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 defined in (164
)
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
The right sector was taken to be frozen at extremality. The resulting chiral limit of the CFT with central charge We will now assume that quantum gravity states form a representation of both a left and a right-moving
Virasoro algebra with generators and
. 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 (67
). 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
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 [106, 160
]. 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
defined in (140
) associated with the
electric current. Using the expressions (74
), we find for the Kerr–Newman black hole the value
As done in [53], we also assume the presence of a right-moving
current algebra, whose zero
eigenmode
is constrained by the level matching condition
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 current and a Virasoro algebra in both
sectors,
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
between bulk modes The conformal weight can be deduced from the transformation of the probe field under the scaling
(24
) in the overlap region
. The scalar field in the overlap region is
. Using the rules of the AdS/CFT dictionary [265], this
behavior is related to the conformal weight as
. One then infers that [160
]
In general, the weight (185) will be complex and real weight will not be integers. However, a curious
fact, described in [129
, 224
], is that for any axisymmetric perturbation (
) of any integer spin
of
the Kerr black hole, the conformal weight (185
) is an integer
Throwing the scalar at the black hole is dual to exciting the CFT by acting with the
operator
. 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]
In order to compare the bulk computations to the CFT result (191), we must match the conformal
weights and the reduced momenta
. The gravity result (178
) agrees with the CFT result (191
) if
and only if we choose
Now, let us notice that the matching conditions (193) – (194
) are “democratic” in that the roles of
angular momentum and electric charge are put on an equal footing, as noted in [79, 71
]. One can then also
obtain the conformal weights and reduced left and right frequencies
using alternative CFT
descriptions such as the
with Virasoro algebra along the gauge field direction, and
the mixed
family of CFTs. We can indeed rewrite (192
) in the alternative form
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
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 (1
) will also be consistent
with a CFT description, as supported by all cases studied beyond the Kerr–Newman black
hole [106
, 73
, 242, 71
, 80
, 46
].
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].
http://www.livingreviews.org/lrr-2012-11 |
Living Rev. Relativity 15, (2012), 11
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