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

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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

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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

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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

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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

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References

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  1. ^ a b Kunduri, Hari K.; Lucietti, James (2009). "A classification of near-horizon geometries of extremal vacuum black holes". Journal of Mathematical Physics. 50 (8): 082502. arXiv:0806.2051. Bibcode:2009JMP....50h2502K. doi:10.1063/1.3190480. ISSN 0022-2488. S2CID 15173886.
  2. ^ a b Kunduri, Hari K; Lucietti, James (2009-11-25). "Static near-horizon geometries in five dimensions". Classical and Quantum Gravity. 26 (24). IOP Publishing: 245010. arXiv:0907.0410. Bibcode:2009CQGra..26x5010K. doi:10.1088/0264-9381/26/24/245010. ISSN 0264-9381. S2CID 55272059.
  3. ^ a b Kunduri, Hari K (2011-05-20). "Electrovacuum near-horizon geometries in four and five dimensions". Classical and Quantum Gravity. 28 (11): 114010. arXiv:1104.5072. Bibcode:2011CQGra..28k4010K. doi:10.1088/0264-9381/28/11/114010. ISSN 0264-9381. S2CID 118609264.
  4. ^ a b Hobson, Michael Paul; Efstathiou, George; Lasenby., Anthony N (2006). General relativity : an introduction for physicists. Cambridge, UK New York: Cambridge University Press. ISBN 978-0-521-82951-9. OCLC 61757089.
  5. ^ a b Frolov, Valeri P; Novikov, Igor D (1998). Black hole physics : basic concepts and new developments. Dordrecht Boston: Kluwer. ISBN 978-0-7923-5145-0. OCLC 39189783.
  6. ^ a b Bardeen, James; Horowitz, Gary T. (1999-10-26). "Extreme Kerr throat geometry: A vacuum analog of AdS2×S2". Physical Review D. 60 (10): 104030. arXiv:hep-th/9905099. Bibcode:1999PhRvD..60j4030B. doi:10.1103/physrevd.60.104030. ISSN 0556-2821. S2CID 17389870.
  7. ^ a b Amsel, Aaron J.; Horowitz, Gary T.; Marolf, Donald; Roberts, Matthew M. (2010-01-22). "Uniqueness of extremal Kerr and Kerr-Newman black holes". Physical Review D. 81 (2): 024033. arXiv:0906.2367. Bibcode:2010PhRvD..81b4033A. doi:10.1103/physrevd.81.024033. ISSN 1550-7998. S2CID 15540019.
  8. ^ Compère, Geoffrey (2012-10-22). "The Kerr/CFT Correspondence and its Extensions". Living Reviews in Relativity. 15 (1). Springer Science and Business Media LLC: 11. arXiv:1203.3561. Bibcode:2012LRR....15...11C. doi:10.12942/lrr-2012-11. ISSN 2367-3613. PMC 5255558. PMID 28179839.