The Dirac field is a coordinate scalar, but it naturally generates an affine centered vector space (spanned by the Dirac currents) at P, which is similar to the tangent space of the four-dimensional manifold at P (spanned by the vectors or). These currents constitute a complete basis, they are of unit length and orthogonal in the sense of Equation (2.5). The continuous field of tetrad is embedded into. Therefore, an infinitesimal change of the (and, eventually, of the) from point P to point is predetermined as,

Also predetermined is the derivative of the scalars, , and it has a very simple meaning. For a given displacement in, the total change can be expanded over a complete system with the coefficients. More precise is the directional deri- vative,

along an arbitrary vector in. By taking, we immediately recognize the connections, with the directional derivative, , along, as objects in principal manifold,

Then. Since we immediately conclude that

viz., the is skew-symmetric in the first two indices.

In what follows, we compute the covariant derivatives of the vector and spinor components with respect to different bases and establish their interrelation.

1. The Dirac tetrad. Starting from Equations (2.7) and (3.3) and following the Cartan’s idea of a moving frame [15] , we can compute the covariant derivative of the components of any vector,

or, in terms of components with respect to the basis,

where are the relative changes of the components and is their total change. We explicitly see that the presence of the physical Dirac field over the principal manifold immediately endows with an affine connection. It also provides a natural definition of parallel transport as a transformation that leaves the components of a vector unchanged with respect to a local basis, even when the local tetrad (or a coordinate hedgehog) changes its orientation from point to point. Equation (3.3) is a special case of Equation (3.6) when. Taking for the components of the vector current, , one can define the covariant derivative of the Dirac field without leaving the principal manifold. Indeed, assuming that

and comparing with Equation (3.6) one readily obtains the equation that determines the connection [16] ,

where and these matrices, depending on, must be considered as primary objects in.

2. Arbitrary tetrads. Knowing the affine connection in the basis of vectors, we can find it in any other basis. Indeed, starting from Equation (3.6) we rewrite covariant derivative in terms of the basis vectors,

where and stands for the expression,. By virtue of Equations

(2.9), we have. Using Equation (3.3), we obtain (by definition, =

0; is a matrix of Lorentz rotation),

These invariants are nothing but the coefficients of rotation of the basic vectors with respect to the basis. Conversely, the equation,

gives the coefficients of rotation of the basic vectors with respect to the basis.

3. Coordinate basis. In the coordinate picture, the basis vectors are assumed to be known in advance. In this case, one can derive the covariant derivative as

where stands for

and (because of the term with) it is transformed as a connection under a change of the coordinates. Alter- natively, we could start with (or just substitute from Equation (3.10)) and obtain another representation of the same connection,

which is now expressed via quantities that explicitly depend on the physical Dirac field. Finally, using Equations (12), we can invert the last two equations to obtain,

which is normally taken as an ad hoc definition of the coefficients of rotation of tetrad vectors when one prefers to stay in. Notably, Equations (3.15) and (3.3) determine the same, although Equation (3.3) app- arently belongs to and has nothing to do with the. This may be considered as an evidence that the vectors and the connections are the auxiliary quantities.

When is a vector and is a tensor (not necessarily determining a metric) then the covariant derivative with respect to is also a tensor [17] . Using Equations (3.12) and (3.15), it is straight- forward to check that if has the form (2.10) then. Indeed, since we have

An idea of how to find this practically, will become clear only in the next paper [8] , where a concrete solution is found. Starting from there, one can take the following path, and, eventually, explicitly determine the.

4. Connections for the Dirac field. Starting from Equation (3.9) for the vector current,

or translating Equation (3.8) into the basis, it is straightforward to obtain the following equation for the matrix ⁵:

where, and nothing implies that must be numerical matrices⁶. If we introduce and and use (3.15), then Equations (3.8) and (3.17) can be rewritten entirely in,

Equations (3.17) and (3.18) indicate that the Dirac matrices are covariantly constant with respect to the “connection” of the Dirac field,. The same is true for other representations as well.

