 Solution

FactorIntroduce $v=t+r_{\ast }$ with $d{r}_{\ast }/dr=f^{-1}$. Since $dt=dv-dr/f$,These are the charged analogue of ingoing Eddington-Finkelstein coordinates. The $(v,r)$ block has determinant $-1$, so the metric and its inverse are nonsingular at $r=r_{+}$. For $M\ge Q$, $r_{+}$ is real and the metric extends analytically through it. The original potential becomes $A=-Qdv/r^2+Qdr/(r^{2}f)$; the radial term is locally exact, and a gauge transformation gives the horizon-regular potential $A=-Qdv/r^2$.