# Near horizon dynamics of three dimensional black holes

## TL;DR

This paper studies the near horizon dynamics of 3D black holes by reducing Einstein gravity to a scalar field theory, revealing insights into black hole charges and soft hair excitations.

## Contribution

It introduces a Hamiltonian reduction to a scalar field theory on the horizon, elucidating the role of zero modes and soft hair in black hole physics.

## Key findings

- Scalar zero modes encode black hole charges
- Near horizon Hamiltonian is a total derivative, indicating softness of modes
- Hierarchy of boundary conditions lifts degeneracy of soft hair

## 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.