# Bekenstein ## Universal upper bound on the entropy-to-energy ratio for bounded systems \[Links: [DOI](https://doi.org/10.1103/PhysRevD.23.287)\] \[Abstract: We present evidence for the existence of a universal upper [[0418 Bekenstein bound|bound]] of magnitude $\frac{2πR}{\hbar c}$ to the entropy-to-energy ratio $\frac{S}{E}$ of an arbitrary system of effective radius $R$. For systems with negligible self-gravity, the bound follows from application of the second law of thermodynamics to a gedanken experiment involving a black hole. Direct statistical arguments are also discussed. A microcanonical approach of Gibbons illustrates for simple systems (gravitating and not) the reason behind the bound, and the connection of $R$ with the longest dimension of the system. A more general approach establishes the bound for a relativistic field system contained in a cavity of arbitrary shape, or in a closed universe. Black holes also comply with the bound; in fact they actually attain it. Thus, as long suspected, black holes have the maximum entropy for given mass and size which is allowed by quantum theory and general relativity.\] # Witten ## Instability of the Kaluza-Klein Vacuum \[Links: [Inspire](https://inspirehep.net/literature/10780)\] \[Abstract: It is argued that the ground state of the [[0169 Kaluza-Klein|Kaluza-Klein]] unified theory is unstable against a process of semiclassical barrier penetration. This is related to the fact that the [[0116 Positive energy theorem|positive energy conjecture]] does not hold for the Kaluza-Klein theory; an explicit counter-example is given. The reasoning presented here assumes that in general relativity one should include manifolds of non-vacuum topology. It is argued that the existence of elementary fermions (not present in the original Kaluza-Klein theory) would stabilize the Kaluza-Klein vacuum.\]