Spinors in cylindrically-symmetric space-time

Within the scope of a static cylindrically-symmetric space-time we studied the role of nonlinear spinor field in the formation of different astrophysical objects. It is found that the energy-momentum tensor (EMT) of the spinor field possesses nontrivial non-diagonal components. This fact imposes three way restrictions on the space-time geometry and/or on the components of the spinor field. It should be noted that such situation occurs in Bianchi type-I cosmological model as well, but while in BI model a specific type of restriction leads to the vanishing spinor field nonlinearity, this is not the case in this model. Moreover, while in a static spherically-symmetric space-time the presence of non-trivial non-diagonal components of EMT of the spinor field has no effect of the space-time geometry, in static cylindrically-symmetric space-time it influences both space-time geometry and spinor field.


I. INTRODUCTION
In recent years spinor field is being used in cosmology by many authors [1][2][3][4][5][6]. The ability of spinor field to simulate different kind of source fields such as perfect fluid, dark energy etc. [7,8] allows one to study the evolution of the Universe at different stages and consider the spinor field as an alternative model of dark energy.
To our knowledge, except FRW model given in Cartesian coordinates, in all other space-time spinor field possesses nontrivial non-diagonal components of the energy-momentum tensor. This very fact imposes severe restrictions on the geometry of space-time and/or on the components of the spinor field [9]. As far as static spherically symmetric space-time is concerned, the presence of non-diagonal components of EMT imposes restrictions on the spinor field only [10].
Since the spinor field was initially introduced in quantum theory, its introduction in general relativity and cosmology gives rise to a number of questions: whether the spinor field is still quantum and, if not, how one can justify a classical spinor. In our opinion, spinors can be treated in two ways: either as Grassmann variables in quantum field theory, or as classical complex projective coordinates in the spirit of Dirac-Sommerfeld-Brioski [11][12][13]. In the second case, they describe the condensation of "quark-antiquarks" and are ordinary classical fields [14].
Note that there were attempts to introduce spinor fields into the Einstein system exploiting both quantum and classical interpretations. As early as 1974, Isham and Nelson considered a Fermi field coupled to the gravitational field given by the Friedmann-Robertson-Walker (FRW) metric [15]. An attempt to treat spinor as a Grassmann variable was undertaken in [16]. Arguing that a quantum matter can be used as a source for the classical field while the quantum aspects of the field itself can be ignored, Dolan has studied the Chiral Fermions and the torsion arising from it within the scope of FRW geometries in the early Universe [ [17].
As it was mentioned earlier, recently spinor field is being used in astrophysics. Most of these works were done within the scope of static spherically-symmetric space-time [10,14,18]. Since a number of astrophysical objects are given by cylindrically-symmetric space-time in this report we plan to consider the spinor field within this model. Note that in a 2014 paper Shikin and Yushenko considered nonlinear spinor field in a static cylindrically-symmetric space-time, but the presence of non-diagonal components were not taken into account there [19]. In this paper I plan to address those problems overlooked there and see if spinor field can be exploited to construct different types of configurations seen in astrophysics.

