BTZ black hole

The BTZ black hole is a three-dimensional vacuum spacetime with negative cosmological constant $\Lambda =-1/L^2$. Locally it has the constant negative sectional curvature of Anti-de Sitter spacetime, but discrete identifications give it a black-hole causal structure. It is a useful example of horizon geometry and thermodynamics without the local gravitational-wave degrees of freedom of four-dimensional general relativity.^[ref]

For $r_{+}>0$ and $0\le r_{-}\le r_{+}$, writeThis convention chooses nonnegative angular momentum. Restoring the three-dimensional gravitational constant $G_3$, its mass and angular momentum areThe outer Killing horizon at $r_{+}$ has angular velocity $r_{-}/(L{r}_{+})$ and surface gravityThus $r_{-}=0$ gives the nonrotating case, while $r_{-}=r_{+}>0$ gives a degenerate Killing horizon. The latter has zero limiting Hawking temperature and nonzero horizon circumference.

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.