Exact self-consistent solutions to nonlinear spinor field equations in Bianchi type-I space-time

This paper provides a self-consistent class of physically meaningful space-independent exact solutions to nonlinear Dirac equations in the framework of Bianchi type-I cosmological models. The piece of the relevant Lagrangian density which gives rise to the nonlinearity is taken as a function of the type lambda S^n, where S stands for the Dirac-Pauli invariant psi^+gamma_0psi and (lambda,n) is a pair of real numbers, with n>0. A suitable form of the spinor field equations is attained by combining the implementation of the spin-connection formalism together with particular choices of the spacetime metrics and flat-space Dirac matrices. Accordingly, the quantity S turns out to depend only upon the product of the metric coefficients which occur in the spatial parts of the arc elements taken into consideration. The authors then carry out the integration of the entire system of equations, likewise obtaining explicit expressions for the components of the pertinent energy-momentum tensors. Roughly speaking, the basic procedure for building up their class of solutions amounts simply to allowing for specific values of lambda and n. The linear limiting case (lambda=0) was initially taken up because it would presumably clarify the role played by the nonlinearity in the processes of evolution of the models at issue. In this case, they arrive at the result that the cosmological backgrounds are anisotropic in addition to having singularities at the earlier stages of the evolution of the complete system. It is stated explicitly that the isotropy nevertheless takes place when one puts into effect the limit as the value of the time coordinate tends to infinity. It appears that the value 2 of n yields the occurrence of a situation involving the Heisenberg-Ivanenko equation, which also entails the "annihilation" of the contributions due to the nonlinearity. When n>2, the asymptotic situation can be regarded as being essentially the same as the ones occurring in the previous cases if the spinor fields are taken to carry rest-mass. In the case where lambda<0, they thus conclude that the solutions are singular at initial times, the result which arises out of taking lambda>0 amounting on the other hand to the existence of regular solutions. They point out that this latter situation appears to be compatible with the violation of the dominant energy condition involved in the Penrose-Hawking theorem. The asymptotically isotropic behaviour of the solutions is also brought about when 1