Movatterモバイル変換

BTZ black hole

From Wikipedia, the free encyclopedia

Black hole solution in 2+1 dimensional topological gravity

TheBTZ black hole, named afterMáximo Bañados,Claudio Teitelboim, andJorge Zanelli, is ablack hole solution for(2+1)-dimensional topological gravity with a negativecosmological constant^{[clarification needed]}.

History

[edit]

In 1992, Bañados, Teitelboim, and Zanelli discovered the BTZ black holesolution (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.^[1] 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.

Properties

[edit]

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

It admits ano hair theorem, fully characterizing the solution by itsADM-mass, angular momentum and charge.
It has the samethermodynamical properties as traditional black hole solutions such as Schwarzschild or Kerr black holes, e.g. its entropy is captured by a law^[which?] directly analogous to theBekenstein bound in (3+1)-dimensions, essentially with the surface area replaced by the BTZ black hole's circumference.
Like theKerr black hole, a rotating BTZ black hole contains an inner and an outer horizon, analogous to anergosphere.

Since (2+1)-dimensional gravity has noNewtonian limit, one might fear^[why?] that the BTZ black hole is not the final state of agravitational 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)-dimensionalquantum gravity.

The case without charge

[edit]

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_{-}}{lr^{2}}}dt\right)^{2}

where $r_{+},~r_{-}$ are the black hole radii and $l {\displaystyle l}$ 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 toanti-de Sitter space. More precisely, it corresponds to anorbifold of theuniversal covering space of AdS_3.^[2]

A rotating BTZ black hole admitsclosed timelike curves.^{[citation needed]}