# 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 gravity with a negative cosmological constant.

## History

In 1992 Bañados, Teitelboim and Zanelli discovered BTZ black hole(Bañados, Teitelboim & Zanelli 1992). At that time, it came as a surprise because it is believed 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, 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:

• It has "no hairs" (No hair theorem) and is fully characterized by ADM-mass, angular momentum and charge.
• It has the same thermodynamical properties as the ordinary black holes, e.g. its entropy is captured by a law directly analogous to the Bekenstein bound in (3+1)-dimensions, essentially with the surface area replaced by the BTZ black hole's circumference.
• Like the Kerr black hole, a rotating BTZ black hole contains an inner and an outer horizon (see also ergosphere).

Since (2+1)-dimensional gravity has no Newtonian limit, one might fear 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

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

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

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

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.