Hawking radiation

Hawking radiation is black-body radiation that is predicted to be released by black holes, due to quantum effects near the black hole event horizon. It is named after the theoretical physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974.

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 through other means are expected to shrink and ultimately vanish. Micro black holes are predicted to be larger emitters of radiation than more massive black holes and should thus shrink and dissipate faster.

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.

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. 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. 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 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.

Discovery

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.

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 that are regular across the horizon. This is necessarily so, since to stay outside a horizon requires acceleration that constantly Doppler shifts the modes.

An outgoing Hawking-radiated photon, if the mode is traced back in time, has a frequency that 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 that 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 that 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 that 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 that 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.

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.

A Schwarzschild black hole has a metric:

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

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

$r=2\,M+{\frac {\rho ^{2}}{\ 8\,M\ }}\,$.

The local metric to lowest order is

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

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

$T\;=\;{\frac {\alpha }{2\pi }}\;=\;{\frac {1}{\,2\pi \rho \,}}\;=\;{\frac {1}{\,4\pi {\sqrt {2M\left(r-2M\right)\,}}\,}}\,$;

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:

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

The inverse temperature redshifted to r′ at infinity is

$T(\infty )={\frac {1}{4\pi {\sqrt {2Mr\,}}}}$

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

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

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

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

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 natural units (G = c = ħ = k_B = 1), the temperature is

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

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

$T={\frac {\hbar \,c^{3}}{\,8\,\pi \,G\,k_{\text{B}}\,M\,}}\;\approx \;1.227\times 10^{+23}{\text{K·kg}}\,\times {\frac {1}{M}}\;=\;6.169\times 10^{-8}{\text{K}}\,\times {\frac {M_{\odot }}{M}}\,$,

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:

$\operatorname {d} S\;=\;{\frac {\operatorname {d} Q}{T}}\;=\;8\pi M\,\operatorname {d} Q\,$.

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

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

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:

$S=\pi \,R^{2}={\frac {\,A\,}{4}}$.

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.

In 2019, Biancalana, Robson and Villari from Heriot-Watt University in Edinburgh (UK), showed that Hawking's radiation temperature is a purely topological quantity that can be calculated very simply by computing the Euler characteristics of the black hole spacetime.

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. 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 his results of black hole lifetimes do not match the modern results which take into account 3 flavors of neutrinos with nonzero masses.

For a mass much larger than 10¹⁷ 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

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

For a mass much smaller than 10¹⁷ g, but much larger than 5×10¹⁴ g, the emission of ultrarelativistic electrons and positrons will accelerate the evaporation, giving a lifetime of

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

If black holes evaporate under Hawking radiation, a solar mass black hole will evaporate over 10⁶⁴ years. A supermassive black hole with a mass of 10¹¹ (100 billion) M_☉ will evaporate in around 2×10¹⁰⁰ years. 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.

Large extra dimensions

The formulae from the previous section are applicable only 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 (10 or 11), 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).

$\tau \sim {\frac {1}{M_{*}}}\left({\frac {M_{\mathrm {BH} }}{M_{*}}}\right)^{\frac {n+3}{n+1}}\,,$

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" ~10⁻²⁶ s.

In loop quantum gravity

A detailed study of the quantum geometry of a black hole event 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.

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, 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.

