 Solution

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.