Hawking radiation

Stephen Hawking

The Hawking radiation is one of the British physicist Stephen Hawking 1975 postulated radiation black holes . It is derived from concepts of quantum field theory and general relativity . A possibility to prove or disprove the existence of the radiation is not in sight according to the current state of the art.

Hawking radiation is also of interest to current research because it could serve as a potential test field for a theory of quantum gravity .

Heuristic considerations led JD Bekenstein to the hypothesis as early as 1973 that the surface of the event horizon could be a measure of the entropy of a black hole ( Bekenstein-Hawking entropy ). Then, according to thermodynamics, it would have to be possible to assign a finite temperature to a black hole and it would have to be in thermal equilibrium with its surroundings. This resulted in a paradox, since it was assumed at the time that no radiation could escape from black holes. Hawking did quantum mechanical calculations and, to his own surprise, found that thermal radiation was to be expected.

Phenomena similar to those in Hawking radiation occur in cosmology ( Gibbons-Hawking effect ) and in accelerated frames of reference ( Unruh effect ).

Clear interpretation

In his publication in 1975 and also in several popular science books, Hawking provided intuitive explanations, which, according to his own statement, should not be taken too literally:

In contrast to classical physics , in quantum electrodynamics (and other quantum field theories ) the vacuum is not an “empty nothing”, but rather allows vacuum fluctuations . Vacuum fluctuations consist of virtual particle - antiparticle pairs. Such pairs can be masses as well as massless particles such as photons . Such vacuum fluctuations also exist in the immediate vicinity of the black hole event horizon. If a particle (or antiparticle) falls into the black hole, the two partners are separated by the event horizon. The partner falling into the black hole carries negative energy, while the second partner, who escapes as a real particle (or antiparticle) into free space, carries positive energy. “According to Einstein's equation E = mc², the energy is proportional to the mass. If negative energy flows into the black hole, its mass is reduced as a result ”.

Elsewhere, Hawking uses a different interpretation of particle-antiparticle pairs to illustrate Hawking radiation: Since a particle or antiparticle of negative energy can also be understood as an antiparticle or particle of positive energy moving backwards in time, one could Interpreting a particle / antiparticle falling into the black hole in such a way that it comes from the black hole and is scattered at the event horizon by the gravitational field in the forward time direction.

Those particles or antiparticles that escape the black hole form Hawking radiation. It is of a thermal nature in the form of black body radiation and is associated with a certain temperature, the so-called Hawking temperature, which is inversely proportional to the mass of the black hole.

Since the vacuum fluctuations are favored by a strong curvature of space-time, this effect is particularly important for black holes of low mass. Low mass black holes are small in size; i.e., they have a smaller Schwarzschild radius . The spacetime surrounding the event horizon is correspondingly more curved. The larger and therefore more massive a black hole, the less it radiates. The smaller a black hole, the higher its temperature and, due to the stronger Hawking radiation, the faster it evaporates.

Large black holes, such as those formed from supernovae , have such low levels of radiation (mostly photons ) that they cannot be detected in the universe. According to this theory, on the other hand, small black holes have significant thermal radiation, which leads to their mass rapidly decreasing. A black hole with a mass of 10 ¹² kilograms - the mass of a mountain - has a temperature of around 10 ¹¹ Kelvin, so that not only photons but also particles with mass such as electrons and positrons are emitted. As a result, the radiation increases further, so that such a small black hole is completely destroyed (evaporated) in a relatively short time. If the mass falls below 1000 tons, the black hole explodes with the energy of several million mega- or teratons of TNT-equivalent . The lifespan of a black hole is proportional to the third power of its original mass and for a black hole with the mass of our sun is approximately 10 ⁶⁴ years. It is therefore beyond all observation limits.

Hawking temperature

Hawking found a formula for the entropy and radiation temperature of a black hole, also known as the Hawking temperature, which is given by: ${\displaystyle S}$ ${\displaystyle T}$ ${\displaystyle T_{\mathrm {H} }}$

${\displaystyle T_{\mathrm {H} }={\frac {\hbar \ c^{3}}{8\pi \,G\,Mk_{\mathrm {B} }}}}$.

