Quantum buoyancy, generalized second law, and higher-dimensional entropy bounds

TL;DR

This paper extends the universal entropy bound to higher-dimensional systems, analyzing how quantum buoyancy affects the applicability of the generalized second law in such contexts, especially as the number of spatial dimensions increases.

Contribution

It derives a generalized entropy-to-energy ratio bound for (D+1)-dimensional systems and studies the impact of quantum buoyancy on the validity of the generalized second law.

Findings

Quantum buoyancy effects increase with the number of spatial dimensions.

The neutral floating point of a body near a black hole horizon depends on the dimension D.

The universal entropy bound remains sufficient for the generalized second law despite higher-dimensional effects.

Abstract

Bekenstein has presented evidence for the existence of a universal upper bound of magnitude $2\pi R/\hbar c$ to the entropy-to-energy ratio $S/E$ of an arbitrary {\it three} dimensional system of proper radius $R$ and negligible self-gravity. In this paper we derive a generalized upper bound on the entropy-to-energy ratio of a $(D+1)$-dimensional system. We consider a box full of entropy lowered towards and then dropped into a $(D+1)$-dimensional black hole in equilibrium with thermal radiation. In the canonical case of three spatial dimensions, it was previously established that due to quantum buoyancy effects the box floats at some neutral point very close to the horizon. We find here that the significance of quantum buoyancy increases dramatically with the number $D$ of spatial dimensions. In particular, we find that the neutral (floating) point of the box lies near the horizon only if its length $b$ is large enough such that $b/b_C>F(D)$, where $b_C$ is the Compton length of the body and $F(D)\sim D^{D/2}\gg1$ for $D\gg1$. A consequence is that quantum buoyancy severely restricts our ability to deduce the universal entropy bound from the generalized second law of thermodynamics in higher-dimensional spacetimes with $D\gg1$. Nevertheless, we find that the universal entropy bound is always a sufficient condition for operation of the generalized second law in this type of gedanken experiments.