One usually expects that conserved charges are captured by highly-symmetric solutions. From the theorems presented in Section 2.8, we infer that in (AdS)–Einstein–Maxwell theory there is no candidate non-trivial near-horizon solution charged under the symmetry ( symmetry when electric charge is present), except for a solution related via a diffeomorphism to the near-horizon geometry. If the conjecture presented in Section 2.8 is correct, there is no non-trivial candidate in the whole theory (1). One can then argue that there will be no solution – even non-stationary – with non-zero mass or angular momentum (or electric charge when a Maxwell field is present) above the background near-horizon geometry, except solutions related via a diffeomorphism.

In order to test whether or not there exist any local bulk dynamics in the class of geometries, which asymptote to the near-horizon geometries (25), one can perform a linear analysis and study which modes survive at the non-linear level after backreaction is taken into account. This analysis has been performed with care for the spin 2 field around the NHEK geometry in [4, 122] under the assumption that all non-linear solutions have vanishing charges (which is justified by the existence of a Birkoff theorem as mentioned in Section 2.8). The conclusion is that there is no linear mode that is the linearization of a non-linear solution. In other words, there is no local spin 2 bulk degree of freedom around the NHEK solution. It would be interesting to investigate if these arguments could be generalized to scalars, gauge fields and gravitons propagating on the general class of near-horizon solutions (25) of the action (1), but such an analysis has not been done at that level of generality.

This lack of dynamics is familiar from the geometry [207], which, as we have seen in Sections 2.2-2.3, is the static limit of the spinning near-horizon geometries. In the above arguments, the presence of the compact was crucial. Conversely, in the case of non-compact horizons, such as the extremal planar AdS–Reissner–Nordström black hole, flux can leak out the boundary and the arguments do not generalize straightforwardly. There are indeed interesting quantum critical dynamics around near-horizon geometries [136], but we will not touch upon this topic here since we concentrate exclusively on compact black holes.

Living Rev. Relativity 15, (2012), 11
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