Solution
= Solution
The inverse radial block is $\begin{pmatrix}0&-1\\-1&f\end{pmatrix}$. 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 $u$.