Equilibrium to Einstein: Entanglement, Thermodynamics, and Gravity

TL;DR

This paper establishes a deep connection between thermodynamics, entanglement, and gravity by deriving gravitational equations from thermodynamic principles applied to causal diamonds and extending entanglement equilibrium to lightcone horizons.

Contribution

It introduces a novel thermodynamic interpretation of gravitational equations and extends entanglement equilibrium to timelike stretched horizons, unifying concepts in gravity and quantum information.

Findings

Derives gravitational equations from the Clausius relation applied to causal diamonds.

Reinterprets the first law of causal diamond mechanics as a reversible thermodynamic process.

Extends entanglement equilibrium to stretched lightcone horizons.

Abstract

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.