Thereby means

• ${\displaystyle \hbar }$the reduced Planck quantum of action ,
• ${\displaystyle c}$the speed of light ,
• ${\displaystyle G}$the gravitational constant ,
• ${\displaystyle M}$the mass of the black hole and
• ${\displaystyle k_{\mathrm {B} }}$the Boltzmann constant .

Often the temperature and entropy in gravitational physics is also specified in such a way that the Boltzmann constant is omitted.

The derivation of the formula for the temperature was carried out in the original work by Hawking using a semiclassical approximation . Since part of the radiation generated is scattered back into the black hole by the gravitational field, black holes are more to be understood as "gray emitters" with a lower radiation intensity than the black body model. The approximations in the derivation only apply to black holes with large mass, since it was assumed that the curvature of the event horizon is negligibly small, so that “ordinary” quantum mechanics in the background spacetime (in the case of the black hole the Schwarzschild metric or its Generalizations) can be operated. For very small black holes , the intensity distribution should deviate significantly from that of a black body, because in this case the quantum mechanical effects become so decisive that the semiclassical approximation no longer applies.

From the found Hawking formula for the temperature was held over (with a formula for the entropy that matched up to the pre-factors derived from Bekenstein with heuristic arguments formula). ${\displaystyle dE=TdS}$ ${\displaystyle E=Mc^{2}}$

Estimates

The Hawking temperature can be derived from the order of magnitude as follows: Wien's law of displacement results in a maximum of black body radiation at wavelengths . In the case of black holes, only the Schwarzschild radius comes into consideration as a unit of length , so that the temperature (in Kelvin) results: ${\displaystyle \lambda \approx {\frac {\hbar c}{k_{\mathrm {B} }T}}}$ ${\displaystyle r_{\mathrm {S} }={\frac {2GM}{c^{2}}}}$ ${\displaystyle \lambda \approx r_{\mathrm {S} }\approx {\frac {GM}{c^{2}}}}$

${\displaystyle T\approx {\frac {\hbar c^{3}}{k_{\mathrm {B} }GM}}\approx 10^{-6}{\frac {M_{\odot }}{M}}}$

with the solar mass . ${\displaystyle M_{\odot }}$

The radiation power can be estimated in a similar way according to the Stefan-Boltzmann law :

${\displaystyle P\approx c{\frac {{(k_{\mathrm {B} }T)}^{4}}{{(\hbar c)}^{3}}}A\approx c{\frac {\hbar c}{{r_{S}}^{4}}}{r_{S}}^{2}\approx {\frac {\hbar c^{6}}{G^{2}M^{2}}}}$

with the area , the Schwarzschild radius and the temperature estimated above . In MKS units the result is: watt ${\displaystyle A\approx 4\pi {r_{S}}^{2}}$ ${\displaystyle r_{S}\approx {\frac {GM}{c^{2}}}}$ ${\displaystyle k_{\mathrm {B} }T\approx {\frac {\hbar c}{r_{S}}}}$ ${\displaystyle P\approx 10^{38}M^{-2}}$

The service life results from the order of magnitude : ${\displaystyle \tau }$ ${\displaystyle P\tau \approx Mc^{2}}$

${\displaystyle \tau \approx {\frac {G^{2}M^{3}}{\hbar c^{4}}}}$

Or when specifying with MKS units:

${\displaystyle \tau \approx 10^{-20}M^{3}}$ Seconds

or years ${\displaystyle \tau \approx 10^{64}\cdot \left({\frac {M}{M_{\odot }}}\right)^{3}}$

Notes on Hawking's original work

Preliminary remarks

Since Hawking's publication in 1975, a number of different methods for deriving the thermal radiation of black holes have been developed, which in various ways confirm and supplement his original results.

Hawking used a massless scalar field in his original work for the sake of simplicity. However, the results can be extended to other particles such as photons and, more generally, to massless fermions . In principle, Hawking radiation also contains particles with mass, but their contribution is suppressed by many orders of magnitude compared to particles with no mass.

Contrary to the pictorial illustrations presented above, Hawking did not use a quantum mechanical perturbation theory in the first two works from 1974 and 1975, as the term “virtual particles” might suggest. If this were the case, the final result would have to depend on the coupling constant of the interaction under consideration, such as e.g. B. the fine structure constant in the electromagnetic interaction, depend. However, the result is already valid for free, non-interacting fields.