Either of Equations (3.8), (3.17) and (3.18) can be solved (algebraically) for the corresponding. The most general solution reads as

where, so far, e and g are arbitrary constants. The term in the connection (19) (or the field) is unquestionably interpreted as the electromagnetic potential. The term (or field) could have been interpreted as another field that interacts with the axial current ⁷. The connection (3.19) commutes with the matrix, so that Equation (3.17) remains the same when. So far, it neither commutes nor anti- commutes with and, viz.

Similar formulae arise for the charge-conjugated connection. Since and,

The commutation relations for the Dirac matrices and are

in and, respectively. We assume that the matrices are associated with the basis in the tan- gent, while matrices belong to the principal manifold. In what follows, we consider Dirac field as the primary matter field; covariant derivatives of its bilinear functions will be computed only using Equations (3.17)-(3.19).

5. Connections in different bases. Equations (3.10) and (3.11) are nothing but the well known formulae for transformation of a linear connection between two non-coordinate (anholonomic) bases. In these bases, all quantities are functions of the point P in the principal manifold, and thus independent of the coordinate basis in the. For example, we readily have the coordinate-independent equation of the parallel transport of a vector along a vector, viz.

If we omit indices and use the notation for matrix (as well as for, for and for) then Equations (3.10) and (3.11) read as

which are the universal expressions⁸ for all kinds of connections associated with local transformations. Equ- ations (3.6) and (3.9), augmented by definition of the derivatives, and, are fix- ing the components of any vector with respect to the (moving) tetrads and. The existence of the field of unitary matrix of the Lorentz transform (and then of an affine connection) appears to be an amazing consequence of the Fierz identities for bilinear forms of the Dirac field. Finally, it is straightforward to check that, once and are the components of vectors and and are scalars, the connection transforms under a further change of the coordinates as

which guarantees that the derivative transforms as a tensor. Transformations (3.10) and (3.11) are re- duced to this formula when the tetrads are formed by the gradient vectors.

By definition, , were index can belong to any of the bases. Therefore, Equation (3.19) has the required general form (3.22) and can be rewritten as in tetrad basis and as in the coordinate.

6. Symmetry of the connection. If we naively assume that the Minkowski signature in Equations (2.4) and (2.5) determines the local metric of an inertial reference frame at point P (with local coordinates) and that of Equations (2.10) is obtained by a local coordinate transformation of the then, being a tensor, the skew-symmetric part of the connection (the tensor of torsion) should be zero. This argument would require, in its turn, that the covariant tetrad vectors be the gradient vectors, , which is by no means self-evident.

There is, however, another reason for the symmetry of, which is hinted by the Cartan’s method of moving frame. The field of tetrad belongs to and can be used as a “ moving frame” for all vectors, including the vectors of infinitesimal displacements. Consider now a closed path through the point and attach the “ natural” tetrad to its points. Then every next point of the path has a position with respect to the tetrad of the previous point. Since the tetrad is changing from point to point, we have no other choice but to specify the transport of a vector as the parallel Fermi transport (in the sense that the components of a vector with respect to the local tetrad do not change) along the chosen path. We will be able to get back to (the image of the path in the moving frame will be closed) with the same and, therefore, with the same tetrad and matrix, which is imperative, if and only if the components of the connection, as they are defined in the coordinate basis of the, are sym- metric in their subscripts. Then the torsion tensor vanishes, and only then will we be able to contract the entire path to the point. Consequently, the following formulae,

can be confidently used for any coordinate scalar.

