Wiggling Throat of Extremal Black Holes

TL;DR

This paper constructs a phase space of near-horizon extremal black hole geometries, revealing an infinite-dimensional symmetry algebra extending Virasoro, with implications for understanding black hole entropy.

Contribution

It explicitly builds the phase space of extremal black hole near-horizon geometries and identifies the associated symplectic symmetry algebra extending Virasoro.

Findings

The phase space is parametrized by a single periodic function on a torus.

The symmetry algebra is an extension of Virasoro with a central charge related to entropy.

Explicit derivation of conserved charges and symplectic structure.

Abstract

We construct the classical phase space of geometries in the near-horizon region of vacuum extremal black holes as announced in [arXiv:1503.07861]. Motivated by the uniqueness theorems for such solutions and for perturbations around them, we build a family of metrics depending upon a single periodic function defined on the torus spanned by the $U(1)$ isometry directions. We show that this set of metrics is equipped with a consistent symplectic structure and hence defines a phase space. The phase space forms a representation of an infinite dimensional algebra of so-called symplectic symmetries. The symmetry algebra is an extension of the Virasoro algebra whose central extension is the black hole entropy. We motivate the choice of diffeomorphisms leading to the phase space and explicitly derive the symplectic structure, the algebra of symplectic symmetries and the corresponding conserved charges. We also discuss a formulation of these charges with a Liouville type stress-tensor on the torus defined by the $U(1)$ isometries and outline possible future directions.