  In this paper we define covariant Galilean transformations in curved
spacetime and find all scalar field theories invariant under this symmetry. The
Slotheon is a Galilean invariant scalar field with a modified propagator such
that, whenever gravity is turned on and energy conditions are not violated, it
moves "slower" than in the canonical set-up. This property is achieved by a
non-minimal derivative coupling of the Slotheon to the Einstein tensor. We
prove that spherically symmetric black holes cannot have Slotheonic hairs. We
then notice that in small derivative regimes the theory has an asymptotic local
shift symmetry whenever the non-canonical coupling dominates over the canonical
one.
