  Using the Newman-Penrose formalism, we obtain the explicit expressions for
the polarization modes of weak, plane gravitational waves with a massive
graviton. Our analysis is restricted for a specific bimetric theory whose term
of mass, for the graviton, appears as an effective extra contribution to the
stress-energy tensor. We obtain for such kind of theory that the extra states
of polarization have amplitude several orders of magnitude smaller than the
polarizations purely general relativity (GR), h_(+) and h_(x), in the
VIRGO-LIGO frequency band. This result appears using the best limit to the
graviton mass inferred from solar system observations and if we consider that
all the components of the metric perturbation have the same amplitude h.
However, if we consider low frequency gravitational waves (e.g., f_(GW) ~
10^(-7) Hz), the extra polarization states produce similar Newman-Penrose
amplitudes that the polarization states purely GR. This particular
characteristic of the bimetric theory studied here could be used, for example,
to directly impose limits on the mass of the graviton from future experiments
that study the cosmic microwave background (CMB).
