We establish the existence of stationary clouds of massive test scalar fields around BTZ black holes. These clouds are zero-modes of the superradiant instability and are possible when Robin boundary conditions (RBCs) are considered at the AdS boundary. These boundary conditions are the most general ones that ensure the AdS space is an isolated system, and include, as a particular case, the commonly considered Dirichlet or Neumann-type boundary conditions (DBCs or NBCs). We obtain an explicit, closed form, resonance condition, relating the RBCs that allow the existence of normalizable (and regular on and outside the horizon) clouds to the system's parameters. Such RBCs never include pure DBCs or NBCs. We illustrate the spatial distribution of these clouds, their energy and angular momentum density for some cases. Our results show that BTZ black holes with scalar hair can be constructed, as the non-linear realization of these clouds.
Ferreira, H. R. C., & Herdeiro, C. A. R. (2017). Stationary scalar clouds around a BTZ black hole. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 773, 129–134. https://doi.org/10.1016/j.physletb.2017.08.017