The BTZ black hole, named after black hole solution for (2+1)dimensional gravity with a negative cosmological constant.
Contents

History 1

Properties 2

The case without charge 3

See also 4

References 5
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. By introducing dilatons, we can have black holes. 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 ADMmass, 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 energymoment 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 AdS_{3} 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 antide Sitter space. More precisely, it corresponds to an orbifold of the universal covering space of AdS_{3}.
A rotating BTZ black hole admits closed timelike curves.
See also
References

Notes

Bibliography

Bañados, Máximo; Teitelboim, Claudio; Zanelli, Jorge (1992), The Black hole in threedimensional spacetime, Phys.Rev.Lett. 69, url=http://arxiv.org/pdf/hepth/9204099v3.pdf

Carlip, Steven (2005), Conformal Field Theory, (2+1)Dimensional Gravity, and the BTZ Black Hole, arxiv url=http://arxiv.org/pdf/grqc/0503022v4.pdf

Carlip, Steven (1995), The (2+1)Dimensional Black Hole, arxiv url=http://arxiv.org/pdf/grqc/9506079.pdf

Bañados, Máximo (1999), Threedimensional quantum geometry and black holes, arxiv url=http://arxiv.org/abs/hepth/9901148v3.pdf

Daisuke, Ida (2000), No Black Hole Theorem in ThreeDimensional Gravity, Phys. Rev. Lett. 85 3758 url=http://arxiv.org/abs/grqc/0005129
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