Thermodynamic equilibrium in relativity: Four-temperature, Killing vectors and Lie derivatives

The main concepts of general relativistic thermodynamics and general relativistic statistical mechanics are reviewed. The main building block of the proper relativistic extension of the classical thermodynamics laws is the four-temperature vector $\beta$, which plays a major role in the quantum framework and defines a very convenient hydrodynamic frame. The general relativistic thermodynamic equilibrium condition demands $\beta$ to be a Killing vector field. We show that a remarkable consequence is that all Lie derivatives of all physical observables along the four-temperature flow must then vanish.