On Page's examples challenging the entropy bound

TL;DR

This paper defends the original entropy bound $S \leq 2\pi ER$ by showing that Page’s proposed counterexamples fail once the energy $E$ is taken to include all essential, gravitating contributions (including wall, zero-point, and carrier energies). Through detailed analyses of a nonlinear scalar field (single and multiwell potentials), an electromagnetic onion, and a coaxial cable loop, the author demonstrates that the complete energy budget keeps the bound intact. The work also argues that at low temperatures the bound still holds for complete systems and that the so-called proliferation of species does not invalidate the bound due to gravitational instabilities in quantum field theory. Together, these results reinforce the bound’s robustness for weakly gravitating, complete systems and clarify its limitations in strongly gravitating or noncomplete contexts, while clarifying the relationship to holographic bounds.

Abstract

According to the entropy bound, the entropy of a complete physical system can be universally bounded in terms of its circumscribing radius and total gravitating energy. Page's three recent candidates for counterexamples to the bound are here clarified and refuted by stressing that the energies of all essential parts of the system must be included in the energy the bound speaks about. Additionally, in response to an oft heard claim revived by Page, I give a short argument showing why the entropy bound is obeyed at low temperatures by a complete system. Finally, I remark that Page's renewed appeal to the venerable ``many species'' argument against the entropy bound seems to be inconsistent with quantum field theory.