Self-gravitating equilibrium with slow steady flow and its consistent form of entropy current

A relativistic self-gravitating equilibrium system with spherical symmetry as well as with steady energy flow is investigated perturbatively around the hydrostatic limit, where the radial component of the fluid velocity field $u^\mu$ is sufficiently small. Each component of vectors and tensors consisting of the system is expanded in different powers, which makes the covariant perturbation approach ineffective. The differential equations to determine the subleading correction of the structure variables are presented. The system retains the current $j^\mu$ accounting for the steady flow, which contributes to the entropy current $s^\mu$ in such a general covariant form that $s^\mu=au^\mu+ bj^\mu$ with $a, b$ unknown parametric functions. To determine them, a new condition is proposed. This condition imposes the entropy current to be of an unconventional form $s^\mu=(s-bj^0)u^\mu/u^0+ bj^\mu$, where $s$ is the entropy density. The remaining parameter $b$ is fixed by the current conservation equation. The perturbative analysis shows that $b$ starts with the quadratic order and its leading term is determined explicitly.