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

- Near-horizon region: ,
- Far region: ,
- Overlap region: .

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

where is the norm of the scalar field in the overlap region with . One can conveniently normalize the scalar field such that it has unit incoming flux . The contribution 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 (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 AdS_{2} 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 AdS_{2} 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 AdS_{2} 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 by

We can now understand that there are two qualitatively distinct solutions for the radial field .
Uncharged fields in AdS_{2} below a critical mass are unstable or tachyonic, as shown by Breitenlohner and
Freedman [55]. Charged particles in an electric field on AdS_{2} have a modified Breitenlohner–Freedman
bound

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

For a massless spin field scattered against a Kerr black hole, exactly the same formula applies, but with . The absorption probability in the case where is imaginary can be found in the original papers [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 [53] 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
http://www.livingreviews.org/lrr-2012-11 |
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