Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 16; Problem P16.7.
Bizarre as it seems, an entropy can be defined for a black hole. From thermodynamics (which we haven’t covered yet, so you’ll need to look elsewhere for a derivation for now), the entropy of a system is defined in terms of its temperature and internal energy by
A black hole’s internal energy is just its mass, so , and we found an expression for its temperature earlier:
We can thus work out the entropy as a function of mass:
where the last line assumes that at . From this, we can see that if we combine two black holes with masses and , the total entropy of the system increases.