The BTZ black hole, named after Máximo Bañados, Claudio Teitelboim, and Jorge Zanelli, is a

black hole A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...

solution for (2+1)-dimensional topological gravity with a negative

cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...

History

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

solution Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Solutio ...

. 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.https://journals.aps.org/prd/abstract/10.1103/PhysRevD.103.064063 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 thermodynamical properties as traditional black hole solutions such as Schwarzschild or Kerr black holes, e.g. its entropy is captured by a law directly analogous to the

Bekenstein bound In physics, the Bekenstein bound (named after Jacob Bekenstein) is an upper limit on the thermodynamic entropy ''S'', or Shannon entropy ''H'', that can be contained within a given finite region of space which has a finite amount of energy—o ...

in (3+1)-dimensions, essentially with the surface area replaced by the BTZ black hole's circumference. * Like the

Kerr black hole The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of gen ...

, a rotating BTZ black hole contains an inner and an outer horizon, analogous to an

ergosphere file:Ergosphere_and_event_horizon_of_a_rotating_black_hole_(no_animation).gif, 300px, In the ergosphere (shown here in light gray), the component ''gtt'' is negative, i.e., acts like a purely spatial metric component. Consequently, timelike or ligh ...

. Since (2+1)-dimensional gravity has no

Newtonian limit In physics, the Newtonian limit is a mathematical approximation applicable to physical systems exhibiting (1) weak gravitation, (2) objects moving slowly compared to the speed of light, and (3) slowly changing (or completely static) gravitationa ...

, one might fear that the BTZ black hole is not the final state of a

gravitational collapse Gravitational collapse is the contraction of an astronomical object due to the influence of its own gravity, which tends to draw matter inward toward the center of gravity. Gravitational collapse is a fundamental mechanism for structure formatio ...

. 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. section 3 Black Holes and Gravitational Collapse. The BTZ solution is often discussed in the realm on (2+1)-dimensional

quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...

The case without charge

The metric in the absence of charge is :

ds^2 =  -\fracdt^2 + \frac + r^2 \left(d\phi - \frac dt \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 = \frac,~~~~~J = \frac

BTZ black holes without any electric charge are locally isometric to anti-de Sitter space. More precisely, it corresponds to an

orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...

of the universal covering space of AdS₃. A rotating BTZ black hole admits

closed timelike curve In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van ...

References

;Notes ;Bibliography * * * * * {{DEFAULTSORT:Btz Black Hole Black holes Quantum gravity Mathematical methods in general relativity

History

Properties

The case without charge

See also

References