Near horizon dynamics of three dimensional black holes

TL;DR

This work shows that near-horizon boundary conditions in AdS$_3$ gravity reduce the bulk theory to a boundary scalar field of Floreanini–Jackiw type, with the zero mode encoding BTZ horizon charges and the Hamiltonian being a total derivative, rendering oscillator modes soft. By organizing a one-parameter hierarchy of boundary conditions (near-horizon, Brown–Henneaux, and a KdV family), the authors derive a spectrum of boundary Hamiltonians $H_N$ built from Gelfand–Dikii polynomials, leading to a KdV-inspired evolution for the horizon current ${rak J}=\Phi^\prime$. A controlled limiting procedure $oldsymbol\varepsilon o0^+$ yields a log-type Hamiltonian that lifts the soft-hair degeneracy and provides a finite-energy spectrum for soft modes, with connections to the fluff proposal and potential implications for black hole microstate counting. The results illuminate how integrable deformations of horizon boundary conditions can regulate soft hair and connect horizon dynamics to known 2D CFT structures (Virasoro, Liouville) and to higher-integrable hierarchies, offering a framework to study horizon entropy and microstructure in three-dimensional gravity.

Abstract

We perform the Hamiltonian reduction of three dimensional Einstein gravity with negative cosmological constant under constraints imposed by near horizon boundary conditions. The theory reduces to a Floreanini-Jackiw type scalar field theory on the horizon, where the scalar zero modes capture the global black hole charges. The near horizon Hamiltonian is a total derivative term, which explains the softness of all oscillator modes of the scalar field. We find also a (Korteweg-de Vries) hierarchy of modified boundary conditions that we use to lift the degeneracy of the soft hair excitations on the horizon.