Black holes are sites of immense gravitational attraction. Classically, the gravitation is so powerful that nothing, not even electromagnetic radiation, can escape from the black hole. It is yet unknown how gravity can be incorporated into quantum mechanics. Nevertheless, far from the black hole the gravitational effects can be weak enough for calculations to be reliably performed in the framework of quantum field theory in curved spacetime. Hawking showed that quantum effects allow black holes to emit exact black body radiation. The electromagnetic radiation is produced as if emitted by a black body with a temperature inversely proportional to the mass of the black hole.

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.

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 7022450000000000000♠4.5×10^{22} kg (about the mass of the Moon, or about 6995129999999999999♠13 µm across) would be in equilibrium at 2.7 K, absorbing as much radiation as it emits. Yet smaller primordial black holes would emit more than they absorb and thereby lose mass.

The trans-Planckian problem is the observation that Hawking's original calculation requires talking about quantum particles in which the wavelength becomes shorter than the Planck length near the black hole's horizon. It is due to the peculiar behavior near a gravitational horizon where time stops as measured from far away. A particle emitted from a black hole with a finite frequency, if traced back to the horizon, must have had an infinite frequency there and a trans-Planckian wavelength.

The Unruh effect and the Hawking effect both talk about field modes in the superficially stationary space-time that change frequency relative to other coordinates which are regular across the horizon. This is necessarily so, since to stay outside a horizon requires acceleration which constantly Doppler shifts the modes.

An outgoing Hawking radiated photon, if the mode is traced back in time, has a frequency which diverges from that which it has at great distance, as it gets closer to the horizon, which requires the wavelength of the photon to "scrunch up" infinitely at the horizon of the black hole. In a maximally extended external Schwarzschild solution, that photon's frequency stays regular only if the mode is extended back into the past region where no observer can go. That region seems to be unobservable and is physically suspect, so Hawking used a black hole solution without a past region which forms at a finite time in the past. In that case, the source of all the outgoing photons can be identified: a microscopic point right at the moment that the black hole first formed.

The quantum fluctuations at that tiny point, in Hawking's original calculation, contain all the outgoing radiation. The modes that eventually contain the outgoing radiation at long times are redshifted by such a huge amount by their long sojourn next to the event horizon, that they start off as modes with a wavelength much shorter than the Planck length. Since the laws of physics at such short distances are unknown, some find Hawking's original calculation unconvincing.

The trans-Planckian problem is nowadays mostly considered a mathematical artifact of horizon calculations. The same effect occurs for regular matter falling onto a white hole solution. Matter which falls on the white hole accumulates on it, but has no future region into which it can go. Tracing the future of this matter, it is compressed onto the final singular endpoint of the white hole evolution, into a trans-Planckian region. The reason for these types of divergences is that modes which end at the horizon from the point of view of outside coordinates are singular in frequency there. The only way to determine what happens classically is to extend in some other coordinates that cross the horizon.

There exist alternative physical pictures which give the Hawking radiation in which the trans-Planckian problem is addressed. The key point is that similar trans-Planckian problems occur when the modes occupied with Unruh radiation are traced back in time. In the Unruh effect, the magnitude of the temperature can be calculated from ordinary Minkowski field theory, and is not controversial.

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.

A Schwarzschild black hole has a metric:

d
s
2
=
−
(
1
−
2
M
r
)
d
t
2
+
1
1
−
2
M
r
d
r
2
+
r
2
d
Ω
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

r
=
2
M
+
ρ
2
8
M
.

The local metric to lowest order is

d
s
2
=
−
ρ
2
16
M
2
d
t
2
+
d
ρ
2
+
d
X
⊥
2
=
−
ρ
2
d
τ
2
+
d
ρ
2
+
d
X
⊥
2
,
which is Rindler in terms of *τ* = *t*/4*M*. 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

T
=
α
2
π
=
1
2
π
ρ
=
1
4
π
2
M
(
r
−
2
M
)
;
this 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:

T
(
r
′
)
=
1
4
π
2
M
(
r
−
2
M
)
1
−
2
M
r
1
−
2
M
r
′
,
which can be simplified as:

T
(
r
′
)
=
1
4
π
2
M
r
(
1
−
2
M
r
′
)
.
The inverse temperature redshifted to r′ at infinity is

T
(
∞
)
=
1
4
π
2
M
r

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

T
(
∞
)
=
1
8
π
M
.
So a field theory defined on a black hole background is in a thermal state whose temperature at infinity is:

T
H
=
1
8
π
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 natural units (*G* = *c* = *ħ* = *k*_{B} = 1), the temperature is

T
H
=
κ
2
π
,
where κ is the surface gravity of the horizon. 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

T
=
ℏ
c
3
8
π
G
M
k
B
(
≈
1.227
×
10
23
kg
M
K
=
6.169
×
10
−
8
K
×
M
⊙
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:

d
S
=
d
Q
T
=
8
π
M
d
Q
.

