First, we saw in Section 2.7 that there is a class of near-extremal solutions (79) obeying the boundary
conditions (111) – (116
) with near-horizon energy
. However, the charge
is a
heat term, which is not integrable when both
and
can be varied. Moreover, upon
scaling the coordinates as
and
using the
generator (24
), one
obtains the same metric as (79
) with
. If one would allow the class of
near-extremal solutions (79
) and the presence of
symmetries in a consistent set of
boundary conditions, one would be forced to fix the entropy
to a constant, in order to define
integrable charges. The resulting vanishing charges would not belong to the asymptotic-symmetry
algebra. Since there is no other obvious candidate for a solution with non-zero near-horizon
energy, we argued in Section 2.9 that there is no such solution at all. If that assumption is
correct, the
algebra would always be associated with zero charges and would not
belong to the asymptotic symmetry group. Hence, no additional non-vanishing Virasoro algebra
could be derived in a consistent set of boundary conditions. For alternative points of view,
see [215, 216, 236, 214].
Second, as far as extremal geometries are concerned, there is no need for a non-trivial or
second Virasoro algebra. As we will see in Section 4.4, the entropy of extremal black holes will be matched
using a single copy of the Virasoro algebra, using the assumption that Cardy’s formula applies. Matching
the entropy of non-extremal black holes and justifying Cardy’s formula requires two Virasoro algebras, as
we will discuss in Section 6.6. However, non-extremal black holes do not admit a near-horizon limit and,
therefore, are not dynamical objects described by a consistent class of near-horizon boundary conditions. At
most, one could construct the near horizon region of non-extremal black holes in perturbation
theory as a large deformation of the extremal near-horizon geometry. This line of thought was
explored in [67
]. In the context of the near-extremal Kerr black hole, it was obtained using
a dimensionally-reduced model such that the algebra of diffeomorphisms, which extends the
algebra, is represented on the renormalized stress-energy tensor as a Virasoro algebra. It
would be interesting to further define and extend these arguments (which go beyond a standard
asymptotic-symmetry analysis) to non-dimensionally-reduced models and to other near-extremal black
holes.
Finally, let us also note that the current discussion closely parallels the lower dimensional example of the
near-horizon limit of the extremal BTZ black hole discussed in [30]. There it was shown that the
near-horizon geometry of the extremal BTZ black hole of angular momentum
is a geometry with
isometry
http://www.livingreviews.org/lrr-2012-11 |
Living Rev. Relativity 15, (2012), 11
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