= Kruskal extension across a simple static horizon
{c}
{title2=$r_*=(2\kappa)^{-1}\log|r-r_H|+h(r)$}
For $ds^2=-fdt^2+dr^2/f+R^2d\Omega^2$, suppose $f$ and $R$ are analytic at $r_H$, $f(r_H)=0$, $f'(r_H)=2\kappa>0$ and $R(r_H)>0$. The <tortoise coordinate> obeys the displayed expansion with analytic $h$. For $U=-e^{-\kappa(t-r_*)}$ and $V=e^{\kappa(t+r_*)}$ 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 $ds^2=f\,dU\,dV/(\kappa^2UV)+R^2d\Omega^2$ has a finite nonzero radial coefficient there. This supplies the local analytic extension and its <bifurcate Killing horizon>.
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