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


== History ==
In 1992 Bañados, Teitelboim and Zanelli discovered BTZ black hole{{harv|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 exist{{Citation needed|date=October 2011}}. By introducing [[dilaton]]s, we can have black holes.{{verify source|date=April 2013}} 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.
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== 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 formalism|ADM-mass]], angular momentum and charge.
*It has the same [[Black hole thermodynamics|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. {{harv|Carlip|1995}} section 3 Black Holes and Gravitational Collapse.
<!--not in any spacetime known to exist-->
 
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
:<math>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                                </math>
where <math>r_+,~r_- </math> are the black hole radii and <math>l </math> is the radius of AdS<sub>3</sub> space. The mass and angular momentum of the black hole is
: <math> M = \frac{r_+^2 + r_-^2}{l^2},~~~~~J = \frac{2r_+ r_-}{l} </math>
 
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 AdS<sub>3</sub>.
 
A rotating BTZ black hole admits [[closed timelike curve]]s.
 
==See also==
*[[Axial symmetry|axi-symmetric spacetime]]
 
== References ==
;Notes
<references/>
 
;Bibliography
*{{Citation | last1=Bañados | first1=Máximo | last2=Teitelboim | first2=Claudio |  last3=Zanelli | first3=Jorge | title=The Black hole in three-dimensional space-time |series=Phys.Rev.Lett. |volume=69 | publisher=[[American Physical Society]] | year=1992 | pages= 1849–1851　}}url=http://arxiv.org/pdf/hep-th/9204099v3.pdf
*{{Citation | last=Carlip | first=Steven | title=Conformal Field Theory, (2+1)-Dimensional Gravity, and the BTZ Black Hole |series=arxiv | year=2005 }}url=http://arxiv.org/pdf/gr-qc/0503022v4.pdf
*{{Citation | last=Carlip | first=Steven | title=The (2+1)-Dimensional Black Hole |series=arxiv | year=1995 }}url=http://arxiv.org/pdf/gr-qc/9506079.pdf
*{{Citation | last=Bañados | first=Máximo | title=Three-dimensional quantum geometry and black holes |series=arxiv | year=1999 }}url=http://arxiv.org/abs/hep-th/9901148v3.pdf
*{{Citation | last=Daisuke | first=Ida | title=No Black Hole Theorem in Three-Dimensional Gravity |series=Phys. Rev. Lett. 85 3758 | year=2000 }} url=http://arxiv.org/abs/gr-qc/0005129
 
{{Time travel}}
 
{{DEFAULTSORT:Btz Black Hole}}
[[Category:Black holes]]
[[Category:Quantum gravity]]
[[Category:Mathematical methods in general relativity]]

Latest revision as of 00:58, 12 March 2014


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