 Solution

Since $dt=dv-dr/f$, the inverse-$F$ radial terms cancel. The ingoing Eddington-Finkelstein coordinates giveAt $r=2m$, $H=1+{\sinh }^2\alpha ={\cosh }^2\alpha >0$. Thus all coefficients are analytic there, and the determinant of the $(v,r)$ metric block is $-1$, not zero. The angular area is also nonzero. This is an analytic extension of a spacetime through the apparent coordinate singularity, into the regular region $0<r<2m$.