The Einstein equation of state as the Clausius relation with an entropy production

TL;DR

This paper extends Jacobson's derivation of Einstein's equations by incorporating non-equilibrium thermodynamics on null hypersurfaces with expansion, using a modified Clausius relation and Noether charge interpretation.

Contribution

It introduces a generalized derivation of Einstein's equation considering entropy production and non-zero expansion, enhancing the thermodynamic approach to gravity.

Findings

Derivation of Einstein's equation with non-zero expansion

Interpretation of thermodynamics via Noether charge

Insight into non-equilibrium gravitational thermodynamics

Abstract

We give a modified derivation of the Einstein equation of state by considering the Clausius relation $T\delta S-\delta N =\delta Q$ on a null hypersurface with a non-vanishing expansion ($\theta \neq 0$), i.e. not in the equilibrium. The derivation corresponds to choosing a specific observer to the hypersurface, and such a generalization gives a hint how we can improve the original derivation by Jacobson. We also give an interpretation of the thermodynamic relation based on the Noether charge method.