The Influence of Gravity on the Boltzmann Entropy of a Closed Universe
Michael K.‐H. Kiessling
Abstract
This contribution inquires into Clausius' proposal that ``the entropy of the world tends to a maximum.'' The question is raised whether the entropy of `the world' actually does have a maximum; and if the answer is ``Yes!,'' what such states of maximum entropy look like, and if the answer is ``No!,'' what this could entail for the fate of the universe. Following R. Penrose, `the world' is modelled by a closed Friedman--Lema\^{\i}tre type universe, in which a three-dimensional spherical `space' is filled with `matter' consisting of $N$ point particles, their large-scale distribution being influenced by their own gravity. As `entropy of matter' the Boltzmann entropy for a (semi-) classical macrostate, and Boltzmann's ergodic ensemble formulation of it for an isolated thermal equilibrium state, are studied. Since the notion of a Boltzmann entropy is not restricted to classical non-relativistic physics, the inquiry will take into account quantum theory as well as relativity theory; we also consider black hole entropy. Model universes having a maximum entropy state and those which don't will be encountered. It is found that the answer to our maximum entropy question is not at all straightforward at the general-relativistic level. In particular, it is shown that the increase in Bekenstein--Hawking entropy of general-relativistic black holes does not always compensate for the Boltzmann entropy of a piece of matter swallowed by a black hole.