= Solution
{c}
In the prescribed <synchronous coordinates>, the <Christoffel symbols> satisfy
$$
\Gamma^a{}_{tt}=\frac12g^{ab}(2\partial_tg_{bt}-\partial_b g_{tt})=0.
$$
Consequently 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> becomes
$$
0=(\rho+p)a_i+(\delta_i{}^a+u_i u^a)\partial_a p=\partial_i p.
$$
On a connected coordinate neighbourhood,
$$
\boxed{p=p(t).}
$$
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.
Back to article page
