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 .

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