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 by
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 [55]. Charged particles in an electric field on AdS2 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
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.
http://www.livingreviews.org/lrr-2012-11 |
Living Rev. Relativity 15, (2012), 11
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