BTZ black hole
Encyclopedia
The BTZ black hole, named after Maximo Banados, Claudio Teitelboim, and Jorge Zanelli, is a black hole
solution for (2+1)-dimensional gravity with a negative cosmological constant
.
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. 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.
The BTZ black hole is remarkably similar to the (3+1)-dimensional black hole. Like the Kerr black hole, a rotating BTZ black hole contains an inner and an outer horizon.
It has "no hairs" (No hair theorem
) and is fully characterized by ADM-mass, angular momentum and charge. It also possesses thermodynamical properties analogous to the (3+1)-dimensional black hole. 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 holes circumference.
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 does arise from collapsing matter.
The BTZ solution is often discussed in the realm on (2+1)-dimensional quantum gravity
.
The metric is
where R is the black hole radius, in the absence of charge and angular momentum.
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 AdS3.
A rotating BTZ black hole admits closed timelike curve
s.
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
solution for (2+1)-dimensional gravity with a negative cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary 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. 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.
The BTZ black hole is remarkably similar to the (3+1)-dimensional black hole. Like the Kerr black hole, a rotating BTZ black hole contains an inner and an outer horizon.
It has "no hairs" (No hair theorem
No hair theorem
The no-hair theorem postulates that all black hole solutions of the Einstein-Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum...
) and is fully characterized by ADM-mass, angular momentum and charge. It also possesses thermodynamical properties analogous to the (3+1)-dimensional black hole. E.g. its entropy is captured by a law directly analogous to the Bekenstein bound
Bekenstein bound
In physics, the Bekenstein bound is an upper limit on the entropy S, or information I, that can be contained within a given finite region of space which has a finite amount of energy—or conversely, the maximum amount of information required to perfectly describe a given physical system down to the...
in (3+1)-dimensions, essentially with the surface area replaced by the BTZ black holes circumference.
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 does arise from collapsing matter.
The BTZ solution is often discussed in the realm on (2+1)-dimensional quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
.
The metric is
where R is the black hole radius, in the absence of charge and angular momentum.
BTZ black holes without any electric charge are locally isometric to anti de Sitter space
Anti de Sitter space
In mathematics and physics, n-dimensional anti de Sitter space, sometimes written AdS_n, is a maximally symmetric Lorentzian manifold with constant negative scalar curvature...
. More precisely, it corresponds to an orbifold
Orbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
of the universal covering space of AdS3.
A rotating BTZ black hole admits closed timelike curve
Closed timelike curve
In mathematical physics, a closed timelike curve is a worldline in a Lorentzian manifold, of a material particle in spacetime that is "closed," returning to its starting point...
s.