Inequivalence of Coset Constructions for Spacetime Symmetries: Coupling
  with Gravity
  We study how the coupling with gravity of theories with non-linearly realized
space-time symmetries is modified when one changes the parametrization of the
coset. As an example, we focus on the so-called Galileon duality: a
reparametrization which maps a Galilean invariant action into another one which
enjoys the same symmetry. Starting with a standard coupling with gravity, with
a parametric separation between the Planck scale and the typical scale of the
coset, one ends up with a theory without such a separation. In particular an
infinite set of higher-dimension operators are relevant when the
superluminality of the Galileon is measurable in the effective theory. This
addresses an apparent paradox since superluminality arises in the dual theory
even when absent in the original one.