From the equations of motion (4.1) one immediately derives two well-known identities. Multiplying the Dirac equation by from the left and its conjugate by from the right and taking their sum we readily obtain that

This equation clearly indicates conservation of the timelike vector current (of probability) of the Dirac field. The second identity is obtained from the Dirac Equation (4.1), which is multiplied by from the left (and its conjugate from the right, and noting that). It indicates that the spacelike axial current is not conserved,

and has the pseudoscalar density as a source. Since is localized not less than, and the vector is spacelike, it defines the radial direction. The existence of such a direction is a distinct characteristic of any loca- lized object.

Similar identities can be derived for the vectors and of Section 2. Using Equations (3.21) and (4.2), we immediately arrive to covariant derivatives of the matrix elements as

Though these vectors are complex and explicitly depend on the phase of, this dependence is compensated in the covariant derivative (4.5) by the gauge transformation of the vector potential. The derivatives of and become

The fields of complex currents look like being “charged” with a charge 2e. From the equations of motion (4.2) and using Equation (4.6), it is straightforward to get and, consequently,

Similarly to the vector of axial current, these vectors are not conserved due to electromagnetic potential.

In order to access the differential identities for the curls of the Dirac currents one has to compute, using the equations of motion, the derivatives of the objects, which are traces of tensors (objects), , , and, respectively. These ten- sors are neither real nor symmetric, and we are not concerned here about their physical interpretation.

1.―a tensor or not? One would expect the absolute differential of, being computed according to the Leibniz rule, be as follows,

and this expression would fix, similarly to Equations (3.9) and (3.12), the components of the tensor with respect to the tetrad. If this expectation turns out justified then the usual covariant derivative will be immediately reproduced as

Contrary to the expectation of (4.8), the answer reads

with the last term of Equation (4.8) missing, and no hope to recover the full geometric expression (4.9) of the covariant derivative of the tensor! Contracting here indices a and c and using equations of motion we would arrive at [7]

with the normal covariant derivative in the l.h.s. The and an abnormal term in the r.h.s. originate

from the commutator of the covariant derivatives,. Its real part is the Lorentz force,

[7] [16] ⁹.

2. Abnormal terms and how they restore the GL(4) covariance. The abnormal term enters another identity that follows from the Dirac equation, which arises after contracting indices a and b in Equation (4.10). On the one hand, we formally have (Cf. footnote⁷. The must be a scalar and the last term in the r.h.s. must be absent.)

On the other hand, by virtue of the Dirac equation, the first term on the r.h.s. of (4.12) becomes

. Alternatively, one can immediately use the equations of motion on the l.h.s. and only then

differentiate,

Comparing the last two equations and using (3.20), we finally find that the abnormal term

vanishes (or at least can be expressed via abnormal field)

thus restoring the covariance of Equation (4.11). Remarkably, the usual covariance in coordinate space is re- stored due to equations of motion. Equation (4.14) yields two nontrivial conditions on the structure of the Dirac currents as follows. The Ricci coefficients are real-valued and skew-symmetric in the first two indices. The r.h.s. of Equation (4.14) is real. Therefore, the imaginary part of Equation (4.14) reads as

In order to facilitate further analysis of the real part of Equation (4.14), let us rewrite its l.h.s. in terms of the axial current. Using the dual representation of the axial current as, and employing the equations of motion we obtain,

where the r.h.s is four times the anti-symmetric Hermitian part of the energy momentum tensor. Therefore, the real part of Equation (4.14) reads as

3. More non-tensors and abnormal terms. Next, consider the stress tensor, mostly following the same protocol and starting from its covariant derivative. We find that

Once again, the last term of Equation (4.8) is missing, and thus we have no confidence that the covariant derivative is a tensor. For the immediate purpose of this work, we only need the equations that emerge after contracting indices a and b in Equation (4.17),

By virtue of the Dirac equations, the first term in the r.h.s. becomes. Alternatively, one can immediately use the equations of motion in the l.h.s. and only then differentiate (matrices and com- mute),

Comparing the last two equations we finally get the equation,

which is complementary to Equation (4.14). Since is skew-symmetric in the first two indices, the imaginary part in the l.h.s. is due to. Since the axial current is a vector, we can rewrite the imaginary part of the last equation as [C.f. footnote⁷],

which is dual to Equation (4.16). The skew-symmetric Hermitian part, , must vanish

since the r.h.s. of Equation (4.20) is an imaginary quantity. Since, , this yields the equation,

which is similar to Equation (4.16) and dual to Equation (4.15).

