Ladder operators and quasinormal modes in Ba\~{n}ados-Teitelboim-Zanelli black holes

TL;DR

This paper explores how ladder operators derived from spacetime conformal symmetries can generate and analyze quasinormal modes of massive scalar fields in BTZ black holes, revealing a systematic way to obtain all overtones from a fundamental mode.

Contribution

It introduces a method using ladder operators based on conformal symmetries to generate all quasinormal modes in BTZ black holes from a fundamental mode.

Findings

Ladder operators can change overtone indices of QNMs.

All QNM overtones can be generated from a fundamental mode.

The method applies under Dirichlet, Neumann, and Robin boundary conditions.

Abstract

We study quasinormal modes (QNMs) of massive Klein-Gordon fields in static Ba\~{n}ados-Teitelboim-Zanelli (BTZ) black holes in terms of ladder operators constructed from spacetime conformal symmetries. Because the BTZ spacetime is locally isometric to the three-dimensional anti-de Sitter spacetime, ladder operators, which map a solution of the massive Klein-Gordon equation into that with different mass squared, can be constructed from spacetime conformal symmetries. In this paper, we apply the ladder operators to the QNMs of the Klein-Gordon equations in the BTZ spacetime. We demonstrate that the ladder operators can change indices of QNM overtones, and all overtone modes can be generated from a fundamental mode when we impose the Dirichlet or Neumann boundary condition at infinity. We also discuss the case with the Robin boundary condition.