Quasi-harmonic approximation

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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 r is fixed in the near-horizon limit.


NHM of extremal Reissner-Nordström black holes

The metric of extremal Reissner-Nordström black hole is

ds2=(12Mr)2dt2+(12Mr)2dr2+r2(dθ2+sin2θdϕ2).

Taking the near-horizon limit

tt~ϵ,rM+ϵr~,ϵ0,

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

ds2=r2M2dt2+M2r2dr2+M2(dθ2+sin2θdϕ2)


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]

ds2=ρK2ΔKΣ2dt2+ρK2ΔKdr2+ρK2dθ2+Σ2sin2θρK2(dϕωKdt)2,
ds2=ΔKρK2(dtMsin2θdϕ)2+ρK2ΔKdr2+ρK2dθ2+sin2θρK2(Mdt(r2+M2)dϕ)2,

where

ρK2:=r2+M2cos2θ,ΔK:=(rM)2,Σ2:=(r+M2)2M2ΔKsin2θ,ωK:=2M2rΣ2.

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

tt~ϵ,rM+ϵr~,ϕϕ~+12Mϵt~,ϵ0,

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

ds21+cos2θ2(r22M2dt2+2M2r2dr2+2M2dθ2)+4M2sin2θ1+cos2θ(dϕ+rdt2M2)2.


NHM of extremal Kerr-Newman black holes

Extremal Kerr-Newman black holes (r+2=M2+Q2) are described by the metric[4][5]

ds2=(12MrQ2ρKN)dt22asin2θ(2MrQ2)ρKNdtdϕ+ρKN(dr2ΔKN+dθ2)+Σ2ρKNdϕ2,

where

ΔKN:=r22Mr+a2+Q2,ρKN:=r2+a2cos2θ,Σ2:=(r2+a2)2ΔKNa2sin2θ.

Taking the near-horizon transformation

tt~ϵ,rM+ϵr~,ϕϕ~+ar02ϵt~,ϵ0,(r02:=M2+a2)

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

ds2(1a2r02sin2θ)(r2r02dt2+r02r2dr2+r02dθ2)+r02sin2θ(1a2r02sin2θ)1(dϕ+2arMr04dt)1.


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]


ds2=(h^ABGAGBF)r2dv2+2dvdrh^ABGBrdvdyAh^ABGArdvdyB+h^ABdyAdyB
=Fr2dv2+2dvdr+h^AB(dyAGArdv)(dyBGBrdv),

where the metric functions {F,GA} are independent of the coordinate r, h^AB denotes the intrinsic metric of the horizon, and yA are isothermal coordinates on the horizon.


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

See also

References

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  1. 1.0 1.1 Hari K Kunduri, James Lucietti. A classification of near-horizon geometries of extremal vacuum black holes. Journal of Mathematical Physics, 2009, 50(8): 082502. arXiv:0806.2051v3 (hep-th)
  2. 2.0 2.1 Hari K Kunduri, James Lucietti. Static near-horizon geometries in five dimensions. Classical and Quantum Gravity, 2009, 26(24): 245010. arXiv:0907.0410v2 (hep-th)
  3. 3.0 3.1 Hari K Kunduri. Electrovacuum near-horizon geometries in four and five dimensions. Classical and Quantum Gravity, 2011, 28(11): 114010. arXiv:1104.5072v1 (hep-th)
  4. 4.0 4.1 Michael Paul Hobson, George Efstathiou, Anthony N Lasenby. General Relativity: An Introduction for Physicists. Cambridge: Cambridge University Press, 2006.
  5. 5.0 5.1 Valeri P Frolov, Igor D Novikov. Black Hole Physics: Basic Concepts and New Developments. Berlin: Springer, 1998.
  6. 6.0 6.1 James Bardeen, Gary T Horowitz. The extreme Kerr throat geometry: a vacuum analog of AdS2×S2. Physical Review D, 1999, 60(10): 104030. arXiv:hep-th/9905099v1
  7. 7.0 7.1 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. arXiv:0906.2367v3 (gr-qc)
  8. Geoffrey Compere. The Kerr/CFT Correspondence and its Extensions. Living Reviews in Relativity, 2012, 15(11): lrr-2012-11 arXiv:1203.3561v2 (hep-th)