Thermodynamic Behavior of Field Equations for f(R) Gravity

M. Akbar; Rong-Gen Cai

doi:10.1016/j.physletb.2007.03.005

paper

Thermodynamic Behavior of Field Equations for f(R) Gravity

arXiv:gr-qc/0612089 · doi:10.1016/j.physletb.2007.03.005

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 $d E = T d S - P d V + T d S ˉ$ , where $T$ is the Hawking temperature and $S = A f^{'} (R) /4 G$ 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 $d S ˉ$ 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, $d E = T d S + W d V + T d S ˉ$ , at the apparent horizon, where $W = (ρ - P) /2$ , $E$ is the energy of perfect fluid with energy density $ρ$ and pressure $P$ inside the apparent horizon. Compared to the case of Einstein's general relativity, an additional term $d S ˉ$ 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.

Latex 16 pages, v2: one reference added

Thermodynamic Behavior of Field Equations for f(R) Gravity

Abstract

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