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 .
In the case of non-extremal black-hole geometries, one can similarly interpret the symmetry breaking using a CFT as follows . 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.
Living Rev. Relativity 15, (2012), 11
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