Black hole entropy

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 {S} of a system is defined in terms of its temperature {T} and internal energy {U} by

\displaystyle  \boxed{\frac{1}{T}\equiv\frac{\partial S}{\partial U}} \ \ \ \ \ (1)

A black hole’s internal energy is just its mass, so {U=M}, and we found an expression for its temperature earlier:

\displaystyle  T=\frac{\hbar}{8\pi k_{B}GM} \ \ \ \ \ (2)

We can thus work out the entropy as a function of mass:

\displaystyle   \frac{\partial S}{\partial M} \displaystyle  = \displaystyle  \frac{8\pi k_{B}G}{\hbar}M\ \ \ \ \ (3)
\displaystyle  S \displaystyle  = \displaystyle  \frac{4\pi k_{B}G}{\hbar}M^{2} \ \ \ \ \ (4)

where the last line assumes that {S=0} at {M=0}. From this, we can see that if we combine two black holes with masses {M_{1}} and {M_{2}}, the total entropy of the system increases.

\displaystyle   S_{tot} \displaystyle  = \displaystyle  \frac{4\pi k_{B}G}{\hbar}\left(M_{1}+M_{2}\right)^{2}\ \ \ \ \ (5)
\displaystyle  \displaystyle  = \displaystyle  \frac{4\pi k_{B}G}{\hbar}\left(M_{1}^{2}+M_{2}^{2}+2M_{1}M_{2}\right)\ \ \ \ \ (6)
\displaystyle  \displaystyle  = \displaystyle  S_{1}+S_{2}+\frac{8\pi k_{B}G}{\hbar}M_{1}M_{2} \ \ \ \ \ (7)

3 thoughts on “Black hole entropy

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