Local thermodynamics and entropy for relativistic hydrostatic equilibrium

By refining the method proposed in Aoki et al. (2021) [5], entropy current and entropy density for a relativistic hydrostatic equilibrium system with spherical symmetry are constructed as a non-Nöther conserved charge in the Einstein gravity with cosmological constant. It is shown that the constructed entropy density satisfies both the local Euler relation and the first law of thermodynamics non-perturbatively with respect to the Newton constant. Finally the established relativistic thermodynamics is applied to one of the systems with uniform energy density as a crude model of a degenerate star and its local thermodynamic observables are determined analytically.