Abstract
We consider quantum field theory near the horizon of an extreme Kerr black hole. In this limit, the dynamics is well approximated by a tower of electrically charged fields propagating in an SL(2,\mathbb{R})SL(2,ℝ) invariant AdS_22 geometry endowed with a constant, symmetry preserving background electric field. At large charge the fields oscillate near the AdS_22 boundary and no longer admit a standard Dirichlet treatment. From the Kerr black hole perspective, this phenomenon is related to the presence of an ergosphere. We discuss a definition for the quantum field theory whereby we ‘UV’ complete AdS_22 by appending an asymptotically two dimensional Minkowski region. This allows the construction of a novel observable for the flux-carrying modes that resembles the standard flat space SS-matrix. We relate various features displayed by the highly charged particles to the principal series representations of SL(2,\mathbb{R})SL(2,ℝ). These representations are unitary and also appear for massive quantum fields in dS_22.
Highlights
The amount of angular momentum that can be acquired by a four-dimensional black hole is bounded by the square of its energy
The geometry acquires an infinitely deep AdS2 throat and the near horizon isometries are enhanced to an S L(2, ) × U(1), which include the conformal group in one-dimension
As suggested by the geometry itself, this problem is intimately connected to quantum field theory of a charged particle propagating in a fixed AdS2 geometry in the presence of a background electric field [32, 47,48,49]
Summary
The amount of angular momentum that can be acquired by a four-dimensional black hole is bounded by the square of its energy. The geometry acquires an infinitely deep AdS2 throat and the near horizon isometries are enhanced to an S L(2, ) × U(1), which include the conformal group in one-dimension Though it is clear how this happens at the level of general relativity, a microscopic understanding of this limit is surprisingly challenging. It is a curious feature that the symmetry group of de Sitter and anti-de Sitter are the same in two-dimensions This may allow for a simpler bridge between our understanding of holography in AdS2 and dS2 [33,34] (see [35,36,37,38,39,40] for related discussions). In this note we construct and analyse quantum fields near the horizon of an extreme Kerr black hole and assess several quantum states which behave interestingly under the symmetries at hand. Certain technical aspects and a discussion on dS2 can be found in the appendix
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