Unified First Law and Thermodynamics of Apparent Horizon in FRW Universe

TL;DR

This work examines how Friedmann equations for FRW cosmologies align with horizon thermodynamics by extending the unified first law to the apparent horizon. It shows that in Einstein gravity and in Lovelock gravity, the Clausius relation δQ = TdS holds on the apparent horizon when heat is defined from matter flux and the entropy is the appropriate horizon entropy (Bekenstein-Hawking for Einstein, Lovelock entropy for higher-curvature theories). In scalar-tensor gravity, the Clausius relation fails to hold in equilibrium form, necessitating an entropy production term and signaling non-equilibrium thermodynamics of spacetime. Overall, the paper provides a unified framework linking cosmological dynamics to horizon thermodynamics and clarifies when equilibrium thermodynamics suffices versus when non-equilibrium treatments are required.

Abstract

In this paper we revisit the relation between the Friedmann equations and the first law of thermodynamics. We find that the unified first law firstly proposed by Hayward to treat the "outer"trapping horizon of dynamical black hole can be used to the apparent horizon (a kind of "inner" trapping horizon in the context of the FRW cosmology) of the FRW universe. We discuss three kinds of gravity theorties: Einstein theory, Lovelock thoery and scalar-tensor theory. In Einstein theory, the first law of thermodynamics is always satisfied on the apparent horizon. In Lovelock theory, treating the higher derivative terms as an effective energy-momentum tensor, we find that this method can give the same entropy formula for the apparent horizon as that of black hole horizon. This implies that the Clausius relation holds for the Lovelock theory. In scalar-tensor gravity, we find, by using the same procedure, the Clausius relation no longer holds. This indicates that the apparent horizon of FRW universe in the scalar-tensor gravity corresponds to a system of non-equilibrium thermodynamics. We show this point by using the method developed recently by Eling {\it et al.} for dealing with the $f(R)$ gravity.