Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 16; Boxes 16.3, 16.4.
Since a black hole can radiate its energy away and the energy of a black hole is equivalent to its mass, it’s possible that a black hole will simply evaporate after a long enough time. We can get an estimate of this time as follows. First, we recall that, for particle-antiparticle pairs that are produced very close to the event horizon, the energy of the particle at infinity tends to a constant:
This is only a crude estimate based on oversimplifying the situation and a more accurate calculation, due to Stephen Hawking, results in
In thermodynamics, a blackbody is a body that absorbs all frequencies of electromagnetic radiation (hence ‘black’) and radiates with a spectrum that depends only its temperature . The wavelength of radiation peaks at a shorter value the higher the temperature (which is why an iron bar, for example, glows red when it gets hotter; at cooler temperatures it radiates in the infra-red). Hawking showed that the energy spectrum of a black hole is actually the same as that of a blackbody with temperature if we set , where is Boltzmann’s constant. We can therefore define the temperature of a black hole as
To use this in calculations, it’s best to convert the constants to relativistic form, where . We start with their values in SI units. For Planck’s constant:
For Boltzmann’s constant:
Thus the temperature is
In terms of the solar mass , we can use so
A one solar mass black hole is thus almost at absolute zero.
The rate at which an object radiates energy is given by the Stefan-Boltzmann law, which states
where is the surface area of the object and is the Stefan-Boltzmann constant, in relativistic units. In the case of a black hole, the energy is just the mass, so
and the area is that of a sphere with a radius , so
Using 3, we then have
where is the time required for the black hole to evaporate completely. We can plug in the numbers, using and to get
In other words, for a one solar mass black hole, we’re not going to see it evaporate any time soon.