4.1 Boundary conditions and asymptotic symmetry algebra

Let us discuss the existence and the construction of a consistent set of boundary conditions that would define “the set of solutions in the near-horizon region of extremal black holes”. Since the near-horizon region is not asymptotically flat or asymptotically anti-de Sitter, one cannot use previous results in those spacetimes to derive the boundary conditions in the near-horizon region. Rather, one has to derive the relevant boundary conditions from first principles. A large literature on the theory of boundary conditions and asymptotic charges exists, see [9Jump To The Next Citation Point, 237Jump To The Next Citation Point, 59Jump To The Next Citation Point, 196, 35Jump To The Next Citation Point, 36Jump To The Next Citation Point] (see also [90Jump To The Next Citation Point] for a review). We will use the Lagrangian methods [35Jump To The Next Citation Point, 36Jump To The Next Citation Point] to address the current problem.

A set of boundary conditions always comes equipped with an asymptotic symmetry algebra. Restricting our discussion to the fields appearing in (1View Equation), the boundary conditions are preserved by a set of allowed diffeomorphisms and U (1) gauge transformations μ (ζ ,Λ ), which act on the fields as

δ g = β„’ g , δ A = β„’ A + ∂ Λ, (ζ,Λ) μνA ζ μνA (ζ,Λ) μ ζ μ μ δ(ζ,Λ)χ = β„’ζχ . (104 )
Asymptotic symmetries are the set of all these allowed transformations that are associated with non-trivial conserved charges. The set of allowed transformations that are associated with zero charges are “pure gauge” or “trivial” transformations. The set of asymptotic symmetries inherits a Lie algebra structure from the Lie commutator of diffeomorphisms and U (1) gauge transformations. Therefore, the asymptotic symmetries form an algebra,
[(ζ ,Λ ),(ζ ,Λ )] ≡ ([ζ ,ζ ],[Λ ,Λ ]), (105 ) m m n n m n m nζ
where [ζm, ζn] is the Lie commutator and
[Λm, Λn]ζ ≡ ζμm∂μ Λn − ζμn∂μΛm. (106 )
Consistency requires that the charge associated with each element of the asymptotic symmetry algebra be finite and well defined. Moreover, as we are dealing with a spatial boundary, the charges are required to be conserved in time. By construction, one always first defines the “infinitesimal variation of the charge” δ𝒬 from infinitesimal variations of the fields around a solution. If δ𝒬 is the exact variation of a quantity 𝒬, the quantity 𝒬 is the well-defined charge and the charges are said to be integrable.

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 [168Jump To The Next Citation Point, 167Jump To The Next Citation Point] 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 d + 1 admit the SO (2, d) asymptotic symmetry algebra for d ≥ 3 [1Jump To The Next Citation Point, 15, 168, 167] and two copies of the Virasoro algebra for d = 2 [58Jump To The Next Citation Point]. 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, 237Jump To The Next Citation Point, 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 r = ∞ and r = − ∞. It was proposed in [156Jump To The Next Citation Point, 159Jump To The Next Citation Point] to build boundary conditions on the boundary r = ∞ such that the asymptotic symmetry algebra contains one copy of the Virasoro algebra generated by

ζ = πœ–(Ο•)∂ − rπœ–′(Ο• )∂ + (subleading terms ), (107 ) πœ– Ο• r Λπœ– = − (f(πœƒ) − e)πœ–(Ο•) + (subleading terms). (108 ) k
Part of the physical motivation behind this ansatz is the existence of a non-zero temperature T Ο• associated with modes corotating with the black hole, as detailed in Section 2.6. This temperature suggests the existence of excitations along ∂ Ο•. The ansatz for Λ πœ– will be motivated in (117View Equation). The subleading terms might be chosen such that the generator ζπœ– is regular at the poles πœƒ = 0,π. This ansatz has to be validated by checking if boundary conditions preserved by this algebra exist such that all charges are finite, well defined and conserved. We will discuss such boundary conditions below. Expanding in modes as11
− inΟ• πœ–(Ο• ) = − e , (109 )
the generators Ln ≡ (ζn, Λn) obey the Virasoro algebra with no central extension
i[Lm, Ln] = (m − n)Lm+n, (110 )
where the bracket has been defined in (105View Equation).

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 (1View Equation), the conserved charges can be found in [97Jump To The Next Citation Point]). 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 [5Jump To The Next Citation Point]). In [156Jump To The Next Citation Point], the following fall-off conditions

