Conical singularity ($d{s}^2=d{R}^2+a^2{R}^{2}d{\vartheta }^2$)

A conical singularity is a failure of smoothness at the apex of a metric cone. If $\vartheta$ has period $P$, small circles have circumference-to-radius ratio $aP$. The two-dimensional apex is smooth precisely when this equals $2\pi$. Euclidean black-hole regularity condition applies this test to the imaginary-time and radial plane.