The Bekenstein-Hawking formula describes the thermodynamic entropy of a black hole with a given mass. It is named after Jacob Bekenstein and Stephen Hawking, who found that the entropy of a black hole is proportional to the area of its event horizon.
A black hole with an area , has an entropy:
For a Schwarzschild black hole of mass , .
This derivation is based on Leonard Susskind's lecture.
Consider a black hole of mass . We wish to increase its mass by a substantial amount, by feeding it with photons. In order to follow the entropy associated with the black hole's growth, we feed it one photon at a time. The event of a photon absorption by the black hole corresponds to a certain amount of information, for example, the solid angle from which the photon has entered the black hole's horizon. We would therefore like to use photons that carry a single entropy unit - a bit. This can be achieved by considering photons that have a wavelength similar to the black hole's radius, and cannot be localized to a specific solid angle when they are absorbed. Note that photons with a wavelength much larger than the black hole's radius will not be absorbed efficiently by the black hole.
Therefore, the wavelength of these photons is given by
The mass of a single photon is given by
Each photon carries a single bit of information, so in order to accumulate a mass , the total entropy is
The right hand side is also equal to the ratio between the surface area of the horizon of a black hole and a the area of a square whose side is a Planck length . If we each such square is capable of storing a single bit, then the entropy simply measures how many such squares are there on the surface of a black hole.
From the entropy it is possible to use the usual thermodynamic relations to calculate the temperature of black holes