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
r
is fixed in the near-horizon limit.
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 The metric of extremal Kerr black hole (
= a =
) 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] ) 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] 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}}
{\displaystyle =-F\,r^{2}dv^{2}+2dvdr+{\hat {h}}_{AB}{\big (}dy^{A}-G^{A}\,rdv{\big )}{\big (}dy^{B}-G^{B}\,rdv{\big )}\,,}
where the metric functions
are independent of the coordinate r,
{\displaystyle {\hat {h}}_{AB}}
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