 Solution

Since $v=t+r_{\star }$, $dt=dv-f^{-1}dr$. Substitution into the radial metric cancels the $d{r}^2/f$ terms. The ingoing Eddington-Finkelstein coordinates giveThe radial block has determinant $-1$, even when $f=0$. Its coefficients are analytic for every $r>0$, so this form extends through $r=r_{+}$ and also through $r=r_{-}$ to $0<r<r_{+}$. Ordinary angular coordinate singularities are removed by angular charts on the three-sphere. The genuine endpoint $r=0$ is not included in this extension.

The field $n=-{\partial }_r$ is an ingoing null tangent. It is future-directed in the original exterior, since its inner product with the future stationary Killing vector field is $-1$, and it remains future-directed by continuation. This fixes the time orientation used in the next part.