Black Hole Thermodynamics

At the heart of black hole thermodynamics is the idea that black holes have entropy. This idea was proposed by Jacob Bekenstein and Stephen Hawking, and hence formula for the entropy of a black hole with a given mass is named after them. 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 $M$, $A=16 \pi (GM/c^2)^2$.

Bekenstein Hawking Entropy

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

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

The blackbody luminosity is given by

$L \approx \lambda^2 \frac{\left(k T\right)^4}{\hbar^3 c^2} \approx \frac{c^5}{G} \frac{\hbar c}{G M^2}$

The time it takes the black hole to evaporate is given by

$t_e \approx \frac{M c^2}{L} \approx \frac{G^2 M^3}{c^4 \hbar}$

Electric Charge

In the previous section we saw that information can be "loaded" into black holes by encoding it into photons and firing them into the black hole. In the case of a charged black hole, some of the black hole energy goes into charging the black hole, in the form of work against the electronstatic force. For this reason, fewer photons can be stored in a charged black hole (compared with an uncharged black hole with the same total energy), and so the entropy is lower. To estimate the reduction in entropy due to electric charge, we can calculate the electrostatic energy, assuming all the charge is concentrated on an infinitely thin shell at the Schwartzschild radius $R \approx GM/c^2$

$U_e \approx \frac{Q^2}{R} \approx \frac{Q^2 c^2}{G M}$

The reduction in dimensionless entropy is comparable to the ratio between the electrostatic and total energy $M c^2$

$\frac{\Delta S}{k} \approx \frac{U_e}{M c^2} \approx \frac{Q^2}{G M^2}$

When low mass charged black holes evaporate, they discharge very fast and then proceed to evaporate as charge free black hole. However, high mass and highly charged black holes lose mass faster than they lose charge, so they eventually turn into a maximally charged black hole. As a maximally charged black hole mass loss is limited by the discharge rate, so both charge and mass decrease very slowly. When the mass crosses some critical value, the black hole proceeds to evaporate as a low mass black hole.

The electric field just outside the horizon increases with the charge and decline with the Schwartzschild radius. Above a certain mass, the electric field outside the horizon is always lower than the Schwinger limit, so pair production is highly supressed. The critical mass is therefore given by

$E \approx \frac{Q}{R^2} \approx \sqrt{\frac{m_e^3 c^4}{r_e \hbar^2}} \Rightarrow M_c \approx \sqrt{\frac{r_e \hbar^2 c^4}{G^3 m_e^3}}$

where $m_e$ is the electron mass and $r_e$ is the classical electron radius. This mass is around a few million solar masses. See Hiscock and Weems 1990 for more details.

Spin

Like electric charge, higher values of spin reduce the entropy. The reason is that for non rotating black holes there are two ways to store information, namely polarisation and phase of the incident photon. To produce a spinning black hole with photons, a portion of them have to have circular polarisation, and so this degree of freedom cannot be used to store information. For low values of angular momentum the rotational energy is given by

$U_r \approx \frac{J^2}{M R^2} \approx \frac{J^2 c^4}{G^2 M^3}$

where $J$ is the angular momentum. The reduction in entropy due to spin is

$\Delta S \approx \frac{U_r}{M c^2} \approx \frac{J^2 c^2}{G^2 M^4} \approx a^2$

where $a$ is the dimensionless spin parameter.

The entropy of a maximally spinning black hole turns out to be half that of a non rotating black hole. One way of thinking about it is that for a non rotating black hole information can be stored both in phase and polarisation of a photon, while for maximally rotating black hole only the phase degree of freedom can be used.

When spinning black holes evaporate, they first spin down on a timescale that is shorter than (but comparable to) the mass loss timescale. After spin-down, the black hole proceeds to evaporate as a non rotating black hole. See Page 1976 for more information.