See also

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

Further reading

• Hawking, Stephen W. (1974). "Black hole explosions?". Nature. 248 (5443): 30–31. Bibcode:1974Natur.248...30H. doi:10.1038/248030a0. → Hawking's first article on the topic
• Page, Don N. (1976). "Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole". Physical Review D. 13 (2): 198–206. Bibcode:1976PhRvD..13..198P. doi:10.1103/PhysRevD.13.198. → first detailed studies of the evaporation mechanism
• Carr, Bernard J.; Hawking, Stephen W. (1974). "Black holes in the early universe". Monthly Notices of the Royal Astronomical Society . 168 (2): 399–415. arXiv:1209.2243. Bibcode:1974MNRAS.168..399C. doi:10.1093/mnras/168.2.399. → links between primordial black holes and the early universe
• Barrau, Aurélien; et al. (2002). "Antiprotons from primordial black holes". Astronomy & Astrophysics. 388 (2): 676–687. arXiv:astro-ph/0112486. Bibcode:2002A&A...388..676B. doi:10.1051/0004-6361:20020313.
• Barrau, Aurélien; et al. (2003). "Antideuterons as a probe of primordial black holes". Astronomy & Astrophysics. 398 (2): 403–410. arXiv:astro-ph/0207395. Bibcode:2003A&A...398..403B. doi:10.1051/0004-6361:20021588.
• Barrau, Aurélien; Féron, Chloé; Grain, Julien (2005). "Astrophysical Production of Microscopic Black Holes in a Low–Planck-Scale World". The Astrophysical Journal. 630 (2): 1015–1019. arXiv:astro-ph/0505436. Bibcode:2005ApJ...630.1015B. doi:10.1086/432033. → experimental searches for primordial black holes thanks to the emitted antimatter
• Barrau, Aurélien; Boudoul, Gaëlle (2002). "Some aspects of primordial black hole physics". arXiv:astro-ph/0212225. {{cite arXiv}}: Cite has empty unknown parameters: |version=, |url=, and |accessdate= (help) → cosmology with primordial black holes
• Barrau, Aurélien; Grain, Julien; Alexeyev, Stanislav O. (2004). "Gauss–Bonnet black holes at the LHC: beyond the dimensionality of space". Physics Letters B. 584 (1–2): 114–122. arXiv:hep-ph/0311238. Bibcode:2004PhLB..584..114B. doi:10.1016/j.physletb.2004.01.019. → searches for new physics (quantum gravity) with primordial black holes
• Kanti, Panagiota (2004). "Black Holes in Theories with Large Extra Dimensions: a Review". International Journal of Modern Physics A. 19 (29): 4899–4951. arXiv:hep-ph/0402168. Bibcode:2004IJMPA..19.4899K. doi:10.1142/S0217751X04018324. → evaporating black holes and extra-dimensions
• Ida, Daisuke; Oda, Kin'ya; Park, Seong-chan (2003). "Rotating black holes at future colliders: Greybody factors for brane fields". Physical Review D. 67 (6): 064025. arXiv:hep-th/0212108. Bibcode:2003PhRvD..67f4025I. doi:10.1103/PhysRevD.67.064025.
• Ida, Daisuke; Oda, Kin'ya; Park, Seong-chan (2005). "Rotating black holes at future colliders. II. Anisotropic scalar field emission". Physical Review D. 71 (12): 124039. arXiv:hep-th/0503052. Bibcode:2005PhRvD..71l4039I. doi:10.1103/PhysRevD.71.124039.
• Ida, Daisuke; Oda, Kin'ya; Park, Seong-chan (2006). "Rotating Black Holes at Future Colliders. III. Determination of Black Hole Evolution". Physical Review D. 73 (12): 124022. arXiv:hep-th/0602188. Bibcode:2006PhRvD..73l4022I. doi:10.1103/PhysRevD.73.124022. → determination of black hole's life and extra dimensions
• Nicolaevici, Nistor (2003). "Blackbody spectrum from accelerated mirrors with asymptotically inertial trajectories". Journal of Physics A . 36 (27): 7667–7677. Bibcode:2003JPhA...36.7667N. doi:10.1088/0305-4470/36/27/317. → consistent derivation of the Hawking radiation in the Fulling–Davies mirror model.
• Smolin, Lee (November 2006). "Quantum gravity faces reality" (PDF). Physics Today. 59 (11): 44–48. Bibcode:2006PhT....59k..44S. doi:10.1063/1.2435646. Archived from the original (PDF) on September 10, 2008. consists of the recent developments and predictions of loop quantum gravity about gravity in small scales including the deviation from Hawking radiation effect by Ansari.
• Ansari, Mohammad H. (2007). "Spectroscopy of a canonically quantized horizon". Nuclear Physics B. 783 (3): 179–212. arXiv:hep-th/0607081. Bibcode:2007NuPhB.783..179A. doi:10.1016/j.nuclphysb.2007.01.009. → studies the deviation of a loop quantized black hole from Hawking radiation. A novel observable quantum effect of black hole quantization is introduced.
• Shapiro, Stuart L.; Teukolsky, Saul A. (1983). Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects. Wiley-Interscience. p. 366. ISBN 978-0-471-87316-7. → Hawking radiation evaporation formula derivation.
• Leonhardt, Ulf; Maia, Clovis; Schuetzhold, Ralf (2010). "Focus on Classical and Quantum Analogs for Gravitational Phenomena and Related Effects". New Journal of Physics . 14 (10): 105032. Bibcode:2012NJPh...14j5032L. doi:10.1088/1367-2630/14/10/105032.

External links

• Hawking radiation calculator tool
• The case for mini black holes A. Barrau & J. Grain explain how the Hawking radiation could be detected at colliders
• Black Holes – Anybody out there?
• University of Colorado at Boulder
• Hawking radiation on arxiv.org
• Hawking radiation observed in laboratory?