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 . 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 . Five years later, B. Saha and G. N. Shikin studied the system of nonlinear spinor and scalar fields with minimal coupling in general relativity . 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 . 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   . 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    .
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.
2. Lagrangian, Metric, Basics Equations and Concepts
The Lagrangian density of the nonlinear spinor and gravitational fields is given by:
where the spinor Lagrangian density is
is the nonlinear part of . It describes the self-interaction of the spinor field. is an arbitrary function depending on the invariant . is the scalar curvature or the trace of Ricci’s
tensor. Then, is Einstein’s gravitational constant, G is Newton’s
gravitational constant and c is the speed of light in vacuum. is the 4-components Dirac’s spinor with 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 , where r stands for the radial component of the spherical symmetric metric. The metric functions, , and are stationary and are functions of alone. They verify the harmonic coordinate condition given by the following expression:
Applying the variational principle with respect to the function and its conjugate , we get the nonlinear spinor field equations under the following form :
Varying the lagrangian (1) with respect to the metric tensor we obtain the general form of Einstein’s equations:
where is the Einstein’s tensor; is the Ricci’s tensor; is the Kronecker’s symbol and is the metric energy-momentum tensor of the spinor field. Then, taking into account (7), we find the components of the tensor in the metric (3) under the coordinate condition (4) as follows:
Prime (') in previous equations denotes differentiation with respect to .
The corresponding metric energy-momentum tensor of the spinor field is
Introducing (5) and (6) into (2), takes the following form:
As the function , substituting (15) into (12), we define the nontrivial components of the metric :
In the precedent equations represent Dirac’s matrices in curved space-time . They are connected with Dirac’s matrices in flat space-time by:
where is Minkowski’s metric and are tetradic 4-vectors.
The relation (18) leads
The matrices are chosen as in 
In (2), (5)-(6) and (12), has the covariant derivative of the spinor meaning  . It is linked to the spinor affine connection matrices by:
As for , it takes the following general form:
where are Christoffel’s symbols. From (21), we find
Einstein’s convention leads
By substituting (20) and (23) into (5) and (6), we have
Considering with , for the components of spinor field, we get the following system of equations from (24)
The functions , , and are connected by the relation
In Section 3, we shall resolve the fundamental fields equations.
3. General Analytical Fundamental Solutions
Summing the set of Equations (26)-(29), we obtain the first-order differential equation for the invariant function as follows:
The Equation (31) possesses the evident solution
The relation (32) reflects the natural link between the nonlinear spinor field of elementary particles and their proper gravitational field.
Using the spinor field Equations (24) and (25), the component of the metric energy-momentum tensor may be rewritten in the form:
In the paragraph to follow, we shall resolve Einstein’s equations. To this end, since , we have . Therefore we obtain the following equation
The transformation of the Equation (34) leads to a Liouville equation type having the solutions :
A and D are integration constants and is a function.
The function has the following form:
where h and are integration constants.
By substituting the expressions (35) and (36) into (4), we get the metric function as follows:
Finally we define the relations between the metric functions , and :
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 have
Taking into account and , from (40) we obtain
We obtain the general solutions of the Equation (41) as follows:
The general solutions (42) depend on the arbitrary function . Thus, setting an analytical concrete form of the function , from (42) we can determine explicitly . Then, knowing , we can find out the metric function from (32). Finally, we can get the functions and from (39).
Considering the invariant , we can establish the regularity properties of the obtained solutions. Studying the distribution of the energy per unit invariant volume , we can establish the localization properties of the solutions.
The following paragraph addresses to the determination of the functions . In this optic, we must solve the set of Equations (26)-(29) in more
compact form if we pass to the functions , with . We have:
With the set of Equations (43)-(46) where let us pass to the system of equations depending on functions of the argument , i.e. , . We obtain for the set of equations as follows:
In sequel, we shall transform the Equations (48)-(51) to the second order differential equations. In this perspective, differentiating the Equation (48) and substituting the expression of the function and the expression of its derivative into the result, we obtain:
Similarly differentiating the Equation (51) and introducing into the result the expression of and the expression of its derivative, we obtain the second-order differential equation for the function :
Doing the same operating on the Equations (49)-(50), we find the second-order differential equations obeyed by the functions and as follows:
By summing (54)-(55) and setting , we obtain the following second-order differential equations of the function :
The Equation (58) may be transformed to:
under the condition with .
