= Variational action for relativistic potential flow
In signature $(+---)$, the matter action $S=\int\sqrt{-g}[p(\epsilon)+n((\nabla\chi)^2-\epsilon^2)/(2\epsilon)]$ uses independent $n,\epsilon,\chi$. Variations of $n,\epsilon,\chi$ give $(\nabla\chi)^2=\epsilon^2$, $p'(\epsilon)=n$, and <particle number conservation in a relativistic fluid> for $u_a=\nabla_a\chi/\epsilon$. Its <Hilbert stress-energy tensor> is $T_{ab}=(\rho+p)u_au_b-pg_{ab}$. The action describes isentropic potential flow, with vanishing <canonical vorticity of a relativistic fluid>, rather than every vortical perfect-fluid solution.
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