Spherical Symmetric Solitons of Interacting Spinor, Scalar and Gravitational Fields in General Relativity

The concept of soliton as regular localized stable solutions of nonlinear differential equations is being widely utilized in pure science for various aims. In present analysis, the soliton concept is used as a model in order to describe the configurations of elementary particles in general relativity. To this end, our study deals with the spherical symmetric solitons of interacting Spinor, Scalar and Gravitational Fields in General Relativity. Thus, exact spherical symmetric general solutions to the interaction of spinor, scalar and gravitational field equations have been obtained. The Einstein equations have been transformed into a Liouville equation type and solved. Let us emphasize that these solutions are regular with localized energy density and finite total energy. In addition, the total charge and spin are limited. Moreover, the obtained solutions are soliton-like solutions. These solutions can be used in order to describe the configurations of elementary particles.


Introduction
The theory of solitons in general relativity was first elaborated by G. N. Shikin in 1995. He formulated the requirements to be fulfilled by solitons [1]. His research work allowed an intensive study on soliton in general relativity by many authors.
In a series of papers, exact plane-symmetric solutions to the spinor and gravita-tional field equations have been obtained. The role of gravitational field and the nonlinear terms in the formation of the field configurations with limited total energy, spin and charge have been thoroughly investigated. Let us emphasize that the total charge and spin of the self-consistent system of spinor and gravitational field equations are unlimited. The divergence of the total charge and spin is related to the non-consideration of the torsion and the properties of the static plane-symmetric utilized [2] [3] [4]. For an excellent review of the interacting scalar and spinor fields in plane-symmetric metric refer to [5] [6] [7]. Note that in [5] [6] [7] the obtained solutions are exact regular with localized energy density and finite total energy. Nevertheless, the total charge and spin are not bounded. The unlimited problem of the charge and the spin is resolved in a series of interesting articles [8]- [13]. The gravitational field is given by a spherically symmetric metric.
The aim of the paper was to study the role of the interaction of nonlinear spinor, scalar and gravitational fields in the formation of configurations with localized energy density and limited energy, spin and charge of the spinor field.
The paper is organized as follows. Section 2 deals with general relativistic equations. The lagrangian of the self-consistent interaction spinor, scalar and gravitational fields has been defined. The gravitational field in our study is given by

General Relativistic Equations
The lagrangian of the self-consistent system of interaction between spinor, scalar and gravitational fields may be written under the following form [5]: L and int L correspond respectively to gravitational field lagrangian, free spinor field lagrangian, free scalar field lagrangian and interaction lagrangian. They are defined as follows: In the sequel, in ordor to simplify the expressions, we shall consider: The grvitational field in our case is given by a spherical symmetric space-time via the metric which is defined under the following form: For simplicity reason, the speed of light has been taken to be unity (c = 1). The metric functions α , β and γ are time and angular coordinates θ and ϕ independent. They are functions of spatial variable ξ alone which is defined as in [8] in the form 1 r ξ = , where r stands for the radial component of the spherical symmetric metric. These metric functions obey the harmonic coordinates conditions: Variation of (1) with respect to the spinor field ψ and its conjugate ψ , we establish the nonlinear spinor field equations as follows: Similarly, varying of (1) with respect to the scalar field we obtain the following scalar field equation: ( ) Then, the general form of Einstein's field equation is: In virtue of (8) and (9), expression (13) becomes: where prime (' ) denotes differentiation with respect to ξ , T ν µ is the energy momentum tensor of the spinor, scalar fields and its interaction, G ν µ is Einstein's tensor, R ν µ is Ricci's tensor and ν µ δ is Kronecker's symbol which is 0 if µ ν ≠ and 1 if µ ν = .
The metric energy-momentum tensor of the interaction of the spinor and scalar fields field can be written as follows: Taking into account (10) and (11), we rewrite m L under the form: Taking into account (19) and (21), let us try to rewrite explicitly the non null components of the metric energy-momentum tensor T ν µ . In this optic, we obtain: In expressions (10), (11)  or .
In the above equations, µ γ are Dirac's matrices in curved space-time. In order to define µ γ , let us use the equalities: With the relation (25), we have: For the matrices a γ in flat space-time, we take [16]: At present, let us define the affine connection matrices of the spinor. To this end, the general form of In (27) Equation (12) has solution: According to Einstein's convention of sommation, we get: Then, using expressions (24), (29) and (30), we can rewrite Equations (10) and (11) as follows: (31), we obtain the following set of equations: The functions 1 V , 2 V , 3 V and 4 V are connected by the relation: The following section deals with the main results.

Main Results
Summing the set of Equations (33)-(36), we infer that the invariant function: satisfies a firt order differential equation as follows: Equation (40) has solution: Expression (41) T may be rewritten in the following form: In the following paragraph, we shall solve Einstein's field equations in order to determine the general expressions of the metric functions α , β and γ and then the link which exits between them.
In this perspective, in virtue of 0 2 0 2 T T = , substraction of Einstein's Equations (14) and (16) A and D are integration constants. T is a function. It is defined under the form: Then, the metric functions ( ) α ξ , ( ) β ξ and ( ) γ ξ are connected as follows: Let us note that Einstein's Equation (15) is a first integral of Equations (14) and (16). It is also a first order differential equation. By substituting, (48) and (42) into (15), we obtain: Equation (50) Setting a concrete form of the function ( ) F S , from (7) we can find ( ) where: Let us re-express Equations (52)-(55) to the function of argument ( ) In these conditions, we have the following set of equations for the functions Let us pass from the set of first-order differential Equation Under the condition Substracting Equations (57) and (60) and taking into account (66), we get: ( ) From expressions (66) and (68), we define the functions 1 W and 4 W as follows: The same operating on Equations (58) and (59) leads to the following expressions: The exact functions  . In addition to this, we have 1 The component 0 j determines the charge density of the spinor field and scalar field in interaction: The charge density is continuous localized function.
The total charge of the interaction system of spinor and scalar fields is: In the integral (84), c ξ denotes the center of the fields configurations.
The spin tensor of the spinor field reads: The analytical expression ,0 , , 1; 2;3 ik S i k = defines the spatial density of the spin vector. In virtue of (85), we have: It follows from (86) that:

Discussion
In this section, we shall analyze the general results obtained in the previous sec-Journal of High Energy Physics, Gravitation and Cosmology By substituting (95) into (51), without losing the generality we can consider massless spinor and scalar fields, according to the theory of Heisenberg [17], we obtain: The function S ψψ = is a continuous and limited function.
Introducing the relations (94) and (96) into (22), we define the energy density of the interaction of spinor and scalar fields: where the function ( ) υ ξ is defined by the following expression: The distribution of the energy density per unit invariant volume is given by the expression:

Concluding Remarks
Taking