5.4 Microscopic greybody factors

In this section we model the emission amplitudes from a microscopic point of view. We will first discuss near-extremal spinning black holes and we will extend our discussion to general charged and/or spinning black holes at the end of this section.

The working assumption of the microscopic model is that the near-horizon region of any near-extremal spinning black hole can be described and therefore effectively replaced by a dual two-dimensional CFT. In the dual CFT picture, the near-horizon region is removed from the spacetime and replaced by a CFT glued along the boundary. Therefore, it is the near-horizon region contribution alone that we expect to be reproduced by the CFT. The normalization σmatch abs defined in (164View Equation) will then be dictated by the explicit coupling between the CFT and the asymptotically-flat region.

Remember from the asymptotic symmetry group analysis in Section 4.1 and 4.3 that boundary conditions were found where the exact symmetry of the near-horizon extremal geometry can be extended to a Virasoro algebra as

U (1)L × SL (2,ℝ)R → VirL × SL (2,ℝ )R. (179 )
The right sector was taken to be frozen at extremality. The resulting chiral limit of the CFT with central charge cJ = 12J sufficed to account for the extremal black-hole entropy.

We will now assume that quantum gravity states form a representation of both a left and a right-moving Virasoro algebra with generators Ln and ¯ Ln. The value of the right-moving central charge will be irrelevant for our present considerations. At near-extremality, the left sector is thermally excited at the extremal left-moving temperature (67View Equation). We take as an assumption that the right-moving temperature is on the order of the infinitesimal reduced Hawking temperature. As discussed in Sections 2.9 and 4.2, the presence of right-movers destabilize the near-horizon geometry. For the Kerr–Newman black hole, we have

2 2 M--+--a-- TL = 4πJ , TR ∼ τH . (180 )

In order to match the bulk scattering amplitude for near-extremal Kerr–Newman black holes, the presence of an additional left-moving current algebra is required [106Jump To The Next Citation Point, 160Jump To The Next Citation Point]. This current algebra is expected from the thermodynamic analysis of charged rotating extremal black holes. We indeed obtained in Section 2.6 and in Section 4.4 that such black holes are characterized by the chemical potential J,e μL defined in (140View Equation) associated with the U (1)e electric current. Using the expressions (74View Equation), we find for the Kerr–Newman black hole the value

3 μe = − Q--. (181 ) L 2J

As done in [53Jump To The Next Citation Point], we also assume the presence of a right-moving U (1) current algebra, whose zero eigenmode J¯0 is constrained by the level matching condition

¯J = L . (182 ) 0 0
The level matching condition is consistent with the fact that the excitations are labeled by three (ω, m, qe) instead of four conserved quantities. The CFT state is then assumed to be at a fixed chemical potential μ R. This right-moving current algebra cannot be detected in the extremal near-horizon geometry in the same way that the right-moving Virasoro algebra cannot be detected, so its existence is conjectural (see, however, [67]). This right-moving current algebra and the matching condition (182View Equation) will turn out to be adequate to match the gravitational result, as detailed below. Note that three-dimensional analogues of this level matching condition appeared in logically independent analyses [95, 94].

Therefore, under these assumptions, the symmetry group of the CFT dual to the near-extremal Kerr–Newman black hole is given by the product of a U (1) current and a Virasoro algebra in both sectors,

(VirL × CurrL) × (VirR × CurrR ). (183 )

In the description where the near-horizon region of the black hole is replaced by a CFT, the emission of quanta is due to couplings

Φbulkπ’ͺ (184 )
between bulk modes Φbulk and operators π’ͺ in the CFT. The structure of the scattering cross section depends on the conformal weights (hL,hR ) and charges (qL,qR) of the operator. The normalization of the coupling is also important for the normalization of the cross section.

The conformal weight hR can be deduced from the transformation of the probe field under the scaling L¯0 = t∂t − r∂r (24View Equation) in the overlap region τH β‰ͺ x β‰ͺ 1. The scalar field in the overlap region is Φ ∼ Φ0(t,πœƒ,Ο• )r− 12+β + Φ1 (t,πœƒ,Ο• )r− 12−β. Using the rules of the AdS/CFT dictionary [265], this behavior is related to the conformal weight as hR−1 − hR Φ ∼ r ,r. One then infers that [160Jump To The Next Citation Point]

hR = 1-+ β. (185 ) 2
The values of the charges (qL,qR) are simply the U (1 ) charges of the probe,
qL = qe, qR = m, (186 )
where the charge qR = m follows from the matching condition (182View Equation). We don’t know any first-principle argument leading to the values of the right-moving chemical potential μ R, the right-moving temperature TR and the left-moving conformal weight hL. We will deduce those values from matching the CFT absorption probability with the gravitational result.

In general, the weight (185View Equation) will be complex and real weight will not be integers. However, a curious fact, described in [129Jump To The Next Citation Point, 224Jump To The Next Citation Point], is that for any axisymmetric perturbation (m = 0) of any integer spin s of the Kerr black hole, the conformal weight (185View Equation) is an integer

hR = 1 + l, (187 )
where l = 0, 1,.... One can generalize this result to any axisymmetric perturbation of any vacuum five-dimensional near-horizon geometry [224]. Counter-examples exist in higher dimensions and for black holes in AdS [129]. There is no microscopic accounting of this feature at present.

