BTZ black hole

The BTZ black hole, named after

In 1992, Bañados, Teitelboim, and Zanelli discovered the BTZ black hole solution (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

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
  which?] 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?] 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

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

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

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

BTZ black holes without any electric charge are locally isometric to

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

See also

• Cosmic string
• MTZ black hole
• AdS black hole

References

Notes