Universal upper bound on the entropy-to-energy ratio for bounded systems

Phys. Rev. D 23, 287 – Published 15 January 1981
Jacob D. Bekenstein


We present evidence for the existence of a universal upper bound of magnitude 2πRc to the entropy-to-energy ratio SE 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

Authors & Affiliations

Jacob D. Bekenstein*

  • Institute for Theoretical Physics, University of California, Santa Barbara, California 93106

  • *On sabbatical leave from Physics Department, Ben Gurion University, Beer Sheva, Israel.


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