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

d
S
=
8
π
M
d
M
=
d
(
4
π
M
2
)
.
The radius of a black hole is twice its mass in natural units, so the entropy of a black hole is proportional to its surface area:

S
=
π
R
2
=
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* = *mc*^{2}).

In 1976 Don Page calculated the power produced, and the time to evaporation, for a nonrotating, non-charged Schwarzschild black hole of mass M. 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. Note that writing in 1976, Page erroneously postulates that neutrinos have no mass and that only two neutrino flavors exist, and therefore miscalculates the black hole lifetimes.

For a mass much larger than 10^{17} grams, Page deduces that electron emission can be ignored, and that black holes of mass M in grams evaporate via massless electron and muon neutrinos, photons, and gravitons in a time τ of

τ
=
8.66
×
10
−
27
[
M
g
]
3
s
.
For a mass much smaller than 10^{17} g, but much larger than 7011500000000000000♠5×10^{14} g, the emission of ultrarelativistic electrons and positrons will accelerate the evaporation, giving a lifetime of

τ
=
4.8
×
10
−
27
[
M
g
]
3
s
.

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:

Stefan–Boltzmann constant:

σ
=
π
2
k
B
4
60
ℏ
3
c
2
Schwarzschild radius:

r
s
=
2
G
M
c
2
Black hole surface gravity at the horizon:

g
=
G
M
r
s
2
=
c
4
4
G
M
Hawking radiation has a blackbody (Planck) spectrum with a temperature T given by:

E
=
k
B
T
=
ℏ
g
2
π
c
=
ℏ
2
π
c
(
c
4
4
G
M
)
=
ℏ
c
3
8
π
G
M
Hawking radiation temperature:

The peak wavelength of this radiation is nearly 16 times the Schwarzschild radius of the black hole. Using Wien's displacement constant *b* = *hc*/4.9651 *k*_{B} = 6997289780000000000♠2.8978×10^{−3} m K:

λ
m
a
x
=
b
T
H
=
8
π
2
4.9651
r
s
=
15.902
r
s

Schwarzschild sphere surface area of Schwarzschild radius *r*_{s}:

A
s
=
4
π
r
s
2
=
4
π
(
2
G
M
c
2
)
2
=
16
π
G
2
M
2
c
4
Stefan–Boltzmann power law:

P
=
A
s
j
⋆
=
A
s
ε
σ
T
4
For simplicity, assume a black hole is a perfect blackbody (*ε* = 1).

Stefan–Boltzmann–Schwarzschild–Hawking black hole radiation power law derivation:

P
=
A
s
ε
σ
T
H
4
=
(
16
π
G
2
M
2
c
4
)
(
π
2
k
B
4
60
ℏ
3
c
2
)
(
ℏ
c
3
8
π
G
M
k
B
)
4
=
ℏ
c
6
15360
π
G
2
M
2
Stefan–Boltzmann–Schwarzschild–Hawking power law:

where P is the energy outflow, ħ 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. Substituting the numerical values of the physical constants in the formula above we obtain P= 7032356200000000000♠3.562×10^{32} W kg^2/M^2.

The power of the Hawking radiation from a solar mass (*M*_{☉}) black hole turns out to be minuscule:

P
=
ℏ
c
6
15360
π
G
2
M
⊙
2
=
9.007
×
10
−
29
W
.

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, cosmic microwave background radiation, or other radiation falls into the black hole, it is possible to calculate how long it would take for the black hole to dissipate:

K
e
v
=
ℏ
c
6
15360
π
G
2
=
3.562
×
10
32
W
kg
2
.
Given that the power of the Hawking radiation is the rate of evaporation energy loss of the black hole:

P
=
−
d
E
d
t
=
K
e
v
M
2
.
Since the total energy E of the black hole is related to its mass M by Einstein's mass–energy formula *E* = *Mc*^{2}:

P
=
−
d
E
d
t
=
−
(
d
d
t
)
M
c
2
=
−
c
2
d
M
d
t
.
We can then equate this to our above expression for the power:

−
c
2
d
M
d
t
=
K
e
v
M
2
.
This differential equation is separable, and we can write:

M
2
d
M
=
−
K
e
v
c
2
d
t
.
The black hole's mass is now a function *M*(*t*) of time t. Integrating over M from *M*_{0} (the initial mass of the black hole) to zero (complete evaporation), and over t from zero to t_{ev}:

∫
M
0
0
M
2
d
M
=
−
K
e
v
c
2
∫
0
t
e
v
d
t
.
The evaporation time of a black hole is proportional to the cube of its mass:

t
e
v
=
c
2
M
0
3
3
K
e
v
=
(
c
2
M
0
3
3
)
(
15360
π
G
2
ℏ
c
6
)
=
5120
π
G
2
M
0
3
ℏ
c
4
=
8.410
×
10
−
17
[
M
0
k
g
]
3
s
.

