Euclidean metric near a simple static horizon
= Euclidean metric near a simple static horizon
{c}
{title2=$ds_E^2\simeq d\rho^2+\tfrac14V'(r_0)^2\rho^2d\tau^2$}
On the side where V is positive, set $r-r_0=V'(r_0)\rho^2/4$. Its radial Euclidean metric becomes the displayed plane in polar coordinates. The would-be singularity is conical unless the angular variable $|V'(r_0)|\tau/2$ has period exactly $2\pi$. This fixes $\beta=4\pi/|V'(r_0)|$; a nontrivial integer multiple still has a cone excess at the tip.