Near-horizon metric

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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[edit]

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[edit]

The metric of extremal Kerr black hole () in Boyer–Lindquist coordinates can be written in the following two enlightening forms,[4][5]


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[edit]

Extremal Kerr–Newman black holes () are described by the metric[4][5]


Taking the near-horizon transformation

and omitting the tildes, one obtains the NHM[7]

NHMs of generic black holes[edit]

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[edit]


  1. ^ 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)
  2. ^ 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)
  3. ^ 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)
  4. ^ a b Michael Paul Hobson, George Efstathiou, Anthony N Lasenby. General Relativity: An Introduction for Physicists. Cambridge: Cambridge University Press, 2006.
  5. ^ a b Valeri P Frolov, Igor D Novikov. Black Hole Physics: Basic Concepts and New Developments. Berlin: Springer, 1998.
  6. ^ 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
  7. ^ 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)
  8. ^ Geoffrey Compere. The Kerr/CFT Correspondence and its Extensions. Living Reviews in Relativity, 2012, 15(11): lrr-2012-11 arXiv:1203.3561v2 (hep-th)