Throwing the scalar Φbulk at the black hole is dual to exciting the CFT by acting with the operator π’ͺ. Reemission is represented by the action of the Hermitian conjugate operator. Therefore, the absorption probability is related to the thermal CFT two-point function [209]

+ − † + − G (t ,t ) = ⟨π’ͺ (t ,t )π’ͺ (0)⟩, (188 )
where t± are the coordinates of the left and right moving sectors of the CFT. At left and right temperatures (TL,TR ) and at chemical potentials (μL, μR) an operator with conformal dimensions (h ,h ) L R and charges (q ,q ) L R has the two-point function
( πT )2hL ( πT )2hR + − G ∼ (− 1)hL+hR ------L--+-- ------R--−-- eiqLμLt +iqR μRt , (189 ) sinh (πTLt ) sinh (πTRt )
which is determined by conformal invariance. From Fermi’s golden rule, the absorption cross section is [53Jump To The Next Citation Point, 106Jump To The Next Citation Point, 160Jump To The Next Citation Point]
∫ + − −iωRt−− iωLt+ [ + − σabs(ωL, ωR) ∼ dt dt e G (t − iπœ–,t − iπœ–) + − ] − G (t + iπœ–,t + iπœ–) . (190 )
Performing the integral in (190View Equation), we obtain15
σ ∼ T2hL−1T 2hR−1(e π&tidle;ωL+π&tidle;ωR ± e−π&tidle;ωL−πω&tidle;R )|Γ (h + i&tidle;ω )|2|Γ (h + i&tidle;ω )|2, (191 ) abs L R L L R R
where
ωL-−--qL-μL- ωR--−-qRμR- &tidle;ωL = 2πTL , ω&tidle;R = 2πTR . (192 )

In order to compare the bulk computations to the CFT result (191View Equation), we must match the conformal weights and the reduced momenta (&tidle;ωL, &tidle;ωR ). The gravity result (178View Equation) agrees with the CFT result (191View Equation) if and only if we choose

hL = 1+ β − |s|, hR = 1+ β, 2 2 &tidle;ωL = Re (qeff), &tidle;ωR = -n-− Re(qeff). (193 ) 2π
The right conformal weight matches with (185View Equation), consistent with SL (2,ℝ )R conformal invariance. The left conformal weight is natural for a spin s field since |hL − hR| = |s|. The value for &tidle;ωL is consistent with the temperature (180View Equation) and chemical potential (181View Equation). Indeed, since the left-movers span the Ο• direction of the black hole, we have ωL = m. We then obtain
2mJ + qeQ3 ω&tidle;L = ----2----2-- = Re (qeff), (194 ) r+ + a
after using the value (175View Equation). The value of &tidle;ωR is fixed by the matching. It determines one constraint between ωR, μR and TR. However, there is a subtlety in the above matching procedure. The conformal weights hL and hR depend on m through β. This m dependence cannot originate from ωL = m since ωL is introduced after the Fourier transform (190View Equation), while hL,hR are already defined in (189View Equation). One way to introduce this m dependence is to assume that there is a right-moving current algebra and that the dual operator π’ͺ has the zero-mode charge qR = m, which amounts to imposing the condition (182View Equation). (It is then also natural to assume that the chemical potential is μR ∼ ΩJ, but the matching does not depend on any particular value for μR [53].) This justifies why a right-moving current algebra was assumed in the CFT. The dependence of the conformal weights in qe is similarly made possible thanks to the existence of the left-moving current with qL = qe. The matching is finally complete.

Now, let us notice that the matching conditions (193View Equation) – (194View Equation) are “democratic” in that the roles of angular momentum and electric charge are put on an equal footing, as noted in [79, 71Jump To The Next Citation Point]. One can then also obtain the conformal weights and reduced left and right frequencies &tidle;ω , &tidle;ω L R using alternative CFT descriptions such as the CFTQ with Virasoro algebra along the gauge field direction, and the mixed SL (2,β„€) family of CFTs. We can indeed rewrite (192View Equation) in the alternative form

Ο•,Q &tidle;ωL = mTe-+-qeT-Ο•-= qχ-−-μL--m-, (195 ) 2πTΟ•Te 2πT QL
where Q TL = RχTe is the left-moving temperature of the CFTQ, Ο•,Q μL is the chemical potential defined in (144View Equation) and qχ = R χqe is the probe electric charge in units of the Kaluza–Klein circle length. The identification of the right-moving sector is unchanged except that now qR = qe. One can trivially extend the matching with the SL (2,β„€ ) family of CFTs conjectured to describe the (near-)extremal Kerr–Newman black hole.

In summary, near-superradiant absorption probabilities of probes in the near-horizon region of near-extremal black holes are exactly reproduced by conformal field theory two-point functions. This shows the consistency of a CFT description (or multiple CFT descriptions in the case where several U (1) symmetries are present) of part of the dynamics of near-extremal black holes. We expect that a general scattering theory around any near-extremal black-hole solution of (1View Equation) will also be consistent with a CFT description, as supported by all cases studied beyond the Kerr–Newman black hole [106Jump To The Next Citation Point, 73Jump To The Next Citation Point, 242, 71Jump To The Next Citation Point, 80Jump To The Next Citation Point, 46Jump To The Next Citation Point].

Finally, let us note finally that the dynamics of the CFTs dual to the Kerr–Newman geometry close to extremality can be further investigated by computing three-point correlation functions in the near-horizon geometry, as initiated in [40, 39].


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