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 $u^{\mu }{\nabla }_{\mu }{u}^{\nu }=0$.