6.6 Entropy matching

We will now argue that the temperatures TL and TR obtained in Section 6 combined with the analysis at extremality in Section 4 lead to a (several) microscopic counting(s) of the black hole entropy of the Kerr, Reissner–Nordström and Kerr–Newman black holes.

Let us assume that there is a two-dimensional CFT (CFTJ) describing the Kerr black hole, a two-dimensional CFT (CFTQ) describing the Reissner–Nordström black hole and a SL (2,ℤ) family of two-dimensional CFTs (CFT (p1,p2,p3), p1,p2,p2 ∈ ℤ) describing the Kerr–Newman black hole. If these CFTs are dual to the black hole, the entropy is reproduced by Cardy’s formula

2 𝒮CFT = π--(cLTL + cRTR ), (232 ) 3
which is valid when TL ≫ 1, TR ≫ 1. As already mentioned in Section 4.4 and argued in [156], the regime T ≫ 1 L, T ≫ 1 R is not a necessary condition for Cardy’s formula to be valid if these CFTs have special properties such as admitting a long string picture, as reviewed in Section 3.3.

Let us discuss the values of the central charges. In a CFT, the difference cR − cL is proportional to the diffeomorphism anomaly of the CFT [188, 187]. One can then argue from diffeomorphism invariance that the two left and right sectors should have the same value for the central charge,

c = c . (233 ) R L
We obtained the value cL at extremality in Section 4.3 and checked that Cardy’s formula reproduces the extremal black-hole entropy. One way to uniquely fix the value cL away from extremality would consist in matching Cardy’s formula (232View Equation) with the Kerr–Newman black-hole entropy
2 2 𝒮KN (ℳ, 𝒥 ,𝒬 ) = π(r+ + a ), (234 )
using (233View Equation) and the values for the temperatures derived in Section 6.4. Therefore, the matching of black-hole entropy is true by construction, which is clearly unsatisfactory. It would be more satisfactory to have an independent computation of cL away from extremality, but such a computation is currently not available.

However, the resulting central charge cL is, however, non-trivial. For the CFTJ, we obtain cL = 12J. For CFTQ, we have cQ = 6Q3 ∕R χ and for the CFT (p ,p ,p ) 1 2 3, we find 3 c(p1,p2) = 6(p1(2J ) + p2Q ∕R χ). Quite remarkably, these central charges are expressed solely in terms of quantized charges. They do not depend on the mass of the black hole. This is a non-trivial feature that has no explanation so far.

The presence of several CFTs dual to the Kerr–Newman black hole is curious but not inconsistent. Each CFT describes part of the low-energy dynamics of probe scalar fields and multiple CFTs are needed in order to reproduce the full dynamics for arbitrary ratios of the probe angular momentum to probe electric charge. Therefore, each CFT description has therefore a range of applicability away from extremality.

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