6] Credit: Andrew Hamilton, JILA

Alternatively, using a set of infalling coordinates in general relativity, one can conceptualize the event horizon as the region beyond which space is infalling faster than the speed of light. (Although nothing can travel through space faster than light, space itself can infall at any speed.)[6] Once matter is inside the event horizon, all of the matter inside falls inexorably 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 (nonrotating and uncharged) is characterized just by its mass and event horizon.[5]^: 37–43

Our current understandings of quantum physics can be used to investigate what may happen in the region around the event horizon. In 1974, British physicist Stephen Hawking used quantum field theory in curved spacetime to show that in theory, the force of gravity at the event horizon was strong enough to cause thermal radiation to be emitted and energy to "leak" into the wider universe from a tiny distance around and outside the event horizon. In effect this energy acted as if the black hole itself was slowly evaporating (although it actually came from outside it).[7]

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 of the black hole (the no-hair theorem). 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 the theory permits no such loss) 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 below).

A black hole of one solar mass (M_☉) has a temperature of only 60 nanokelvins (60 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 4.5×10²² kg (about the mass of the Moon, or about 133 μm across) would be in equilibrium at 2.7 K, absorbing as much radiation as it emits.

Hawking's discovery followed a visit to Moscow in 1973, where the Soviet scientists Yakov Zel'dovich and Alexei Starobinsky convinced him that rotating black holes ought to create and emit particles, while Russian physicist Vladimir Gribov believed that even a non-rotating black hole should emit radiation. When Hawking did the calculation, he found to his surprise that it was true.[8] In 1972, Jacob Bekenstein conjectured that the black holes should have an entropy,[9] where by the same year, he proposed no-hair theorems. Bekenstein's discovery and results are commended by Stephen Hawking, which also led him to think about radiation due to this formalism.

According to the physicist Dmitri Diakonov, there was an argument between Zeldovich and Vladimir Gribov at the Zeldovich Moscow 1972–1973 seminar. Zeldovich believed that only a rotating black hole could emit radiation, while Gribov believed that even a non-rotating black hole emits radiation due to the laws of quantum mechanics.[10][11] This account is confirmed by Gribov's obituary in the Physics-Uspekhi by Vitaly Ginzburg and others.[12][13]

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.[14][15]

A Schwarzschild black hole has a metric

${\displaystyle (\mathrm {d} s)^{2}=-\left(1-{\tfrac {2M}{r}}\right)\,(\mathrm {d} t)^{2}+{\frac {1}{\left(1-{\frac {2M}{r}}\right)}}\,(\mathrm {d} r)^{2}+r^{2}\,(\mathrm {d} \Omega )^{2}.}$

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

${\displaystyle r=2M+{\frac {\rho ^{2}}{8M}}.}$

The local metric to lowest order is

${\displaystyle (\mathrm {d} s)^{2}=-\left({\frac {\rho }{4M}}\right)^{2}\,(\mathrm {d} t)^{2}+(\mathrm {d} \rho )^{2}+(\mathrm {d} X_{\perp })^{2}=-\rho ^{2}\,(\mathrm {d} \tau )^{2}+(\mathrm {d} \rho )^{2}+(\mathrm {d} X_{\perp })^{2},}$

which is Rindler in terms of τ = t/4M. The metric describes a frame that is accelerating to keep from falling into the black hole. The local acceleration, α = 1/ρ, diverges as ρ → 0.

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

${\displaystyle T={\frac {\alpha }{2\pi }}={\frac {1}{2\pi \rho }}={\frac {1}{4\pi {\sqrt {2Mr\left(1-{\frac {2M}{r}}\right)}}}},}$

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:

${\displaystyle T(r')={\frac {1}{4\pi {\sqrt {2Mr\left(1-{\frac {2M}{r}}\right)}}}}{\sqrt {\frac {1-{\frac {2M}{r}}}{1-{\frac {2M}{r'}}}}}={\frac {1}{4\pi {\sqrt {2Mr\left(1-{\frac {2M}{r'}}\right)}}}}.}$

The inverse temperature redshifted to r′ at infinity is

${\displaystyle T(\infty )={\frac {1}{4\pi {\sqrt {2Mr}}}},}$

and r is the near-horizon position, near 2M, so this is really

${\displaystyle T(\infty )={\frac {1}{8\pi M}}.}$

Thus a field theory defined on a black-hole background is in a thermal state whose temperature at infinity is

