Stability of the massive graviton around a BTZ black hole in three dimensions

TL;DR

This paper examines the stability of BTZ black holes under massive graviton perturbations in three-dimensional gravity theories, analyzing both parity-even and parity-odd cases through Schrödinger equations and quasinormal modes.

Contribution

It provides a detailed stability analysis of BTZ black holes in different massive gravity theories, highlighting differences between parity-even and parity-odd cases.

Findings

Parity-even gravity shows stability consistent with the Breitenlohner-Freedman bound.

Parity-odd gravity's stability condition conflicts with s-mode analysis.

The study links stability properties between different massive gravity theories.

Abstract

We investigate the massive graviton stability of the BTZ black hole obtained from three dimensional massive gravities which are classified into the parity-even and parity-odd gravity theories. In the parity-even gravity theory, we perform the $s$-mode stability analysis by using the BTZ black string perturbations, which gives two Schr\"odinger equations with frequency-dependent potentials. The $s$-mode stability is consistent with the generalized Breitenlohner-Freedman bound for spin-2 field. It seems that for the parity-odd massive gravity theory, the BTZ black hole is stable when the imaginary part of quasinormal frequencies of massive graviton is positive. However, this condition is not consistent with the $s$-mode stability based on the second-order equation obtained after squaring the first-order equation. Finally we explore the black hole stability connection between the parity-odd and parity-even massive gravity theories.