The situation is similar to the BTZ black hole in 2+1 gravity that has a
symmetry, which is spontaneously broken by the identification of the angular coordinate. This breaking of
symmetry can be interpreted in that case as placing the dual CFT to the BTZ black hole in a density
matrix with left and right-moving temperatures dictated by the
group element
generating the
identification of the geometry [210
].
In the case of non-extremal black-hole geometries, one can similarly interpret the symmetry breaking
using a CFT as follows [68]. First, we need to assume that before the identification, the near region
dynamics is described by a dual two-dimensional CFT, which possesses a ground state that is invariant
under the full symmetry. This is a strong assumption, since there are several
(apparent) obstacles to the existence of a ground state, as we already discussed in the case of extremal
black holes; see Section 4.4. Nevertheless, assuming the existence of this vacuum state, the two
conformal coordinates
can be interpreted as the two null coordinates on the plane
where the CFT vacuum state can be defined. At fixed
, the relation between conformal
coordinates
and Boyer–Lindquist
coordinates is, up to an
-dependent scaling,
The quantum state describing this accelerating strip of Minkowski spacetime is obtained from the
invariant Minkowski vacuum by tracing over the quantum state in the region
outside the strip. The result is a thermal density matrix at temperatures
. Hence, under the
assumption of the existence of a CFT with a vacuum state, non-extremal black holes can be described as a
finite temperature
mixed state in a dual CFT.
It is familiar from the three-dimensional BTZ black hole that the identifications required to obtain
extremal black holes are different than the ones required to obtain non-extremal black holes [27, 210]. Here
as well, the vector fields (214) – (215
) are not defined in the extremal limit because the change of
coordinates (210
) breaks down. Nevertheless, the extremal limit of the temperatures
and
match with the temperatures defined at extremality in Section 5.4. More precisely, the
temperatures
and
defined in (222
), (223
) and (225
) match with the temperatures defined at
extremality
and
, respectively, where
and
are
defined in (74
). This is consistent with the interpretation that states corresponding to extremal
black holes in the CFT can be defined as a limit of states corresponding to non-extremal black
holes.
http://www.livingreviews.org/lrr-2012-11 |
Living Rev. Relativity 15, (2012), 11
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