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.
http://www.livingreviews.org/lrr-2012-11 |
Living Rev. Relativity 15, (2012), 11
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