The near-horizon metric (NHM) refers to the near-horizon limit of the global metric of a black hole. NHMs play an important role in studying the geometry and topology of black holes, but are only well defined for extremal black holes.[1][2][3] NHMs are expressed in Gaussian null coordinates, and one important property is that the dependence on the coordinate
is fixed in the near-horizon limit.
NHM of extremal Reissner–Nordström black holes
The metric of extremal Reissner–Nordström black hole is
![{\displaystyle ds^{2}\,=\,-{\Big (}1-{\frac {M}{r}}{\Big )}^{2}\,dt^{2}+{\Big (}1-{\frac {M}{r}}{\Big )}^{-2}dr^{2}+r^{2}\,{\big (}d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}{\big )}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a7dc71372a7e6afa4688f7d5df5c91108bfe5d)
Taking the near-horizon limit
![{\displaystyle t\mapsto {\frac {\tilde {t}}{\epsilon }}\,,\quad r\mapsto M+\epsilon \,{\tilde {r}}\,,\quad \epsilon \to 0\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ef51c3d076ffd3e62a3c1b58f47cf515c0b64d)
and then omitting the tildes, one obtains the near-horizon metric
![{\displaystyle ds^{2}=-{\frac {r^{2}}{M^{2}}}\,dt^{2}+{\frac {M^{2}}{r^{2}}}\,dr^{2}+M^{2}\,{\big (}d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}{\big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e159a778122a99a208804d4abdf04ff022c83b1)
NHM of extremal Kerr black holes
The metric of extremal Kerr black hole (
) in Boyer–Lindquist coordinates can be written in the following two enlightening forms,[4][5]
![{\displaystyle ds^{2}\,=\,-{\frac {\rho _{K}^{2}\Delta _{K}}{\Sigma ^{2}}}\,dt^{2}+{\frac {\rho _{K}^{2}}{\Delta _{K}}}\,dr^{2}+\rho _{K}^{2}d\theta ^{2}+{\frac {\Sigma ^{2}\sin ^{2}\theta }{\rho _{K}^{2}}}{\big (}d\phi -\omega _{K}\,dt{\big )}^{2}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77c3365ca825d722cc6934a8b2371ad0422b64ce)
![{\displaystyle ds^{2}\,=\,-{\frac {\Delta _{K}}{\rho _{K}^{2}}}\,{\big (}dt-M\sin ^{2}\theta d\phi {\big )}^{2}+{\frac {\rho _{K}^{2}}{\Delta _{K}}}\,dr^{2}+\rho _{K}^{2}d\theta ^{2}+{\frac {\sin ^{2}\theta }{\rho _{K}^{2}}}{\Big (}Mdt-(r^{2}+M^{2})d\phi {\Big )}^{2}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ff41c8730d01e5cf9fc00428a9e9aea0fdcebe)
where
![{\displaystyle \rho _{K}^{2}:=r^{2}+M^{2}\cos ^{2}\theta \,,\;\;\Delta _{K}:={\big (}r-M{\big )}^{2}\,,\;\;\Sigma ^{2}:={\big (}r^{2}+M^{2}{\big )}^{2}-M^{2}\Delta _{K}\sin ^{2}\theta \,,\;\;\omega _{K}:={\frac {2M^{2}r}{\Sigma ^{2}}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6dbffd74c157cbb3e17395014663265befeec1d)
Taking the near-horizon limit[6][7]
![{\displaystyle t\mapsto {\frac {\tilde {t}}{\epsilon }}\,,\quad r\mapsto M+\epsilon \,{\tilde {r}}\,,\quad \phi \mapsto {\tilde {\phi }}+{\frac {1}{2M\epsilon }}{\tilde {t}}\,,\quad \epsilon \to 0\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78793b2ce58436e71a2a67b5fd575c8c84f77644)
and omitting the tildes, one obtains the near-horizon metric (this is also called extremal Kerr throat[6] )
![{\displaystyle ds^{2}\simeq {\frac {1+\cos ^{2}\theta }{2}}\,{\Big (}-{\frac {r^{2}}{2M^{2}}}\,dt^{2}+{\frac {2M^{2}}{r^{2}}}\,dr^{2}+2M^{2}d\theta ^{2}{\Big )}+{\frac {4M^{2}\sin ^{2}\theta }{1+\cos ^{2}\theta }}\,{\Big (}d\phi +{\frac {rdt}{2M^{2}}}{\Big )}^{2}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eddddc4492105838fef9238aacbfe07a1c0f2176)
NHM of extremal Kerr–Newman black holes
Extremal Kerr–Newman black holes (
) are described by the metric[4][5]
