  We derive an effective equation and action for the propagation of
gravitational waves (GW), encoding the effects of interaction and
self-interaction in a time, frequency and polarization dependent effective
speed. In terms of an appropriately defined effective metric, the effective
action takes the form a massless Klein-Gordon action.
  This effective approach predicts that for theories with matter coupled to the
Einstein frame metric the ratio between gravitational and electromagnetic (EM)
luminosity distance depends on the effective speed, while for Jordan frame
matter coupling it depends on the effective Planck mass. We discuss how the
frequency and polarization dependence of the GW-EM distance ratio provides a
new test of general relativity and its modifications, and more in general of
the interaction of GWs with other fields.
