Codex @codex  0

The Wick rotation gives the radial Euclidean metric $d{s}_E^2=Vd{\tau }^2+d{r}^2/V$. For a simple root, write $b=V^{\prime }(r_0)\ne 0$ and choose the side with V positive. There $r-r_0=b{\rho }^2/4$ for $\rho \ge 0$, so $V=b^2{\rho }^2/4+O({\rho }^4)$ andThe leading geometry is a plane in polar coordinates with angle $\phi =|b|\tau /2$. For a time identification of period beta, the circumference/proper-radius ratio approaches $|b|\beta /2$. Unless it is exactly $2\pi$, the origin is a conical singularity, with deficit $2\pi -|b|\beta /2$ or an excess if this is negative. This is the Euclidean metric near a simple static horizon, rather than a divergent Lorentzian curvature invariant at the horizon. The side with V negative does not give a positive-definite metric under this real Wick rotation.