Exact self-consistent solutions to the interacting spinor and scalar field equations in Bianchi type-I space-time

This paper is initially concerned with describing the dynamics of a system of interacting Dirac and uncharged d'Alembert fields within the framework of Bianchi type-I spacetimes. Roughly speaking, the interacting piece of the relevant Lagrangian density involves the coupling of the usual product of derivatives of the scalar field with a function Phi that depends explicitly upon an interaction parameter and the Dirac-Pauli invariant. This function is prescribed in such a way that the interaction gets switched off when the limit, as the parameter tends to zero, is effectively allowed for. The authors accordingly write the spacetime metric in a diagonalized form upon working out their least-action principle, with the functional dependence of the spatial metric coefficients being taken to involve only the time coordinate. Einstein's equations thus constitute a reduced set of four statements whose left-hand sides carry only suitable combinations of time-derivative pieces involving those coefficients. The equation for the former Dirac field turns out to carry an additional term which is obtained from the interacting piece by simply taking the derivative of Phi with respect to the corresponding conjugate field. An explicit expression for the pertinent energy-momentum tensor, which reflects the structural features of the Lagrangian density utilized, is also provided. Subsequently, the authors make use of the vierbein formalism to obtain a space-independent class of exact self-consistent solutions for the interacting fields being dealt with. An apparently familiar result arising out of this situation is that the Dirac-Pauli invariant appears to be proportional to the inverse of the square root of the absolute value of the determinant formed by the metric coefficients. It is emphasized that one needs to use only three of the cosmological equations upon carrying through the entire integration process, the other equation being then taken into consideration in connection with the verification of the correctness of the solutions. The authors point out that it is worthwhile to investigate the minimal coupling limiting case insofar as it presumably may shed some light on the problem concerning the specification of the role played by the interaction in the evolution of the cosmological models at issue.