A set of boundary conditions always comes equipped with an asymptotic symmetry algebra. Restricting
our discussion to the fields appearing in (1), the boundary conditions are preserved by a set of
allowed diffeomorphisms and
gauge transformations
, which act on the fields as
Imposing consistent boundary conditions and obtaining the associated asymptotic symmetry algebra requires a careful analysis of the asymptotic dynamics of the theory. If the boundary conditions are too strong, all interesting excitations are ruled out and the asymptotic symmetry algebra is trivial. If they are too weak, the boundary conditions are inconsistent because transformations preserving the boundary conditions are associated to infinite or ill-defined charges. In general, there is a narrow window of consistent and interesting boundary conditions. There is not necessarily a unique set of consistent boundary conditions.
There is no universal algorithm to define the boundary conditions and the set of asymptotic symmetries.
One standard algorithm used, for example, in [168, 167
] consists in first promoting all exact symmetries of
the background solution as asymptotic symmetries and second acting on solutions of interest with the
asymptotic symmetries in order to generate tentative boundary conditions. The boundary conditions are
then restricted in order to admit consistent finite, well defined and conserved charges. Finally, the set of
asymptotic diffeomorphisms and gauge transformations, which preserve the boundary conditions are
computed and one deduces the full asymptotic symmetry algebra after computing the associated conserved
charges.
As an illustration, asymptotically anti-de Sitter spacetimes in spacetime dimensions admit the
asymptotic symmetry algebra for
[1
, 15, 168, 167] and two copies of the Virasoro
algebra for
[58
]. Asymptotically-flat spacetimes admit as asymptotic symmetry algebra
the Poincaré algebra or an extension thereof depending on the precise choice of boundary
conditions [9, 231, 147, 237
, 13, 12, 16, 37, 38, 92, 261]. From these examples, we learn that the
asymptotic symmetry algebra can be larger than the exact symmetry algebra of the background spacetime
and it might in some cases contain an infinite number of generators. We also notice that several choices of
boundary conditions, motivated from different physical considerations, might lead to different asymptotic
symmetry algebras.
Let us now motivate boundary conditions for the near-horizon geometry of extremal black holes. There
are two boundaries at and
. It was proposed in [156
, 159
] to build boundary conditions
on the boundary
such that the asymptotic symmetry algebra contains one copy of the Virasoro
algebra generated by
Finding consistent boundary conditions that admit finite, conserved and integrable Virasoro charges and
that are preserved by the action of the Virasoro generators is a non-trivial task. The details of these
boundary conditions depend on the specific theory at hand because the expression for the conserved charges
depend on the theory. (For the action (1), the conserved charges can be found in [97
]). Specializing in the
case of the extremal Kerr black hole in Einstein gravity, the problem of finding consistent boundary
conditions becomes more manageable but is still intricate (see discussions in [5
]). In [156
], the following
fall-off conditions
Let us now generalize these arguments to the electrically-charged Kerr–Newman black hole in
Einstein–Maxwell theory. First, the presence of the chemical potential suggests that some dynamics
are also present along the gauge field. The associated conserved electric charge
can be shown to be
canonically associated with the zero-mode generator
with gauge parameter
. It is
then natural to define the current ansatz
Let us also discuss what happens in higher dimensions (). The presence of several independent
planes of rotation allows for the construction of one Virasoro ansatz and an associated Frolov–Thorne
temperature for each plane of rotation [203, 173, 21
, 225, 83]. More precisely, given
compact
commuting Killing vectors, one can consider an
family of Virasoro ansätze by considering all
modular transformations on the
torus [201
, 76
]. However, preliminary results show that there is no
boundary condition that allows simultaneously two different Virasoro algebras in the asymptotic symmetry
algebra [21
]. Rather, there are mutually-incompatible boundary conditions for each choice of Virasoro
ansatz.
Since two circles form a torus invariant under
modular transformations, one can then
form an ansatz for a Virasoro algebra for any circle defined by a modular transformation of the
and
-circles. More precisely, we define
The occurrence of multiple choices of boundary conditions in the presence of multiple
symmetries raises the question of whether or not the (AdS)–Reissner–Nördstrom black hole admits
interesting boundary conditions where the
gauge symmetry (which is canonically associated to the
conserved electric charge
) plays the prominent role. One can also ask these questions for the general
class of (AdS)–Kerr–Newman black holes.
It was argued in [159, 204
] that such boundary conditions indeed exist when the
gauge field can be promoted to be a Kaluza–Klein direction of a higher-dimensional spacetime,
or at least when such an effective description captures the physics. Denoting the additional
direction by
with
, the problem amounts to constructing boundary conditions
in five dimensions. As mentioned earlier, evidence points to the existence of such boundary
conditions [21
, 201]. The Virasoro asymptotic-symmetry algebra is then defined using the ansatz
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Living Rev. Relativity 15, (2012), 11
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