:BTZ black hole

In 1992, Bañados, Teitelboim, and Zanelli discovered the BTZ black hole solution {{harv|Bañados|Teitelboim|Zanelli|1992}}. This came as a surprise, because 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.{{cite journal | url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.103.064063 | doi=10.1103/PhysRevD.103.064063 | title=Black holes of ( 2+1 )-dimensional f(R) gravity coupled to a scalar field | year=2021 | last1=Karakasis | first1=Thanasis | last2=Papantonopoulos | first2=Eleftherios | last3=Tang | first3=Zi-Yu | last4=Wang | first4=Bin | journal=Physical Review D | volume=103 | issue=6 | page=064063 | arxiv=2101.06410 | bibcode=2021PhRvD.103f4063K | s2cid=231632352 }} But thanks to the negative cosmological constant in the BTZ black hole, it is able to have remarkably similar properties to the 3+1 dimensional Schwarzschild and Kerr black hole solutions, which model real-world black holes.

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

• It admits a no hair theorem, fully characterizing the solution by its ADM-mass, angular momentum and charge.
• It has the same thermodynamical properties as traditional black hole solutions such as Schwarzschild or Kerr black holes, e.g. its entropy is captured by a law{{which|date=March 2017}} 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, analogous to an ergosphere.

Since (2+1)-dimensional gravity has no Newtonian limit, one might fear{{why|date=March 2017}} 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|loc=section 3 Black Holes and Gravitational Collapse}}

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

The metric in the absence of charge is

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

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

: $M\; =\; \backslash frac\{r\_+^2\; +\; r\_-^2\}\{l^2\},~~~~~J\; =\; \backslash frac\{2r\_+\; r\_-\}\{l\}$

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 AdS_3.{{Cite book |last=Kraus |first=Per |chapter=Lectures on Black Holes and the AdS3/CFT2 Correspondence |date=20 September 2006 |title=Supersymmetric Mechanics - Vol. 3 |chapter-url=https://link.springer.com/chapter/10.1007/978-3-540-79523-0_4 |journal=Springer Publications |series=Lecture Notes in Physics |volume=3|pages=1–55 |doi=10.1007/978-3-540-79523-0_4 |isbn=978-3-540-79522-3 }}

A rotating BTZ black hole admits closed timelike curves.{{Citation needed|date=September 2020}}

• Cosmic string
• MTZ black hole
• AdS black hole

;Notes

;Bibliography

• {{Citation | last1=Bañados | first1=Máximo | last2=Teitelboim | first2=Claudio | last3=Zanelli | first3=Jorge | title=The Black hole in three-dimensional space-time | journal=Phys. Rev. Lett. |volume=69 | issue=13 | date=28 September 1992 | pages=1849–51 | doi=10.1103/PhysRevLett.69.1849 | pmid=10046331 | arxiv=hep-th/9204099v3 | bibcode=1992PhRvL..69.1849B| s2cid=18095488 }}
• {{Citation | last=Carlip | first=Steven | title=Conformal Field Theory, (2+1)-Dimensional Gravity, and the BTZ Black Hole | journal=Classical and Quantum Gravity | year=2005 | volume=22 | issue=12 | pages=R85–R123 | arxiv=gr-qc/0503022v4 |bibcode = 2005CQGra..22R..85C |doi = 10.1088/0264-9381/22/12/R01 | s2cid=115762178 }}
• {{Citation | last=Carlip | first=Steven | title=The (2+1)-Dimensional Black Hole | journal=Classical and Quantum Gravity | year=1995 | volume=12 | issue=12 | pages=2853–2879 | arxiv=gr-qc/9506079 |bibcode = 1995CQGra..12.2853C |doi = 10.1088/0264-9381/12/12/005 | s2cid=119508585 }}
• {{Citation | last=Bañados | first=Máximo | title=Three-dimensional quantum geometry and black holes | journal=Trends in Theoretical Physics II | series=AIP Conference Proceedings | year=1999 | volume=484 | pages=147–169 | doi=10.1063/1.59661 | arxiv=hep-th/9901148v3 | bibcode=1999AIPC..484..147B | s2cid=7598959 | url=https://cds.cern.ch/record/377704/files/9901148.pdf }}
• {{Citation | first=Daisuke | last=Ida | title=No Black Hole Theorem in Three-Dimensional Gravity | journal=Phys. Rev. Lett. | volume=85 | issue=18 | pages=3758–60 | date=30 October 2000 | doi=10.1103/PhysRevLett.85.3758 | arxiv=gr-qc/0005129 | pmid=11041920|bibcode = 2000PhRvL..85.3758I | s2cid=38770795 }}

{{DEFAULTSORT:Btz Black Hole}}

Category:Black holes

Category:Quantum gravity

Category:Mathematical methods in general relativity