  General relativity predicts the existence of only two tensorial gravitational
wave polarizations, while a generic metric theories of gravity can possess up
to four additional polarizations, including two vector and two scalar ones.
These vector/scalar polarizations are in general generated by the intrinsic new
vector/scalar degrees of freedom of the specific theories of gravity. In this
paper, we show that, with the violation of the Lorentz symmetry in the
framework of the standard model extension, the additional nontensorial
polarizations can be directly excited by the two tensorial degrees of freedom.
We consider the diffeomorphism invariant standard model extension in the
gravity sector with the Lorentz-violating coefficients $\hat{\boldsymbol
s}^{(d)\mu\rho\nu\sigma}$ of the even mass dimension $d\ge4$. In addition to
the extra polarizations induced by the tensor modes, the gravitational wave in
this theory travels at a speed depending on the propagation direction,
experiences dispersion if and only if $d\ge6$, and possesses neither velocity
nor amplitude birefringence. The excitement of the extra polarizations is also
chiral. The antenna pattern functions of interferometers due to such kind of
gravitational waves are generally linear combinations of those for all
polarizations. Detected by pulsar timing arrays and the Gaia satellite, the
stochastic gravitational wave background in this model could induce couplings
among cross correlations, of the redshifts of photons and the astrometric
deflections of the positions of pulsars, for different polarizations. These
characteristics enable the use of interferometers, pulsar timing arrays and
Gaia mission to constrain this model.
