Nonequilibrium Thermodynamics of Spacetime

It has previously been shown that the Einstein equation can be derived from the requirement that the Clausius relation $dS=\delta Q/T$ hold for all local acceleration horizons through each spacetime point, where $dS$ is one-quarter the horizon area change in Planck units and $\delta Q$ and $T$ are the energy flux across the horizon and the Unruh temperature seen by an accelerating observer just inside the horizon. Here we show that a curvature correction to the entropy that is polynomial in the Ricci scalar requires a nonequilibrium treatment. The corresponding field equation is derived from the entropy balance relation $dS=\delta Q/T+d_{i}S$, where $d_{i}S$ is a bulk viscosity entropy production term that we determine by imposing energy-momentum conservation. Entropy production can also be included in pure Einstein theory by allowing for shear viscosity of the horizon.