Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.7.
The entropy of a black hole is
Taking as the energy of a black hole, we can write this as
The temperature is therefore
which agrees with our earlier result from general relativity.
For a solar mass black hole, this gives a value of
Not only are black holes dark, they are also very cold.
The entropy-versus-energy curve 2 is a parabola so its slope increases as increases. As we’ve seen, this means that a black hole has negative heat capacity, and thus decreases in temperature as more energy (mass) is added. This is also clear from 3, since .