Geometrical Finiteness, Holography, and the Bañados-Teitelboim-Zanelli Black Hole

Abstract

We show how a theorem of Sullivan provides a precise mathematical statement of a 3d holographic principle, that is, the hyperbolic structure of a certain class of 3d manifolds is completely determined in terms of the corresponding Teichmüller space of the boundary. We explore the consequences of this theorem in the context of the Euclidean BTZ black hole in three dimensions.

Noncommutative duality and fermionic quasinormal modes of the BTZ black hole

“…As discussed above, another manifestation of the holography for the BTZ black hole appears through its connection with Sullivan's theorem [49,58,59]. It was shown in [60] JHEP06 (2017)107 that instead of using the boundary conditions at infinity, certain monodromy conditions could be imposed on the solutions of a massless KG equation in the background of a BTZ black hole to give exactly the same QNM frequencies.…”

“…Further evidence for holography in the case of the BTZ comes from Sullivan's theorem, which says that for a certain class of manifolds, there is a 1-1 correspondence of the hyperbolic structure as encoded in the metric and the conformal structure of the boundary [57]. It has been shown that the Sullivan's theorem is applicable for the BTZ black hole [49,58,59], which provides an exact kinematical statement of holography for the BTZ. Furthermore, using certain monodromy conditions which can be derived using the Sullivan's theorem, it is possible to calculate the so called nonquasinormal frequencies for the BTZ black hole, which have a form that is identical to the usual QNM frequencies for the BTZ [60,61].…”

The Ricci Flow on Noncommutative Two-Tori

“…It is an asymptotically AdS 2+1 space obtained as a global quotient of anti-de Sitter space by a discrete group of isometries Γ ⊂ SO(2, 2) generated by a single loxodromic element. In Euclidean gravity, the Euclidean version of the BTZ black hole is given by a quotient X q = H 3 /(q Z ) of 3-dimensional real hyperbolic space H 3 (the Euclidean version of AdS 2+1 ) by a subgroup Γ = q Z of PSL(2, C) generated by an element q ∈ C * with |q| < 1, acting on H 3 by (z, y) → (qz, |q|y) in the realization of H 3 as the upper half space C × R * + , see [7], [39], [45]. The space X q obtained in this way describes a spinning black hole whenever q is not purely real.…”

Geometric finiteness, holography and quasinormal modes for the warped AdS 3 black hole

“…For the Euclidean BTZ black hole, a mathematically precise kinematical description of holography was obtained [4,12] using the idea of geometric finiteness [13,14]. The demonstration of the holography for the Euclidean BTZ was based on a mathematical theorem due to Sullivan, which states that for a geometrically finite manifold with boundary, there is a one-to-one correspondence between the hyperbolic structure of the interior and the conformal structure of the boundary [13,14].…”

“…BTZ black hole [1,2] provides an example of the AdS/CFT correspondence [3,4,5]. It has been suggested that there is a unique CFT associated with the BTZ black hole, which provides a toy model of quantum gravity [6].…”