![{\displaystyle ds^{2}=-{\Big (}1-{\frac {2Mr-Q^{2}}{\rho _{KN}}}\!{\Big )}dt^{2}-{\frac {2a\sin ^{2}\!\theta \,(2Mr-Q^{2})}{\rho _{KN}}}dtd\phi +\rho _{KN}{\Big (}{\frac {dr^{2}}{\Delta _{KN}}}+d\theta ^{2}{\Big )}+{\frac {\Sigma ^{2}}{\rho _{KN}}}d\phi ^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96fe69d3edc1a02e4eb395926fa1ec0ccb033c6d)
where
![{\displaystyle \Delta _{KN}\,:=\,r^{2}-2Mr+a^{2}+Q^{2}\,,\;\;\rho _{KN}\,:=\,r^{2}+a^{2}\cos ^{2}\!\theta \,,\;\;\Sigma ^{2}\,:=\,(r^{2}+a^{2})^{2}-\Delta _{KN}a^{2}\sin ^{2}\theta \,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/936fa881642b46bfebd4560ad90bc081e9bf49aa)
Taking the near-horizon transformation
![{\displaystyle t\mapsto {\frac {\tilde {t}}{\epsilon }}\,,\quad r\mapsto M+\epsilon \,{\tilde {r}}\,,\quad \phi \mapsto {\tilde {\phi }}+{\frac {a}{r_{0}^{2}\epsilon }}{\tilde {t}}\,,\quad \epsilon \to 0\,,\quad {\Big (}r_{0}^{2}\,:=\,M^{2}+a^{2}{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9f47ccca2c3b0ff0c3198773dcd1cd3123e6485)
and omitting the tildes, one obtains the NHM[7]
![{\displaystyle ds^{2}\simeq {\Big (}1-{\frac {a^{2}}{r_{0}^{2}}}\sin ^{2}\!\theta {\Big )}\left(-{\frac {r^{2}}{r_{0}^{2}}}dt^{2}+{\frac {r_{0}^{2}}{r^{2}}}dr^{2}+r_{0}^{2}d\theta ^{2}\right)+r_{0}^{2}\sin ^{2}\!\theta \,{\Big (}1-{\frac {a^{2}}{r_{0}^{2}}}\sin ^{2}\!\theta {\Big )}^{-1}\left(d\phi +{\frac {2arM}{r_{0}^{4}}}dt\right)^{2}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/153ec1fc3990e6562bd54b26c727e73dbdf40ced)
NHMs of generic black holes
In addition to the NHMs of extremal Kerr–Newman family metrics discussed above, all stationary NHMs could be written in the form[1][2][3][8]
![{\displaystyle ds^{2}=({\hat {h}}_{AB}G^{A}G^{B}-F)r^{2}dv^{2}+2dvdr-{\hat {h}}_{AB}G^{B}rdvdy^{A}-{\hat {h}}_{AB}G^{A}rdvdy^{B}+{\hat {h}}_{AB}dy^{A}dy^{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de502b8d84be7360b5e11d9ab427f1f3e55255bf)
where the metric functions
are independent of the coordinate r,
denotes the intrinsic metric of the horizon, and
are isothermal coordinates on the horizon.
Remark: In Gaussian null coordinates, the black hole horizon corresponds to
.
See also
References
- ^ a b Hari K Kunduri, James Lucietti. A classification of near-horizon geometries of extremal vacuum black holes. Journal of Mathematical Physics, 2009, 50(8): 082502. arXiv:0806.2051v3 (hep-th)
- ^ a b Hari K Kunduri, James Lucietti. Static near-horizon geometries in five dimensions. Classical and Quantum Gravity, 2009, 26(24): 245010. arXiv:0907.0410v2 (hep-th)
- ^ a b Hari K Kunduri. Electrovacuum near-horizon geometries in four and five dimensions. Classical and Quantum Gravity, 2011, 28(11): 114010. arXiv:1104.5072v1 (hep-th)
- ^ a b Michael Paul Hobson, George Efstathiou, Anthony N Lasenby. General Relativity: An Introduction for Physicists. Cambridge: Cambridge University Press, 2006.
- ^ a b Valeri P Frolov, Igor D Novikov. Black Hole Physics: Basic Concepts and New Developments. Berlin: Springer, 1998.
- ^ a b James Bardeen, Gary T Horowitz. The extreme Kerr throat geometry: a vacuum analog of AdS2×S2. Physical Review D, 1999, 60(10): 104030. arXiv:hep-th/9905099v1
- ^ a b Aaron J Amsel, Gary T Horowitz, Donald Marolf, Matthew M Roberts. Uniqueness of Extremal Kerr and Kerr–Newman Black Holes. Physical Review D, 2010, 81(2): 024033. arXiv:0906.2367v3 (gr-qc)
- ^ Geoffrey Compere. The Kerr/CFT Correspondence and its Extensions. Living Reviews in Relativity, 2012, 15(11): lrr-2012-11 arXiv:1203.3561v2 (hep-th)