Classical stability of BTZ black holes in new massive gravity

Abstract

We study the stability of the BTZ black hole in the new massive gravity. This is a nontrivial task because the linearized equation around the BTZ black hole background is a fourth-order differential equation. Away from the critical point of ${m}^{2}{\ensuremath{\ell}}^{2}=1/2$, this fourth-order equation is split into two second-order equations: one describes a massless graviton, and the other is designed for a massive graviton, which could be obtained from the Fierz-Pauli action. In this case, calculating quasinormal modes leads to confirming the stability of the BTZ black hole. At the critical point, we derive two left- and right-logarithmic quasinormal modes from the logarithmic conformal field theory. Finally, we identify two $s$ massive modes propagating on the black hole background through the conventional black hole stability analysis.