BTZ black hole

The BTZ black hole is a quotient of three-dimensional Anti-de Sitter spacetime. In lapse-and-shift coordinates its outer and inner horizon radii are $r_{+}$ and $r_{-}$, and its outer-horizon surface gravity is

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.

Forthe transformations $dv=dt+dr/f$ and $d\chi =d\varphi +\Omega dr/f$ givewhich is regular at every horizon where $f$ vanishes while $\Omega$ remains finite.

For the nonrotating BTZ metric in ingoing coordinates, future null normals to a circle of fixed $v,r$ may be chosen as $\ell ={\partial }_v+(f/2){\partial }_r$ and $n=-{\partial }_r$. Their expansions are ${\theta }_{(\ell )}=f/(2r)$ and ${\theta }_{(n)}=-1/r$, so both are negative inside the outer horizon.

The BTZ black hole can be null-geodesically incomplete at $r=0$ even though every local curvature invariant is finite there. The endpoint comes from the global anti-de Sitter quotient rather than curvature blowup.