# BTZ black hole

The BTZ black hole, named after Máximo Bañados, Claudio Teitelboim, and Jorge Zanelli, is a black hole solution for (2+1)-dimensional topological gravity with a negative cosmological constant[clarification needed].

## History

In 1992 Bañados, Teitelboim and Zanelli discovered the BTZ black hole solution (Bañados, Teitelboim & Zanelli 1992). At that time[clarification needed], it came as a surprise because it is believed[according to whom?] that no black hole solutions are shown to exist for a negative cosmological constant and BTZ black hole has remarkably similar properties to the 3+1 dimensional black hole, which would exist in our real universe.

When the cosmological constant is zero, a vacuum solution of (2+1)-dimensional gravity is necessarily flat (the Weyl tensor vanishes in three dimensions, while the Ricci tensor vanishes due to the Einstein field equations, so the full Riemann tensor vanishes), and it can be shown that no black hole solutions with event horizons exist[citation needed]. By introducing dilatons, we can have black holes.[verification needed] We do have conical angle deficit solutions, but they don't have event horizons. It therefore came as a surprise when black hole solutions were shown to exist for a negative cosmological constant.

## Properties

The similarities to the ordinary black holes in 3+1 dimensions:

Since (2+1)-dimensional gravity has no Newtonian limit, one might fear[why?] that the BTZ black hole is not the final state of a gravitational collapse. It was however shown, that this black hole could arise from collapsing matter and we can calculate the energy-moment tensor of BTZ as same as (3+1) black holes. (Carlip 1995) section 3 Black Holes and Gravitational Collapse.

The BTZ solution is often discussed in the realm on (2+1)-dimensional quantum gravity.

## The case without charge

The metric in the absence of charge is

${\displaystyle ds^{2}=-{\frac {(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{l^{2}r^{2}}}dt^{2}+{\frac {l^{2}r^{2}dr^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}}+r^{2}\left(d\phi -{\frac {r_{+}r_{-}}{lr^{2}}}dt\right)^{2}}$

where ${\displaystyle r_{+},~r_{-}}$ are the black hole radii and ${\displaystyle l}$ is the radius of AdS3 space. The mass and angular momentum of the black hole is

${\displaystyle M={\frac {r_{+}^{2}+r_{-}^{2}}{l^{2}}},~~~~~J={\frac {2r_{+}r_{-}}{l}}}$

BTZ black holes without any electric charge are locally isometric to anti-de Sitter space. More precisely, it corresponds to an orbifold of the universal covering space of AdS3.

A rotating BTZ black hole admits closed timelike curves.