Hawking radiation

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Today it is believed that many galaxies tend to have a black hole in their center, including ours.

The Hawking radiation is radiation theoretically produced near the event horizon of a black hole and is entirely due to quantum-type effects. Hawking radiation is named after the British physicist Stephen Hawking, who first postulated its existence in 1974 by describing the properties of such radiation and obtaining some of the first results in quantum gravity. Hawking's work followed his visit to Moscow in 1973, where Soviet scientists Yakov Zeldovich and Aleksei Starobinsky showed him that, according to the quantum mechanical uncertainty principle, rotating black holes should create and emit particles.

Hawking radiation reduces the mass and rotational energy of black holes and is therefore also known as "black hole evaporation". Because of this, black holes that do not gain mass by other means are expected to shrink and eventually disappear. Micro black holes are predicted to be greater emitters of radiation than more massive black holes, and therefore should shrink and dissipate more rapidly.

In June 2008, NASA launched the Fermi Space Telescope, which is searching for the gamma-ray terminal flashes expected from the evaporation of some primordial black hole. In the event that the speculative large extra dimension theories are correct, CERN's Large Hadron Collider can create microblack holes and observe their evaporation. No micro black holes have been observed at CERN.

Later Paul Davies and Bill Unruh proved that an accelerating observer or Rindler observer in flat Minkowski spacetime would also detect Hawking-type radiation.

Origin of Hawking radiation

One of the consequences of Heisenberg's uncertainty principle is the quantum fluctuations of the vacuum. These consist of the creation, for very brief moments, of particle-antiparticle pairs from the vacuum. These particles are "virtual," but the black hole's intense gravity makes them real. Such pairs rapidly disintegrate with each other, returning the energy borrowed for their formation. However, at the edge of the event horizon of a black hole, the probability that one member of the pair forms from the inside and the other from the outside is non-zero, so one member of the pair could escape from the black hole. black hole; if the particle manages to escape, the energy will come from the black hole. That is, the black hole will have to lose energy to compensate for the creation of the two particles that it separated. This phenomenon has as consequences the net emission of radiation by the black hole and the decrease in its mass.

According to this theory, a black hole loses mass, at a rate inversely proportional to it, due to a quantum effect. That is, a low-mass black hole will disappear faster than a more massive one. Specifically, a black hole of subatomic dimensions would disappear almost instantly.

It is worth mentioning that the decrease in mass of a black hole by Hawking radiation would only be perceptible on time scales comparable to the age of the universe and only in microscopically sized black holes remnant perhaps from the time immediately after the Big Bang. If this is so, today we could see very small black hole explosions, something for which there is no evidence.

Issuance process

A black hole emits thermalized Hawking radiation, according to a distribution identical to that of the black body corresponding to a temperature ${displaystyle T_{H}}$ . Which, expressed in terms of the Planck units, turns out to be:

(1a) ${displaystyle T_{H}={frac {alpha _{H}}{2pi }}}$

Where ${displaystyle alpha _{H},}$ is a parameter related to gravity on the surface of the horizon. Similarly, an observer of Rindler with a uniform acceleration perceives a thermalized radiation associated with a black body temperature:

(2nd) ${displaystyle T_{R}={frac {alpha _{R}}{2pi }}}$

Where ${displaystyle alpha _{R},}$ is the acceleration in Planck units, obviously the expression (1a) and (2nd) are formally identical expressed in Planck units.

If we rewrite the two previous equations in conventional units, the Hawking radiation for a Schwarzschild hole and the Unruh radiation for an accelerated observer are:

${displaystyle T_{H}={hbar ,c^{3} over 8pi GMk},qquad T_{R}={frac {hbar a}{2pi ck}}}$

where:

hbar

It's Planck's reduced constant.

c is the speed of light

k is the Boltzmann constant

G the gravitational constant

M It's the mass of a black hole.

a is the acceleration of Rindler's observer.

Applying the previous equations to the solar case, if it were to become a black hole, it would have a radiation temperature of only 60 nK (nanokelvin). This radiation temperature is significantly lower than the temperature due to microwave background radiation, which is greater than 2.7 K, so if Hawking radiation exists, it could be undetectable.

