Time-dependent spinor field in a static cylindrically symmetric space-time

Within the scope of a static cylindrically symmetric space-time we study the behavior of a nonlinear spinor field that depends on time and radial coordinates. It is found that the presence of nontrivial non-diagonal components of the energy-momentum tensor (EMT) imposes some restrictions on both the metric functions and the spinor field. While for the time independent spinor field there occur three way restrictions resembling the Bianchi type-I Universe, in this case things become more complicated.