4. A full set of prerequisites for the covariance. Considered together, Equations (4.15) and (4.22) constitute a linear system of eight equations for the six unknowns,. In general, the rank of its matrix equals 6. Therefore, it can only have a trivial solution. Since are the invariants of a true tensor, , we have the tensor equation,

Equations (4.16) and (4.21) constitute the system of 8 equations for 10 unknown quantities, and. These equations also explicitly depend on a choice of the auxiliary field of tetrad, which is unacceptable. Insisting on independence as a physical (and then mathematical) requirement and realizing that does not exist as a physical field, we must put ¹⁰. Then we have the system of 8 homogeneous equations for only 6 unknowns with a trivial solution,

which is similar to Equations (4.23) that we had for the vector current.

More identities are readily obtained along the same guidelines as Equation (4.14). Namely, duplicating (4.12)-

(4.14), we compute and directly and using equations of motion. Adding up the results we obtain that

Computing in the same way the dual quantities, and, we end up with

which once again is a system of 8 equations for six unknowns with only a trivial solution. Since is skew- symmetric in the first two indices and is not zero, we arrive at

which, by virtue of (4.6), results in

The differential identities (4.15), (4.23) and (4.28) for the Dirac currents are written down in the covariant tensor form and can be transformed further into tetrad representation with respect to any tetrad. Therefore, it is indeed possible to overcome the Cartan’s veto [C.f. footnote 4] relying on the second reservation in Cartan’s statement.

To analyze the lines of the vector current, the two obtained earlier equations, (4.3) and (4.23),

must be examined together. When the invariant density of the Dirac (spinor) matter is positive, , the vector field is strictly timelike; its tangent unit vector is,. Therefore, Equation (4.23) becomes

Contracting this equation with, and using Equation (5.1) we find that

which is a necessary and sufficient condition for the congruence to be normal [17] [18] . Namely, there exists such a function, , that the vector field is orthogonal to the family of surfaces ,

where satisfies the complete system of three equations, , , and is a coordinate scalar. Contracting Equation (5.3) with we get

where is the derivative in the direction of the arc. Contraction of Equ- ation (5.3) with yields

which indicates that congruences of lines, defined by the system of equations, , must experience

permanent bending (acceleration) whenever the invariant density of the Dirac field is not uniformly distributed. The spatial gradient of cannot vanish for any localized state.

Additional information can be extracted from Equation (4.3),. Then, by definition,

Hence, we can rewrite (5.6) as

which shows that the r.h.s. of Equation (5.9), which contains only geometric objects, is a component of a gradient. Together with condition (5.4) this constitutes a necessary and sufficient condition that the function defined by Equation (5.5) is an harmonic function [17] ,

The parameter of is the definition of the world time. For the harmonic function, , the conditions of integrability for system (5.5) of partial differential equations reads as [17]

Comparing it with (5.9) we find that, so that the world time and the “proper time” are related by

Furthermore, since and system possesses the proper time, we can rewrite Equation (5.9) as which could have been inferred directly from Equation (4.15). Then, the harmonic nature of immediately follows from the current conservation,. Since is the total differential and the vector current belongs, in fact, to the principal manifold, so does the interval of the world time,

and this interval does not depend on the path of integration (the time variable is a holonomic coordinate).

Now, we can draw the major conclusion: The proper time, , flows more slowly than the world time, , whenever Dirac matter has a magnified density. Because of the wave nature of the Dirac field, its localization is inevitable. Since the congruence appeared to be normal, the hypersurfaces represent space at different times. The states can be considered stationary only with respect to; one can hope to find them only after replacing by in the operator of energy!

