Hydro & Thermo Dynamics at Causal Boundaries, Examples in 3d Gravity

We study 3-dimensional gravity on a spacetime bounded by a generic 2-dimensional causal surface. We review the solution phase space specified by 4 generic functions over the causal boundary, construct the symplectic form over the solution space and the 4 boundary charges and their algebra. The boundary charges label boundary degrees of freedom. Three of these charges extend and generalize the Brown-York charges to the generic causal boundary, are canonical conjugates of boundary metric components and naturally give rise to a fluid description at the causal boundary. Moreover, we show that the boundary charges besides the causal boundary hydrodynamic description, also admit a thermodynamic description with a natural (geometric) causal boundary temperature and angular velocity. When the causal boundary is the asymptotic boundary of the 3d AdS or flat space, the hydrodynamic description respectively recovers an extension of the known conformal or conformal-Carrollian asymptotic hydrodynamics. When the causal boundary is a generic null surface, we recover the null surface thermodynamics of [1] which is an extension of the usual black hole thermodynamics description.