II. BASIC EQUATIONS
The action we choose in the form where κ = 8πG is Einstein's gravitational constant, R is the scalar curvature and L sp is the spinor field Lagrangian given by [2] To maintain the Lorentz invariance of the spinor field equations the self-interaction (nonlinear term) F = F(K) is constructed as some arbitrary functions of invariants generated from the real bilinear forms. On account of Fierz equality in (2) we set K = K(I, for K we obtain one of the following expressions {I, J, I + J, I − J}. Here I = S 2 and J = P 2 are the invariants of bilinear spinor forms with S =ψψ and P = ıψγ 5 ψ being the scalar and pseudo-scalar, respectively. In (2) λ is the self-coupling constant.
The covariant derivatives of spinor field takes the form [2] where Γ µ is the spinor affine connection which can be defined as [2] The γ matrices obey the following anti-commutation rules Let us consider the cylindrically symmetric space-time give by where γ, α, β and µ are the functions of u only.
The tetrad we will choose in the form The nontrivial Christoffel symbols for (5) are Then from (4) we find the following Γ µ 's: The spinor field equations corresponding to (2) are [2] On account of (7) from (2) one finds that L sp = λ (2KF K − F) .
Let the spinor field be a function of u only, then in view of (6) the spinor field equations can be written as where prime denotes differentiation with respect to u. Here we define The energy-momentum tensor of the spinor field is defined as [2] T From (10) one finds the non-trivial components of the energy-momentum tensor of the the spinor field with A η =ψγ 5γ η ψ being the pseudovector. It can be noticed that T 0 0 + T 1 1 = mS + 2λ (F − KF K ) and T 0 0 − T 1 1 = − (mS + 2KF K ) might be positive or negative under certain conditions. From (8) we find the following system of equations for the spinor field invariants Equation (12) leads to the following relation between the bilinear spinor forms: In case of K = I, i.e., G = 0 from (12a) we find If K = J, then in case of a massless spinor field from (12b) we find Let us consider the case when K = I +J. In this case b 1 = b 2 = 1. Then on account of expression for D and G from (12a) and (12b) for the massless spinor field we find Finally in case when K = I − J, i.e. b 1 = −b 2 = 1 from (12a) and (12b) for the massless spinor field we find which yields The Einstein tensor corresponding to the metric (5) possesses only diagonal components. So let us first consider the diagonal equations of Einstein system Subtraction of (20d) from (20b) yields with the solution Analogically, subtracting (20d) from (20c) one finds In view of (9), (22) and (23) one finds Thus, γ, β and µ can be found in terms of α and τ. Let us find the equation for τ. Summation of (20b), (20c), (20d) and 3 times (20a) gives But for non-diagonal components of the EMT of the spinor field we have non-trivial expressions. Equating these expressions to zero from (11c), (11d) and (11e) we obtain the following constrains The foregoing expressions give rise to three possibilities: It should be noted that in a Bianchi type-I space-time there occur such possibilities. In that case the assumption (27a) leads to the vanishing spinor field nonlinearity. In a static cylindrically symmetric space-time that is not necessarily the case. Unfortunately, right now we cannot exactly solve the equation for defining either τ or α. So we have to assume some coordinate conditions. There might be a few. In what follows, we consider the case with K = I, as in this case it is possible to consider massive spinor. Further we set S = K 0 e −τ and K = K 2 0 e −2τ . Case 1 Let us first consider the harmonic radial coordinate [20] α = γ + β + µ. (28) In view of (9) Eq. (25) takes the form Let us consider the case when F is a power law function of K, i.e. F = K n+1 . Inserting S = K 0 e −τ and K = K 2 0 e −2τ into (25) on account of α = τ we find with the first integral So the solution can be given in quadrature Let us consider some simple cases those allow exact solution.
Fisrt we study the Heisenberg-Ivanenko type nonlinearity when F(K) = K. It can be obtained by setting n = 0 in (31). In this case (31) takes the form which finally gives For a general power law type nonlinearity we study the massless spinor field. Setting m = 0 in (31) we have with the solution For a more general solution to the Einstein equations with massive and nonlinear spinor field as source we rewrite it in the form of Cauchy: This system can be solved numerically. In Fig 1 and 2 we have plotted the metric functions γ(u), α(u), β (u), µ(u) for different types of nonlinearities, namely, n = 0 (Heisenberg-Ivanenko case) and n = 4. For simplisity, we set the following values for other parameters C 1 = 1, C 2 = 2 and m = 1. The initial values were taken to be τ(0) = 0.3, γ(0) = 0.03, β (0) = 0.2, µ(0) = 0.07 and ν(0) = 0.2. As we see from the graphics, with the increase of the value of n the difference between the metric functions increases.

III. CONCLUSIONS
Within the scope of a static cylindrically-symmetric space-time we studied the role of nonlinear spinor field in the formation of different astrophysical objects. It is found that the energymomentum tensor (EMT) of the spinor field possesses nontrivial non-diagonal components. This fact imposes three way restrictions on the space-time geometry and/or on the components of the spinor field. It should be noted that such situation occurs in Bianchi type-I cosmological model as well, but while in BI model a specific type of restriction leads to the vanishing spinor field nonlinearity, this is not the case in this model. Moreover, while in a static spherically-symmetric space-time the presence of non-trivial non-diagonal components of EMT of the spinor field has no effect on the space-time geometry, in static cylindrically-symmetric space-time it influences both the space-time geometry and the components of the spinor field. Moreover, the equality T 0 0 = T 2 2 in our view can play crucial role in the formation of some astrophysical configurations. It should be noted that the expressions (T 0 0 + T 1 1 ) and (T 0 0 − T 1 1 ) can be both positive and negative, depending on the type of nonlinearity. In our view this fact will provide some very interesting results which we plan to study in our upcoming papers.