Dynamical horizon entropy and equilibrium thermodynamics of generalized gravity theories

TL;DR

This paper explores the connection between thermodynamics and field equations in generalized gravity theories on dynamical horizons, proposing a universal thermodynamic framework that incorporates entropy, energy, and the second law.

Contribution

It introduces a unified approach to horizon thermodynamics in various gravity theories, including Gauss-Bonnet and $f(R)$ gravity, extending the second law and equilibrium identities.

Findings

Entropy as Noether charge satisfies the second law.

Generalized Gibbs equations are formulated for different gravity theories.

Equilibrium thermodynamics holds in static spacetimes for $f(R)$ gravity.

Abstract

We study the relation between the thermodynamics and field equations of generalized gravity theories on the dynamical trapping horizon with sphere symmetry. We assume the entropy of dynamical horizon as the Noether charge associated with the Kodama vector and point out that it satisfies the second law when a Gibbs equation holds. We generalize two kinds of Gibbs equations to Gauss-Bonnet gravity on any trapping horizon. Based on the quasi-local gravitational energy found recently for $f(R)$ gravity and scalar-tensor gravity in some special cases, we also build up the Gibbs equations, where the nonequilibrium entropy production, which is usually invoked to balance the energy conservation, is just absorbed into the modified Wald entropy in the FRW spacetime with slowly varying horizon. Moreover, the equilibrium thermodynamic identity remains valid for $f(R)$ gravity in a static spacetime. Our work provides an alternative treatment to reinterpret the nonequilibrium correction and supports the idea that the horizon thermodynamics is universal for generalized gravity theories.