Equilibrium to Einstein: Entanglement, Thermodynamics, and Gravity

Here we develop the connection between thermodynamics, entanglement, and gravity. By attributing thermodynamics to timeslices of a causal diamond, we show that the Clausius relation $T\Delta S_{\text{rev}}=Q$, where $\Delta S_{\text{rev}}$ is the reversible entropy change, gives rise to the non-linear gravitational equations of motion for a wide class of diffeomorphism invariant theories. We then compare the Clausius relation to the first law of causal diamond mechanics (FLCD), a geometric identity and necessary ingredient in deriving Jacobson's entanglement equilibrium proposal -- the entanglement entropy of a spherical region with a fixed volume is maximal in vacuum. Specifically we show that the condition of fixed volume can be understood as subtracting the irreversible contribution to the thermodynamic entropy. This provides a "reversible thermodynamic process" interpretation of the FLCD, and that the condition of entanglement equilibrium may be regarded as equilibrium thermodynamics for which the Clausius relation holds. Finally, we extend the entanglement equilibrium proposal to the timelike stretched horizons of future lightcones, providing an entanglement interpretation of stretched lightcone thermodynamics.