Thermodynamics of Spacetime: The Einstein Equation of State

The Einstein equation is derived from the proportionality of entropy and the horizon area together with the fundamental relation $\delta Q=T\mathrm{dS}$. The key idea is to demand that this relation hold for all the local Rindler causal horizons through each spacetime point, with $\delta Q$ and $T$ interpreted as the energy flux and Unruh temperature seen by an accelerated observer just inside the horizon. This requires that gravitational lensing by matter energy distorts the causal structure of spacetime so that the Einstein equation holds. Viewed in this way, the Einstein equation is an equation of state.