Horizon Thermodynamics on Nice Slices of the Causal Diamond

There is a deep link between gravity and thermodynamics; in a precise way gravity can be derived from entanglement entropy in conformal field theories. However, this depends crucially on properties of horizons, and asymptotic symmetries of phase space. To explore how this relation behaves under dynamical processes, we consider covariant gravitational phase space enhanced with bulk conformal symmetry. As is well known, the Noether–Wald entropy has an explicit form in terms of the Abbott–Deser–Tekin conserved surface charges of gauge theories. We find a new vector contribution to the Abbott–Deser–Tekin charges that arises for conformal symmetries. In applying this to the causal diamond, we derive a general relation for surface gravity, based on the conformal invariance of horizons, that allows us to find slices where the zeroth law holds, as well as the degree to which a first law arises on the phase space.