Comment on entropy bounds and the generalized second law

In a gedanken experiment in which a box initially containing energy E and entropy S is lowered toward a black hole and then dropped in, it was shown by Unruh and Wald that the generalized second law of black hole thermodynamics holds, without the need to assume any bounds on S other than the bound that arises from the fact that entropy at a given energy and volume is bounded by that of unconstrained thermal matter. The original analysis by Unruh and Wald made the approximation that the box was “thin,” but they later generalized their analysis to thick boxes (in the context of a slightly different process). Nevertheless, Bekenstein has argued that, for a certain class of thick boxes, the buoyancy force of the “thermal atmosphere” of the black hole is negligible, and that his previously postulated bound on $S/E$ is necessary for the validity of the generalized second law. In arguing for these conclusions, Bekenstein made some assumptions about the nature of unconstrained thermal matter and the location of the “floating point” of the box. We show here that under these assumptions, Bekenstein’s bound on $S/E$ follows automatically from the fact that S is bounded by the entropy of unconstrained thermal matter. Thus, a box of matter which violates Bekenstein’s bound would violate the assumptions made in his analysis, rather than violate the generalized second law. Indeed, we prove here that no universal entropy bound need be hypothesized in order to ensure the validity of the generalized second law in this process.