= Solution
Since $v=t+r_\star$, $dt=dv-f^{-1}dr$. Substitution into the radial metric cancels the $dr^2/f$ terms. The <ingoing Eddington-Finkelstein coordinates> give
$$
\boxed{ds^2=-f(r)dv^2+2\,dv\,dr+r^2d\Omega_3^2.}
$$
The 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.
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