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.

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