Here, we have to deal with the system of equations,

which is similar to Equations (5.2) that we had for the vector current. The only difference is that the axial current has a source. Since there is no flux of vector current in this direction (the amount of matter inside

a closed surface remains the same), we associate the radial direction with the axial current,.

Next, observe that by virtue of the Fierz identity (2.3), , we can parameterize,. Then the second Equation (5.13) takes form

On the one hand, by definition,. On the other hand, according to Equ-

ation (5.7), we have. Substituting these expressions into Equation (5.14) we ob- tain an important relation,

The first of Equations (5.13), being contracted with, yields

so that the congruence of lines is normal and there exists such a family of hypersurfaces that

where satisfies the complete system of three equations, , , and is a coordinate scalar. In the same way as before [cf. (5.6), (5.7)], contracting the first of Equations (5.13) with and, we will get

and this is compatible with the condition for integrability, , of the system (5.17)

only when. Next, we may compute the second derivative of. Using Equation (5.7) and Equation (5.27) below, we arrive at

From here we find that if, then is the solution of an inhomogeneous wave equation,

for the “ potential” with the source density proportional to the mass parameter m of the Dirac equation and pseudoscalar density (in static limit, it becomes the Poisson equation). Not surprisingly, this source is equal to the derivative of the invariant density in the direction of the axial current. If the invariant density was not changing in a “radial direction”, the whole idea of a localized object would be vague. Similarly to (5.5) and (5.11), we have

From here, we conclude that the differential form is integrable and the “radial distance”,

does not depend on the integration path (the coordinate variable is holonomic).

Here, we must deal with four equations (4.6) and (4.28). Taking (an alter-

native choice with will be discussed later), starting from Equation (4.6), and duplicating the deri-

vation of Equation (5.8) we arrive at the equations,

Since by the second Equation (5.18) we have and, these equations com- pletely define and,

Putting further in Equations (4.28) and, and duplicating the scheme of Equation (5.3)-(5.7), we obtain,

Giving index A in Equations (5.24) and (5.26) all possible values, we get the following constraints,

Equations (5.28) and (5.22) are mutually compatible only when and

i.e., when the vectors of the geodesic curvature and of the congruences [0] and [3] of the vector and axial currents have no projections on the lines of the congruences [1] and [2] of the charged currents. Together with the previously obtained Equations (5.8), (5.18) and (5.22), they give all in terms of deri- vatives of the invariant density and electromagnetic potentials. Namely, since, we also have, which together with the first Equation (5.27) entails that

The second of these equations means that the congruence [3] is geodesic¹¹. Quite remarkably, this conclusion about static character of the configuration that satisfies Dirac equations of motion is reached only after all the differential identities are considered together. The additional constraints that follow from Equations (5.23) and (5.25), when indices A and B are given all possible values, are as follows,

Combined with the previous results (Equation (5.4), particularly) they yield,

The last of these equations is the necessary and sufficient condition for the congruences of lines, and being canonical of the congruence [18] . This property appears to be yet another consequence of the Dirac equation of motion, which thus guarantees that the orthogonal tetrad is Fermi-transported. Finally, comparing Equations (5.16) and (5.34) we find that

By now, we have succeeded to find simple expressions for all coefficients of rotation of the basis of the normalized Dirac currents. This is the last step in the design of the matter-induced affine geometry. From this point, one can rely on the common tools of the differential geometry. We can divide the not vanishing components of into two distinct groups:

1) Five geodesic curvatures ( the with only two distinct indices),

2) Only two of the with all three different indices are nonzero. These are

3) The coefficients, which depend on the potential, are of the same form

so that presence of electromagnetic field causes rotation of the Dirac tetrad in the (12)―tangent plane. This inter-action makes it impossible, in general, to match Dirac equation with the all-orthogonal system of hyper- surfaces¹².