The time that the black hole takes to dissipate is:

where *M*_{0} is the mass of the black hole.

The lower classical quantum limit for mass for this equation is equivalent to the Planck mass, *m*_{P}.

Hawking radiation evaporation time for a Planck mass quantum black hole:

t
e
v
=
5120
π
G
2
m
P
3
ℏ
c
4
=
5120
π
t
P
=
5120
π
ℏ
G
c
5
=
8.671
×
10
−
40
s
where *t*_{P} is the Planck time.

For a black hole of one solar mass (*M*_{☉} = 7030198892000000000♠1.98892×10^{30} kg), we get an evaporation time of 7067209800000000000♠2.098×10^{67} years—much longer than the current age of the universe at 7010137990000000000♠(13.799±0.021)×10^{9} years.

t
e
v
=
5120
π
G
2
M
⊙
3
ℏ
c
4
=
6.617
×
10
74
s
.
But for a black hole of 7011100000000000000♠10^{11} kg, the evaporation time is 2.667 billion 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, it must have a temperature greater than that of the present-day blackbody radiation of the universe of 2.7 K = 6977368500592010000♠2.3×10^{−4} eV. This implies that M must be less than 0.8% of the mass of the Earth – approximately the mass of the Moon.

Cosmic microwave background radiation universe temperature:

T
u
=
2.725
K
Hawking total black hole mass:

M
H
≤
ℏ
c
3
8
π
G
k
B
T
u
≤
4.503
×
10
22
kg
M
H
M
⊕
=
7.539
×
10
−
3
=
0.754
%
where *M*_{⊕} is the total Earth mass.

In common units,

P
=
3.563
45
×
10
32
[
k
g
M
]
2
W
t
e
v
=
8.407
16
×
10
−
17
[
M
0
k
g
]
3
s
≈
2.66
×
10
−
24
[
M
0
k
g
]
3
y
r
M
0
=
2.282
71
×
10
5
[
t
e
v
s
]
1
3
k
g
≈
7.2
×
10
7
[
t
e
v
y
r
]
1
3
k
g
So, for instance, a 1-second-life black hole has a mass of 7005227999999999999♠2.28×10^{5} kg, equivalent to an energy of 7022205000000000000♠2.05×10^{22} J that could be released by 7006500000000000000♠5×10^{6} megatons of TNT. The initial power is 7021684000000000000♠6.84×10^{21} W.

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 approaches Planck mass and Planck radius.
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.

The formulae from the previous section are only applicable if the laws of gravity are approximately valid all the way down to the Planck scale. In particular, for black holes with masses below the Planck mass (~6992100000000000000♠10^{−8} kg), they result in impossible lifetimes below the Planck time (~6957100000000000000♠10^{−43} s). This is normally seen as an indication that the Planck mass is the lower limit on the mass of a black hole.

In a model with large extra dimensions, the values of Planck constants can be radically different, and the formulae for Hawking radiation have to be modified as well. In particular, the lifetime of a micro black hole with a radius below the scale of the extra dimensions is given by equation 9 in Cheung (2002) and equations 25 and 26 in Carr (2005).

τ
∼
1
M
∗
(
M
B
H
M
∗
)
n
+
3
n
+
1
,
where *M*_{∗} is the low energy scale, which could be as low as a few TeV, and n is the number of large extra dimensions. This formula is now consistent with black holes as light as a few TeV, with lifetimes on the order of the "new Planck time" ~6974100000000000000♠10^{−26} s.

A detailed study of the quantum geometry of a black hole horizon has been made using loop quantum gravity. Loop-quantization reproduces the result for black hole entropy originally discovered by Bekenstein and Hawking. Further, it led to the computation of quantum gravity corrections to the entropy and radiation of black holes.

Based on the fluctuations of the horizon area, a quantum black hole exhibits deviations from the Hawking spectrum that would be observable were X-rays from Hawking radiation of evaporating primordial black holes to be observed. The quantum effects are centered at a set of discrete and unblended frequencies highly pronounced on top of Hawking radiation spectrum.

Under experimentally achievable conditions for gravitational systems this effect is too small to be observed directly. However, in September 2010 an experimental set-up created a laboratory "white hole event horizon" that the experimenters claimed was shown to radiate an optical analog to Hawking radiation, although its status as a genuine confirmation remains in doubt. Some scientists predict that Hawking radiation could be studied by analogy using sonic black holes, in which sound perturbations are analogous to light in a gravitational black hole and the flow of an approximately perfect fluid is analogous to gravity.