  We analyze the polarization content of gravitational waves in Horndeski
theory. Besides the familiar plus and cross polarizations in Einstein's General
Relativity, there is one more polarization state which is the mixture of the
transverse breathing and longitudinal polarizations.The additional mode is
excited by the massive scalar field. In the massless limit, the longitudinal
polarization disappears, while the breathing one persists. The upper bound on
the graviton mass severely constrains the amplitude of the longitudinal
polarization, which makes its detection highly unlikely by the ground-based or
space-borne interferometers in the near future. However, pulsar timing arrays
might be able to detect the polarization excited by the massive scalar field.
Since additional polarization states appear in alternative theories of gravity,
the measurement of the polarizations of gravitational waves can be used to
probe the nature of gravity. In addition to the plus and cross states, the
detection of the breathing polarization means that gravitation is mediated by
massless spin 2 and spin 0 fields, and the detection of both the breathing and
longitudinal states means that gravitation is propagated by the massless spin 2
and massive spin 0 fields.
