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Hawking radiation, also known as
Hawking–Bekenstein radiation,
[1] or
Hawking–Zel'dovich radiation,
[2] is
blackbody radiation that is predicted to be released by
black holes, due to
quantum effects near the
event horizon. It is named after the physicist
Stephen Hawking, who provided a theoretical argument for its existence in 1974,
[3][4] and
Jacob Bekenstein, who predicted that black holes should have a finite
entropy.
[5]
Hawking's work followed his visit to
Moscow in 1973 where the Soviet scientists
Yakov Zel'dovich and
Alexei Starobinsky showed him that, according to the quantum mechanical
uncertainty principle,
rotating black holes should create and emit particles.
[6] Hawking radiation reduces the mass and energy of black holes and is therefore also known as
black hole evaporation. Because of this, black holes that do not gain mass through other means are expected to shrink and ultimately vanish.
Micro black holes are predicted to be larger emitters of radiation than larger black holes and should shrink and dissipate faster.
[7]
In June 2008,
NASA launched the
Fermi space telescope, which is searching for the terminal gamma-ray flashes expected from evaporating
primordial black holes. In the event that speculative
large extra dimension theories are correct,
CERN's Large Hadron Collider may be able to create micro black holes and observe their evaporation. No such micro black hole has ever been observed at CERN.
[8][9][10][11]
In September 2010, a signal that is closely related to black hole Hawking radiation (see
analog gravity)
was claimed to have been observed in a laboratory experiment involving
optical light pulses. However, the results remain unverified and
debatable.
[12][13] Other projects have been launched to look for this radiation within the framework of
analog gravity.
Overview
Black holes are sites of immense
gravitational attraction. Classically, the gravitation generated by the
gravitational singularity inside a black hole 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.
[14]
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.
[15]
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.
[14]
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
[citation needed], 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. Yet smaller
primordial black holes would emit more than they absorb and thereby lose mass.
[14]
Trans-Planckian problem
The
trans-Planckian problem is the issue that Hawking's original calculation includes
quantum particles where the
wavelength becomes shorter than the
Planck length
near the black hole's horizon. This is due to the peculiar behavior
there, 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, and therefore 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.
[16][17][18][19]
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.
[citation needed]
The key point is that similar trans-Planckian problems occur when the
modes occupied with Unruh radiation are traced back in time.
[20] In the Unruh effect, the magnitude of the temperature can be calculated from ordinary
Minkowski field theory, and is not controversial.
Emission process
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.
[20]
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
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
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:
which can be simplified as:
The inverse temperature redshifted to
r′ at infinity is
and
r is the near-horizon position, near
2M, so this is really:
So a field theory defined on a black hole background is in a thermal state whose temperature at infinity is:
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 = kB = 1), the temperature is
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
where
ħ is the
reduced Planck constant,
c is the
speed of light,
kB 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:
The heat energy that enters serves to increase the total mass, so:
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:
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.
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 E = mc2).
1976 Page numerical analysis
In 1976
Don Page calculated the power produced, and the time to evaporation, for a nonrotating, non-charged
Schwarzschild black hole of mass
M.
[21] 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
For a mass much smaller than 10
17 g, but much larger than
5×1014 g, the emission of
ultrarelativistic electrons and positrons will accelerate the evaporation, giving a lifetime of
A crude analytic estimate
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:
Schwarzschild radius:
Hawking radiation has a
blackbody (Planck) spectrum with a temperature
T given by:
Hawking radiation temperature:
-
|
For a one solar mass black hole, the peak Hawking radiation temperature is:
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 kB =
2.8978×10−3 m K:
Schwarzschild
sphere surface area of
Schwarzschild radius rs:
Stefan–Boltzmann power law:
For simplicity, assume a black hole is a perfect blackbody (
ε = 1).
Stefan–Boltzmann–Schwarzschild–Hawking black hole radiation power law derivation:
This yields the Bekenstein–Hawking luminosity of a black hole, under
the assumption of pure photon emission (no other particles are emitted)
and under the assumption that the horizon is the radiating surface:
-
|
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.
Substituting the numerical values of the physical constants in the formula for luminosity we obtain P=
3.562×1032 W kg^2/M^2. The power of the Hawking radiation from a
solar mass (
M☉) black hole turns out to be minuscule:
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:
Given that the power of the Hawking radiation is the rate of evaporation energy loss of the black hole:
Since the total energy
E of the black hole is related to its mass
M by Einstein's mass–energy formula
E = Mc2:
We can then equate this to our above expression for the power:
This differential equation is separable, and we can write:
The black hole's mass is now a function
M(t) of time
t. Integrating over
M from
M0 (the initial mass of the black hole) to zero (complete evaporation), and over
t from zero to
tev:
The evaporation time of a black hole is proportional to the cube of its mass:
The time that the black hole takes to dissipate is:
-
|
where
M0 is the mass of the black hole.
The lower classical quantum limit for mass for this equation is equivalent to the
Planck mass,
mP.
Hawking radiation evaporation time for a Planck mass quantum black hole:
-
|
where
tP is the
Planck time.
For a black hole of one
solar mass (
M☉ =
1.98892×1030 kg), we get an evaporation time of
2.098×1067 years—much longer than the current
age of the universe at
(13.799±0.021)×109 years.
[22]
But for a black hole of
1011 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 =
2.3×10−4 eV. This implies that
M must be less than 0.8% of the mass of the
Earth[23] – approximately the mass of the
Moon.
Cosmic microwave background radiation universe temperature:
Hawking total black hole mass:
-
|
where
M⊕ is the total
Earth mass.
In common units,
So, for instance, a 1-second-life black hole has a mass of
2.28×105 kg, equivalent to an energy of
2.05×1022 J that could be released by
5×106 megatons of TNT. The initial power is
6.84×1021 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 (~
10−8 kg), they result in impossible lifetimes below the Planck time (~
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)
[24] and equations 25 and 26 in Carr (2005).
[25]
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" ~
10−26 s.
In loop quantum gravity
A detailed study of the quantum geometry of a black hole horizon has been made using
loop quantum gravity.
[26] 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.
[27]
The quantum effects are centered at a set of discrete and unblended
frequencies highly pronounced on top of Hawking radiation spectrum.
[28]
Experimental observation
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,
[29] although its status as a genuine confirmation remains in doubt.
[30] 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.
[31][32]