= BTZ black hole
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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>.<The Black Hole in Three Dimensional Spacetime by Máximo Bañados and others>[Equations (1)–(3) and final geometric discussion]{ref}
For $r_+>0$ and $0\leq r_-\leq r_+$, write
$$
ds^2=-N^2dt^2+N^{-2}dr^2+r^2(d\phi-\Omega\,dt)^2,
\qquad N^2=\frac{(r^2-r_+^2)(r^2-r_-^2)}{L^2r^2},\qquad
\Omega=\frac{r_+r_-}{Lr^2},\qquad \phi\sim\phi+2\pi.
$$
This convention chooses nonnegative <angular momentum>. Restoring the three-dimensional gravitational constant $G_3$, its <mass> and <angular momentum> are
$$
M=\frac{r_+^2+r_-^2}{8G_3L^2},\qquad J=\frac{r_+r_-}{4G_3L}.
$$
The outer <Killing horizon> at $r_+$ has <angular velocity> $r_-/(Lr_+)$ and <surface gravity>
$$
\kappa=\frac12(N^2)'(r_+)=\frac{r_+^2-r_-^2}{L^2r_+}.
$$
Thus $r_-=0$ gives the nonrotating case, while $r_-=r_+>0$ gives a <degenerate Killing horizon>. The latter has zero limiting <Hawking temperature> and nonzero horizon circumference.
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