${\displaystyle T_{\text{H}}={\frac {1}{8\pi M}}.}$

From the black-hole temperature, it is straightforward to calculate the black-hole entropy S. The change in entropy when a quantity of heat dQ is added is

${\displaystyle \mathrm {d} S={\frac {\mathrm {d} Q}{T}}=8\pi M\,\mathrm {d} Q.}$

The heat energy that enters serves to increase the total mass, so

${\displaystyle \mathrm {d} S=8\pi M\,\mathrm {d} M=\mathrm {d} (4\pi M^{2}).}$

The radius of a black hole is twice its mass in Planck units, so the entropy of a black hole is proportional to its surface area:

${\displaystyle S=\pi R^{2}={\frac {A}{4}}.}$

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 onto a bounding surface.

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 E = mc²). 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,[16][17]^{: 176–177} and Hawking estimated that any black hole formed in the early universe with a mass of less than approximately 10¹⁵ g would have evaporated completely by the present day.[18]

In 1976, Don Page refined this estimate by calculating the power produced, and the time to evaporation, for a non-rotating, non-charged Schwarzschild black hole of mass M.[16] The time for the event horizon or entropy of a black hole to halve is known as the Page time.[19] The calculations are complicated by the fact that a black hole, being of finite size, is not a perfect black body; the absorption cross section goes down in a complicated, spin-dependent manner as frequency decreases, especially when the wavelength becomes comparable to the size of the event horizon. Page concluded that primordial black holes could only survive to the present day if their initial mass were roughly 4×10¹¹ kg or larger. Writing in 1976, Page using the understanding of neutrinos at the time erroneously worked on the assumption that neutrinos have no mass and that only two neutrino flavors exist, and therefore his results of black hole lifetimes do not match the modern results which take into account 3 flavors of neutrinos with nonzero masses. A 2008 calculation using the particle content of the Standard Model and the WMAP figure for the age of the universe yielded a mass bound of (5.00±0.04)×10¹¹ kg.[20]

If black holes evaporate under Hawking radiation, a solar mass black hole will evaporate over 10⁶⁴ years which is vastly longer than the age of the universe.[21] A supermassive black hole with a mass of 10¹¹ (100 billion) M_☉ will evaporate in around 2×10¹⁰⁰ years.[22] Some monster black holes in the universe are predicted to continue to grow up to perhaps 10¹⁴ M_☉ during the collapse of superclusters of galaxies. Even these would evaporate over a timescale of up to 10¹⁰⁶ years.[21]

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 formulas 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), several equations can be derived.

The Hawking radiation temperature is:[4][23][24]

${\displaystyle T_{\mathrm {H} }={\frac {\hbar c^{3}}{8\pi GMk_{\mathrm {B} }}}}$

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:[24][23]

${\displaystyle P={\frac {\hbar c^{6}}{15360\pi G^{2}M^{2}}}}$

where P is the luminosity, i.e., the radiated power, ħ is the reduced Planck constant, c is the speed of light, G is the gravitational constant and M is the mass of the black hole. It is worth mentioning that the above formula has not yet been derived in the framework of semiclassical gravity.

The time that the black hole takes to dissipate is:[24][23]

${\displaystyle t_{\mathrm {ev} }={\frac {5120\pi G^{2}M^{3}}{\hbar c^{4}}}={\frac {480c^{2}V}{\hbar G}}\approx 2.1\times 10^{67}\,{\text{years}}\ \left({\frac {M}{M_{\odot }}}\right)^{3},}$

where M and V are the mass and (Schwarzschild) volume of the black hole. A black hole of one solar mass (M_☉ = 2.0×10³⁰ kg) takes more than 10⁶⁷ years to evaporate—much longer than the current age of the universe at 1.4×10¹⁰ years.[25] But for a black hole of 10¹¹ kg, the evaporation time is 2.6×10⁹ years. This is why some astronomers are searching for signs of exploding primordial black holes.

However, since the universe contains the cosmic microwave background radiation, in order for the black hole to dissipate, the black hole must have a temperature greater than that of the present-day blackbody radiation of the universe of 2.7 K.A study suggests that M must be less than 0.8% of the mass of the Earth[26] – approximately the mass of the Moon.

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, when 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.

• Hawking radiation calculator tool
• The case for mini black holes A. Barrau & J. Grain explain how the Hawking radiation could be detected at colliders