Temperature of a black hole

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.7.

The entropy of a black hole is

\displaystyle  S=\frac{8\pi^{2}GM^{2}}{hc}k \ \ \ \ \ (1)

Taking {U=Mc^{2}} as the energy of a black hole, we can write this as

\displaystyle  S=\frac{8\pi^{2}Gk}{hc^{5}}U^{2} \ \ \ \ \ (2)

The temperature is therefore

\displaystyle  T=\left(\frac{\partial S}{\partial U}\right)^{-1}=\frac{hc^{5}}{16\pi^{2}GkU}=\frac{hc^{3}}{16\pi^{2}GkM} \ \ \ \ \ (3)

which agrees with our earlier result from general relativity.

For a solar mass black hole, this gives a value of

\displaystyle   T \displaystyle  = \displaystyle  \frac{\left(6.62\times10^{-34}\right)\left(3\times10^{8}\right)^{3}}{16\pi^{2}\left(6.67\times10^{-11}\right)\left(1.38\times10^{-23}\right)\left(2\times10^{30}\right)}\ \ \ \ \ (4)
\displaystyle  \displaystyle  = \displaystyle  6\times10^{-8}\mbox{ K} \ \ \ \ \ (5)

Not only are black holes dark, they are also very cold.

The entropy-versus-energy curve 2 is a parabola so its slope {\frac{\partial S}{\partial U}} increases as {U} 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 {T\propto\frac{1}{M}}.

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