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. 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
and omitting the tildes, one obtains the near-horizon metric (this is also called extremal Kerr throat )
NHM of extremal Kerr-Newman black holes
Taking the near-horizon transformation
and omitting the tildes, one obtains the NHM
NHMs of generic black holes
Remark: In Gaussian null coordinates, the black hole horizon corresponds to .
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