SPHERICAL SYMMETRIC SOLITONS OF SPINOR FIELDIN GRAVITATIONAL THEORY

This paper deals with soliton-like solutions as model in order to describe the configurations of elementary particles through the interaction of their fields in general relativity. To this end, we have obtainedexact sphericalsymmetric soliton-like solutions to the nonlinear spinor field equations, taking into account the proper gravitational field of elementary particles. The nonlinear terms in the spinor field lagrangian are given by an arbitrary function L N depending on the invariant I T = S 2 − P 2 with 𝑆 = ѱ ѱ and P = i ѱ γ 5 ѱ. It is shown that, under certain choice of the nonlinear terms in the spinor lagrangian, the solutions are regular with a localized energy density, limited total energy only if m=0 (m is the mass

This paper deals with soliton-like solutions as model in order to describe the configurations of elementary particles through the interaction of their fields in general relativity. To this end, we have obtainedexact sphericalsymmetric soliton-like solutions to the nonlinear spinor field equations, taking into account the proper gravitational field of elementary particles. The nonlinear terms in the spinor field lagrangian are given by an arbitrary function L N depending on the invariant I T = S 2 − P 2 with = ѱ ѱ andP = iѱ γ 5 ѱ. It is shown that, under certain choice of the nonlinear terms in the spinor lagrangian, the solutions are regular with a localized energy density, limited total energy only if m=0 (m is the mass parameter in the spinor field equations). In addition, the total charge and the total spin are bounded. The solutions to Heisenberg-Ivanenko nonlinear equation have been also obtained.Let us emphasize that Heisenberg-Ivanenkononlinear spinor field equation possesses soliton-like solutions. Later, the influence of the nonlinear terms in the formation of the fields configurations with limited total energy have been examined. We noted that, in linear case, the obtained solutionsare regular andhaving an unlimited energy density. Nevertheless, exact solutions, including soliton-like configurations existin flat space-time.

…………………………………………………………………………………………………….... Introduction:-
At present, the nonlinear generalization of classical field theory remains one of the possible ways to overcome the difficulties of the theory, which considers elemetary particles as mathematical points. In this approch, elementaryparticles are modeled by soliton-like solutions of corresponding nonlinear equations [1]. The soliton is a regular solution of nonlinear differential equationswith localized energy density, a bounded total energy and stable. It is widely present in many branches in pure science and used for differents purposes. The concept of soliton is thoroughly dealt in a series of papers. A.H. Taub has defined the characterics of inhomogenous plane-symmetric metric of the space-timein [2,3]. Thus, considering plane-symmetric metric of the space-time under the form: ds 2 = e 2γ dt 2 − e 2α dx 2 −e 2β (dy 2 +dz 2 ) where the metric fuctions g 00 = e 2γ , g 11 = −e 2α andg 22 = g 33 = −e 2β are time independent, plane-symmetric solutions have been obtained in a series of articles [4,5]. The authors, in all these activities, investigated the ISSN: 2320-5407 Int. J. Adv. Res. 8(06), 1331-1340 1332 influence of nonlinear terms in the nonlinear fields equations. They also examined the role of the proper gravitational field of elementary particles by solving Einstein's and spinor field equations in the flat space-time. Let us emphasize that the obtained solutions are singular to the metric considered because the components of Riemann-Christoffel tensor are limited but the total charge Q and the total spin S 1 diverge.Later, spherical symmetric solitonlike solutions to the nonlinear spinor filed equations in General Relativity Theory have been obtained in series remarkable papers [6,7,8]. It is demonstrated that the solutions obtained possess a finite value of total charge and total spin. These results confirm the importance of the metric and its geometric properties in the configuration of the structure of elementary particles by soliton model in General Relativity Theory. The symmetries of the gravitational field plays an important role in general relativity. Its importance in the the theory of general relativity has been dealt with byKatzin, Lavine and Davis in a series of papers. An excellent review on fundamental symmetry of the space-time of the general relativity defined by the vanishing Lie derivative of the Riemann curvature tensor may be found in [9]. Then, for details literature on groups of curvature collineation in Riemannian space-times, which admit fields of parallel, refer to [10]. As for the applications of Lie derivative to symmetries, geodesic mappings and first integrals in Riemannian spaces, see [11] and references therein.
The purpose of the paper is to obtain the spherical symmetric solitons in microcosmof spinor field in general relativity, taking into account the owngravitational field of elementary particles. This paper is organized as follows. First, in section 1, we briefly did the literature review on soliton. Section 2 is instented for the model and fundamental equations. Thus, applying the variational principle and usual algebraic manipulations, we established the fundamental equations. The nonlinearity in the spinor field lagrangian is given by an arbitrary function of the invariant I T =I S − I P where I S = (ѱ ѱ) 2 and I P = (iѱ γ 5 ѱ) 2 . Section 3 deals with the results through the fundamental solutions. The section 4 addresses to the discussion. To this end, we have chosen a concrete form of nonlinear terms in the lagrangian density. In the same section, we proved thecontribution of the nonlinearity in the configuration of the geometrical structure of elementary particles. Lastly, Section 5 presents concluding remarks and future work.

