For that purpose, it is useful to define the dimensionless horizon radius such that the outer horizon is at . The two other singular points of the radial equation (159) are the inner horizon and spatial infinity . One then simply partitions the radial axis into two regions with a large overlap as
The overlap region is guaranteed to exist thanks to (147).
In the near-extremal regime, the absorption probability gets a contribution from each region as
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 (79), which is, as detailed in Section 2.3, a warped and twisted product of . The presence of poles in the hypergeometric equation at and requires one to choose the AdS2 base of the near-horizon geometry to be
One can consider the non-diagonal term appearing in the geometry (79) as a electric field twisted along the fiber spanned by 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 twist in the four-dimensional geometry, and one coming from the original gauge field. By invariance, these two gauge fields are given by2 and and are the electric charge couplings. One can rewrite more simply the connection as , where is the effective total charge coupling and is a canonically-normalized effective gauge field. The equation for a charged scalar field with mass is then 14
We can now understand that there are two qualitatively distinct solutions for the radial field . Uncharged fields in AdS2 below a critical mass are unstable or tachyonic, as shown by Breitenlohner and Freedman . Charged particles in an electric field on AdS2 have a modified Breitenlohner–Freedman bound[233, 184]. Let us define
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 [53, 106, 160]. The net result is as follows. A massive, charge , spin field with energy and angular momentum and real scattered against a Kerr–Newman black hole with mass and charge has near-region absorption probability[235, 257].
We will now show that the formulae (178) are Fourier transforms of CFT correlation functions. We will not consider the scattering of unstable fields with imaginary in this review. We refer the reader to  for arguments on how the scattering absorption probability of unstable spin 0 modes around the Kerr black hole match with dual CFT expectations.
Living Rev. Relativity 15, (2012), 11
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