Nonlinear Spinor Fields in Bianchi type-$VI_0$ spacetime

Within the scope of Bianchi type-$VI_0$ spacetime we study the role of spinor field on the evolution of the Universe. It is found that the presence of nontrivial non-diagonal components of energy-momentum tensor of the spinor field plays vital role on the evolution of the Universe. As a result of their mutual influence there occur two different scenarios. In one case the invariants constructed from the bilinear forms of the spinor field become trivial, thus giving rise to a massless and linear spinor field Lagrangian. According to the second scenario massive and nonlinear terms do not vanish and depending on the sign of coupling constants we have either an expanding mode of expansion or the one that after obtaining some maximum value contracts and ends in big crunch generating spacetime singularity. This result shows that the spinor field is highly sensitive to the gravitational one.