#### Abstract

We present evidence for the existence of a universal upper bound of magnitude $\frac{2\pi R}{\hslash c}$ to the entropy-to-energy ratio $\frac{S}{E}$ of an arbitrary system of effective radius $R$. For systems with negligible self-gravity, the bound follows from application of the second law of thermodynamics to a gedanken experiment involving a black hole. Direct statistical arguments are also discussed. A microcanonical approach of Gibbons illustrates for simple systems (gravitating and not) the reason behind the bound, and the connection of $R$ with the longest dimension of the system. A more general approach establishes the bound for a relativistic field system contained in a cavity of arbitrary shape, or in a closed universe. Black holes also comply with the bound; in fact they actually attain it. Thus, as long suspected, black holes have the maximum entropy for given mass and size which is allowed by quantum theory and general relativity.

DOI: http://dx.doi.org/10.1103/PhysRevD.23.287

- Received 7 July 1980
- Revised 25 August 1980
- Published in the issue dated 15 January 1981

© 1981 The American Physical Society