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

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

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.

  • Received 7 July 1980

DOI:https://doi.org/10.1103/PhysRevD.23.287

©1981 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.

Comments & Replies

Comment on a universal upper bound on the entropy-to-energy ratio for bounded systems

Don N. Page
Phys. Rev. D 26, 947 (1982)

Possible violations of the entropy-to-energy-ratio bound

S. D. Unwin
Phys. Rev. D 26, 944 (1982)

Specific entropy and the sign of the energy

Jacob D. Bekenstein
Phys. Rev. D 26, 950 (1982)

Bounds on the entropy

P. F. González-Díaz
Phys. Rev. D 27, 3042 (1983)

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Issue

Vol. 23, Iss. 2 — 15 January 1981

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