195.202.245.43: Potential edit

2014-01-30T11:16:51Z

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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 hole|extremal]] black holes.<ref name=hk1>Hari K Kunduri, James Lucietti. ''A classification of near-horizon geometries of extremal vacuum black holes''. Journal of Mathematical Physics, 2009, '''50'''(8): 082502. [http://arxiv.org/abs/0806.2051 arXiv:0806.2051v3 (hep-th)]</ref><ref name=hk2>Hari K Kunduri, James Lucietti. ''Static near-horizon geometries in five dimensions''. Classical and Quantum Gravity, 2009, '''26'''(24): 245010. [http://arxiv.org/abs/0907.0410 arXiv:0907.0410v2 (hep-th)]</ref><ref name=hk3>Hari K Kunduri. ''Electrovacuum near-horizon geometries in four and five dimensions''. Classical and Quantum Gravity, 2011, '''28'''(11): 114010. [http://arxiv.org/abs/1104.5072 arXiv:1104.5072v1 (hep-th)]</ref> NHMs are expressed in Gaussian null coordinates, and one important property is that the dependence on the coordinate <math>r</math> is fixed in the near-horizon limit.

==NHM of extremal Reissner-Nordström black holes==

The metric of [[Reissner-Nordström metric|extremal Reissner-Nordström]] black hole is

:<math>ds^2\,=\,-\Big(1-\frac{2M}{r}\Big)^2\,dt^2+\Big(1-\frac{2M}{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>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>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 [[Kerr metric|extremal Kerr]] black hole (<math>M=a=J/M</math>) in [[Boyer-Lindquist coordinates]] can be written in the following two enlightening forms,<ref name=ref1>Michael Paul Hobson, George Efstathiou, Anthony N Lasenby. ''General Relativity: An Introduction for Physicists''. Cambridge: Cambridge University Press, 2006.</ref><ref name=ref2>Valeri P Frolov, Igor D Novikov. ''Black Hole Physics: Basic Concepts and New Developments''. Berlin: Springer, 1998.</ref>

:<math>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>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>\rho_K^2:=r^2+M^2\cos^2\theta\,,\;\; \Delta_K:=\big(r-M\big)^2\,,\;\; \Sigma^2:=\big(r+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<ref name=ref3>James Bardeen, Gary T Horowitz. ''The extreme Kerr throat geometry: a vacuum analog of AdS2×S2''. Physical Review D, 1999, '''60'''(10): 104030. [http://arxiv.org/abs/hep-th/9905099 arXiv:hep-th/9905099v1]</ref><ref name=ref4>Aaron J Amsel, Gary T Horowitz, Donald Marolf, Matthew M Roberts. ''Uniqueness of Extremal Kerr and Kerr-Newman Black Holes''. Physical Review D, 2010, '''81'''(2): 024033. [http://arxiv.org/abs/0906.2367 arXiv:0906.2367v3 (gr-qc)]</ref>

:<math>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''<ref name="ref3" /> )

:<math>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 metric|Kerr-Newman]] black holes (<math>r_+^2=M^2+Q^2</math>) are described by the metric<ref name="ref1" /><ref name="ref2" />
:<math>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>\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>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<ref name="ref4" />

:<math>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)^{-1}\,.</math>

==NHMs of generic black holes==

In addition to the NHMs of extremal Kerr-Newman family metrics discussed above, all [[Stationary black hole|stationary]] NHMs could be written in the form<ref name="hk1" /><ref name="hk2" /><ref name="hk3" /><ref>Geoffrey Compere. ''The Kerr/CFT Correspondence and its Extensions''. Living Reviews in Relativity, 2012, '''15'''(11): [http://www.livingreviews.org/lrr-2012-11 lrr-2012-11] [http://arxiv.org/abs/1203.3561 arXiv:1203.3561v2 (hep-th)]</ref>

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

Remark: In Gaussian null coordinates, the black hole horizon corresponds to <math>r=0</math>.

==See also==

*[[Extremal black hole]]
*[[Reissner-Nordström metric]]
*[[Kerr metric]]
*[[Kerr-Newman metric]]

==References==
{{reflist}}

[[Category:General relativity]]
[[Category:Black holes]]

Quasi-harmonic approximation - Revision history

195.202.245.43: Potential edit