Using Equations (5.36)-(5.37) and employing Equation (2.5) as, , we obtain,

It is essential that the only directional derivative that survived all constrains is, and even it can be expressed via pseudoscalar density. Therefore, the practical computation of the connection does not re- quire any reference to a coordinate background. The congruence of integral lines of the vector field is both normal and geodesic. This is the only geodesic of the principal manifold, and it is inherited by the hyper- surfaces of the constant world time. The congruences constitute a canonical system with respect to the congruence. Therefore the entire tetrad is Fermi-transported along the the lines of the radial congruence. Equations (5.36)-(5.39) assume a localized configuration with maximum of invariant density in its interior and a naturally right-handed spatial trihedron. If there is a minimum, then the signs of tetrad components in coefficients of rotation (5.36)-(5.37) (and only there!) must be reverted.

Since we are aiming at the discovery of the localized solutions, a coordinate picture may become most app- ropriate, and it is useful to know in advance what the admissible coordinate net may look like. Solely for this purpose, we study here whether the congruences of the Dirac currents in can form at least some of the four 3-d coordinate hypersurfaces and of the six 2-d coordinate surfaces. Once found, these surfaces will be studied in detail as submanifolds embedded into endowed with the connections identified above.

1. Hypersurfaces S₍₁₂₃₎ and S₍₁₂₀₎. From visual inspection of the Poisson brackets (6.1), among the four equ- ations, , there are two integrable systems of three equations that define two hypersurfaces and two integrable system of two equations that define two surfaces in the coordinate space. Namely, three com-

mutators between the, and [Equations (6.1 d,e,f)] are the linear combinations of these operators

alone. Therefore, the function (as well as any function) is the first integral of the complete (Jacobian) system of three equations,

The parameter enumerates the family of hypersurfaces, which are spanned by the streamlines of the vector fields, and and have as the normal. Equations (6.1 b,c,d) indicate that three equ- ations,

also constitute an integrable system with a first integral (or any function); the latter represents hypersurfaces of the constant “radius” ρ when. These are spanned by the integral lines

of the vector fields, and and have as the spacelike normal.

2. Surfaces S₍₁₂₎ and S₍₀₃₎. Next, by Equation (6.1 d) the system of equations

is integrable. Its two first integrals, and, determine a two-dimensional surface

spanned by the streamlines of the vector fields and having the normal vectors.

The first integrals of the system (6.5) are known because both of its equations are satisfied by and. Once and are algebraically independent, these are the two first integrals of the system (5), and the 2-d surface is uniquely fixed by the values of constants and, which enumerate the surfaces of a constant “radius” at a given “world time”.

Finally, according to Equation (6.1 a) the commutator between and is proportional to. There- fore, the system of equations

is integrable. It has two first integrals, and, which determine a two-dimensional surface spanned by the streamlines of the vector fields and. The two normal vectors of these surfaces are the linear combinations. One of the first integrals of the second Equation (6.6) is

, i.e. we have. Also, one of the first integrals of the first Equation (6.6) is

, i.e.. Since the congruences of integral lines of the fields and are

normal―(cf. Section 5), we have and, as well as

and. In terms of the new independent variables,

, the system (6.6) immediately acquires the normal (Jacobian) form,

Its second equation is equivalent to the system of three ODEs,

which has three first integrals, , ,. In terms of the new independent variables, , the system (6.7) reads as

where. Since is independent of, we have one PDE in three variables, which is equivalent to the system of two ODEs. The variables and form an orthogonal coordinate basis on every 2-d surface (enumerated by the values of and).

Conditions for simultaneous integrability of the PDEs for the streamlines of the Dirac currents prompted the existence of the (hyper)surfaces in and, most importantly, in. Here, in order to understand their shape, we look at them as submanifolds of the principal manifold.

