Near-horizon metric explained

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.

NHM of extremal Reissner–Nordström black holes

The metric of extremal Reissner–Nordström black hole is

ds ^{2=- ( 1- (M)/(r)} ) ^{2dt} ^{2+ ( 1- (M)/(r)} ) ^{-2} dr ^{2+r} ^{2(d\theta} ^{2+\sin} ^{2\thetad\phi} ^{2} ) .

Taking the near-horizon limit

t\mapsto (\tilde{t)/() }\,,\quad r\mapsto M+\epsilon\,\tilde\,,\quad \epsilon\to 0\,,

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

ds ^{2=- (r ^{2})/(M ^{2})} dt ^{2+ (M ^{2})/(r ^{2})} dr ^{2+M} ^{2(d\theta} ^{2+\sin} ^{2\thetad\phi} ^{2} )

NHM of extremal Kerr black holes

The metric of extremal Kerr black hole (

M = a = J/M ) in Boyer–Lindquist coordinates can be written in the following two enlightening forms,^{[4] [5]} ds ^{2=- (\rho ^{2 \Delta} _{K})/(\Sigma ^{2})} dt ^{2+ (\rho ^{2} _{K})/(\Delta _{K})} dr ^{2 d\theta} _{K} ^{2+ (\Sigma ^{2\sin} ^{2\theta})/(\rho ^{2} _{K})} ( d\phi - \omega _{K} dt ) ^{2,} ds ^{2=- (\Delta _{K})/(\rho ^{2} _{K})} ( dt - M\sin ^{2\theta} d\phi ) ^{2+ (\rho ^{2} _{K})/(\Delta _{K})} dr ^{2} _{K} d\theta ^{2+ (\sin ^{2\theta})/(\rho ^{2} _{K})} ( Mdt - (r ^{2+M} ^{2)d\phi} ) ^{2,}

where

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

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

t\mapsto (\tilde{t)/() }\,,\quad r\mapsto M+\epsilon\,\tilde\,,\quad \phi\mapsto \tilde+\frac\tilde\,,\quad \epsilon\to 0\,,

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

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

NHM of extremal Kerr–Newman black holes

Extremal Kerr–Newman black holes (

r ^{2= M} _{+} ^{2+Q} ^{2} ) are described by the metricds ^{2=- ( 1- (2 Mr - Q ^{2})/(\rho _{KN})} ) dt ^{2- (2 a\sin ^{2\theta(2Mr-Q} ^{2)})/(\rho _{KN})} dt d\phi + \rho _{KN} ( (dr ^{2})/(\Delta _{KN}) + d\theta ^{2 ) + (\Sigma ^{2})/(\rho _{KN})} d\phi ^{2,}

where

\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\mapsto (\tilde{t)/() }\,,\quad r\mapsto M+\epsilon\,\tilde\,,\quad \phi\mapsto \tilde+\frac\tilde\,,\quad \epsilon\to 0\,,\quad \Big(r^2_0\,:=\,M^2+a^2\Big)

and omitting the tildes, one obtains the NHM

ds ^{2\simeq} ( 1- (a ^{2})/(r ^{2} _{0}) \sin ^{2\theta} ) \left( - (r ^{2})/(r ^{2} _{0}) dt ^{2+ (r ^{2} _{0})/(r ^{2})} dr ^{2+r} ^{2} ^{0 d\theta} \right) + r ^{2 \theta ( 1- (a ^{2})/(r ^{2} _{0})} _{0 \sin} \sin ^{2\theta)} ^{-1} \left( d\phi + (2 arM)/(r ^{4} _{0}) 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^[8]

ds ^{2= (\hat{h}} _{AB} G ^{A} G ^{B-F)r} ^{2} dv ^{2+2dvdr-} \hat{h} _{AB} G ^{B} r dv dy ^{A} - \hat{h} _{AB} G ^{Ar} dv dy ^{B + \hat{h}} _{AB} dy ^{A} dy ^{B}
=- Fr ^{2} dv ^{2+2 dvdr + \hat{h}} _{AB} ( dy ^{A-G} ^{Ar} dv ) ( dy ^{B-G} ^{Br} dv ) ,

where the metric functions

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

Remark: In Gaussian null coordinates, the black hole horizon corresponds to

r =0 .

See also

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

Notes and References

1. Kunduri . Hari K. . Lucietti . James . A classification of near-horizon geometries of extremal vacuum black holes . Journal of Mathematical Physics . 50 . 8 . 2009 . 0022-2488 . 10.1063/1.3190480 . 082502. 0806.2051. 2009JMP....50h2502K . 15173886 .
2. Kunduri . Hari K . Lucietti . James . Static near-horizon geometries in five dimensions . Classical and Quantum Gravity . IOP Publishing . 26 . 24 . 2009-11-25 . 0264-9381 . 10.1088/0264-9381/26/24/245010 . 245010. 0907.0410. 2009CQGra..26x5010K . 55272059 .
3. Kunduri . Hari K . Electrovacuum near-horizon geometries in four and five dimensions . Classical and Quantum Gravity . 28 . 11 . 2011-05-20 . 0264-9381 . 10.1088/0264-9381/28/11/114010 . 114010. 1104.5072. 2011CQGra..28k4010K . 118609264 .
4. Book: Hobson . Michael Paul . George . Efstathiou. Anthony N . Lasenby.. General relativity : an introduction for physicists . Cambridge University Press . Cambridge, UK New York . 2006 . 978-0-521-82951-9 . 61757089 .
5. Book: Frolov . Valeri P. Igor D . Novikov . Black hole physics : basic concepts and new developments . Kluwer . Dordrecht Boston . 1998 . 978-0-7923-5145-0 . 39189783 .
6. Bardeen . James . Horowitz . Gary T. . Extreme Kerr throat geometry: A vacuum analog of AdS₂×S² . Physical Review D . 60 . 10 . 1999-10-26 . 0556-2821 . 10.1103/physrevd.60.104030 . 104030. hep-th/9905099. 1999PhRvD..60j4030B . 17389870 .
7. Amsel . Aaron J. . Horowitz . Gary T. . Marolf . Donald . Roberts . Matthew M. . Uniqueness of extremal Kerr and Kerr-Newman black holes . Physical Review D . 81 . 2 . 2010-01-22 . 1550-7998 . 10.1103/physrevd.81.024033 . 024033. 0906.2367. 2010PhRvD..81b4033A . 15540019 .
8. Compère . Geoffrey . The Kerr/CFT Correspondence and its Extensions . Living Reviews in Relativity . Springer Science and Business Media LLC . 15 . 1 . 2012-10-22 . 2367-3613 . 10.12942/lrr-2012-11 . 11. 28179839 . 5255558 . free. 1203.3561. 2012LRR....15...11C .