The original work, however, is based on a calculation, the essential terms of which mainly contribute to Hawking radiation in the vicinity of the event horizon. The wave function of the already mentioned massless scalar field can also be split into two parts, the first part being scattered into the outer space and the second part into the inner space of the black hole. The second part is therefore causally separated from the outer space of the black hole in the distant future.

Explanations

Hawking works in a semi-classical approximation , i. i.e., he considers a free quantum field theory on a classical, weakly curved spacetime. Essentially, the global structure of space-time and, in particular, the existence of an event horizon are relevant .

Hawking assumes a spherically symmetrical collapse of a mass M, i. i.e. it does not assume a purely static Schwarzschild metric . However, due to the Birkhoff theorem, the latter applies exactly in the outer space of the collapse. The details of the interior solution are irrelevant for the argument.

Hawking begins with the canonical quantization of free fields based on a generalized Fourier expansion . These Fourier modes are special solutions of the Klein-Gordon equation for massless scalar fields on spacetime geometry. The necessary decomposition of the Fourier modes into positive and negative frequencies as well as the resulting classification of particles and antiparticles is not clear due to the spacetime geometry. In the course of quantization, an observer can mathematically define creation and annihilation operators that are valid for him as well as a vacuum state ( Fock state ) that is valid for him , in which no particles and antiparticles exist according to his classification. While this observer dependency in the Minkowski space-time is ultimately irrelevant for the creation and annihilation operators as well as for the vacuum state, it leads to inequivalent vacuum states in the presence of an event horizon.

Mathematically, there is a transformation, the so-called Bogoljubov transformation , which converts the creation and annihilation operators of both observers into one another. Hawking first fixes a vacuum state as well as the creation and annihilation operators for the distant past. In this state the expectation value of the particle number operator (defined for the distant past) disappears. He then determines the Bogolyubov transform for the creation and annihilation operators for the distant future. For this purpose, the scattering of the Fourier modes at the collapsing black hole is essentially calculated. The proportion relevant for Hawking radiation comes from the scattering of the modes within the collapsing body. With this the expectation value of the particle number operator (defined for the distant future) in the original vacuum state (defined for the distant past) can be calculated. It turns out that this expected value does not vanish! The observer in the distant future does not see the vacuum state that is valid for him, but a state in which particles and antiparticles (with regard to its definition) are actually contained. The thermal nature of the spectrum follows from the exact shape of the Bogolyubov transform.

The physical core of Hawking's argument is therefore as follows: The collapse and the presence of a horizon lead to inequivalent vacuum states. While the evolution of time in a flat space-time leaves the vacuum invariant, in a space-time with a black hole it is subject to a “scattering process” that converts the initial vacuum into a thermal state.

Details

Hawking considers the free Klein-Gordon equation

${\displaystyle -g^{ab}\nabla _{a}\nabla _{b}\phi =0}$

of a massless scalar field.

He now introduces the two hypersurfaces and , which represent the outer space of the black hole in the distant, asymptotic past (-) and the distant future (+). On these hypersurfaces there are complete systems of functions and by means of which the field operator can be represented as a Fourier sum of generators and annihilators: ${\displaystyle {\mathcal {I}}^{-}}$ ${\displaystyle {\mathcal {I}}^{+}}$ ${\displaystyle f_{i}}$ ${\displaystyle p_{i}}$ ${\displaystyle \phi }$

${\displaystyle \phi =\sum _{i}[f_{i}a_{i}+{\bar {f}}_{i}a_{i}^{\dagger }]}$

${\displaystyle \phi =\sum _{i}[p_{i}b_{i}+{\bar {p}}_{i}b_{i}^{\dagger }]+\ldots }$

“…” Stands for a further function system on the light-like hypersurface of the event horizon. In principle, this is necessary in order to obtain a clearly solvable initial value problem, but it is not important for the further calculation.

