It is important to note that one is free to redefine the near-horizon limit parameter as for any . This transformation scales inversely proportionally to . Therefore, the near-horizon geometry admits the enhanced symmetry generator for a detailed derivation. The general near-horizon solution compatible with an symmetry is then given by
For some supersymmetric theories, the values are generically completely fixed by the electric () and magnetic () charges of the black hole and do not depend continuously on the asymptotic value of the scalar fields in the asymptotic region – the scalar moduli. This is the attractor mechanism [141, 250, 140]. It was then realized that it still applies in the presence of certain higher-derivative corrections [199, 200, 198]. The attractor mechanism was also extended to non-supersymmetric extremal static black holes [139, 240, 150, 179]. As a consequence of this mechanism, the entropy of these extremal black hole does not depend continuously on any moduli of the theory.7 The index that captures the entropy can still have discrete jumps when crossing walls of marginal stability in the scalar moduli space [227, 118]. This allows one to account for their black-hole entropy by varying the moduli to a weakly-coupled description of the system without gravity, where states with fixed conserved charges can be counted. Therefore, the attractor mechanism led to an explanation [18, 111] of the success of previous string theory calculations of the entropy of certain nonsupersymmetric extremal black holes [181, 172, 110, 260, 131, 132].
As will turn out to be useful in the development of the Reissner–Nordström correspondence, let us discuss the near-horizon geometry (21) under the assumption that one gauge field can be lifted as a Kaluza–Klein vector to a higher-dimensional spacetime, as discussed in Section 1.2. In the simple model (2), the change of gauge is implemented as the change of coordinates . Using the definition of the electrostatic potential (10) at extremality, it is straightforward to obtain that in the geometry (2) the horizon is generated by the vector field . The change of coordinates (17) combined with with defined in (19) then maps this vector to
Living Rev. Relativity 15, (2012), 11
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