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 r {\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
d s 2 = − ( 1 − M r ) 2 d t 2 + ( 1 − M r ) − 2 d r 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) . {\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
t ↦ t ~ ϵ , r ↦ M + ϵ r ~ , ϵ → 0 , {\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
d s 2 = − r 2 M 2 d t 2 + M 2 r 2 d r 2 + M 2 ( d θ 2 + sin 2 θ d ϕ 2 ) {\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 ( M = a = J / M {\displaystyle M=a=J/M} ) in Boyer–Lindquist coordinates can be written in the following two enlightening forms,
d s 2 = − ρ K 2 Δ K Σ 2 d t 2 + ρ K 2 Δ K d r 2 + ρ K 2 d θ 2 + Σ 2 sin 2 θ ρ K 2 ( d ϕ − ω K d t ) 2 , {\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}\,,}
d s 2 = − Δ K ρ K 2 ( d t − M sin 2 θ d ϕ ) 2 + ρ K 2 Δ K d r 2 + ρ K 2 d θ 2 + sin 2 θ ρ K 2 ( M d t − ( r 2 + M 2 ) d ϕ ) 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
ρ K 2 := r 2 + M 2 cos 2 θ , Δ K := ( r − M ) 2 , Σ 2 := ( r 2 + M 2 ) 2 − M 2 Δ K sin 2 θ , ω K := 2 M 2 r Σ 2 . {\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
t ↦ t ~ ϵ , r ↦ M + ϵ r ~ , ϕ ↦ ϕ ~ + 1 2 M ϵ t ~ , ϵ → 0 , {\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 )
d s 2 ≃ 1 + cos 2 θ 2 ( − r 2 2 M 2 d t 2 + 2 M 2 r 2 d r 2 + 2 M 2 d θ 2 ) + 4 M 2 sin 2 θ 1 + cos 2 θ ( d ϕ + r d t 2 M 2 ) 2 . {\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 ( r + 2 = M 2 + Q 2 {\displaystyle r_{+}^{2}=M^{2}+Q^{2}} ) are described by the metric
d s 2 = − ( 1 − 2 M r − Q 2 ρ K N ) d t 2 − 2 a sin 2 θ ( 2 M r − Q 2 ) ρ K N d t d ϕ + ρ K N ( d r 2 Δ K N + d θ 2 ) + Σ 2 ρ K N d ϕ 2 , {\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
Δ K N := r 2 − 2 M r + a 2 + Q 2 , ρ K N := r 2 + a 2 cos 2 θ , Σ 2 := ( r 2 + a 2 ) 2 − Δ K N a 2 sin 2 θ . {\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
t ↦ t ~ ϵ , r ↦ M + ϵ r ~ , ϕ ↦ ϕ ~ + a r 0 2 ϵ t ~ , ϵ → 0 , ( r 0 2 := M 2 + a 2 ) {\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
d s 2 ≃ ( 1 − a 2 r 0 2 sin 2 θ ) ( − r 2 r 0 2 d t 2 + r 0 2 r 2 d r 2 + r 0 2 d θ 2 ) + r 0 2 sin 2 θ ( 1 − a 2 r 0 2 sin 2 θ ) − 1 ( d ϕ + 2 a r M r 0 4 d t ) 2 . {\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
d s 2 = ( h ^ A B G A G B − F ) r 2 d v 2 + 2 d v d r − h ^ A B G B r d v d y A − h ^ A B G A r d v d y B + h ^ A B d y A d y B {\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}}
= − F r 2 d v 2 + 2 d v d r + h ^ A B ( d y A − G A r d v ) ( d y B − G B r d v ) , {\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 { F , G A } {\displaystyle \{F,G^{A}\}} are independent of the coordinate r, h ^ A B {\displaystyle {\hat {h}}_{AB}} denotes the intrinsic metric of the horizon, and y A {\displaystyle y^{A}} are isothermal coordinates on the horizon. Remark: In Gaussian null coordinates, the black hole horizon corresponds to r = 0 {\displaystyle r=0} .
See also Extremal black hole Reissner–Nordström metric Kerr metric Kerr–Newman metric
References