1 | For independent arguments pointing to the presence of conformal symmetry around arbitrary black holes, see [63, 245, 180]. We will not discuss these approaches here. | |
2 | Nevertheless, let us mention that some classes of black holes admit a vanishing horizon area and zero temperature limit such that the ratio is finite. Such extremal vanishing horizon (EVH) black holes admit near-horizon limits, which contain (singular) identifications of AdS_{3} that can be used for string model building [157, 98, 116, 130, 115]. Most of the ideas developed for the Kerr/CFT correspondence and its extensions can be developed similarly for EVH black holes [243]. | |
3 | That has been proven for any non-extremal black hole in Einstein gravity coupled to any matter obeying the weak energy condition with hyperbolic equations of motion and asymptotically-flat boundary conditions [161, 163, 254, 86, 143]. The proof has been extended to extremal black holes, to higher dimensions and to anti-de Sitter asymptotics in [171, 170, 85]. | |
4 | The original proofs were limited to non-extremal black holes, which have a bifurcation surface [66, 163]. The proof for extremal black holes can now be found in [170]. | |
5 | Nevertheless, one can describe the process of spontaneous creation of extremal black holes in an electromagnetic field as an analogue to the Schwinger process of particle creation [126]. | |
6 | We thank the anonymous referee for pointing out this reference. | |
7 | In some special cases, there may be some continuous dependence of the near-horizon parameters on the scalar moduli, but the entropy is constant under such continuous changes [17]. | |
8 | We fix the range of as . Since the original black hole has topology and no conical singularities, the functions , also obey regularity conditions at the north and south poles Similar regularity requirements apply for the scalar and gauge fields. | |
9 | In singular limits where both the temperature and horizon area of black holes can be tuned to zero, while keeping the area-over-temperature–ratio fixed, singular near-horizon geometries can be constructed. Such singular near-horizon geometries contain a local AdS_{3} factor, which can be either a null self-dual orbifold or a pinching orbifold, as noted in [33, 29, 135, 23] (see [116] for a comprehensive study of the simplest three-dimensional model and [243] for a partial classification of four-dimensional vanishing area near-horizon solutions of (1)). | |
10 | Our conventions for the infinitesimal charges associated with symmetries is as follows: the energy is , the angular momentum is and the electric charge is . In other words, the electric charge is associated with the gauge parameter . The first law then reads . | |
11 | The sign choice in this expansion is motivated by the fact that the central charge to be derived in Section 4.3 will be positive with this choice. Also, the zero mode is canonically associated with the angular momentum in our conventions. | |
12 | Compère, in preparation, (2012). | |
13 | We thank Tom Hartman for helping deriving this central charge during a private communication. | |
14 | There is a ambiguity in the definition of parameters since Eq. (171) is invariant upon replacing by . We simply chose one of the two identifications. | |
15 | The two-point function (189) has a branch cut, and as a result, one must find a way to fix the choice of relative sign between the two exponentials in (191). The sign is fixed by matching the gravitational computation to be , where is the spin of the corresponding field. | |
16 | Note that at extremality , so the central charge at extremality (129) could as well be written as . However, away from extremality, matching the black hole entropy requires that the central charge be expressed in terms of the quantized charge . | |
17 | Alternatively, it was suggested in [73, 71] that one can describe the dynamics of the scalar field in the near-horizon region using the truncated expansion of around at second order. However, the resulting function has, in addition to the pole , a fake pole , which is not associated with any geometric or thermodynamic feature of the solution. Therefore, the physical meaning of this truncation is unclear. |
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Living Rev. Relativity 15, (2012), 11
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