Soliton-Like Spherical Symmetric Solutions of the Nonlinear Spinor Field equations in General Relativity

In this research work, we opt for the static spherical symmetric metric. Thus, taking into account the own gravitational field of elementary particles, we have obtained exact static spheric symmetric solutions of the nonlinear spinor and gravitational fields equations. The nonlinear terms in the spinor lagrangian density characterize the self-interaction of a spinor field. We have investigated in detail equations with power and polynomial nonlinearities. In this case, we have obtained exact regular solutions which have a localized energy density and limited total energy (soliton-like solutions) only if the mass parameter in the spinor field equations is equal to zero. In additional to this, the total charge and the total spin are bounded. We have also shown that in the linear case, soliton-like solutions are absent. But in the flat space-time, the obtained solutions are soliton-like configurations. Therefore, the proper gravitational field of elementary particles, the geometrical properties of the metric and the nonlinear terms in the lagrangian density play a crucial role in the purpose to get the regular solutions with localized energy density and limited total energy.