Exact Soliton-Like Spherical Symmetric Solutions of the Heisenberg-Ivanenko Type Nonlinear Spinor Field Equation in Gravitational Theory

In the present research work, we have obtained the exact spherical symmetric solutions of Heisenberg-Ivanenko nonlinear spinor field equations in the Gravitational Theory. The nonlinearity in the spinor Lagrangian is given by an arbitrary function which depends on the invariant generated from the bi-linear spinor form . We admit the static spherical symmetric metric. It is shown that a soliton-like configuration has a localized energy density and a finite total energy. In addition, The total charge and total spin are also finite. Role of the metric i.e. the proper gravitational field of elementary particles in the formation of the field configurations with limited total energy, spin and charge has been examined by solving the field equations in flat space-time. It has been established that the obtained solutions are soliton-like configuration with bounded energy density and finite total energy. In order to clarify the role of the nonlinearity in this model, we have obtained exact stat-ical symmetric solutions to the above spinor field equations in the linear case corresponding to Dirac’s linear equation. It is proved that soliton-like solutions are absent.


Introduction
The description of elementary particles by considering the nonlinear phenomena and taking into account the proper gravitational field has been one of the most century popular topics. Indeed, the theory which considers elementary particles as material points has shortcomings. With this theory, it is impossible to obtain a finite value of mass, charge and spin of elementary particles. In this approach, elementary particles are modeled by soliton-like solutions corresponding to nonlinear equations. In 1995, G. N. Shikin investigated the basics of soliton concept in General Relativity [1]. In 1998, A. Adomou and G. N. Shikin have obtained exact plane-symmetric solutions to the spinor field equations with nonlinear terms, which are arbitrary functions of the invariant S ψψ = [2]. Five years later, B. Saha and G. N. Shikin studied the system of nonlinear spinor and scalar fields with minimal coupling in general relativity [3]. In 2012, V. Adanhoumè, A. Adomou, F. P. Codo and M.N. Hounkonnou extended the research work [Gravitation and Cosmology, Vol. 4, 1998, pp.107-113] to exact spherical symmetric soliton-like solutions [4]. In a series of remarkable papers appearing in 2019, Alain Adomou, Jonas Edou and Siaka Massou obtained soliton-like solutions of the nonlinear spinor field equations in plane-symmetric metric and spherical symmetric metric [5] [6] [7]. In all these cases, the solutions are regular, the energy density is localized and the total energy is finite. In addition, the charge density and total spin are not bounded in the plane-symmetric metric but they are localized in the spherical symmetric metric. The geometrical symmetries of the space-time are very important in the gravitational theory. Let us emphasize that, the role of symmetries in general theory of relativity has been introduced by Katzin, Lavine and Davis in a series of papers [8] [9] [10] [11].
The aim of this paper is to describe the configuration of elementary particles by the soliton model in general relativity. In addition, the paper deals with the role of the proper gravitational field and the nonlinear terms in the formation of field configurations with limited total energy, spin and charge.
The paper is organized as follows. In Section 2, we established the basics equations and concepts using the variational principle and usual algebraic manipulations. Section 3 addresses the general analytical fundamental solutions. In Section 4, we analyzed and discussed in detail the principal results. Finally, concluding remarks are outlined in Section 5.

Lagrangian, Metric, Basics Equations and Concepts
The Lagrangian density of the nonlinear spinor and gravitational fields is given by: is Einstein's gravitational constant, G is Newton's gravitational constant and c is the speed of light in vacuum. ψ is the 4-components Dirac's spinor with ψ its conjugate. In the sequel, we shall deal with the metric.
In this present analysis, we admit the static spherical symmetric metric in the form: For simplicity reason, the velocity of light is chosen to equal to unity (c = 1) in natural unity. We define spatial variable as in 1 r ξ = , where r stands for the radial component of the spherical symmetric metric. The metric functions, α , β and γ are stationary and are functions of ξ alone. They verify the harmonic coordinate condition given by the following expression:
Applying the variational principle with respect to the function ψ and its conjugate ψ , we get the nonlinear spinor field equations under the following form [12]: Varying the lagrangian (1) with respect to the metric tensor g µν we obtain the general form of Einstein's equations: where G ν µ is the Einstein's tensor; R ν µ is the Ricci's tensor; ν µ δ is the Kronecker's symbol and T ν µ is the metric energy-momentum tensor of the spinor field. Then, taking into account (7), we find the components of the tensor G ν µ in the metric (3) under the coordinate condition (4) as follows: Prime (') in previous equations denotes differentiation with respect to ξ .
The corresponding metric energy-momentum tensor of the spinor field is ( ) Introducing (5) and (6) into (2), p S L takes the following form: As the function ( ) ψ ψ ξ = , substituting (15) into (12), we define the nontrivial components of the metric T ν µ : In the precedent equations µ γ represent Dirac's matrices in curved space-time [12]. They are connected with Dirac's matrices in flat space-time a γ by: The matrices µ γ are chosen as in [4] In (2), (5)- (6) and (12), µ ∇ has the covariant derivative of the spinor meaning [13] [14]. It is linked to the spinor affine connection matrices As for µ Γ , it takes the following general form: , for the components of spinor field, we get the following system of equations from (24) The functions 1 V , 2 V , 3 V and 4 V are connected by the relation In Section 3, we shall resolve the fundamental fields equations.

