Entropy bounds and the second law for black holes

A counterexample to the entropy bound proposed by Unruh and Wald and by Page is described. The bound states that the entropy S of any system with energy E and volume V cannot exceed the entropy of an equal volume and energy of unconfined thermal radiation. The bound is found to be violated by thermal field systems whose various dimensions differ by an order of magnitude or more, the violation occurring at intermediate energies. Unruh and Wald used the bound in an argument establishing the validity of the generalized second law when a box containing some entropy is lowered to near a black hole, then opened, and withdrawn open. If the box is thin in one direction, the failure of the bound for its contents makes it appear that a violation of the second law is possible. We show that, in fact, for a thin box the buoyancy effects on which Unruh and Wald's argument relies cancel out. As a result, the second law is fulfilled despite the failure of the bound. It appears from the second law that the bound must nevertheless hold when applied to box plus contents, but a direct proof of this is still lacking. We also consider the alternative entropy bound S2 (2R is the largest dimension of the system) proposed earlier. For field systems it is shown to fail at very low energies, but to be valid for complete systems (i. e. , box plus confined fields). Further, we show S2 is a necessary condition for fulfillment of the second law when a thin system is dropped into a black hole (even in the face of buoyancy) in the sense that if the bound failed, the law would be violated.