Hawking Radiation

Classically, black holes are black.

Quantum mechanically, black holes radiate, with a radiation known as Hawking radiation, after the British physicist Stephen Hawking who first proposed it.

The animation at top left cartoons the Hawking radiation from a black hole of the size shown at bottom left. The blobs are supposed to be individual photons. Notice, first, that the photons have ‘sizes’ (wavelengths) comparable to the size of the black hole, and, second, that the Hawking radiation is not very bright — the black hole emits roughly one photon every light crossing time of the black hole. So a black hole observed by its Hawking radiation looks fuzzy, a quantum mechanical object.

This is one animation that I did not compute mathematically. How do you draw a quantum mechanical object, whose appearance depends not only on the object but also on the way the observer chooses to observe it? I figured my impressionism was good enough here.

Hawking radiation has a blackbody (Planck) spectrum with a temperature \(T\) given by \[ k T = {\hbar g \over 2\pi c} = {\hbar c \over 4\pi r_s} \ , \] where \(k\) is Boltzmann’s constant, \(\hbar = h / (2\pi)\) is Planck’s constant divided by \(2\pi\), and \(g = G M / r_s^2\) is the surface gravity at the horizon, the Schwarzschild radius \(r_s\), of the black hole of mass \(M\). Numerically, the Hawking temperature is \(T = 4 \times 10^{-20} g \, \mbox{Kelvin}\) if the gravitational acceleration g is measured in Earth gravities (gees).

The Hawking luminosity \(L\) of the black hole is given by the usual Stefan-Boltzmann blackbody formula \[ L = A \sigma T^4 \] where \(A = 4\pi r_s^2\) is the surface area of the black hole, and \(\sigma = \pi^2 k^4 / ( 60 c^2 \hbar^3 )\) is the Stefan-Boltzmann constant. If the Hawking temperature exceeds the rest mass energy of a particle type, then the black hole radiates particles and antiparticles of that type, in addition to photons, and the Hawking luminosity of the black hole rises to \[ L = A ( n_\textrm{eff} / 2 ) \sigma T^4 \ , \] where \(n_\textrm{eff}\) is the effective number of relativistic particle types, including the two helicity types (polarizations) of the photon.

Black holes for which astronomical evidence exists have masses ranging from stellar-sized black holes of a few solar masses, up to supermassive black holes in the nuclei of galaxies, such as the \(6 \times 10^9\) solar mass black hole at the centre of the galaxy Messier 87. The Hawking radiation from such black holes is minuscule. The Hawking temperature of a \(30\) solar mass black hole is a tiny \(2 \times 10^{-9} \, \mbox{Kelvin}\), and its Hawking luminosity a miserable \(10^{-31} \, \mbox{Watts}\). Bigger black holes are colder and dimmer: the Hawking temperature is inversely proportional to the mass, while the Hawking luminosity is inversely proportional to the square of the mass.

Answer to the quiz question 7: No, the x-ray emission from x-ray binary star systems is not Hawking radiation. The x-rays come not from the black hole (or neutron star), but from a circling accretion disk of hot gas. The accretion disk is heated to x-ray emitting temperatures by the release of gravitational energy as the gas spirals toward the black hole (or neutron star). The Hawking radiation from the black hole is exquisitely tiny by comparison.

Claus Kiefer (1998) ‘‘Towards a Full Quantum Theory of Black Holes’’ (gr-qc/9803049) gives a pedagogical review of Hawking radiation and other quantum aspects of black holes.

Black holes get the energy to radiate Hawking radiation from their rest mass energy. So if a black hole is not accreting mass from outside, it will lose mass by Hawking radiation, and will eventually evaporate. For astronomical black holes, the evaporation time is prodigiously long — about \(10^{61}\) times the age of the Universe for a \(30\) solar mass black hole. However, the evaporation time is shorter for smaller black holes (evaporation time \(t\) is proportional to \(M^3\)), and black holes with masses less than about \(10^{11} \, \mbox{kg}\) (the mass of a small mountain) can evaporate in less than the age of the Universe. The Hawking temperature of such mini black holes is high: a \(10^{11} \, \mbox{kg}\) black hole has a temperature of about \(10^{12} \, \mbox{Kelvin}\), equivalent to the rest mass energy of a proton. The gravitational pull of such a mini black hole would be about \(1\) gee at a distance of \(1\) metre.

It is not well established what an evaporating mini black hole would actually look like in realistic detail. The Hawking radiation itself would consist of fiercely energetic particles, antiparticles, and gamma rays. Such radiation is invisible to the human eye, so optically the evaporating black hole might look like a dud. However, it is also possible that the Hawking radiation, rather than emerging directly, might power a hadronic fireball that would degrade the radiation into particles and gamma rays of less extreme energy, possibly making the evaporating black hole visible to the eye. Whatever the case, you would not want to go near an evaporating mini black hole, which would be a source of lethal gamma rays and energetic particles, even if it didn’t look like much visually.

The animation at left is a fanciful depiction of the final moments of the evaporation of a hypothetical mini black hole. In the final second of its existence, the mini black hole radiates about \(1000\) tonnes of rest mass energy. Such an explosion is large by human standards, but modest by astronomical standards. An evaporating black hole would be detectable from Earth only if it went off within the solar system, or at best no further away than the nearest star.

What’s the cross shape? Telescopic diffraction spikes, added for artistic effect. Compare these beautiful Hubble Space Telescope pictures of the Orion nebula or of stars in the Messier 4 globular star cluster.