Higher derivative gravity: field equation as the equation of state

TL;DR

This paper extends the thermodynamic derivation of gravitational field equations to general diffeomorphism invariant theories of gravity, introducing a new entropy density that allows deriving equations of motion from horizon thermodynamics.

Contribution

It proposes a novel entropy density form enabling the derivation of field equations for any diffeomorphism invariant gravity theory via thermodynamic principles.

Findings

Derived field equations from thermodynamics for general gravity theories.

Introduced a new entropy density related to Noether charge.

Unified thermodynamic derivation applicable to a broad class of theories.

Abstract

One of the striking features of general relativity is that the Einstein equation is implied by the Clausius relation imposed on a small patch of locally constructed causal horizon. Extension of this thermodynamic derivation of the field equation to more general theories of gravity has been attempted many times in the last two decades. In particular, equations of motion for minimally coupled higher curvature theories of gravity, but without the derivatives of curvature, have previously been derived using a thermodynamic reasoning. In that derivation the horizon slices were endowed with an entropy density whose form resembles that of the Noether charge for diffeomorphisms, and was dubbed the Noetheresque entropy. In this paper, we propose a new entropy density, closely related to the Noetheresque form, such that the field equation of any diffeomorphism invariant metric theory of gravity can be derived by imposing the Clausius relation on a small patch of local causal horizon.