Soft Heisenberg hair on black holes in three dimensions

Abstract

Three-dimensional Einstein gravity with negative cosmological constant admits stationary black holes that are not necessarily spherically symmetric. We propose boundary conditions for the near horizon region of these black holes that lead to a surprisingly simple near horizon symmetry algebra consisting of two affine u(1) current algebras. The symmetry algebra is essentially equivalent to the Heisenberg algebra. The associated charges give a specific example of "soft hair" on the horizon, as defined by Hawking, Perry and Strominger. We show that soft hair does not contribute to the Bekenstein-Hawking entropy of Banados-Teitelboim-Zanelli black holes and "black flower" generalizations. From the near horizon perspective the conformal generators at asymptotic infinity appear as composite operators, which we interpret in the spirit of black hole complementarity. Another remarkable feature of our boundary conditions is that they are singled out by requiring that the whole spectrum is compatible with regularity at the horizon, regardless the value of the global charges like mass or angular momentum. Finally, we address black hole microstates and generalizations to cosmological horizons.Comment: 6p

New entropy formula for Kerr black holes

“…In this section we review material that appeared in [1,9], whose conventions and notations we use. We start with the near horizon expansion of non-extremal black holes (or cosmologies).…”

“…The near horizon boundary conditions of [1,9] allow for arbitrary variations of the horizon radius function γ and the rotation function ω but no variations of Rindler acceleration α, i.e., δγ 0 δω and δα = 0. For constant Rindler acceleration the equations of motion imply conservation of these functions in time, ∂ t γ = 0 = ∂ t ω, which are near horizon analogs of the holographic Ward identities ∂ ∓ T ±± = 0 for the usual Brown-Henneaux boundary conditions [20].…”

“…Formula (1) was first derived for black holes in 3-dimensional Anti-de Sitter space within Einstein gravity [1] inspired by related earlier discussions of near horizon boundary conditions [6,7] and the concept of soft hair [8]. During the past 18 months the entropy formula (1) was shown to apply also to flat space cosmologies in Einstein gravity [9], to black holes (flat space cosmologies) in higher spin theories [10] ( [11]) where Bekenstein-Hawking fails, and to higher derivative theories with gravitational Chern-Simons term [12] where Wald's formula fails.…”

“…Besides the entropy itself also Cardyology simplifies [1,9,13] including log-corrections [14], while semi-classical considerations allow for Hardyology [15][16][17][18], by which we mean a counting of explicitly constructed semi-classical black hole microstates that combinatorially reduces to partitions of large integers. Nearly all results so far apply only to three spacetime dimensions, with the exception of [18] that applies Hardyology to extremal Kerr black holes.…”

Near-horizon BMS symmetries as fluid symmetries

“…We will study the fluid interpretation of 3d flat gravity in a companion paper [30]. In the future, it would be interesting to use the fluid perspective to understand the symmetry groups appearing at Rindler horizons [31], with negative cosmological constant [32,33], and in more general theories of gravity (e.g., [34,35]). The remainder of this paper is organized as follows.…”