Black Hole Thermodynamics

Last updated
Save as PDF

Page ID: 1281

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Drop a box of gas into a black hole, and you end up with no gas and a bigger black hole. But a box of gas has entropy, and the second law of thermodynamics tells us that entropy cannot decrease. Based on thought experiments of this sort, and on the known result that the area of a black hole event horizon can never decrease, Jacob Bekenstein proposed in the early 1970s that a black hole should have an entropy proportional to its area.

This argument was at first dismissed because it was believed that black holes were truly "black," that is, that they emitted no radiation. This would mean their temperature was absolute zero -- heat flows from "hot" to "cold," and if radiation can only flow into a black hole and not out, the black hole must be colder than any possible surroundings. But we also know from thermodynamics that an object at zero temperature cannot have a changing entropy.

All this changed, though, when Hawking discovered that black holes actually do radiate, with a black body spectrum. (See "Hawking radiation.") The laws of thermodynamics were quickly extended to include black holes, which are ascribed an entropy equal to one-fourth of their horizon area (in Planck units).

In ordinary thermodynamic systems, though, thermal properties appear as a consequence of statistical mechanics, that is, as a collective result of the behavior of large numbers of "microscopic" degrees of freedom. The temperature of a box of gas, for instance, reflects the kinetic energy of the molecules of gas, and the entropy measures the number of accessible physical states. There have been many attempts to formulate a "statistical mechanical" explanation of black hole thermodynamics, but it's safe to say that a complete picture has not yet been found.

For a recent review, see here.

Contributors and Attributions

Steve Carlip (Physics, UC Davis)