Extremal Rotating Black Holes in the Near-Horizon Limit: Phase Space and Symmetry Algebra

TL;DR

This paper constructs a classical phase space for near-horizon extremal geometries of rotating black holes, revealing a rich symmetry algebra with implications for understanding black hole microstates.

Contribution

It introduces the NHEG phase space with maximal symmetry algebra, including Virasoro structures, for vacuum solutions in arbitrary dimensions.

Findings

In 4D, the phase space has a unique structure with a Virasoro algebra.

In higher dimensions, the symmetry algebra contains multiple Virasoro subalgebras.

The central charge of the algebra is proportional to the black hole entropy.

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 $SL(2,\mathbb R)\times U(1)^{d-3}$ isometries which has vanishing $SL(2,\mathbb R)$ 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. This phase space and in particular its symmetries might serve as a basis for a semiclassical description of extremal rotating black hole microstates.