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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

kT = hbar g / (2 pi c) = hbar c / (4 pi rs)

where k is Boltzmann's constant, hbar = h / (2 pi) is Planck's constant divided by 2 pi, and g = G M / rs2 is the surface gravity at the horizon, the Schwarzschild radius rs, of the black hole of mass M. Numerically, the Hawking temperature is T = 4 × 10-20 g 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 T4

where A = 4 pi rs2 is the surface area of the black hole, and sigma = pi2 k4 / (60 c2 hbar3) 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 (neff / 2) sigma T4

where neff 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 3×109 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 2×10-9 Kelvin, and its Hawking luminosity a miserable 10-31 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.

15 Apr 2003 update. Adam Helfer (2003) ``Do black holes radiate?'' (gr-qc/0304042) opens with the statement: ``The prediction that black holes radiate due to quantum effects is often considered one of the most secure in quantum field theory in curved space-time. Yet this prediction rests on two dubious assumptions ...''. This delightfully readable review paper does an excellent job of convincing the reader that Hawking radiation is still far from being an established prediction of the quantum physics of black holes. The paper gives the clearest exposition of Hawking radiation that I know of, emphasizing the physical concepts while simplifying the mathematics to its barest essentials (not that the mathematics is simple even in stripped form).

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Updated 19 Apr 1998