Hawking radiation

Black holes are astrophysical objects of interest due to their immense gravitational attraction. A black hole occurs when more than a certain amount of matter and/or energy is located within a small enough space. Given a large enough mass in a small enough space, the gravitational forces become large enough that within a nearby region of space, nothing - not even light - can escape from inside that region to the wider universe. The boundary of that region is known as the event horizon because an observer outside it cannot observe, become aware of, or be affected by events within the event horizon. This region is the black hole's boundary, in effect.

It is unknown what exactly happens to the mass inside a black hole. It is possible that a gravitational singularity forms at the center - a point of zero size and infinite density - or perhaps quantum effects prevent that happening. However, in either case, the event horizon is at some distance from any such point, so the force of gravity is a little weaker at the event horizon (although still exceedingly strong). That means that regardless of the interior conditions, our current understandings of physics can be used to predict what may happen in the region of 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 energy to "leak" into the wider universe within a tiny distance of the event horizon. In effect this energy acted as if the black hole itself was slowly evaporating (although it actually came from outside it).[4]

Hawking's insight was based on a phenomenon of quantum physics known as virtual particles, and their behaviour near the event horizon. Even in empty space, subatomic "virtual" particles and antiparticles come briefly into existence, then mutually annihilate and vanish again. Close to a black hole, this manifests as pairs of photons.[2] One of these photons might be pulled beyond the event horizon, leaving the other to escape into the wider universe. Careful analysis showed that if this happened, quantum effects would cause a "partner wave" carrying negative energy to be created and also pass into the black hole, reducing the black hole's total mass, or energy.[2] In effect to an observer it would appear as if the gravitational force had somehow allowed the black hole's energy to be reduced and the energy of the wider universe to be increased. Hence black holes must gradually lose energy and evaporate over time.[2] Considering the thermal properties of black holes, and conservation laws affecting this process, Hawking calculated that the visible outcome would be a very low level of exact black-body radiation - electromagnetic radiation produced as if emitted by a black body with a temperature inversely proportional to the mass of the black hole.[2]

Physical insight into 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. As the particle–antiparticle pair was produced by the black hole's gravitational energy, the escape of one of the particles lowers the mass of the black hole.[5]

An alternative 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 while the other escapes. In order to preserve total energy, the particle that fell into the black hole must have had a negative energy (with respect to an observer far away from the black hole). This causes the black hole to lose mass, and, to an outside observer, it would appear that the black hole has just emitted a particle. In another model, the process is a quantum tunnelling effect, whereby particle–antiparticle pairs will form from the vacuum, and one will tunnel outside the event horizon.

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. When Hawking did the calculation, he found to his surprise that even non-rotating black holes produce radiation.[6] In parallel, in 1972, Jacob Bekenstein conjectured that the black holes should have an entropy,[7] 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.

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.[8]

A Schwarzschild black hole has a metric:

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

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

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

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:

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

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:

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

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

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

This can be expressed in a cleaner way in terms of the surface gravity of the black hole; this is the parameter that determines the acceleration of a near-horizon observer. In Planck units (G = c = ħ = k_B = 1), the temperature is

${\displaystyle T_{\text{H}}={\frac {\kappa }{\,2\pi \,}}\,}$,

where κ is the surface gravity of the horizon (in units of lightspeed per Planck-time squared). So a black hole can only be in equilibrium with a gas of radiation at a finite temperature. Since radiation incident on the black hole is absorbed, the black hole must emit an equal amount to maintain detailed balance. The black hole acts as a perfect blackbody radiating at this temperature.

In SI units, the radiation from a Schwarzschild black hole is blackbody radiation with temperature

${\displaystyle T={\frac {\hbar \,c^{3}}{8\pi Gk_{\text{B}}M}}\;\approx \;1.2\times 10^{23}{\text{K}}\,\times {\frac {1\,{\text{kg}}}{M}}\;=\;6\times 10^{-8}{\text{K}}\,\times {\frac {M_{\odot }}{M}}\,}$,

where ħ is the reduced Planck constant, c is the speed of light, k_B is the Boltzmann constant, G is the gravitational constant, M_☉ is the solar mass, and M is the mass 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 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:

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

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,[9][10]^: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.[11]

In 1976, Don Page refined this estimate by calculating the power produced, and the time to evaporation, for a nonrotating, non-charged Schwarzschild black hole of mass M.[9] The time for the event horizon or entropy of a black hole to halve is known as the Page time.[12] 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 4×10¹⁴ g 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)×10¹⁴ g.[13]

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.[14] A supermassive black hole with a mass of 10¹¹ (100 billion) M_☉ will evaporate in around 2×10¹⁰⁰ years.[15] 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.[14]

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:[3][16][17]

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

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:[17][16]

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

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:[17][16]

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

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 14×10⁹ years.[18] 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. In 2020, Chou proposed a theory if a Pluto-mass rotating radiating primordial black hole, the Hawking radiation temperature will be 9.42 K, higher than 2.7 K CMB.[19] Other study suggest that M must be less than 0.8% of the mass of the Earth[20] – 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.

• Black hole starship
• Blandford–Znajek process and Penrose process, other extractions of black-hole energy
• Gibbons–Hawking effect
• Thorne–Hawking–Preskill bet
• Unruh effect

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