  We study the degrees of freedom of the Proca theory, non-minimally coupled to
gravity. In the Minkowski background, this theory propagates five degrees of
freedom - a massive longitudinal mode, two massive vector ones, and two
massless tensor modes. At first sight, the non-linear coupling between the
metric perturbations and the vector field indicates that both longitudinal and
tensor modes become strongly coupled, at the same scale. This would imply that
no matter how small the photon mass is if non-minimal coupling is taken into
account, gravitational waves would necessarily be strongly coupled. We show
that the way out of this inconsistency is through the introduction of the
disformal coupling to the metric perturbations that resemble the vector-type
disformal transformations. This way, the unphysical coupling between the two
types of modes can be avoided, rendering the model consistent. As a result, we
show that only the longitudinal modes enter a strong coupling regime, while
both tensor and transverse modes remain weakly coupled at all scales up to the
Planck length. Finally, we show how this cures recently reported runaway modes
in curved spacetime.
