  The invariance of physical observables under disformal transformations is
considered. It is known that conformal transformations leave physical
observables invariant. However, whether it is true for disformal
transformations is still an open question. In this paper, it is shown that a
pure disformal transformation without any conformal factor is equivalent to
rescaling the time coordinate. Since this rescaling applies equally to all the
physical quantities, physics must be invariant under a disformal
transformation, that is, neither causal structure, propagation speed nor any
other property of the fields are affected by a disformal transformation itself.
This fact is presented at the action level for gravitational and matter fields
and it is illustrated with some examples of observable quantities. We also find
the physical invariance for cosmological perturbations at linear and high
orders in perturbation, extending previous studies. Finally, a comparison with
Horndeski and beyond Horndeski theories under a disformal transformation is
made.
