Abstract
The physical space-time in general relativity is the four-dimensional pseudo-Riemannian space V
4 with the metric signature (+, +, +, ā). We assume that the space V
4 referred to a coordinate system with the covariant vector basis 
and variables x
i, iā=ā1, 2, 3, 4.
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Notes
- 1.
This transformation group of the vectors of the proper basis eĢ1, eĢ2 and eĢ4 is related to a group of the gauge transformations of the spinor components Ļ A in the basis e a:
$$\displaystyle \begin{aligned} \psi ^{\prime A } = \alpha \psi ^A-\beta \psi ^{ + A }. {} \end{aligned} $$(*)Here Ī±, Ī² are arbitrary complex numbers, satisfying the condition \(\dot \alpha \alpha - \dot \beta \beta =1 \). Transformations (*) were considered in Chap. 3 (see (3.170)).
- 2.
In an arbitrary coordinate system received from synchronous system by admissible transformations (6.14), this integral has the form
$$\displaystyle \begin{aligned} C_u^2 \, g^{ 44}\cos ^2\eta =-(u^4) ^2, \end{aligned}$$where g 44 is the component of the metric tensor.
- 3.
- 4.
As known, the exponential function of the matrix argument is defined as
$$\displaystyle \begin{aligned} \exp \alpha \gamma ^5 =I+\sum _{n=1}^{\infty} \frac 1{n\, !}\,(\alpha \gamma ^5)^n. \end{aligned}$$Bearing in mind that (Ī³ 5)2ā=āāI, we find
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \exp\alpha\gamma ^5=I\left( 1-\frac 1{2\, !}\,\alpha ^2 +\frac 1{4\, !}\,\alpha ^4-\cdots \right) \\ &\displaystyle &\displaystyle \qquad \qquad \quad \qquad \qquad \quad \qquad +\gamma ^5\left( \alpha -\frac 1{3\, !}\,\alpha ^3+\frac 1{5\, !}\,\alpha ^5-\cdots\right) =I\cos \alpha +\gamma ^5\sin\alpha . \end{array} \end{aligned} $$ - 5.
It is easy to show that Eq. (6.70) does not invariant[83] under the Pauli group
$$\displaystyle \begin{aligned} \psi^{\prime}{{}^A} = \alpha \psi ^A-\mathrm{i} \beta \gamma ^{5A}{}_B\psi ^{ + B }, \quad \overset.{\alpha }\alpha + \overset.{\beta }\beta =1. \end{aligned}$$However, as is well-known [17], quantized equations
$$\displaystyle \begin{aligned} \gamma ^i\nabla _i\psi + \lambda: S_i\overset{*}{\gamma }{}^i\psi : =0 \end{aligned}$$are invariant under the Pauli group. Thus, a symmetry group of nonlinear spinor equations can change at their quantization. We note in this regard that the opinion is sometimes expressed about impossibility changing the symmetry group of the equations at them quantization (see e.g. the article [17] devoted to properties of nonlinear spinor equations).
- 6.
- 7.
- 8.
If Ī m depends only on the fluid density Ļ, there is no electromagnetic field E iā=āB iā=ā0 and an additional equation ā i M iā=ā0 is fulfilled, then the spinor equations (6.132) for \(\lambda (\rho ) = - \dfrac g{ 2mc\rho }\dfrac {d\varLambda _m }{d\rho }\) coincide with Eqs. (6.70). Therefore, Eqs. (6.70) describe a spin fluid, defined by the equations
$$\displaystyle \begin{gathered} \partial _j\widetilde P_i{}^j=0,\quad \partial _i\rho u^i=0,\quad \partial _iM^i=0,\\ \rho \frac {d{}}{d\tau }\left( \frac 1\rho M_i\right)= g\widetilde F_{ij}M^j,\quad \widetilde F_{ij}=\overset{*}{F}_{ij}+ \frac cg\varepsilon _{ijks}u^k\partial ^s\eta ,\\ \widetilde P_i{}^j=P_i{}^j+\frac cg\big( M^su_iu^j-M_iu^su^j+ M^j\delta _i^s\big) \partial _s\eta , \end{gathered} $$in which P i j and \(\overset {*}{F}_{ij}\) are defined according to (6.129) (without terms with an electromagnetic field). The quantity Ī· in this case is considered as a Lagrange multiplier corresponding to the equation ā i M iā=ā0; the additional terms in \(\breve P_i{ }^j \) and FĢij are related with the introduction into the Lagrangian of the term cg ā1 M i ā i Ī· with the Lagrange multiplier Ī·. These equations determine hydrodynamic analogy of the theory of the elementary particles described by Eq. (6.70). In particular, for the Heisenberg equation (Ī»ā=āconst) we have Ī mā=āāĪ»mcg ā1 Ļ 2 and for the pressure p we obtain pā=āāĪ»mcg ā1 Ļ 2. A condition of positivity of the pressure (or sound velocity) for gā<ā0 gives Ī»ā>ā0.
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Zhelnorovich, V.A. (2019). Exact Solutions of the Nonlinear Spinor Equations. In: Theory of Spinors and Its Application in Physics and Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-27836-6_6
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