gtt = π’ͺ(r2), g tΟ• = kΓ(πœƒ)γ(πœƒ)2r + π’ͺ(1),
gtπœƒ = π’ͺ(1 r), gtr = π’ͺ( 1_ r2), gϕϕ = O(1),
gΟ•πœƒ = π’ͺ(1 r), gΟ•r = π’ͺ(1 r), gπœƒr = π’ͺ( 1_ r2), (111)
gπœƒπœƒ = Γ(πœƒ)α(πœƒ)2 + π’ͺ(1 r), grr = Γ (πœƒ) r2 + π’ͺ( 1_ r3),
were proposed as a part of the definition of boundary conditions. The zero energy excitation condition
δ𝒬 ∂t = 0, (112 )
was imposed as a supplementary condition. We will discuss in Section 4.2 the relaxation of this condition. A non-trivial feature of the boundary conditions (111) – (112View Equation) is that they are preserved precisely by the Virasoro algebra (107View Equation), by ∂ t and the generator (24View Equation) (as pointed out in [5Jump To The Next Citation Point]) and subleading generators. (Note that these boundary conditions are not preserved by the action of the third SL (2,ℝ) generator (29).) It was shown in [156Jump To The Next Citation Point] that the Virasoro generators are finite given the fall-off conditions and well defined around the background NHEK geometry. It was shown in [5Jump To The Next Citation Point] that the Virasoro generators are conserved and well defined around any asymptotic solution given that one additionally regularizes the charges using counter-term methods [96Jump To The Next Citation Point]. Therefore, up to some technical details that remain to be fully understood, it can be claimed that consistent boundary conditions admitting (at least) a Virasoro algebra as asymptotic symmetry algebra exist. The set of trivial asymptotic symmetries comprise two of the SL (2, ℝ) generators. It is not clear if the boundary conditions could be enhanced in order to admit all SL (2,ℝ ) generators as trivial asymptotic symmetries.

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 Te suggests that some dynamics are also present along the gauge field. The associated conserved electric charge 𝒬e can be shown to be canonically associated with the zero-mode generator J = (0,− 1) 0 with gauge parameter Λ = − 1. It is then natural to define the current ansatz

Jn = (0,− e−inΟ•), (113 )
which obeys the commutation relations
i[Lm, Jn] = − nJm+n, i[Jm, Jn] = 0. (114 )
The non-trivial step consists in establishing the existence of boundary conditions such that the Virasoro and the current charges are well defined and conserved. Ongoing work is in progress in that direction.12 One can simplify the problem of constructing boundary conditions by imposing the following additional constraints
δ 𝒬∂t = 0, δ𝒬e = 0, (115 )
which discard the current algebra. Such a simplification was used in [159Jump To The Next Citation Point] and the following boundary conditions were proposed (up to the term eβˆ•k, which was omitted in [159Jump To The Next Citation Point])
e- 1- At = π’ͺ (r), A Ο• = f(πœƒ) − k + π’ͺ (r), 1 A πœƒ = π’ͺ (1 ), Ar = π’ͺ (-2 ), (116 ) r
which are preserved upon acting with the Virasoro generator (107View Equation) – (108View Equation). In particular, the choice of the compensating gauge transformation Λ πœ– (108View Equation) is made such that
β„’ A + ∂ Λ = O (r−1). (117 ) ζπœ– Ο• Ο• πœ–
It can be shown that the Virasoro generators are finite under these boundary conditions.

Let us also discuss what happens in higher dimensions (d > 4). 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, 21Jump To The Next Citation Point, 225, 83]. More precisely, given n compact commuting Killing vectors, one can consider an SL (n,β„€ ) family of Virasoro ansätze by considering all modular transformations on the U (1)n torus [201Jump To The Next Citation Point, 76Jump To The Next Citation Point]. However, preliminary results show that there is no boundary condition that allows simultaneously two different Virasoro algebras in the asymptotic symmetry algebra [21Jump To The Next Citation Point]. Rather, there are mutually-incompatible boundary conditions for each choice of Virasoro ansatz.

Since two U(1) circles form a torus invariant under SL (2,β„€ ) modular transformations, one can then form an ansatz for a Virasoro algebra for any circle defined by a modular transformation of the Ο•1 and Ο• 2-circles. More precisely, we define

′ ′ Ο• 1 = p1Ο•1 + p2Ο•2, Ο• 2 = p3 Ο•1 + p4Ο•2, (118 )
where p1p4 − p2p3 = 1 and we consider the vector fields
(p1,p2) − inΟ•′1 −inΟ•′1 Ln = − e ∂Ο•′1 − ire ∂r + (subleading terms ). (119 )
The resulting boundary conditions have not been thoroughly constructed, but evidence points to their existence [21Jump To The Next Citation Point, 201Jump To The Next Citation Point].

The occurrence of multiple choices of boundary conditions in the presence of multiple U (1) symmetries raises the question of whether or not the (AdS)–Reissner–Nördstrom black hole admits interesting boundary conditions where the U (1) gauge symmetry (which is canonically associated to the conserved electric charge Q) plays the prominent role. One can also ask these questions for the general class of (AdS)–Kerr–Newman black holes.

It was argued in [159Jump To The Next Citation Point, 204Jump To The Next Citation Point] that such boundary conditions indeed exist when the U(1) 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 ∂χ with χ ∼ χ + 2πR χ, the problem amounts to constructing boundary conditions in five dimensions. As mentioned earlier, evidence points to the existence of such boundary conditions [21Jump To The Next Citation Point, 201]. The Virasoro asymptotic-symmetry algebra is then defined using the ansatz

Q − inRχχ- − inRχχ- Ln = − R χe ∂χ − ire ∂r + (subleading terms ) (120 )
along the gauge Kaluza–Klein direction. The same reasoning leading to the SL (2,β„€ ) family of Virasoro generators (119View Equation) would then apply as well. The existence of such a Virasoro symmetry around the Kerr–Newman black holes is corroborated by near-extremal scattering amplitudes as we will discuss in Section 5, and by the hidden conformal symmetry of probes, as we will discuss in Section 6.
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