Entropy, energy and temperature–length inequality for Friedmann universes

Abstract

In this paper, we continue the study of the physical consequences of our modified black hole entropy formula in expanding spacetimes. In particular, we apply the new formula to apparent horizons of Friedmann expanding universes with zero, negative and positive spatial curvature. As a first result, we found that apart from the static Einstein solution, the only Friedmann spacetimes with constant (zero) internal energy are the ones with zero spatial curvature. This happens because, in the computation of the internal energy [Formula: see text], the contribution due to the nonvanishing Hubble flow must be added to the usual Misner–Sharp energy giving, for zero curvature spacetimes, a zero value for [Formula: see text]. This fact does not hold when curvature is present. After analyzing the free energy [Formula: see text], we obtain the correct result that [Formula: see text] is stationary only for physical systems in isothermal equilibrium, i.e. a de Sitter expanding universe. This result permits us to trace back a physically reasonable hypothesis concerning the origin of the early and late times de Sitter phase of our universe. Finally, we deduce an interesting temperature–length inequality similar to the time–energy uncertainty of ordinary quantum mechanics but with temperature instead of time coordinate. Remarkably, this relation is independent of the gravitational constant [Formula: see text] and can thus be explored also in nongravitational contexts.