Thermodynamic Behavior of Field Equations for f(R) Gravity

TL;DR

This paper explores the thermodynamic properties of $f(R)$ gravity, showing that its field equations at black hole horizons and cosmological horizons can be expressed in thermodynamic forms, revealing non-equilibrium thermodynamics features.

Contribution

It extends the thermodynamic interpretation of gravitational field equations to $f(R)$ gravity, including additional entropy production terms indicating non-equilibrium thermodynamics.

Findings

Field equations at black hole horizons resemble thermodynamic relations with entropy production.

At FRW horizons, field equations include additional entropy terms compared to Einstein gravity.

The results support the view that $f(R)$ gravity exhibits non-equilibrium thermodynamic behavior.

Abstract

Recently it has shown that Einstein's field equations can be rewritten into a form of the first law of thermodynamics both at event horizon of static spherically symmetric black holes and apparent horizon of Friedmann-Robertson-Walker (FRW) universe, which indicates intrinsic thermodynamic properties of spacetime horizon. In the present paper we deal with the so-called $f(R)$ gravity, whose action is a function of the curvature scalar $R$. In the setup of static spherically symmetric black hole spacetime, we find that at the event horizon, the field equations of $f(R)$ gravity can be written into a form $dE = TdS - PdV + Td\bar{S}$, where $T$ is the Hawking temperature and $S=Af'(R)/4G$ is the horizon entropy of the black hole, $E$ is the horizon energy of the black hole, $P$ is the radial pressure of matter, $V$ is the volume of black hole horizon, and $d\bar S$ can be interpreted as the entropy production term due to nonequilibrium thermodynamics of spacetime. In the setup of FRW universe, the field equations can also be cast to a similar form, $dE=TdS +WdV +Td\bar S$, at the apparent horizon, where $W=(\rho-P)/2$, $E$ is the energy of perfect fluid with energy density $\rho$ and pressure $P$ inside the apparent horizon. Compared to the case of Einstein's general relativity, an additional term $d\bar S$ also appears here. The appearance of the additional term is consistent with the argument recently given by Eling {\it et al.} (gr-qc/0602001) that the horizon thermodynamics is non-equilibrium one for the $f(R)$ gravity.