Self-gravitating equilibrium with slow steady flow and the correct form of entropy current

TL;DR

The paper addresses relativistic self-gravitating equilibrium with spherical symmetry and a steady energy flow, perturbing around the hydrostatic limit where the radial velocity $u^r$ is small. It extends the Tolman–Oppenheimer–Volkov framework by including a steady current $j^$ that contributes to an anisotropic pressure and modifies the structure equations; off-diagonal perturbations necessitate a noncovariant treatment. A key contribution is the construction of the entropy current via a non-Noether conserved current approach, which leads to an unconventional form $s^= (s-b j^0)/u^0\,u^ + b j^$, with the remaining function $b$ fixed by current conservation through a perturbative matching condition that ties the entropy current to the thermodynamic entropy density. The leading nontrivial term for $b$ appears at quadratic order in the perturbations, highlighting how steady flow reshapes the entropy balance and suggesting revisions to the conventional entropy-current–entropy-density relation in relativistic fluids.

Abstract

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 $u^μ$ 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 $j^μ$ accounting for the steady flow, which contributes to the entropy current $s^μ$ in such a general covariant form that $s^μ=au^μ+ bj^μ$ 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 $s^μ=(s-bj^0)u^μ/u^0+ bj^μ$, 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.