Entropy, energy and temperature-length inequality for Friedmann universes

TL;DR

This paper explores the thermodynamic properties of Friedmann universes using a modified black hole entropy formula, revealing conditions for constant internal energy, equilibrium states, and deriving a temperature-length inequality with broad implications.

Contribution

It introduces a new entropy formula applied to Friedmann universes, analyzing internal energy, free energy, and deriving a novel temperature-length inequality independent of G.

Findings

Zero internal energy only in flat Friedmann universes (excluding static Einstein solutions).

De Sitter universe is the only equilibrium state with stationary free energy.

Derived a temperature-length inequality analogous to quantum uncertainty, independent of G.

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 $U$, the contribution due to the non-vanishing Hubble flow must been added to the usual Misner-Sharp energy giving, for zero curvature spacetimes, a zero value for $U$. This fact does not hold when curvature is present. After analyzing the free energy $F$, we obtain the correct result that $F$ 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 on the gravitational constant $G$ and can thus be explored also in non gravitational contexts.