Astronomy:Near-horizon metric

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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 [math]\displaystyle{ r }[/math] is fixed in the near-horizon limit.

NHM of extremal Reissner–Nordström black holes

The metric of extremal Reissner–Nordström black hole is

[math]\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)\,. }[/math]

Taking the near-horizon limit

[math]\displaystyle{ t\mapsto \frac{\tilde{t}}{\epsilon}\,,\quad r\mapsto M+\epsilon\,\tilde{r}\,,\quad \epsilon\to 0\,, }[/math]

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

[math]\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) }[/math]

NHM of extremal Kerr black holes

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

[math]\displaystyle{ ds^2\,=\,-\frac{\rho_K^2\Delta_K}{\Sigma^2}\,dt^2+\frac{\rho_K^2}{\Delta_K}\,dr^2+\rho_K^2d\theta^2+\frac{\Sigma^2\sin^2\theta}{\rho_K^2}\big( d\phi-\omega_K\, dt \big)^2\,, }[/math]
[math]\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\,, }[/math]

where

[math]\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}\,. }[/math]

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

[math]\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\,, }[/math]

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

[math]\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^2d\theta^2 \Big)+\frac{4M^2\sin^2\theta}{1+\cos^2\theta}\,\Big(d\phi +\frac{rdt}{2M^2}\Big)^2\,. }[/math]

NHM of extremal Kerr–Newman black holes

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

[math]\displaystyle{ ds^2=-\Big(1-\frac{2Mr-Q^2}{\rho_{KN}} \!\Big)dt^2-\frac{2a\sin^2\!\theta\,(2Mr-Q^2)}{\rho_{KN}}dt d\phi +\rho_{KN}\Big(\frac{dr^2}{\Delta_{KN}} + d\theta^2\Big)+\frac{ \Sigma^2 }{\rho_{KN}}d\phi^2, }[/math]

where

[math]\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\,. }[/math]

Taking the near-horizon transformation

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

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

[math]\displaystyle{ ds^2\simeq \Big(1-\frac{a^2}{r_0^2}\sin^2\!\theta \Big)\left(-\frac{r^2}{r^2_0}dt^2+\frac{r^2_0}{r^2}dr^2+r^2_0d\theta^2 \right)+r^2_0\sin^2\!\theta\,\Big(1-\frac{a^2}{r_0^2} \sin^2\!\theta\Big)^{-1}\left( d\phi+\frac{2arM}{r^4_0}dt \right)^{2}\,. }[/math]

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]

[math]\displaystyle{ 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 }[/math]
[math]\displaystyle{ =-F\,r^2 dv^2+2dvdr+\hat{h}_{AB}\big(dy^A-G^A\,r dv \big)\big(dy^B-G^B\,r dv \big)\,, }[/math]

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

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

See also

References

  1. 1.0 1.1 Kunduri, Hari K.; Lucietti, James (2009). "A classification of near-horizon geometries of extremal vacuum black holes". Journal of Mathematical Physics 50 (8): 082502. doi:10.1063/1.3190480. ISSN 0022-2488. Bibcode2009JMP....50h2502K. 
  2. 2.0 2.1 Kunduri, Hari K; Lucietti, James (2009-11-25). "Static near-horizon geometries in five dimensions". Classical and Quantum Gravity (IOP Publishing) 26 (24): 245010. doi:10.1088/0264-9381/26/24/245010. ISSN 0264-9381. Bibcode2009CQGra..26x5010K. 
  3. 3.0 3.1 Kunduri, Hari K (2011-05-20). "Electrovacuum near-horizon geometries in four and five dimensions". Classical and Quantum Gravity 28 (11): 114010. doi:10.1088/0264-9381/28/11/114010. ISSN 0264-9381. Bibcode2011CQGra..28k4010K. 
  4. 4.0 4.1 Hobson, Michael Paul; Efstathiou, George; Lasenby., Anthony N (2006). General relativity : an introduction for physicists. Cambridge, UK New York: Cambridge University Press. ISBN 978-0-521-82951-9. OCLC 61757089. 
  5. 5.0 5.1 Frolov, Valeri P; Novikov, Igor D (1998). Black hole physics : basic concepts and new developments. Dordrecht Boston: Kluwer. ISBN 978-0-7923-5145-0. OCLC 39189783. 
  6. 6.0 6.1 Bardeen, James; Horowitz, Gary T. (1999-10-26). "Extreme Kerr throat geometry: A vacuum analog of AdS2×S2". Physical Review D 60 (10): 104030. doi:10.1103/physrevd.60.104030. ISSN 0556-2821. Bibcode1999PhRvD..60j4030B. 
  7. 7.0 7.1 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. doi:10.1103/physrevd.81.024033. ISSN 1550-7998. Bibcode2010PhRvD..81b4033A. 
  8. Compère, Geoffrey (2012-10-22). "The Kerr/CFT Correspondence and its Extensions". Living Reviews in Relativity (Springer Science and Business Media LLC) 15 (1): 11. doi:10.12942/lrr-2012-11. ISSN 2367-3613. PMID 28179839. Bibcode2012LRR....15...11C.