1. The method. For the sake of brevity, we will use the Latin capitals to label the entire tetrad basis (or). In the context of the current work this is the basis of the ambient space. The capitals will label the tangent tetrad vectors of a 3-d or 2-d submanifold. The capitals will be used to label the normal vectors. Then the induced metric of a submanifold is and, by virtue of definition (2.11), the first quadratic form of the surface is (pseudo)-Euclidean,.

Since we are interested in submanifolds that are spanned by the integral lines of the tetrad vectors, the Gauss and Weingarten decompositions of the covariant derivatives of tangent and normal (with respect to a sub- manifold) tetrad vectors immediately follow from Equations (3.2),

where all the 's listed in Equations (5.36)-(5.37) are known explicitly¹³. The first term, , in the r.h.s. of the Gauss decomposition (6.10) is the connection of the intrinsic tangent space of the submanifold. The second term, (with two tangent and one normal indices), is the second fundamental form of the submani- fold with respect to the normal. The first term in Weingarten decomposition (6.11), , (the shape form with two tangent and one normal indices) is similar to the second fundamental form in (6.10); both account for the rotation of the tetrad in the (PA) plane when it is displaced in a tangent direction. The second term of Equation (6.11), , with two normal and one tangent indices, is the covariant derivative of the normal components of a vector in a tangent direction of the submanifold. It accounts for the rotation of the (AB)―plane of the two normals under infinitesimal displacement in tangent direction.

Now, since there is no question of how a submanifold is embedded into the ambient space with explicitly known tetrad vectors, we are in position to study the internal geometry of various coordinate surfaces, as submanifolds of the principal manifold. Besides the second fundamental form, we will use the Riemann curvature tensor in ambient space and in subspaces,

With these preliminaries, we are in the position to consider all subspaces on-by-one.

2. The hypersurface S₍₁₂₃₎ represents space at a given time. It has three spacelike tangent vectors, , and a single timelike normal vector. The coefficients of the single second fundamental form are and. The second fundamental form, ,

is proportional to first fundamental form, of the,

Therefore, the is a totally umbilical submanifold¹⁴ with zero mean normal curvature. The latter means that is a totally geodesic submanifold; it inherits its sole geodesic from the ambient. From the perspective of the ambient space, the hypersurface has no curvature, it is extrinsically flat. The extrinsic part vanishes together with the connections. The intrinsic Riemann curvature of the has six different (modulo sign) components; it is given by the terms of (6.12) with all indices in tangent space of the,

where coincide, by appearance, with the tetrad components of the electromagnetic field tensor rewritten in the basis. It should be remembered that all the here came from the components of the Ricci coefficients of rotation (5.38).

3. The hypersurface S₍₁₂₀₎ represents the surface of a given “radius” at all times. It has two spacelike and one timelike tangent vectors, , and a single spacelike normal vector. The coefficients of the second fundamental form are and. The second fun-

damental form, , is proportional to the first fundamental form

of the,

Therefore, the hypersurface is also a totally umbilical submanifold with the mean curvature. By virtue of Equations (6.2), the vector of (mean) geodesic curvature H is constant and parallel throughout every hypersurface.

The intrinsic part of the Riemann curvature of the hypersurface has only the following components,

identical with those of. The extrinsic parts are due to, i.e., the connections that contain normal component,

Since congruences, and are canonical with respect to the normal congruence, their lines are the lines of curvature of the hypersurface. If at some point of we have, then the directions of, and become the asymptotic directions.

