Third gravitational wave polarization mode in Rastall theory and analogy with f(R) theories

The recent starting of the gravitational wave (GW) astronomy with the events GW150914, GW151226, GW170104, and the very recent GW170814 and GW170817 seems to be fundamental not only in order to obtain new intriguing astrophysical information from our surrounding Universe, but also in order to discriminate among Einstein's general theory of relativity (GTR) and alternative gravitational theories. At the present time, despite the cited events, and in particular the last ones, which are the events GW170814 and GW170817, have put very strong constraint on the GTR, extended theories of gravity have not been completely ruled out. Here we discuss, in our knowledge for the first time in the literature, GWs in the Rastall theory of gravity. In fact, the Rastall theory recently obtained a renovated interest in the literature. We show that there is a profound analogy between GWs in f(R) theories of gravity and GWs in the Rastall theory. This will permit us to linearize the Rastall field equations and to find the corresponding GWs. We will also study the motion of the test masses due to GWs in this theory which could help, in principle, to discriminate between the GTR, f(R) theories and the Rastall theory of gravity.