Entropy bounds and black hole remnants

Phys. Rev. D 49, 1912 – Published 15 February 1994
Jacob D. Bekenstein


We rederive the universal bound on entropy with the help of black holes while allowing for Unruh-Wald buoyancy. We consider a box full of entropy lowered toward and then dropped into a Reissner-Nordström black hole in equilibrium with thermal radiation. We avoid the approximation that the buoyant pressure varies slowly across the box, and compute the buoyant force exactly. We find, in agreement with independent investigations, that the neutral point generically lies very near the horizon. A consequence is that in the generic case the Unruh-Wald entropy restriction is neither necessary nor sufficient for enforcement of the generalized second law. Another consequence is that generically the buoyancy makes only a negligible contribution to the energy bookkeeping, so that the original entropy bound is recovered if the generalized second law is assumed to hold. The number of particle species does not figure in the entropy bound, a point that has caused some perplexity. We demonstrate by explicit calculation that, for an arbitrarily large number of particle species, the bound is indeed satisfied by cavity thermal radiation in the thermodynamic regime, provided vacuum energies are included. We also show directly that thermal radiation in a cavity in D-dimensional space also respects the bound regardless of the value of D. As an application of the bound we show that it strongly restricts the information capacity of the posited black hole remnants, so that they cannot serve to resolve the information paradox.

DOI: http://dx.doi.org/10.1103/PhysRevD.49.1912

  • Received 14 June 1993
  • Published in the issue dated 15 February 1994

© 1994 The American Physical Society

Authors & Affiliations

Jacob D. Bekenstein

  • Department of Physics, University of California at Santa Barbara, Santa Barbara, California 93106
  • The Racah Institute of Physics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel

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