Nontensorial gravitational wave polarizations from the tensorial degrees of freedom: Linearized Lorentz-violating theory of gravity

General relativity predicts the existence of only two tensorial gravitational wave polarizations, while a generic metric theory of gravity can possess up to four additional polarizations, including two vectors 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 s & eth;d & THORN;mu rho nu sigma of the even mass dimension d >= 4. 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 >= 6, 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 kinds 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 missions to constrain this model.