Black hole evaporation

When the particles escape, the black hole loses a small amount of its energy and therefore decreases in size, because it corresponds to a smaller collapsed mass. This continuous emission of Hawking radiation is sometimes called "black hole evaporation" and it is a process by which the hole gradually loses size until it disappears completely, although it is a very, very long process. An approximation of the "evaporation" can be obtained assuming that the black hole is a perfect black body, in which case the emission rate corresponds to the Stefan-Boltzmann law, we have:

${displaystyle {frac {{text{d}}E}{{text{d}}t}}=-(4pi R^{2})sigma T^{4}}$

using that energy is approximately ${displaystyle Eapprox mc^{2}}$ , the radius of a black hole of Schwarzschild came given by ${displaystyle R=2GM/c^{2}}$ Stefan-Boltzmann's constant is ${displaystyle sigma =(pi ^{2}k_{rm {B}}^{4})/(60hbar ^{3}c^{2})}$ and temperature by ${displaystyle T=hbar c^{3}/(8pi GMk_{B})}$ You have to. effective mass the black hole evolves as:

${displaystyle {frac {{text{d}}M}{{text{d}}t}}=-{frac {hbar ^{3}c^{4}}{15360pi G^{2}}}{frac {1}{M^{2}}}={frac {K}{M^{2}}},qquad Rightarrow M(t)=left(M_{0}^{3}-3Ktright)^{1/3}}$

This is only an approximation, but gives an idea of the order magnitude for the evaporation time, in terms of the initial mass ${displaystyle M_{0}}$ from the black hole. The small value of the constant ${displaystyle K=0,44cdot 10^{-52} {text{kg}}^{3}/{text{s}}}$ , makes the process very slow, and the total evaporation time would be given by:

${displaystyle t_{tot}={frac {M_{0}^{3}}{3K}}}$

For a supermassive black hole like Sagittarius A* that occupies the center of the Milky Way, the evaporation time would be:

${displaystyle t_{tot}={frac {M_{0}}{3K}}={frac {(8,57cdot 10^{37} {text{kg}})^{3}}{0,44cdot 10^{-52} {text{kg}}^{3}/{text{s}}}}approx 1,51cdot 10^{155} {text{años}}}$

Numerical analysis of the process

In 1976 Don Page calculated the power produced, and the evaporation time, for a Schwarzschild black hole of solar mass, without rotation or charge. The calculations are complicated due to the fact that a black hole, being of finite size, is not a perfect black body. The absorption cross section decreases in a complicated (spin dependent) way as the frequency decreases, especially when the wavelength becomes comparable to the size of the event horizon. Note that, when writing his paper in 1976, Page erroneously postulated that there are only two flavors of neutrinos and that they are massless, so his results for the life of black holes do not match modern results that take into account the three "flavors" neutrino with masses other than zero.

For a mass much greater than 10¹⁷ grams, Page deduces that the emission of electrons can be ignored, and that black mass holes M (in grams in the formula) evaporate through neutrinos (muonic) ${displaystyle nu _{mu }}$ and electronics ${displaystyle nu _{e}}$ ), photons $gamma$ and gravitons without mass in a time $tau$ of

{displaystyle tau =8.66times 10^{-27};left[{frac {M}{mathrm {g} }}right]^{3};mathrm {s} ,.}

For masses smaller than 10¹⁷ g, but much larger than 5 x 10¹⁴ g, ultrarelativistic emission of electrons and positrons will speed up evaporation, giving resulting in a lifetime of

{displaystyle tau =4.8times 10^{-27};left[{frac {M}{mathrm {g} }}right]^{3};mathrm {s} ,.}

If a black hole evaporates through Hawking radiation, a black solar mass hole (1 ${displaystyle M_{bigodot }}$ ) will evaporate in 10⁶⁴ years.

A supermassive black hole with a mass of 10¹¹ (100 billion) ${displaystyle M_{bigodot }}$ will evaporate in about 2×10¹⁰⁰ years.

It is predicted that some exceptionally large black holes will continue to grow up to perhaps 10¹⁴ ${displaystyle M_{bigodot }}$ during the collapse of galaxy superclusters. Even these holes will end up evading on a time scale above 10¹⁰⁶ years.

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