AdS black hole

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In theoretical physics, an anti-de Sitter (AdS) black hole is a black hole solution of general relativity or its extensions which represents an isolated massive object, but with a negative cosmological constant. Such a solution asymptotically approaches anti-de Sitter space at spatial infinity, and is a generalization of the Kerr vacuum solution, which asymptotically approaches Minkowski spacetime at spatial infinity.

In 3+1 dimensions, the metric is given by

d s 2 = − ( k 2 r 2 + 1 − C r ) d t 2 + 1 k 2 r 2 + 1 − C r d r 2 + r 2 d Ω 2 {\displaystyle ds^{2}=-\left(k^{2}r^{2}+1-{\frac {C}{r}}\right)dt^{2}+{\frac {1}{k^{2}r^{2}+1-{\frac {C}{r}}}}dr^{2}+r^{2}d\Omega ^{2}}

where t is the time coordinate, r is the radial coordinate, Ω are the polar coordinates, C is a constant and k is the AdS curvature.

In general, in d+1 dimensions, the metric is given by

d s 2 = − ( k 2 r 2 + 1 − C r d − 2 ) d t 2 + 1 k 2 r 2 + 1 − C r d − 2 d r 2 + r 2 d Ω 2 {\displaystyle ds^{2}=-\left(k^{2}r^{2}+1-{\frac {C}{r^{d-2}}}\right)dt^{2}+{\frac {1}{k^{2}r^{2}+1-{\frac {C}{r^{d-2}}}}}dr^{2}+r^{2}d\Omega ^{2}}

According to the AdS/CFT correspondence, if gravity were quantized, an AdS black hole would be dual to a thermal state on the conformal boundary. In the context of say, AdS/QCD, this would correspond to the deconfinement phase of the quark–gluon plasma.

## See also

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* BTZ black hole
* Black brane

## References

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