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 $ A $, has an entropy:

$ S_{BH} = \frac{k c^3 A}{4G \hbar} $

For a Schwarzschild black hole of mass $ M $, $ A=16 \pi (GM/c^2)^2 $.

Derivation Edit

This derivation is based on Leonard Susskind's lecture.

Consider a black hole of mass $ M $. 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

$ \lambda \approx GM/c^2 $

The mass of a single photon is given by

$ \delta M = \frac{\hbar}{ c \lambda } = \frac{\hbar c}{G M} $

Each photon carries a single bit of information, so in order to accumulate a mass $ M $, the total entropy is

$ \frac{S}{k} = \frac{M}{\delta M} = \frac{GM^2}{\hbar c} = \frac{c^3 A}{G \hbar} $

where $ k $ is the Boltzmann constant. 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 $ l_p \approx \sqrt{h G/c^3} $. 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

$ k T \approx \left(\frac{1}{c^2} \frac{d S}{d M}\right)^{-1} \approx \frac{h c^3}{G M} $