General Analytical Fundamental Solutions
Summing the set of Equations (26)-(29), we obtain the first-order differential equation for the invariant function ( ) In the paragraph to follow, we shall resolve Einstein's equations. To this end, Therefore we obtain the following equa- The transformation of the Equation (34) A and D are integration constants and T is a function.
The function T has the following form: where h and 1 ξ are integration constants.
By substituting the expressions (35) and (36) into (4), we get the metric function ( ) α ξ as follows: Finally we define the relations between the metric functions ( ) α ξ , ( ) Equation (9) looks like the first integral of the Equations (8) and (10). It is also a first order differential equation. Then, introducing (33) and (39) into (9) We obtain the general solutions of the Equation (41) as follows: The general solutions (42)  We have: With the set of Equations (43) In sequel, we shall transform the Equations (48)-(51) to the second order differential equations. In this perspective, differentiating the Equation (48) Similarly differentiating the Equation (51) Doing the same operating on the Equations (49)-(50), we find the second-order differential equations obeyed by the functions ( ) By summing (54)-(55) and setting 1 4 U W W = + , we obtain the following second-order differential equations of the function ( ) The Equation (58) may be transformed to: under the condition The Equation (59)  ( ) ( ) where 1 a and 1 b are integration constants.
Solving analogously the Equations (56) and (57) In these conditions, it then follows from the expressions (66) and (67)  Considering the cases where Let us note that we can also obtain the expressions of the functions ( )  (63) and (70), we can use the minus sign before the integral. By doing so, we don't lose generality [15]. We pass to the functions The following section deals with the analysis of the general results obtained previously by considering the concrete nonlinear terms of the arbitrary function ( ) S F I in the Lagrangian density.

Discussions
In this section, we choose the concrete type of nonlinear spinor field equations under the form: where λ is nonlinearity parameter and Introducing (82) into (16), the energy density is defined as follows Let us note that the energy density is bounded when It takes the val- In virtue of (86), the energy density per unit invariant volume is defined in the following way: where the function ( ) ζ ξ has the form: Let us emphasize that the Equation (80) In the case of Heisenberg-Ivanenko type nonlinear equation, the components Journal of Applied Mathematics and Physics of the spinor current vector may be rewritten in the following way: As in this study the configuration is static, the components 1 j , 2 j and 3 j are evident. But only the component 0 j is nonzero. With this assumption, we Note that the spin tensor of the spinor field has a finite value and positive because the integrand is positive. We can conclude that the Heisenberg-Ivanenko type nonlinear equation possesses soliton-like configuration with finite value of the total charge and the total spin. In addition, the metrics functions are stationary and regular. Therefore, these solutions must be used to describe the configuration of elementary particles with mass. In sequel, as mentioned in [2], we shall clarify the influence of nonlinear terms in the nonlinear field equations in the formation of regular localized soliton-like solutions. To this end, we must resolve Dirac's equation and compare its solutions with solutions to nonlinear spinor equations.
The invariant function is given by the following expression: As for the metric functions we have: A. Adomou et al.
From (16), the energy density is Let us emphasize that in the space-time without gravitation, the obtained solutions are soliton-like configurations. Indeed, the solutions are regular with localized energy density and finite total energy. In addition to this, the metric functions are stationary. The consideration of the proper gravitational field and the geometrical proper of the metric are crucial and necessary to obtain the regular solutions having the soliton configuration type. We devote the last section to concluding remarks.

Concluding Remarks
In this manuscript, taking into account the proper gravitational field of elementary particles, we obtained the general solutions of Einstein and nonlinear spinor field equations. We analyzed in particular the Heisenberg-Ivanenko type nonlinear spinor field equations. We note that the solutions of Heisenberg-Ivanenko equation are regular and possess a bounded energy density and limited total energy. Similarly, the metric functions are stationary. The total charge and the total spin are finite quantities as well. We demonstrated that the soliton-like solutions exist in flat space-time and absent in linear case. The nonlinearity of the spinor field vanishes in the space-time without gravitation. Therefore, we note that, the gravitational field is nonlinear by nature and its nonlinearity induces the nonlinearity of the spinor field. In order to extend the present analysis, the forthcoming paper will address to Dirac equation of spinor field in curved space-time in gravitational theory: spherical symmetric soliton-like solutions.