  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.