Lagrangian, Metric, Basic Fields Equations:-
In the present analysis, the lagrangian of the self-consistent system of spinor and gravitational fields is defined as follows: where R is the scalar curvature, = 8πG c 4 is Einstein's gravitational constant, G is Newton's gravitational constant and c is the speed of light in the vacuum. L N = H (I T ) is an arbitrary function of the invariant functionI T corresponding to the real bilinear formI T = T μν T μν = ѱ σ μν ѱ g μα g νβ ѱ σ αβ ѱ . L N is the nonlinear term of the spinor lagrangiancharacterizing theself-interaction of the spinor field. The following paragraph deals with the metric of the space-time used in this manuscript.
In this paper, instead of the static plane-symmetric metric chosen in [12], we opt for the staticspherical symmetric metric defined in the pseudo-riemannian varieties by the following expression: The signature of the metric is (+1, -1, -1, -1) and the velocity of light is chosen to be unity in natural unities (c=1). The metric functions α, β and γ are time and coordinates angular θ and φ independent. They depend only on the variable ξ = 1 r where r stands for the radial component of the static spherical symmetric metric defined above. They obey the harmonic coordinates condition in the form: The following paragraph deals withthe Einstein's equations, the spinor field equations and the energy-momentum tensor. From the lagrangian (1), through the variational principle and usual algebraic manipulations, we obtain the components of Einstein's tensor and the spinor field equations for the functions ѱ and ѱ in the metric (2) under the coordinate condition (3) as follows: As for the nonlinear spinor field equations, we have: The corresponding energy-momentum tensor of the spinor fieldis: From the spinor field equations (8) and (9), the expression of L S P may be written under the form: With the expression (11), the nontrivial components of the energy-momentum tensor are: In (8)-(10), ∇ μ denotes the covariant derivative of spinor field. As defined in [13], ∇ μ is connected to the functions ѱ and ѱ as follows: where Γ μ ( ) denotes the spinor affine connection matrices. The general expression of Γ μ ( ) is given bythe following equality: In (15) The Dirac's matrices γ δ in flat space-time are chosen as in [14,15]. The relations (15) and (16) lead to : Accordindg to the Einstein's convention, from (16) and (17), we have : Substituting (14) and (18) into (8) and (9), the spinor field equations may be rewritten as follows We get the set following of equations from (19), setting ψ(ξ) = V δ ( );δ = 1, 2, 3 and 4: Let us emphasize that the resolution of the system of equations (22)-(25) consists to determine D(S,P), G(S,P), S and P as functions of e α(ξ) . The previous set of equations has allowed to define the system of equations of the invariant functions S= ψ ψ, P = i ψ γ 5 ψand = ψ γ 5 γ 1 ψ. The following section deals with the results. 1334

1336
Note that from the relation = 1 −2 ( ) , we can study the regular properties of the obtained solutions. We can verify the localisation of the energy density and the energy per unit invariant volume 0 0 3 − . Finally, we can find the total energy of the spinor field and establish the localisation properties of the solutions. In the sequel, we analyze the influence of the nonlinearity of the spinor field of the general solutions obtained previously for concrete nonlinear terms.

Discussion:-
The principle aim, in thissection, is to show the importance of the nonlinearity of the spinor field in order to obtain the regular solutions with bounded energy density and finite total energy (Soliton model). To this end, we choose the nonlinear terms in the spinor field density lagrangian under the following polynomial form: = , > 1(60) with the parameter of nonlinearity and n the power of nonlinearity. Let us emphasize that we consider and analyzeseparately the four cases: (19), we obtain Dirac's linear equation as follows: According to (59), by assuming that 2+ 4+3 = 1, we get ; . (62) From = 1 −2 ( ) , we deduce : According to (12), 0 0 = 0(65) The energy density is not localized in the case under consideration. Let us note that in the linear case soliton-like solutions are absent. Therefore, the nonlinear terms in the nonlinear field equations are very important in the formation of regular localized soliton-like solutions. Moreover, it is necessary to introduce nonlinear terms that characterize the field interactions in the lagrangian. The similar results are obtained in [4]. In the sequel, we deal with Heisenberg-Ivanenko type nonlinear spinor field equation. . In these conditions, we get: As for the metric functions, they are defined under the forms:  Res. 8(06), 1331-1340 1337 The expression of the energy density 0 0 defined by (70) With the condition 1 = 1 2 > 0, we deduce from (37) and (72) the expression of the metric function as follows: Taking into account (55) and (73), we get: The energy density is definedby: Taking into account (55) and (8), we get: As for the energy density of the spinor field, we have where 1 , 2 , 3 , 4 , 5 , 6 , 7 and 8 are integration constants. For details about the determination of the analytic formsof the functions , refer to [7] and references therein. Let us find the charge density and the chronometric invariant spin tensor. Using solutions (88)-(91), as we consider the field configuration to be a static one, the spatial components 1 , 2 3 of the spinor current = vanish. The non-null component 0 is defined as follows: dξ (99) From (93), we conclude that the charge density is localized when ξ ∈ 0, ξ c as well as the charge density unit invariant volumeρ ξ 3 −g . Thus, the total charge of the spinor fieldis bounded. Then, according to (93) and (98), the expression for chronometric invariant tensor of spin S chI 23,0 coincides with the charge density ρ ξ one. Therefore, the conclusion made for ρ ξ and Q will be valid for S chI 23,0 andS 1 . Moreover, S chI 23,0 islocalized and S 1 is limited.

Concluding Remarks:-
In the present research work, we obtained the exact static spherical symmetric solutions to the nonlinear spinor field and Einstein's equations. We investigatedin detail equations with power and polynomial nonlinearities. The obtained solutions are regular with localized energy densityand finite total energy. In addition, the total charge Q and the total spinS 1 are also finite. The forthcoming paper will deal with interaction spinor and scalar fields equations in gravitational theory.

Conflicts of Interest:-
The authors declare no conflits of interest regarding the publication of this paper.