In a perturbatively analyzed relativistic self-gravitating equilibrium with steady flow, the entropy current takes the form (s - b j^0) u^μ / u^0 + b j^μ with b starting at quadratic order and fixed by current conservation.
Hartle,Gravity: An Introduction to Einstein ’s General Relativity
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Self-gravitating equilibrium with slow steady flow and its consistent form of entropy current
In a perturbatively analyzed relativistic self-gravitating equilibrium with steady flow, the entropy current takes the form (s - b j^0) u^μ / u^0 + b j^μ with b starting at quadratic order and fixed by current conservation.