4.2 Absence of SL (2,ℝ) asymptotic symmetries

The boundary conditions discussed so far do not admit solutions with non-trivial charges under the SL (2,ℝ ) exact symmetry group of the background geometry generated by ζ0,±1 (29). In fact, the boundary conditions are not even invariant under the action of the generator ζ1. One could ask the question if such an enlargement of boundary conditions is possible, which would open the possibility of enlarging the asymptotic-symmetry group to include the SL (2,ℝ ) group and even a Virasoro extension thereof. We will now argue that such enlargement would result in trivial charges, which would not belong to the asymptotic-symmetry group.

First, we saw in Section 2.7 that there is a class of near-extremal solutions (79View Equation) obeying the boundary conditions (111) – (116View Equation) with near-horizon energy near−ext ∕δ𝒬 ∂t = T δ𝒮ext. However, the charge ∕δ𝒬 ∂t is a heat term, which is not integrable when both T near−ext and 𝒮ext can be varied. Moreover, upon scaling the coordinates as t → t∕ α and r → αr using the SL (2,ℝ ) generator (24View Equation), one obtains the same metric as (79View Equation) with T near−ext → T near−ext∕α. If one would allow the class of near-extremal solutions (79View Equation) and the presence of SL (2,ℝ ) symmetries in a consistent set of boundary conditions, one would be forced to fix the entropy 𝒮ext to a constant, in order to define integrable charges. The resulting vanishing charges would not belong to the asymptotic-symmetry algebra. Since there is no other obvious candidate for a solution with non-zero near-horizon energy, we argued in Section 2.9 that there is no such solution at all. If that assumption is correct, the SL (2,ℝ ) algebra would always be associated with zero charges and would not belong to the asymptotic symmetry group. Hence, no additional non-vanishing Virasoro algebra could be derived in a consistent set of boundary conditions. For alternative points of view, see [215, 216, 236, 214].

Second, as far as extremal geometries are concerned, there is no need for a non-trivial SL (2,ℝ ) or second Virasoro algebra. As we will see in Section 4.4, the entropy of extremal black holes will be matched using a single copy of the Virasoro algebra, using the assumption that Cardy’s formula applies. Matching the entropy of non-extremal black holes and justifying Cardy’s formula requires two Virasoro algebras, as we will discuss in Section 6.6. However, non-extremal black holes do not admit a near-horizon limit and, therefore, are not dynamical objects described by a consistent class of near-horizon boundary conditions. At most, one could construct the near horizon region of non-extremal black holes in perturbation theory as a large deformation of the extremal near-horizon geometry. This line of thought was explored in [67Jump To The Next Citation Point]. In the context of the near-extremal Kerr black hole, it was obtained using a dimensionally-reduced model such that the algebra of diffeomorphisms, which extends the SL (2,ℝ ) algebra, is represented on the renormalized stress-energy tensor as a Virasoro algebra. It would be interesting to further define and extend these arguments (which go beyond a standard asymptotic-symmetry analysis) to non-dimensionally-reduced models and to other near-extremal black holes.

Finally, let us also note that the current discussion closely parallels the lower dimensional example of the near-horizon limit of the extremal BTZ black hole discussed in [30Jump To The Next Citation Point]. There it was shown that the near-horizon geometry of the extremal BTZ black hole of angular momentum J is a geometry with SL (2,ℝ ) × U(1) isometry

l2( dr2 r ) ds2 = -- − r2dt2 + -2--+ 2J(dϕ + √---dt)2 , (121 ) 4 r 2J
with ϕ ∼ ϕ + 2π, which is known as the self-dual AdS3 orbifold [100]. It was found in [30Jump To The Next Citation Point] that the asymptotic symmetry group consists of one chiral Virasoro algebra extending the U (1) symmetry along ∂ϕ, while the charges associated with the SL (2,ℝ ) symmetry group are identically zero. These observations are consistent with the analysis of four-dimensional near-horizon geometries (25View Equation), whose constant 𝜃 sections share similar qualitative features with the three-dimensional geometries (121View Equation). It was also shown that an extension of the boundary conditions exists that is preserved by a second Virasoro algebra extending the SL (2,ℝ ) exact-symmetry algebra [24]. All associated charges can be argued to be zero, but a non-trivial central extension still appears as a background charge when a suitable regularization is introduced. However, the regularization procedure does not generalize to the four-dimensional geometries mainly because two features of the three-dimensional geometry (121View Equation) are not true in general (∂ t is null in (121View Equation) and ∂ − 𝜖∂ t ϕ with 𝜖 ≪ 1 is a global timelike Killing vector).
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