Causal structure of black holes in shift-symmetric Horndeski theories

In theories with derivative (self-)interactions, the propagation of perturbations on nontrivial field configurations is determined by effective metrics. Generalized scalar-tensor theories belong in this class and this implies that the matter fields and gravitational perturbations do not necessarily experience the same causal structure. Motivated by this, we explore the causal structure of black holes as perceived by scalar fields in the Horndeski class. We consider linearized perturbations on a fixed background metric that describes a generic black hole. The effective metric that determines the propagation of these perturbations does not generally coincide with the background metric (to which matter fields couple minimally). Assuming that the metric and the scalar respect stationarity and that the surface gravity of the horizon is constant, we prove that Killing horizons of the background metric are always Killing horizons of the effective metric as well. Hence, scalar perturbations cannot escape the region that matter fields perceive as the interior of the black hole. This result does not depend on asymptotics but only on local considerations and does not make any reference to no-hair theorems. We then demonstrate that, when one relaxes the stationarity assumption for the scalar, solutions where the horizons of the effective and the background metrics do not match can be found in the decoupling limit.