Past exam of the general relativity II course of the University of Oxford / 2021 / 1 / d / ii

In the prescribed synchronous coordinates, the Christoffel symbols satisfyConsequently the four-acceleration of $u={\partial }_t$ vanishes. For a spatial index $i$, $u_i=0$, so the spatial component of the relativistic Euler equation becomesOn a connected coordinate neighbourhood,A nonzero pressure is allowed, but its spatial gradient must vanish if the fluid's four-velocity follows these freely falling coordinate lines. No division by $p$ or $\rho +p$ is needed.