Black Holes: Phase Transitions

It is not so easy to make a black hole in a laboratory, yet we can use thought experiments to try to figure out what would happen if we were able to do so. We can be pretty confident that those thought experiments are meaningful, given the utter simplicity of the structure of black holes. Unlike normal stars or planets or even neutron stars, which have a wealth of detailed structure (complex magnetic fields for one thing), a classical black hole can be described by only a few parameters. This is the meaning of the statement that a black hole has no hair.

A Charged Black Hole in a Box

With the discovery of Stephen Hawking in 1974 that black holes will evaporate through quantum effects, it became possible to conduct thermodynamic thought experiments in which a black hole is enclosed in a box, in equilibrium with its own radiation. A simple Schwarzschild black hole is not that interesting, because it has no free parameters, apart from its mass. Although you can change the size of the box and the total energy of the system, the first-order phase transition that you can observe is always qualitatively the same.

The two other free parameters of a black hole are its angular momentum and its charge. It is difficult to use the angular momentum, since a rotating black hole cannot be brought into complete equilibrium with an enclosing box if the box is too large, since that would require the box to rotate faster than the speed of light. That is why I chose to enclose a charged black hole in a box, in a paper that I wrote a few years after Hawking's discovery:

I made use of the fact that the lightest charged particles, electrons and positrons, have a finite non-zero mass. If we make the hole and the box large enough, the Hawking temperature will be small enough to exclude the emission of charged particles during the thought experiments. This simplifies the theoretical treatment considerably, since we only have to deal with a neutral gas of particles in equilibrium with the charged black hole.

In this simple system I found a wealth of thermodynamical effects, including the possibility of first-order phase transitions and a meaningful way to talk about phase diagrams and a critical point. Various heat capacities could be calculated, some of which exhibited discontinuities, again hinting at the presence of phase transitions.