Kruskal extension across a simple static horizon ($r_{\ast }=(2\kappa {)}^{-1}\log |r-r_H|+h(r)$)

For $d{s}^2=-fd{t}^2+d{r}^2/f+R^{2}d{\Omega }^2$, suppose $f$ and $R$ are analytic at $r_H$, $f(r_H)=0$, $f^{\prime }(r_H)=2\kappa >0$ and $R(r_H)>0$. The tortoise coordinate obeys the displayed expansion with analytic $h$. For $U=-e^{-\kappa (t-r_{\ast })}$ and $V=e^{\kappa (t+r_{\ast })}$ in the right exterior, $-UV=(r-r_H)e^{2\kappa h(r)}$ after fixing an additive constant. The inverse function theorem makes $r$ analytic in $UV$ through zero, while $d{s}^2=fdUdV/({\kappa }^{2}UV)+R^{2}d{\Omega }^2$ has a finite nonzero radial coefficient there. This supplies the local analytic extension and its bifurcate Killing horizon.