A General Maximum Entropy Principle for Self-Gravitating Perfect Fluid

We consider a self-gravitating system consisting of perfect fluid in a spherically symmetric spacetime. Using the Gibbs-Duhem relation, we extremize the total entropy S under the constraints that the total number and energy of particles are fixed. We show that the extrema of S coincides precisely with the relativistic Tolman-Oppenheimer-Volkoff (TOV) equation of hydrostatic equilibrium. Furthermore, we show that the maximum entropy principle is also valid for a charged perfect fluid. Our work provides a strong evidence for the fundamental relationship between general relativity and ordinary thermodynamics.