Variational action for relativistic potential flow

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