5.3 Macroscopic greybody factors close to extremality

The Sturm–Liouville problem (159View Equation) cannot be solved analytically. However, in the regime of near-extremal excitations (145View Equation) – (146View Equation) an approximative solution can be obtained analytically using asymptotic matched expansions: the wave equation is solved in the near-horizon region and in the far asymptotically-flat region and then matched along their common overlap region.

For that purpose, it is useful to define the dimensionless horizon radius x = (r − r+)∕r+ such that the outer horizon is at x = 0. The two other singular points of the radial equation (159View Equation) are the inner horizon x = − τH and spatial infinity x = ∞. One then simply partitions the radial axis into two regions with a large overlap as

The overlap region is guaranteed to exist thanks to (147View Equation).

In the near-extremal regime, the absorption probability σabs gets a contribution from each region as

near match σabs = σabs σabs , (163 ) near --dEabs-∕dt-- σabs = |Ψ (x = xB)|2, (164 ) 2 σmatch = |Ψ-(x-=--xB)|-, (165 ) abs dEin∕dt
where 2 |Ψ (x = xB )| is the norm of the scalar field in the overlap region with τH ≪ xB ≪ 1. One can conveniently normalize the scalar field such that it has unit incoming flux dEin∕dt = 1. The contribution σmatch abs is then simply a normalization that depends on the coupling of the near-horizon region to the far region.

In the near-horizon region, the radial equation reduces to a much simpler hypergeometric equation. One can in fact directly obtain the same equation from solving for a probe in a near-extremal near-horizon geometry of the type (79View Equation), which is, as detailed in Section 2.3, a warped and twisted product of 2 AdS2 × S. The presence of poles in the hypergeometric equation at x = 0 and x = − τH requires one to choose the AdS2 base of the near-horizon geometry to be

2 ds2(2) = − x(x + τH)dt2 + ---dx-----. (166 ) x(x + τH )

One can consider the non-diagonal term 2 Γ (𝜃)γ(𝜃)krdtdϕ appearing in the geometry (79View Equation) as a U (1 ) electric field twisted along the fiber spanned by d ϕ over the AdS2 base space. It may then not be surprising that the dynamics of a probe scalar on that geometry can be expressed equivalently as a charged massive scalar on AdS2 with two electric fields: one coming from the U (1) twist in the four-dimensional geometry, and one coming from the original U(1) gauge field. By SL (2,ℝ ) invariance, these two gauge fields are given by

A1 = α1xdt, A2 = α2xdt. (167 )
The coupling between the gauge fields and the charged scalar is dictated by the covariant derivative
𝒟 = ∇ − iq1A1 − iq2A2 = ∇ − iqeffA, (168 )
where ∇ is the Levi-Civita connection on AdS2 and q1 and q2 are the electric charge couplings. One can rewrite more simply the connection as qeffA, where qeff = q1α1 + q2α2 is the effective total charge coupling and A = xdt is a canonically-normalized effective gauge field. The equation for a charged scalar field Φ (t,x ) with mass μeff is then
2 2 𝒟 Φ − μeffΦ = 0. (169 )
Taking Φ(t,r) = e−iωeffτHtΦ (x), we then obtain the following equation for Φ(x),
[ 2 ] ∂x(x (x + τH )∂x) + (ωeffτH-+-qeffx)--− μ2 Φ (x) = 0. x(x + τH) eff
Using the field redefinition
s∕2 -x- s∕2 s Φ(x) = x (τH + 1 ) R (x ), (170 )
we obtain the equivalent equation,
x (x + τ )∂2Rs + (1 + s)(2x + τ )∂ Rs + V (x)Rs = 0, (171 ) H x H x
where the potential is
2 V (x ) = (ax-+-bτH)--−-is(2x-+-τH)(ax-+-bτH-)− c. (172 ) x(x + τH )
Here, the parameters a,b,c are related to μ eff, q eff and ω eff as14
is a = qeff + is, b = ωeff + --, c = μ2eff − s. (173 ) 2
Finally, comparing Eq. (171View Equation) with (159View Equation), where the potential Vs(r) is approximated by the near-horizon potential, we obtain that these equations are identical, as previously announced, after identifying the parameters as
n is ωe ff = ---− --, 4π 2 qeff = 2r+ω − qeQ − is, (174 ) μ2eff = Asaω,l,m − 2am ω − s2 + μ2 (r2+ + a2) − 2ims.
Moreover, using the expression of the frequency (150View Equation) near extremality, one can write the effective charge in the convenient form
qeff = -m---+ -qe--− is, (175 ) 2πT ϕ 2πTe
where the extremal Frolov–Thorne temperatures Te and Tϕ are defined in (74View Equation).

We can now understand that there are two qualitatively distinct solutions for the radial field Rs(x ). Uncharged fields in AdS2 below a critical mass are unstable or tachyonic, as shown by Breitenlohner and Freedman [55]. Charged particles in an electric field on AdS2 have a modified Breitenlohner–Freedman bound

2 1- 2 m BF = − 4 + qeff , (176 )
in which the square mass is lifted up by the square charge. Below the critical mass, charged scalars will be unstable to Schwinger pair production [233, 184]. Let us define
β2 ≡ μ2eff − m2BF. (177 )
Stable modes will be characterized by a real β ≥ 0, while unstable modes will be characterized by an imaginary β. This distinction between modes is distinct from superradiant and non-superradiant modes. Indeed, from the definition of n (152View Equation), superradiance happens at near-extremality when n < 0.

We can now solve the equation, impose the boundary conditions, compute the flux at the horizon and finally obtain the near-horizon absorption probability. The computation can be found in [53Jump To The Next Citation Point, 106Jump To The Next Citation Point, 160Jump To The Next Citation Point]. The net result is as follows. A massive, charge e, spin s = 0, 12 field with energy ω and angular momentum m and real β > 0 scattered against a Kerr–Newman black hole with mass M and charge Q has near-region absorption probability

(TH )2β (e n2 − (− 1)2se− n2) ( 1 ) σneabasr∼ --------------2---------|Γ --+ β − s + iRe(qeff) |2 ( Γ (2β) 2) 1- (-n- ) 2 |Γ 2 + β + i 2π − Re(qeff) | . (178 )
For a massless spin s = 1,2 field scattered against a Kerr black hole, exactly the same formula applies, but with e = Q = 0. The absorption probability in the case where β is imaginary can be found in the original papers [235, 257].

We will now show that the formulae (178View Equation) are Fourier transforms of CFT correlation functions. We will not consider the scattering of unstable fields with β imaginary in this review. We refer the reader to [53Jump To The Next Citation Point] for arguments on how the scattering absorption probability of unstable spin 0 modes around the Kerr black hole match with dual CFT expectations.

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