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Asymptotically flat gravitating spinor field solutions. Step 2 - the compatibility of Dirac equations in a curve and a flat spaces

Using the fact that a spin connection is defined to an accuracy of a vector it is shown that the spin connection should be modified in such a manner that Dirac equation in a curve space would be compatible with Dirac equation in a flat space.


I. DIRAC EQUATION IN A CURVE SPACE
This note is the continuation of the Ref. [1]. There it was mentioned that at the moment in general relativity asymptotically flat solutions for a spinor field are unknown although do exist asymptotically flat solutions for all known fields: scalar and gauge fields. We continue the investigations in this direction and we will consider Dirac equation in a curve space. We will show that a spin connection should be modified by the addition of a vector in order to obtain correct Dirac equation in a flat space.
A covariant derivative of a spinor ψ is and the covariant derivative of a Dirac conjugated spinorψ is where a spinor connection Γ µ is defined from following equation (here we follow to Ref. [2], section 1.7.2) where γā are Dirac matrixes in a flat (Minkowski) space, γ µ = e μ a γā are Dirac matrixes in a curve space. For the definition of tetrad, spin connection and so on we follow to Ref. [2]. The inverse tetrad e μ a satisfies eā µ e ν a = δ ν µ .
whereā,b = 0, 1, 2, 3 are Lorentz indices; µ, ν are world indices. The metric tensor g µν in a curve space is related to the Minkowski metric ηāb through the tetrad The solution of (3) is where A µ is a spinor-tensor quantity with one vector index. Substituting (7) to (3) gives us the equation for A µ − γāA µ γā + 4A µ = 0.
The simplest solution of this equation is: A µ is an arbitrary vector multiple of I. Usually A µ = 0. Here we would like to show that A µ should be nonzero that the Dirac equation in a curve space would be consistent with the Dirac equation in a flat space. In order to find such solution we rewrite (7) in following form here there is not summation overc. In order to find Ac we calculate the term γcΓc We choose It is easy to show that (12) is the solution of (8). Let us rewrite the term Finally we choose the spin connection Γ µ in the form hereΓ µ means that we use the Fock -Ivanenko coefficients ωābc withā =b,b =c,ā =c. Consequently one can write Dirac equation in a curve space in following form for the Dirac conjugated spinorψ, hereψ ← − ∂ µ = ∂ µψ . Usually Dirac equation is written as In order to compare equations (15) and (17) we will write Dirac operators D 1 , D 2 for both equations (15) and (17) in Minkowski space for the spherical coordinate system. For the calculations of ωāb µ we use following definitions from [2] (section 1.5.4) ωāb µ = e ᾱ a e β b ω αβµ = −ωbā µ .
For this case ωābc = 0, withā,b =c (30) which we will use in (18). For the first case (18) we have that is agreed with Dirac equation for an electron in hydrogen atom. But for the second case (19) we have that obviously is not the necessary equation.

II. ENERGY -MOMENTUM TENSOR
According to Ref. [3] the energy -momentum tensor for the classical spinor field is [4] here a (µ b ν) = (1/2)(a µ b ν + a ν b µ ) means the symmetrization. For our goal we change ∇ →∇ The calculation of the energy -momentum tensorTāb with the spinor ansatz (23) and modified spin connection (14) gives usT00 The current is From (37) and (42) we see that there is the energy and charge densities. From (43) we see that there is the current J 3 along ϕ direction leading toT14 component of energy -momentum (39). One can say thatT14 = 0 and J 3 = 0 is the consequence of a spin distinct from zero.

III. CONCLUSIONS
Thus we have shown that the standard definition of the spin connection in a curve space should be modified that the Dirac equation in a curve space would have correct limit in going from a curve space to a flat space. The modification of the spin connection to lie in the fact that to the standard spin connection is a vector added. So that the modified spin connection has Fock -Ivanenko coefficents with unequal indices only.