Extremal rotating black holes in the near-horizon limit: Phase space and symmetry algebra

Abstract

We construct the NHEG phase space, the classical phase space of Near-Horizon Extremal Geometries with fixed angular momenta and entropy, and with the largest symmetry algebra. We focus on vacuum solutions to d dimensional Einstein gravity. Each element in the phase space is a geometry with SL(2, R) × U (1) d−3 isometries which has vanishing SL(2, R) and constant U (1) charges. We construct an on-shell vanishing symplectic structure, which leads to an infinite set of symplectic symmetries. In four spacetime dimensions, the phase space is unique and the symmetry algebra consists of the familiar Virasoro algebra, while in d > 4 dimensions the symmetry algebra, the NHEG algebra, contains infinitely many Virasoro subalgebras. The nontrivial central term of the algebra is proportional to the black hole entropy. The conserved charges are given by the Fourier decomposition of a Liouville-type stress-tensor which depends upon a single periodic function of d − 3 angular variables associated with the U (1) isometries. This phase space and in particular its symmetries can serve as a basis for a semiclassical description of extremal rotating black hole microstates.

Near-horizon extremal geometries: coadjoint orbits and quantization

“…It was shown that the Virasoro algebra of Near Horizon Extreme Kerr consists of symplectic symmetries 1 [25,26] that do not actually require fall-off conditions for the metric; instead one can construct a family of mutually diffeomorphic but physically inequivalent solutions that span a phase space of "boundary gravitons".…”

“…In this work we study the symplectic symmetry group of the near-horizon region of a large class of extremal black holes in diverse dimensions, specifically the Near-Horizon Extremal Geometries (NHEGs) of [25,26,33]. These are solutions of (n + 4)-dimensional vacuum Einstein equations with SL(2, R)×U(1) n+1 isometry.…”

“…Each NHEG orbit can be seen as a set of physically inequivalent field configurations dressing a given background, 1 The name "symplectic symmetry" was coined in [24] for cases where the pre-symplectic density ω vanishes. Symplectic (as opposed to asymptotic) symmetries are such that surface charges can be defined on any codimension two compact spacelike surface -not necessarily at infinity [25][26][27][28]. Apart from NHEGs, examples of symplectic symmetries include Brown-Henneaux transformations of Bañados geometries [27,[29][30][31] and ADM charges [28].…”

“…In this section we review the Near-Horizon Extremal Geometries studied in [25,26,33]; they solve the (n + 4)-dimensional vacuum Einstein equations and have an SL(2, R)×U(1) n+1 isometry group.…”

“…For the NHEG (2.1), an interesting family of vector fields generating symplectic symmetries is given by [25] …”