Near-horizon metric

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 ${\displaystyle r}$ is fixed in the near-horizon limit.

NHM of extremal Reissner–Nordström black holes

The metric of extremal Reissner–Nordström black hole is

${\displaystyle ds^{2}\,=\,-{\Big (}1-{\frac {M}{r}}{\Big )}^{2}\,dt^{2}+{\Big (}1-{\frac {M}{r}}{\Big )}^{-2}dr^{2}+r^{2}\,{\big (}d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}{\big )}\,.}$

Taking the near-horizon limit

${\displaystyle t\mapsto {\frac {\tilde {t}}{\epsilon }}\,,\quad r\mapsto M+\epsilon \,{\tilde {r}}\,,\quad \epsilon \to 0\,,}$

and then omitting the tildes, one obtains the near-horizon metric

${\displaystyle ds^{2}=-{\frac {r^{2}}{M^{2}}}\,dt^{2}+{\frac {M^{2}}{r^{2}}}\,dr^{2}+M^{2}\,{\big (}d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}{\big )}}$

NHM of extremal Kerr black holes

The metric of extremal Kerr black hole (${\displaystyle M=a=J/M}$) in Boyer–Lindquist coordinates can be written in the following two enlightening forms,^[4][5]

${\displaystyle ds^{2}\,=\,-{\frac {\rho _{K}^{2}\Delta _{K}}{\Sigma ^{2}}}\,dt^{2}+{\frac {\rho _{K}^{2}}{\Delta _{K}}}\,dr^{2}+\rho _{K}^{2}d\theta ^{2}+{\frac {\Sigma ^{2}\sin ^{2}\theta }{\rho _{K}^{2}}}{\big (}d\phi -\omega _{K}\,dt{\big )}^{2}\,,}$

${\displaystyle ds^{2}\,=\,-{\frac {\Delta _{K}}{\rho _{K}^{2}}}\,{\big (}dt-M\sin ^{2}\theta d\phi {\big )}^{2}+{\frac {\rho _{K}^{2}}{\Delta _{K}}}\,dr^{2}+\rho _{K}^{2}d\theta ^{2}+{\frac {\sin ^{2}\theta }{\rho _{K}^{2}}}{\Big (}Mdt-(r^{2}+M^{2})d\phi {\Big )}^{2}\,,}$

where

${\displaystyle \rho _{K}^{2}:=r^{2}+M^{2}\cos ^{2}\theta \,,\;\;\Delta _{K}:={\big (}r-M{\big )}^{2}\,,\;\;\Sigma ^{2}:={\big (}r^{2}+M^{2}{\big )}^{2}-M^{2}\Delta _{K}\sin ^{2}\theta \,,\;\;\omega _{K}:={\frac {2M^{2}r}{\Sigma ^{2}}}\,.}$

Taking the near-horizon limit^[6][7]

${\displaystyle t\mapsto {\frac {\tilde {t}}{\epsilon }}\,,\quad r\mapsto M+\epsilon \,{\tilde {r}}\,,\quad \phi \mapsto {\tilde {\phi }}+{\frac {1}{2M\epsilon }}{\tilde {t}}\,,\quad \epsilon \to 0\,,}$

and omitting the tildes, one obtains the near-horizon metric (this is also called extremal Kerr throat^[6] )

${\displaystyle ds^{2}\simeq {\frac {1+\cos ^{2}\theta }{2}}\,{\Big (}-{\frac {r^{2}}{2M^{2}}}\,dt^{2}+{\frac {2M^{2}}{r^{2}}}\,dr^{2}+2M^{2}d\theta ^{2}{\Big )}+{\frac {4M^{2}\sin ^{2}\theta }{1+\cos ^{2}\theta }}\,{\Big (}d\phi +{\frac {rdt}{2M^{2}}}{\Big )}^{2}\,.}$

NHM of extremal Kerr–Newman black holes

Extremal Kerr–Newman black holes (${\displaystyle r_{+}^{2}=M^{2}+Q^{2}}$) are described by the metric^[4][5]

${\displaystyle ds^{2}=-{\Big (}1-{\frac {2Mr-Q^{2}}{\rho _{KN}}}\!{\Big )}dt^{2}-{\frac {2a\sin ^{2}\!\theta \,(2Mr-Q^{2})}{\rho _{KN}}}dtd\phi +\rho _{KN}{\Big (}{\frac {dr^{2}}{\Delta _{KN}}}+d\theta ^{2}{\Big )}+{\frac {\Sigma ^{2}}{\rho _{KN}}}d\phi ^{2},}$

where

${\displaystyle \Delta _{KN}\,:=\,r^{2}-2Mr+a^{2}+Q^{2}\,,\;\;\rho _{KN}\,:=\,r^{2}+a^{2}\cos ^{2}\!\theta \,,\;\;\Sigma ^{2}\,:=\,(r^{2}+a^{2})^{2}-\Delta _{KN}a^{2}\sin ^{2}\theta \,.}$

Taking the near-horizon transformation

${\displaystyle t\mapsto {\frac {\tilde {t}}{\epsilon }}\,,\quad r\mapsto M+\epsilon \,{\tilde {r}}\,,\quad \phi \mapsto {\tilde {\phi }}+{\frac {a}{r_{0}^{2}\epsilon }}{\tilde {t}}\,,\quad \epsilon \to 0\,,\quad {\Big (}r_{0}^{2}\,:=\,M^{2}+a^{2}{\Big )}}$

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

${\displaystyle ds^{2}\simeq {\Big (}1-{\frac {a^{2}}{r_{0}^{2}}}\sin ^{2}\!\theta {\Big )}\left(-{\frac {r^{2}}{r_{0}^{2}}}dt^{2}+{\frac {r_{0}^{2}}{r^{2}}}dr^{2}+r_{0}^{2}d\theta ^{2}\right)+r_{0}^{2}\sin ^{2}\!\theta \,{\Big (}1-{\frac {a^{2}}{r_{0}^{2}}}\sin ^{2}\!\theta {\Big )}^{-1}\left(d\phi +{\frac {2arM}{r_{0}^{4}}}dt\right)^{2}\,.}$

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]

${\displaystyle ds^{2}=({\hat {h}}_{AB}G^{A}G^{B}-F)r^{2}dv^{2}+2dvdr-{\hat {h}}_{AB}G^{B}rdvdy^{A}-{\hat {h}}_{AB}G^{A}rdvdy^{B}+{\hat {h}}_{AB}dy^{A}dy^{B}}$
${\displaystyle =-F\,r^{2}dv^{2}+2dvdr+{\hat {h}}_{AB}{\big (}dy^{A}-G^{A}\,rdv{\big )}{\big (}dy^{B}-G^{B}\,rdv{\big )}\,,}$

where the metric functions ${\displaystyle \{F,G^{A}\}}$ are independent of the coordinate r, ${\displaystyle {\hat {h}}_{AB}}$ denotes the intrinsic metric of the horizon, and ${\displaystyle y^{A}}$ are isothermal coordinates on the horizon.

Remark: In Gaussian null coordinates, the black hole horizon corresponds to ${\displaystyle r=0}$.

See also

• Extremal black hole
• Reissner–Nordström metric
• Kerr metric
• Kerr–Newman metric