Hawking then defines the vacuum state

${\displaystyle a_{i}|0_{-}\rangle =0}$

with respect to that of incoming particles. ${\displaystyle {\mathcal {I}}^{-}}$

The general connection between the two families of producers and annihilators is now the Bogolyubov transformation

${\displaystyle p_{i}=\sum _{j}\left[\alpha _{ij}f_{j}+\beta _{ij}{\bar {f}}_{j}\right]}$

${\displaystyle b_{i}=\sum _{j}\left[{\bar {\alpha }}_{ij}a_{j}-{\bar {\beta }}_{ij}a_{j}^{\dagger }\right]}$

Hawking shows in the following that the scattering of the incoming modes at the black hole leads to an observer on the state having a non-vanishing particle content ${\displaystyle {\mathcal {I}}^{-}}$ ${\displaystyle {\mathcal {I}}^{+}}$ ${\displaystyle |0_{-}\rangle }$

${\displaystyle \langle 0_{-}|b_{i}^{\dagger }b_{i}|0_{-}\rangle =\sum _{ij}|\beta _{ij}|^{2}>0}$

attributes. The generation rate of the particles follows directly from the coefficients of the Bogoljubov transformation. These mix up the destroyers with a share of producers . ${\displaystyle \beta _{ij}}$ ${\displaystyle {\mathcal {I}}^{-}}$ ${\displaystyle {\mathcal {I}}^{+}}$

The modes are scattered both on the outer Schwarzschild geometry and on the geometry of the interior of the collapsing star. The latter gives a non-trivial contribution to the modes, which then cause the special form of the Bogolyubov coefficients. ${\displaystyle p_{i}}$

The contribution of a mode with radial frequency is included ${\displaystyle \omega }$

${\displaystyle p_{i}\sim {\frac {1}{e^{2\pi \omega /\kappa }-1}}}$

with . This means that there is thermal radiation with temperature (in natural units ) according to the Bose-Einstein statistics. ${\displaystyle \kappa =1/4M}$ ${\displaystyle T_{H}=1/8\pi M}$

Hawking roughly explains that there is a course for fermions

${\displaystyle p_{i}\sim {\frac {1}{e^{2\pi \omega /\kappa }+1}}}$

is to be expected according to the Fermi-Dirac statistics.

The contribution of particles with mass is exponentially suppressed, as in this case the frequency or the mass must be taken into account accordingly . ${\displaystyle \omega ^{2}\sim E^{2}=m^{2}+p^{2}}$

Conclusions and Outlook

The prediction of Hawking radiation is based on the combination of the effects of quantum mechanics and general relativity and thermodynamics. Since a standardization of these theories ( quantum theory of gravitation ) has not yet been successful, such predictions are always fraught with a certain uncertainty.

With the thermal radiation, the black hole loses energy and thus mass. So it "shrinks" over time. However, due to their large mass, black holes of stellar origin have a lower temperature than the cosmic background radiation , which is why these black holes absorb thermal energy from their surroundings. In this case, the black hole cannot shrink, because the absorption of radiation energy increases the mass according to Einstein's mass-energy equivalence formula. Only when the ambient temperature has fallen below the temperature of the black hole does the hole lose mass due to radiation emission.

What happens to a black hole at the "end of its life" is partly unclear. According to Hawking, an explosive evaporation process of the black hole takes place there. The approximation of a weak curvature of space-time used in the original derivation is no longer valid. In particular, the so-called information paradox occurs. The question arises as to what happens when the black hole "evaporates" with the original information about those quantum objects that fell into the black hole when it was formed. According to certain requirements from quantum mechanics (unitarity), it is to be expected that this information will be retained over time. However, this question cannot be investigated in the context of Hawking's approximation, since the collapsing matter is treated in a purely classical manner and only the resulting Hawking radiation itself is treated quantum mechanically.

