= Conical singularity of negative-mass BTZ geometry
{title2=$\delta=2\pi(1-\sqrt{-M})$}
For the nonrotating <BTZ black hole> family with $M<0$ and angular period $2\pi$, the circumference of a small circle divided by its proper radius tends to $2\pi\sqrt{-M}$. A smooth axis requires this ratio to be $2\pi$, so only $M=-1$ gives regular <Anti-de Sitter spacetime>. Other values have a <conical singularity>, with deficit angle $\delta$ or an angular excess when $\delta<0$. The lapse $r^2/\ell^2-M$ is strictly positive, so outward <radial null geodesics> escape from arbitrarily near the axis: the defect is a <naked singularity>, despite finite local curvature away from it.
Back to article page
