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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.^[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.

Claudio Teitelboim, with Máximo Bañados and Jorge Zanelli, derived the BTZ black hole and investigated its thermodynamics. Their calculation related temperature and entropy to a three-dimensional black hole's horizon, with entropy proportional to horizon circumference.^[ref]

In three spacetime dimensions the horizon cross-section is a curve, so its length replaces the familiar two-dimensional horizon area.

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.

Jorge Zanelli co-discovered a black-hole solution in three-dimensional spacetime with Máximo Bañados and Claudio Teitelboim. The BTZ black hole exists with a negative cosmological constant and has global horizon and thermodynamic properties despite the simpler local geometry of three-dimensional gravity.^[ref]

The example separates global spacetime structure from the local curvature description.

Máximo Bañados co-developed the BTZ black hole with Claudio Teitelboim and Jorge Zanelli in 1992. It showed that three-dimensional gravity with a negative cosmological constant admits black-hole states characterized by conserved quantities and horizons.^[ref]

This supplied a simpler setting for investigating global gravitational and thermodynamic questions without assuming that all black-hole phenomena require four-dimensional spacetime.

Put $r_{+}=\ell \sqrt{M}$ and $f=(r^2-r_{+}^2)/{\ell }^2$. The apparent failure of the nonrotating BTZ black hole coordinates at $r=r_{+}$ is a coordinate singularity. Define the tortoise coordinate and ingoing time byThe Ingoing BTZ coordinates giveAll coefficients are analytic at $r_{+}>0$, and the determinant is $-r^2$, so the metric tensor extends nondegenerately through that radius. The Killing vector field $K={\partial }_v$ agrees with ${\partial }_t$ in the exterior. On $r=r_{+}$, $K^2=-f=0$ and $K_a=(dr{)}_a$. It is therefore both normal and tangent to this null hypersurface, proving that it is a Killing horizon.

One may include both horizon branches and their bifurcation surface explicitly. Set $\kappa =r_{+}/{\ell }^2$, $u=t-r_{\ast }$ and, initially in the exterior, $U=-e^{-\kappa u}$, $V=e^{\kappa v}$. ThenThis analytic extension of a spacetime is regular near $UV=0$, and $K=\kappa (V{\partial }_V-U{\partial }_U)$ extends analytically. The surfaces $U=0$ and $V=0$ form the extended Killing horizon at $r=r_{+}$; $K$ vanishes only at their regular bifurcation circle.