An intensification of the information paradox of black holes comes from Joseph Polchinski and colleagues (Feuerwand-Paradoxon, English: Firewall ). According to this hypothesis, there would be no inner black holes, it would be limited by the wall of fire. The equivalence principle would also be violated by the wall of fire hypothesis, since an observer falling into the black hole would very well notice a difference when crossing the event horizon; he would burn on the wall of fire. Ultimately, the reason for their existence would be a proposition of quantum mechanics, according to which entanglement can only ever exist between two particles. In the case of black holes, on the one hand, a pair of particles would be correlated, one partner of which disappears in the black hole, and, on the other hand, entanglement with other particles in the Hawking radiation. According to Polchinski and colleagues, there is a gradual transfer of quantum entanglement from the surroundings of the event horizon into Hawking radiation to the outside, which ultimately leads to a singularity in the form of a wall of fire inside the black hole, where the temperature diverges. An alternative was proposed by Juan Maldacena and Leonard Susskind in their EPR-ER hypothesis (EPR stands for Einstein-Podolsky roses and quantum entangled particle pairs, ER for Einstein-Rosen bridges , special wormholes) of the equivalence of quantum entanglement and wormholes between the particle pairs , expanded after the discovery of traversable wormholes by Ping Gao, Daniel Louis Jafferis and Aron C. Wall. The information paradox is solved by connecting the individual quantum entangled particles of Hawking radiation to their partners via wormholes ( octopus picture). The wormholes in turn causally connect two black holes inside, whose Hawking radiation is quantum entangled across the wormhole. The wall of fire paradox is avoided, since outside the black holes the contact of the particles still has to take place via spacetime.

literature

• Robert Brout , S. Massar, R. Parentani, P. Spindel: A Primer for black hole quantum physics , Physics Reports, Volume 260, 1995, p. 329. arxiv : 0710.4345
• Stephen W. Hawking, Particle creation by black holes , Commun. Math. Phys., Vol. 43, 1975, pp. 199-220
• Don N. Page: Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole . In: Physical Review D . 13, No. 2, 1976, pp. 198-206. bibcode : 1976PhRvD..13..198P . doi : 10.1103 / PhysRevD.13.198 . → First detailed study of the processes involved in the evaporation of black holes
• Robert M. Wald : General Relativity , University of Chicago Press, Chicago, 1984.
• Matt Visser: Essential and essential features of Hawking radiation . In: Cornell University . January 2001. arxiv : hep-th / 0106111 .

Web links

• Parentani, Spindel: Hawking Radiation, Scholarpedia
• Andreas Müller: Hawking radiation, lexicon of astronomy, spectrum
• R. Brout, S. Massar, R. Parentani, Ph. Spindel: A Primer for Black Hole Quantum Physics, Arxiv.org

Individual evidence

1. ↑ ^{a b c} Stephen W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975), 199–220 (PDF; 2.8 MB)
2. ↑ Stephen Hawking: A Brief History of Time , pp. 141 f., Rowohlt Taschenbuch Verlag, 2005, 25th edition, ISBN 3-499-60555-4
3. ↑ Stephen Hawking: The universe in a nutshell , p. 153, Hoffmann and Campe, 2001, ISBN 3-455-09345-0
4. Jump up ↑ Stephen Hawking, The Quantum Mechanics of Black Holes, Scientific American, January 1977
5. ↑ Hawking, Black hole explosions? , Letters to Nature, Volume 248, March 1, 1974, pp. 30-31
6. ^ Roman Sexl, Hannelore Sexl: White Dwarfs - Black Holes, Vieweg 1979, p. 83
7. ^ JB Hartle, SW Hawking, "Path-integral derivation of black-hole radiance", Phys. Rev. D 13, (1976), p. 2188
8. ↑ ^{a b} R.M. Wald, "On particle creation by black holes", Comm. Math. Phys. 1975, Volume 45, Issue 1, pp. 9-34
9. ↑ Article on www.scholarpedia.org
10. ↑ Ahmed Almheiri, Donald Marolf, Joseph Polchinski, James Sully: Black Holes: Complementarity or Firewalls? J. High Energy Phys. 2, 062 (2013), arxiv : 1207.3123 .
11. ^ The firewall hypothesis was supported by Leonard Susskind , The Transfer of Entanglement: The Case for Firewalls, Arxiv 2012
12. ↑ Ping Dao, Daniel Jafferis, Aron Wall, Traversable Wormholes via a Double Trace Deformation , Arxiv 2016
13. ↑ Natalie Wolchover: Newfound Wormhole Allows Information to Escape Black Holes , Quanta Magazine, October 23, 2017
14. ↑ Juan Maldacena, Douglas Stanford, Zhenbin Yang Diving into transversable wormholes, Arxiv 2017