- NHM of extremal Reissner–Nordström black holes
- NHM of extremal Kerr black holes
- NHM of extremal Kerr–Newman black holes
- NHMs of generic black holes
- See also
- References
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
Taking the near-horizon limit
and then omitting the tildes, one obtains the near-horizon metric
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]
where
Taking the near-horizon limit[6][7]
and omitting the tildes, one obtains the near-horizon metric (this is also called extremal Kerr throat[6] )
NHM of extremal Kerr–Newman black holes
Extremal Kerr–Newman black holes (
) are described by the metric[4][5]
where
Taking the near-horizon transformation
and omitting the tildes, one obtains the NHM[7]
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]
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
- Extremal black hole
- Reissner–Nordström metric
- Kerr metric
- Kerr–Newman metric
References
2. ^1 Hari K Kunduri, James Lucietti. Static near-horizon geometries in five dimensions. Classical and Quantum Gravity, 2009, 26(24): 245010. [https://arxiv.org/abs/0907.0410 arXiv:0907.0410v2 (hep-th)]
3. ^1 Hari K Kunduri. Electrovacuum near-horizon geometries in four and five dimensions. Classical and Quantum Gravity, 2011, 28(11): 114010. [https://arxiv.org/abs/1104.5072 arXiv:1104.5072v1 (hep-th)]
4. ^1 Michael Paul Hobson, George Efstathiou, Anthony N Lasenby. General Relativity: An Introduction for Physicists. Cambridge: Cambridge University Press, 2006.
5. ^1 Valeri P Frolov, Igor D Novikov. Black Hole Physics: Basic Concepts and New Developments. Berlin: Springer, 1998.
6. ^1 James Bardeen, Gary T Horowitz. The extreme Kerr throat geometry: a vacuum analog of AdS2×S2. Physical Review D, 1999, 60(10): 104030. [https://arxiv.org/abs/hep-th/9905099 arXiv:hep-th/9905099v1]
7. ^1 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. [https://arxiv.org/abs/0906.2367 arXiv:0906.2367v3 (gr-qc)]
8. ^Geoffrey Compere. The Kerr/CFT Correspondence and its Extensions. Living Reviews in Relativity, 2012, 15(11): lrr-2012-11 [https://arxiv.org/abs/1203.3561 arXiv:1203.3561v2 (hep-th)]
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