Gravitational equilibrium with steady flow and relativistic local thermodynamics

A relativistic self-gravitating equilibrium system with steady flow as well as spherical symmetry is discovered. The energy-momentum tensor contains the contribution of a current related to the flow and the metric tensor does an off-diagonal component to balance with the flow momentum. The presence of the off-diagonal component of the metric implies the radial motion of the reference frame, which gives rise to a problem how the relativistic effect is included in thermodynamic observables for such a general relativistic system. This problem is solved by taking an instantaneously rest frame in which geometric thermodynamic observables read as previously and giving them the special relativistic effect emerged from the inverse transformation to the original frame pointwise. The solution of the thermodynamic observables in accord with the laws of thermodynamics and the theory of relativity is presented. Finally the relativistic structure equations for the equilibrium are derived, from which the general relativistic Poisson equation as well as the heat conduction one are developed exactly.