]> The Colour of a Black Hole

The Colour of a Black Hole

A black hole is black because it reliably absorbs light of all frequencies. As a result, it also radiates energy as a black body of a specific temperature, that's proportional to the inverse of its mass (or of its radius). A small black hole thus has a high temperature, which means it radiates light at high frequencies; and each photon radiated away has energy proportional to its frequency. However, a black hole of a given mass cannot radiate away a photon with greater energy than the mass of the black hole times the square of the speed of light, since this is all the energy the black hole has. Consequently, the emission spectrum of a black hole is clipped, at the frequency corresponding to the black hole's mass; for a small enough black hole, this clipping shall actually cut off its emission spectrum at a frequency where it would otherwise be radiating significantly. This means it no longer has a pure black body radiation spectrum and thus is no longer black. This likely won't last long, as small black holes evaporate very very fast; but it means that a black hole, in its last instant, does actually have a colour, albeit one that changes rapidly.

It is also conceivable that this has implications for the very last photons that the black hole emits: there must be more than one of these, as a photon always carries momentum (proportional to its frequency), relative to any rest frame, and there is some frame of reference – for the space around the black hole at a modest distance – with respect to which the black hole is at rest just before it finally evaporates. In that rest frame, the dying black hole has no net momentum, so the final burst of photons it produces must have none, yet each photon carries some momentum; so there must be several (most obviously two) whose momenta add up to zero. This means that the highest energy photon that can be in this final burst carries at most half the mass of the final black hole, at half the frequency of the cut-off just mentioned. The black hole could, however, emit a single photon of any lower frequency, leaving the black hole with a recoil velocity and a reduced mass. The tricky part is that, each time it does this, its temperature goes up while the cut-off frequency goes down; which makes the fact of the cut-off more significant, in terms of reducing the total power output of the evaporating black hole, hence its rate of evaporation.

The power radiated away, at frequencies between f and f+φ for small φ, by a black body at temperature T is given by:

in which c is the speed of light, h is Planck's constant and k is Boltzmann's constant. Each photon of such radiation carries energy h.f, so the rate of emission of photons in this frequency range is found by omitting a factor of h.f from the numerator. This is all per unit area of the black body; and our black body is a sphere whose radius is proportional to the mass, r = 2.G.M/c/c, with total surface area 4.π.r.r. We have k.T = ℏ/κ/M, where κ = G/c/c/c and ℏ = h/2/π; so h.f/k/T = 2.π.κ.M.f and r = 2.κ.c.M, whence h.f/k/T = π.f.r/c, so that the total rate of emission of photons in our frequency band is

The typical time between such emissions is the inverse of that:

except that the rate shall be zero if h.f > M.c.c = r.c/κ/2, i.e for f > r.c/κ/h/2. Now, h.κ = G.h/c/c/c is the square of the Planck length (which is very very short indeed), so introduce R = r / √(h.κ) as the radius over the Planck length, as a dimensionless size parameter. Substituting in the cut-off frequency for f, the typical time between emissions of photons within φ of the cutt-off is thus

so when r is the Planck length (so M is half the Planck mass) this time is just (exp(π/2) −1) / (2.π.φ) = 0.60645631/φ and the cuf-off frequency is half the Planck frequency (which is huge, because its inverse is the Planck time), so the probability per unit time of the black hole evaporating remains respectable. Note, however, that this is when the black hole's mass is down to half the Planck mass; which is the mass of a droplet of water a respectable fraction of a millimeter across. It may not be much, but we can definitely believe in much smaller masses. For these, the rate of emission of photons close below the cut-off shall get small rapidly with decreasing R; so we can expect evaporation to slow down below Planck mass (where it speeded up rapidly on the way down to that mass).

All of this is back-of-envelope physics; a proper analysis would involve going into Hawking's derivation (with which I am not familiar) of the thermodynamic properties of black holes and the standard derivation of the black body spectrum to work out how the black hole's mass – as limit on mass hence energy hence frequency of emitted photons – affects the spectrum of the radiation the black hole emits. It would not surprise me if the cut-off is gradual, for example; or induces a discrete spectrum superimposed on (at lower frequencies) the normal black-body distribution.

In any case, the cut-off doesn't become significant until the modal frequency of the Hawking radiation is very high, putting the colouring it causes to the black hole firmly in (or beyond) the range of ultra-hard gamma rays and X rays. For the colouring to fall in the visible spectrum, the black hole's mass would need to get all the way down to the femto-gram range (around 3e-16 g in fact, give or take a factor of less than two); this is 7e-30 times the Planck mass, putting the radius at 14e-30 times the Planck length. At that scale, the clipping of the rate of radiation would be dramatic; and the black hole's mass is a few thousandths of that of the electron, so it would be unable to emit anything except photons and (perhaps) neutrinos. Still, the emission spectrum of a black hole does get clipped, which does mean it has a colour; it's not quite black, although you'll be hard pressed to notice this if the black hole is at all detectable. Even a Planck mass black hole would be hard to notice, if it weren't exploding; one with its cut-off in the visible spectrum would be hard to notice even if we were looking right at it when it evaporated.

Another factor that might complicate matters is that (if I remember correctly) it's usually harder for a body to emit radiation whose wavelength exceeds the body's physical dimensions; which might lead to some further limitation on the black hole's ability to radiate at low frequencies. The modal wavelength of Hawking radiation is roughly the black hole's diameter; for a black hole small enough that the modal frequency is excluded, the wavelengths of what remains of the black hole's spectrum are all significantly longer than the length scale of the black hole itself. This might lead to black holes this small being unable to radiate away the last of their mass, despite having a nominally very high temperature.

The up-shot of all this is that it's conceivable (at a back-of-envelope level of analysis; as noted above, a proper analysis would be needed to have any certainty) that the evaporation of a black hole could leave behind a very tiny black hole that's so small that it can't radiate away the last of its remaining mass as energy. If that is the case, such black holes could be floating around all over the universe, if there's some way that the early universe could have spawned them. We wouldn't notice them much, because their emission (hence also absorption) spectrum is limited and they're too tiny to capture other matter (its uncertainty of position would allow only negligible probability of being near enough to the event horizon to be captured). The only effect these would have is in the aggregate; each has tiny mass, but enough of them floating around would constitute a mass density in space that's otherwise unobservable – just exactly the property Dark Matter has. Of course, this remains mere speculation unless there's a viable model of tiny black holes not evaporating and a plausible mechanism for the early universe to have produced lots of them.

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