 Solution

Smoothness requires the polar angle $|V^{\prime }(r_0)|\tau /2$ to have period exactly $2\pi$. Therefore Euclidean time must be identified byA nontrivial integer multiple of this fundamental period still gives an excess-angle cone at the tip; it is not a smooth plane there. The static time has already been normalized at infinity because V tends to one.

For distinct charged roots, writing $d=\sqrt{M^2-Q^2}$ gives ${\beta }_{\pm }=\pi r_{\pm }^3/d$. These unequal periods cannot simultaneously remove both simple-root cones using one common time identification; the outer and inner static Euclidean regions are in any case separated by a band where V is negative. The physically relevant asymptotically flat exterior uses ${\beta }_{+}$. The neutral outer horizon gives ${\beta }_{+}=2\pi \sqrt{2M}$.