Critical collapse in 2+1 dimensional AdS spacetime: quasi-CSS solutions and linear perturbations

We construct a one-parameter family of exact time-dependent solutions to 2+1 gravity with a negative cosmological constant and a massless minimally coupled scalar field as source. These solutions present a continuously self-similar (CSS) behaviour near the central singularity, as observed in critical collapse, and an asymptotically AdS behaviour at spatial infinity. We consider the linear perturbation analysis in this background, and discuss the crucial question of boundary conditions. These are tested in the special case where the scalar field decouples and the linear perturbations describe exactly the small-mass static BTZ black hole. In the case of genuine scalar perturbations, we find a growing mode with a behavior characteristic of supercritical collapse, the spacelike singularity and apparent horizon appearing simultaneously and evolving towards the AdS boundary. Our boundary conditions lead to the value of the critical exponent $\gamma = 0.4$.