  Non-linear realizations of spacetime symmetries can be obtained by a
generalization of the coset construction valid for internal ones. The physical
equivalence of different representations for spacetime symmetries is not
obvious, since their relation involves not only a redefinition of the fields
but also a field-dependent change of coordinates. A simple and relevant
spacetime symmetry is obtained by the contraction of the 4D conformal group
that leads to the Galileon group. We analyze two non-linear realizations of
this group, focusing in particular on the propagation of signals around
non-trivial backgrounds. The aperture of the lightcone is in general different
in the two representations and in particular a free (luminal) massless scalar
is mapped in a Galileon theory which admits superluminal propagation. We show
that in this theory, if we consider backgrounds that vanish at infinity, there
is no asymptotic effect: the displacement of the trajectory integrates to zero,
as can be expected since the S-matrix is trivial. Regarding local measurements,
we show that the puzzle is solved taking into account that a local coupling
with fixed sources in one theory is mapped into a non-local coupling and we
show that this effect compensates the different lightcone. Therefore the two
theories have a different notion of locality. The same applies to the different
non-linear realizations of the conformal group and we study the particular case
of a cosmologically interesting background: the Galilean Genesis scenarios.