The Equation (59) possesses the first integral
If , then the Equation (60) has the solution
If , the solution of the equation of (60) is given by:
The difference of Equations (48) and (51), taking into account of (61) and (62), gives:
where and are integration constants.
Solving analogously the Equations (56) and (57), we obtain the following expressions for as follows:
In these conditions, it then follows from the expressions (66) and (67) that:
where , and are integration constants.
Considering the cases where and , let us determine the expressions of the functions . We get for the functions the following expressions:
with and .
Let us note that we can also obtain the expressions of the functions by choosing and . In addition, in the expressions (63) and (70), we can use the minus sign before the integral. By doing so, we don’t lose generality . We pass to the functions by multiplying the
functions obtained in the expressions (71)-(74) by as follows:
The following section deals with the analysis of the general results obtained previously by considering the concrete nonlinear terms of the arbitrary function in the Lagrangian density.
In this section, we choose the concrete type of nonlinear spinor field equations under the form:
where is nonlinearity parameter and is the invariant function. It is convenient to separately analyze the two cases and .
• Firstly, we address the Ivanenko-Heisenberg type nonlinear spinor field equation corresponding to and . From (24) we have:
By substituting into (42) and assuming that , without loss of generality, we obtain:
The invariant function is given by the following expression:
From (32), we deduce the expression of the metric function . Then, from (39) we deduce the expressions of the functions 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 value when . Then, it takes the value when and .
In virtue of (86), the energy density per unit invariant volume is defined in the following way:
where the function has the form:
From (88) the energy density per unit invariant volume of Heisenberg-Ivanenko type equation of a nonlinear spinor field is localized when . Therefore, the total energy is finite.
Let us find an explicit form of the function , . By doing so, we deduce from (63) and (70) the expressions of the functions and by substituting the obtained expression for from (82) into (52) as follows:
Then, the expressions of and are substituted into (71)-(74) to produce the explicit form of the functions that we multiply by
to get explicitly as follows:
Let us emphasize that the Equation (80) has soliton-like solutions. Here, the existence of the soliton-like configurations with localized energy density, finite total energy in Heisenberg-Ivanenko type nonlinear equation is an interesting result.
Using the solutions (90)-(93) we can determine the components of the spinor current vector under the general form
In the case of Heisenberg-Ivanenko type nonlinear equation, the components of the spinor current vector may be rewritten in the following way:
As in this study the configuration is static, the components , and are evident. But only the component is nonzero. With this assumption, we get , , and . From the component , we define the charge density or the chronometric invariant of the spinor field as follows:
where is defined by the expression (89) and
The charge density is localized when . The total charge of the spinor field in the Heisenberg-Ivanenko type nonlinear equation is:
being the center of the field configuration and
From (104) the total charge is finite when .
Let us deal with the spin tensor of the nonlinear spinor field. Its general form is:
Using (106), the spatial density of the spin tensor is:
Thus, we have
The relation (109) leads to the definition of the chronometric invariant of the spatial density as follows:
Thus, the projection of the spin vector on the radial axis has the form:
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 , 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.
• In the linear case, the nonlinearity parameter is equal to zero i.e. . Introducing into the Equation (24) leads to the following expression:
Then, according to (42), we get
The invariant function is given by the following expression:
As for the metric functions we have:
From (16), the energy density is
Let us note that the invariant function and the metric functions , , , are regular. Then, the invariant function is bounded when . But the energy density is unlimited. Here, it is clear that soliton-like solutions do not exist in the linear case. Moreover, from these results it is shown that to obtain the regular localized soliton-like solutions, the nonlinear terms are very important. In the following paragraph, we shall consider the case when the influence of the gravitational field is not taken into account for proving the importance of the proper gravitational field of elementary particles in the configuration of their geometrical structures.
In order to determine the role of the own gravitational field in the formation of regular localized solutions of soliton-like type to Heisenberg-Ivanenko type nonlinear spinor field equations, it is necessary to consider solutions to the Equation (5) in flat space-time when in (2). It then follows that in the space-time without gravitation the set of equations for the functions becomes
From , we obtain the expressions of the functions and as follows
Taking into account the expression (123) and (16) we obtain the expression of the energy density as follows:
In this condition the distribution of the spinor field energy density per unit invariant volume takes the form
The total energy has finite value and given by the following expression:
In the flat space-time, we get the explicit expressions of the functions and under the form:
Let us find the explicit form of the functions . In this perspective, by Substituting (127) into (90)-(93) the system of Equations (119)-(122) has the following solutions:
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.
5. 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.
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