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

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 Schrödinger 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.