 Solution

The inverse radial block is $\left(\begin{array}{cc}0& -1\\ -1& f\end{array}\right)$. Since $u_{,a}$ has only its $u$ component nonzero, $g^{ab}{u}_{,a}{u}_{,b}=g^{uu}=0$. Therefore $\boxed{\nabla u=-{\partial }_r\text{ is null}}$; its integral curves lie on the outgoing null hypersurfaces of constant $u$.