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In the framework of the Einstein-aether theory we consider a cosmologicalmodel, which describes the evolution of the unit dynamic vector field withactivated rotational degree of freedom. We discuss exact solutions of theEinstein-aether theory, for which the space-time is of the G\"odel-type, thevelocity four-vector of the aether motion is characterized by a non-vanishingvorticity, thus the rotational vectorial modes can be associated with thesource of the universe rotation. The main goal of our paper is to study themotion of test relativistic particles with vectorial internal degree of freedom(spin or polarization), which is coupled to the unit dynamic vector field. Theparticles are considered as the test ones in the given space-time background ofthe G\"odel-type; the spin (polarization) coupling to the unit dynamic vectorfield is modeled using exact solutions of three types. The first exact solutiondescribes the aether with arbitrary Jacobson's coupling constants; the secondone relates to the case, when the Jacobson's constant responsible for thevorticity is vanishing; the third exact solution is obtained using threeconstraints for the coupling constants. The analysis of the exact expressions,which are obtained for the particle momentum and for the spin (polarization)four-vector components, shows that the interaction of the spin (polarization)with the unit vector field induces a rotation, which is additional to thegeodesic precession of the spin (polarization) associated with the universerotation as a whole.

Einstein-aether theory: Dynamics of relativistic particles with spin or polarization in a G\"odel-type universe