4. Surface S₍₁₂₎ is the surface of a given “radius” at a given time and can be viewed as a hypersurface of either or with the normals or, respectively. It has two spacelike tangent vectors, , and two normal vectors, , timelike and spacelike. Accordingly, there are two second fundamental forms, and, with the following coefficients,

,. The first fundamental form of is

, and the two second fundamental forms are

Therefore, the 2-d surface is a totally umbilical submanifold with the mean curvature , which is determined by the Dirac field within principal manifold. The Gaussian curvature is positive. Such a surface can only be the sphere with the radius of curvature [19] [20] . (It is a plane, when, but then must be uniform and. Here, the spherical shape is a dynamic symmetry since it originates from equations of motion.). Nearly the most important property of sub- manifolds follows from the compatibility conditions (5.29) and Equation (6.2), which indicate that the invariant densities are constant along every 2-d surface,. The mean curvature H is constant along as well. The normal connection for this submanifold can be only due to the components and of the connection, but these vanish identically, , so that both normal vector fields (and the mean curvature vector) are parallel with respect to the tangent displacements along,. The Riemann curvature of has only one component, and it can be de- composed in two parts. The intrinsic one, , is given by the terms of (6.12) with all indices in tangent space of. The only nonzero connections here are and, so that sectional curvature of the,

is entirely due to the tangent tetrad components of the electromagnetic field. The extrinsic part, , is due to the connections from the second fundamental form and

5. The surface S₍₀₃₎ represents a given “angular direction” at all “radial” distances and at all times. It has one spacelike and one timelike tangent vectors, , and two spacelike normal vectors,. Here, we also have two second fundamental forms, and, with the following coefficients, ,. The first fundamental form of the is

and both second fundamental forms are just zero,

The submanifold is totally umbilical with the mean curvature, and as such is a totally geodesic submanifold. The shape form of is zero. The normal connection for the coordinate surface (and only for this surface) does not vanish,

solely due to the external potential, ,. A displacement in the directions of and, rotates the tetrad in plane (12). The Riemannian sectional curvature of the is induced by an ambient space,

According to Equation (6.2), system (6.5) of PDEs admits, along with the first integrals and of hypersurfaces and, respectively, the first integrals, and, which must be functions of the former ones, and vice versa,

being, ultimately, the known functions of the Dirac field. Potentially, one can obtain the functions and purely algebraically,without even solving system (6.5) of PDEs. Every 2-d surface is fixed not only by the constants and, but also, e.g., by and, which indicates that surface belongs to the principal manifold without any reference to a coordinate. These observations are compli- mentary to the main idea of this work that Dirac field naturally determines the moving frame. Here, the two scalars, e.g., and, can replace the coordinates and (similarly to the hodograph transformation in hydrodynamics). From Equation (6.2) with tetrad index one can see that neither of the scalars depends on the time variable (or). Therefore, these quantities depend only on the radial variable (or, equivalently, on the affine parameter).

1. Radial lines. When a geodesic line is given in the parametric form, , the unit tangent vector is. The affine parameter of the radial geodesic lines is, but it differs from the parameter of the hypersurfaces, which determines distance (5.21) at some moment of the world time (5.12). In terms of the variable, the ODE for geodesic line with the tangent vector is

where the connection is defined by Equation (3.14). The ODE for a geodesic line in terms of the physical variable that can be obtained by means of a simple transformation, , and reads as

where the r.h.s. does not contain derivatives of the Dirac field and it clearly manifests that the (not unit) tangent vector and its change are parallel along the “radial” geodesic curve.

2. The lines of the world time. The acceleration of the unit tangent vector of the lines of the vector current is

and it has only the radial component (precisely the same as radial geodesic (6.25)), which equals in magnitude but has opposite sign with respect to the mean curvature vector of surface and hypersurface. The ODE for the trajectory reads as

Obviously, the line of the vector current that passes through a point with the radial coordinate never leaves the the surface. Therefore, there is no flux of the charge density in the outside direction, which is an indirect but indisputable evidence of localization.

3. The coordinate net over S₍₁₂₎. Finally, the lines of the Dirac currents and are also bound to the surface. Indeed, for the curves and we have

so that they have the same normal component of the mean curvature vector, and they are bent within surface even when the components.

To summarize, all the currents passing in a tangent direction through a point on hypersurface of a given radius never leave this surface.