6 Hidden Symmetries of Non-Extremal Black Holes

In Section 4 we described evidence showing that the asymptotic growth of states of extremal rotating or charged black holes is controlled by a chiral half of a two-dimensional CFT, at least in the semi-classical limit. We also reviewed in Section 5 how the near-horizon dynamics of probes can be reproduced by manipulating near-chiral CFT two-point functions in the near-extremal limit. These analyses strongly rely on the existence of a decoupled near-horizon geometry for all extremal or near-extremal black holes. Away from extremality, one cannot decouple the horizon from the surrounding geometry. Therefore, it is unclear whether any of the previous considerations will be useful in describing non-extremal geometries.

It might then come as a surprise that even away from extremality, conformal invariance is present in the dynamics of probe scalar fields around the Kerr black hole in a specific regime (at low energy and close enough to the black hole as we will make more precise below) [68Jump To The Next Citation Point]. In that regime, the probe scalar field equation can be written in a SL (2,ℝ) × SL (2,ℝ ) invariant fashion in a region close enough to the horizon. Such a local hidden symmetry is non-geometric but appears in the probe dynamics. The 2π periodic identification of the azimuthal angle ϕ breaks globally-conformal symmetry. Using the properties of this representation of conformal invariance, one can then argue that the Kerr black hole is described by a CFT with specific left and right-moving temperatures [68Jump To The Next Citation Point]

√ --------- M 2 M 4 − J2 TL = ----, TR = -----------. (196 ) 2πJ 2πJ
The well-known low-energy scattering amplitudes coincide with correlators of a two-dimensional CFT with these temperatures. Finally, quite remarkably, the entropy of the Kerr black hole is then reproduced by Cardy’s formula if one assumes that the CFT has left and right-moving central charges equal to the value cL = cR = 12J, which matches with the value for the left-moving central charge (129View Equation) derived at extremality.16

These observations are consistent with the interpretation of a 2d CFT dual to the Kerr black hole, but the existence of such a CFT is conjectural. For example, there is no known derivation of two Virasoro algebras with central charges cL = cR = 12J from the non-extremal Kerr geometry. Asymptotic symmetry group methods are not directly applicable here because the horizon is not an isolated system. Therefore, it is unclear how these Virasoro algebras could be derived in Kerr. However, as argued in [68Jump To The Next Citation Point], the resulting picture shows a remarkable cohesiveness and only future research can prove or disprove such a CFT interpretation.

Given the successful generalization of the extremal Kerr/CFT correspondence to several independent extremal black hole/CFT correspondences in gravity coupled to matter, as we reviewed above, it is natural to test the ideas proposed in [68Jump To The Next Citation Point] to more general black holes than the Kerr geometry. First, hidden symmetry can be found around the non-extremal Reissner–Nordström black hole [81Jump To The Next Citation Point, 77Jump To The Next Citation Point] under the assumption that the gauge field can be understood as a Kaluza–Klein gauge field, as done in the extremal case [160]. One can also generalize the analysis to the Kerr–Newman black hole [263, 74, 78]. In complete parallel with the existence of an SL (2,ℤ) family of CFT descriptions, there is a class of hidden SL (2, ℝ) × SL (2,ℝ ) symmetries of the Kerr–Newman black hole related with SL (2,ℤ) transformations [75]. What has not been noted in the literature so far is that each member of the SL (2,ℤ ) family of CFTs describes only probes with a fixed ratio of probe angular momentum to probe charge as we will discuss in detail in Section 6.4. Therefore, one needs a family of CFTs to fully describe the dynamics of low energy, low charge and low mass probes. Remarkably, for all cases where a hidden local conformal invariance can be described, the non-extremal black-hole entropy matches with Cardy’s formula using the central charges cR = cL and using the value cL in terms of the quantized conserved charges derived at extremality. Five-dimensional asymptotically-flat black holes were also discussed in [189, 80].

In attempting to generalize the hidden symmetry arguments to four-dimensional black holes in AdS one encounters an apparent obstruction, as we will discuss in Section 6.2. It is expected that hidden symmetries are present at least close to extremality, as illustrated by five-dimensional analogues [46Jump To The Next Citation Point]. However, the structure of the wave equation is more intricate far from extremality because of the presence of complex poles, which might have a role to play in microscopic models [102Jump To The Next Citation Point].

Quite surprisingly, one can also find a single copy of hidden SL (2,ℝ ) symmetry around the Schwarzschild black hole [45], which turns out to be globally defined. As a consequence, no dual temperature can be naturally defined in that case. This hidden symmetry can be understood as a special case of a generalized notion of hidden conformal symmetry around the Kerr geometry [202]. At present, it is unclear how these hidden symmetries fit in the general picture of the Kerr/CFT correspondence since the derivations of the central charges of the CFT dual to Kerr, Reissner–Nordström or Kerr–Newman black holes are done at extremality, which clearly cannot be done in the Schwarzschild case.

All arguments presented in the literature so far have been derived for a probe scalar field. It is not clear if any of these arguments can be generalized to higher-spin fields, and, if such, a generalization would give the same values for the left and right-moving CFT temperatures. It would certainly be interesting to understand whether this is a technical obstruction that can be overcome or whether it is a fundamental limitation in the CFT descriptions.

Hidden symmetries in asymptotically-flat spacetimes only appear in a region close enough to the black hole. It has been suggested that one deform the geometry far from the black hole such that hidden symmetries appear in the entire resulting geometry [108Jump To The Next Citation Point, 107Jump To The Next Citation Point]. The resulting “subtracted” geometries are not asymptotically flat and are supported by additional matter fields [108, 107, 101]. The nature of these geometries and their role in the Kerr/CFT correspondence remains to be clarified. We will therefore not cover these constructions in this review.

In what follows, we present a summary of the derivation of the hidden symmetries of the Kerr–Newman black hole and we discuss their CFT interpretation. We will limit our presentation to the approach of [68Jump To The Next Citation Point] but we will generalize the discussion to the Kerr–Newman black hole, which contains several new interesting features. In particular, we will show that each member of the conjectured SL (2,ℤ) family of CFTs controls part of the dynamics of low energy, low charge and low mass probes. We do not review the matching of absorption probabilities with CFT correlation functions. This matching is very similar to the analysis already performed in Section 5 at near-extremality and it follows from local conformal invariance. As noted in [68Jump To The Next Citation Point], the only difference is that in the present context the region close enough to the horizon is not geometrically a near-horizon region, but it does not affect the discussion.

 6.1 Scalar wave equation in Kerr–Newman
 6.2 Scalar wave equation in Kerr–Newman–AdS
 6.3 Near-region scalar-wave equation
 6.4 Local SL (2,ℝ ) × SL (2,ℝ) symmetries
 6.5 Symmetry breaking to U (1)L × U(1)R
 6.6 Entropy matching

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