Geodesic motion of uncharged dust
= Geodesic motion of uncharged dust
For $T^{\mu\nu}=\rho u^\mu u^\nu$ and constant normalized $u_\mu u^\mu=c^2$, contraction of $\nabla_\mu T^{\mu\nu}=0$ with $u_\nu$ gives $\nabla_\mu(\rho u^\mu)=0$. At nonzero <density> the remaining projection gives $u^\mu\nabla_\mu u^\nu=0$.