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

Siaka Massou,Jonas Edou,Alain Adomou

doi:10.11648/j.ijamtp.20190504.14

Abstract

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.

Highlights

In many models elaborated in the pure science in order to describe the configuration of elementary particles, the gravitational theory is absent
We can cite for example super symmetry super strings (SUSY) model, the standar model (SM) and the theories of great unification (GUT)
Poplawski [2], it proved that the metric-affine lagrangian of Ponomarev and Obukhov for the unified gravitational and electromagnetic field is linear in the Ricci scalar and quadratic in the tensor of homothetic curvature

Summary

Introduction

In many models elaborated in the pure science in order to describe the configuration of elementary particles, the gravitational theory is absent. We can cite for example super symmetry super strings (SUSY) model, the standar model (SM) and the theories of great unification (GUT). These models of elementary particles adequately describe obervations. The elementary particles have been extensively examined at the level of special relativity, firstly, in the context of Abelian theories and later in non-Abelian theories. In this case, the influence of gravitational theory is not taken into account. It proved that as in the Ponomarev-Obukhov [3] formulation, the generally covariant Dirac lagrangian gives rise to the standard spinor source for the Einstein-Maxwell equations, while the spinor field obeys the nonlinear Heisenberg-Ivanenko equation with

Results

Discussion

Conclusion