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

We give a modified derivation of the Einstein equation of state by considering the Clausius relation $TδS-δN =δQ$ on a null hypersurface with a non-vanishing expansion ($θ\neq 0$), 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.

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