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Near-horizon metric

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Near-horizon limit of the global metric of a black hole

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

References

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  2. ^ Kunduri, Hari K; Lucietti, James (2009-11-25). "Static near-horizon geometries in five dimensions". Classical and Quantum Gravity. 26 (24). IOP Publishing: 245010. arXiv:0907.0410. Bibcode:2009CQGra..26x5010K. doi:10.1088/0264-9381/26/24/245010. ISSN 0264-9381. S2CID 55272059.
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  6. ^ Bardeen, James; Horowitz, Gary T. (1999-10-26). "Extreme Kerr throat geometry: A vacuum analog of AdS2×S". Physical Review D. 60 (10): 104030. arXiv:hep-th/9905099. Bibcode:1999PhRvD..60j4030B. doi:10.1103/physrevd.60.104030. ISSN 0556-2821. S2CID 17389870.
  7. ^ Amsel, Aaron J.; Horowitz, Gary T.; Marolf, Donald; Roberts, Matthew M. (2010-01-22). "Uniqueness of extremal Kerr and Kerr-Newman black holes". Physical Review D. 81 (2): 024033. arXiv:0906.2367. Bibcode:2010PhRvD..81b4033A. doi:10.1103/physrevd.81.024033. ISSN 1550-7998. S2CID 15540019.
  8. Compère, Geoffrey (2012-10-22). "The Kerr/CFT Correspondence and its Extensions". Living Reviews in Relativity. 15 (1). Springer Science and Business Media LLC: 11. arXiv:1203.3561. Bibcode:2012LRR....15...11C. doi:10.12942/lrr-2012-11. ISSN 2367-3613. PMC 5255558. PMID 28179839.
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