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For a barotropic fluid in hydrostatic equilibrium along a Killing vector, assume $\rho +p>0$. Integrating $dp/(\rho +p)=-d\log N$ gives the displayed first integral, constant on each connected fluid region. For the equation of state $p=w\rho$ with constant $w>0$, it gives $\rho N^{(1+w)/w}=C_1$.

Let $N=\sqrt{K^a{K}_a}>0$ on the fluid region and choose the Killing vector field $K$ to be future-directed. The normalized four-velocity isThe Killing equation implies both $K^b{\nabla }_{b}N=0$ andConsequently its Killing-flow four-acceleration isFor a scalar, the Lie derivative is ${\mathcal{L}}_{K}p=K^b{\nabla }_{b}p$. Thus the assumed stationarity of the pressure gives $U^b{\nabla }_{b}p=0$. These are the ingredients needed for hydrostatic equilibrium along a Killing vector; hypersurface orthogonality of $K$ is not needed.