The Einstein equations for generalized theories of gravity and the thermodynamic relation $\delta Q = T \delta S$ are equivalent

TL;DR

This paper demonstrates that the equations of motion in generalized gravity theories are equivalent to a thermodynamic relation, extending Jacobson's idea that gravity can be viewed as thermodynamics, with entropy obeying the second law under certain conditions.

Contribution

It generalizes the thermodynamic derivation of Einstein's equations to broader gravity theories using a new definition of Noether charge entropy.

Findings

Equations of motion are equivalent to the thermodynamic relation $ abla Q = T abla S$.

Noether charge entropy satisfies the second law if the null energy condition holds.

Supports the view that gravity emerges from vacuum thermodynamics.

Abstract

We show that the equations of motion of generalized theories of gravity are equivalent to the thermodynamic relation $\delta Q = T \delta S$. Our proof relies on extending previous arguments by using a more general definition of the Noether charge entropy. We have thus completed the implementation of Jacobson's proposal to express Einstein's equations as a thermodynamic equation of state. Additionally, we find that the Noether charge entropy obeys the second law of thermodynamics if the matter energy momentum tensor obeys the null energy condition. Our results support the idea that gravitation on a macroscopic scale is a manifestation of the thermodynamics of the vacuum.