Classical stability of BTZ black hole 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 $m^2 l^2=1/2$, 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 confirm 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.