Recently it was pointed out that in shift-symmetric scalar-tensor theories a black hole can have nontrivial scalar hair which depends linearly on time. We develop black hole perturbation theory for such solutions and compute the quadratic action of odd parity perturbations. We show that around all the solutions known so far with such time-dependent scalar hair the perturbations trigger instabilities or are presumably strongly coupled.

• Received 30 October 2015

DOI:https://doi.org/10.1103/PhysRevD.93.064078