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(71)

The linearized expression for the action integral for just Einstein term could be obtained in this manner:

(72)

Both underlined terms vanish thru the means of partial integrations, so the final form for the action integral for the Einstein terms has this form:

(73)

We can see from this that Einstein term is of the same form―in terms of order of magnitude and in terms of variables (and) as the linearized form of the Lagrangian in expr. (71).

Combining all sub-Lagrangians together―expression (71) and (73)―we can write the final expression for the linear, with respect the factor approximation of the Lagrangian of Total-Matter:

(74)

The Lagrangian above represents exactly the physical phenomenon that we postulated at the beginning of this paper. The equations of motion for Matter, which includes mass-particle (tensor), electromagnetic field (vector field) as well as two other vector fields (and) do not depend on the metric and/or grav-vector. Their description is well determined by set of equations written in flat Minkowski space.

The description of metric and Grav-vector could be presented as a correction and to a constant (flat) space and constant grav-field. The equations for corrections has a traditional flat space field description with each one of them having a source defined by mass-matter and vector fields, electromagnetic field and.

The terms grouped in the Lagrangian represent a coupling (inter-dependence) of the metric and. In the “Gravitation” section it will be shown that these equations could be decoupled using a “gage” requirements for the metric, which is defined up to 4 functions associated with arbitrarily system of coordinates. And by doing so we will effectively separate out a “chosen” or “rest” system of coordinate from all inertial system of coordinates.

We will address in greater details the equations of motion for each variable in their corresponding sections―“Gravitation” for and, “Electromagnetic and Other Fields” for, , and “Mass-Matter (Elementary Particles)” for.

Before we proceed writing the equations for each of these fields we would like make few general comments:

1) The Lagrangian (74) is a in effect the Einstein’s Lagrangian where Grav- vector is a dark matter and the magnitude of the Grav-vector serving as gravitational constant.

It worth pointing out that the form of the Lagrangian of the Total-Matter is identical to the standard General Relativity written in slightly different manner, where we multiply the Lagrangian of General relativity by a constant, express the Lagrangian and thus the Energy-Momentum tensor in units and write approximation for metric tensor in a form:

(75)

2) According to this theoretical derivations using Affine Unification this Lagrangian is a linear approximation for a system of masses (like our Solar system and smaller), where Grav-field created by the system is a small addition to the already existing Grav-field created by the masses outside of our system―such as our Galaxy, or group of Galaxies, or the entire Universe. It is for that reasons, the considerations of Universe models taken Einstein equations as a starting point (with or without Dark Matter) might not be a justified approach. We will address the issue again in the section “Cosmology”.

3) We derive this Lagrangian from the Unified Affine description by imposing certain requirements on parameters―see expr. (30)―in order for this theory to be non contradicting to the common experimental facts and to common sense. But as we continue investigating the equations of motions for each unknown variable we might (and more likely will) need to impose more requirements on. What is important here is that these requirements simply outlaw some terms in the final Lagrangian. However, we are totally controlled by the “affine derivation” to the degree what type of terms is allowed to be present as a part of the Lagrangian.

4) Regarding vanishing of “unwanted” by means of setting some requirements on the -constants we would like to point out to the fact that there is of course much easier way of removing unwanted terms in the general Lagrangian. In stead doing of the calculation and finding the right -s, thru solving all the equation-conditions that one places on -s, we can simple postulate that this term in the Lagrangian of the Total-Matter is zero. For example, in stead of searching for -s such that, we could take a Lagrangian almost at will and then place a requirement that the terms of the Lagrangian proportional to is zero. Express in the “Lagrange coefficients” this would effectively add one more unknown function (call it) as a dynamic variable. Between removing all terms proportional to (n = 4, 3, 2, 1, and −1) and some unacceptable terms inside the Lagrangian for the Matter and Lagrangian for the gravitational field and there will be somewhere around dozen or so such new unknown functions.

The problem with such approach is its universality. This method could be apply to any Lagrangian and thus totally leaving unanswered the question how this Lagrangian actually looks. Another words, we have no idea which Lagrangian (out of thousand or so) we should really choose. The approach that we took in this paper (defining the -s) is much more complex precisely because it actually defines (selects) the right Lagrangian. However, if we outrun number of available constants or get a contradiction in the constants’ values, it is much more realistic to envision a mid-point situation that most requirements (such for example that involve the vector fields, and) for vanishing “unwanted” terms can be met thru the form of Lagrangian (-s) and thru the form of Total-Matter (-s). The remaining non-vanishing terms would be related to only one group of functions (or and) could be vanished by introduction of Lagrange-coefficients.

7. Equations for Gravitational Field and Metric

This section deals with equations of motion for the square-vector and the metric Let us again write the flat-space Lagrangian of the Total-Matter:

(76)

where are set of constants linearly depending on the -s constants of expr. (69), which by themselves are functions of the -constants of the Lagrangian, and where are the terms that all came from decomposition (writing) of covariant derivatives of the Lagrangian―see expr. (59)―on three groups: a) square of partial derivatives of; b) square if partial derivatives of and c) the product of.

The first group with some simple manipulations (like partial integration) could be written in this form:

(77)

Since in Minkowski space the unit vector, the equations above could be written in somewhat more explicit (if not simpler) form where we use “0” as time coordinate.

(78)

Although for derivation of equations for by variation of is better to use expr. (76) and switch to Minkowski coordinate at the very end. Thus variation of the term we get The same form of writing (zero instead of time coordinate) can be applied to the others sub-Lagrangians of the. For, example the term can be written as this:

(79)

where by square bracket with appropriate label (in parentheses) we indicated to which sub-Lagrangian which terms belong. The “interaction” part of the Lagrangian―terms―could also be written (thru partial integrations) in the form. Using this form of writing the Gravitational sub-Lagrangians, and can be written in this form:

(80)

(81)

(82)

In the expr above we for the sake of uniformity replaced with The above Lagrangian is a quadratic with respect to the first derivative of either metric or grav-field, which will produce linear system of equations for each variable. However due to their interactions―Lagrangian―the equations of motion will have “interactive” form, meaning that equations for grav-vector will include metric and vice-versa. In symbolic form these equations could be written as:

(83)

This “interaction” could be removed is we request that metric tensor satisfies some conditions―the gage. It is not difficult to see that contains 12 terms:

(84)

The number of terms can be reduced to 8, if we consider a “gage” procedure, which comes form the following considerations. Variation of the gravitational Lagrangian (expr. (74)) by produce a sets of 10 equations for space metric in Minkowski space. However this system has 4 more functions than needed. Metric has only 6 independent functions as in equations above have all 10. This can be fixed by imposing 4 additional conditions on the metric functions―so call “gage”―which takes a form of first order partial derivatives of the metric tensor. In Landau [36] for example the gage has this form. This gage condition reflects the fact that the Minkowski space―as asymptotic at infinity―is not uniquely defined, and the linear equations of “gage” removes this uncertainty.

Imposing such a gage we select one particular system of coordinates (or one group of systems of coordinates―like inertial one) as prefer one, compared to all others. One would hope that such selection is governed by a some general physical principle―like, for example, a system in which the laws of conservation have simple form. In our case we choose the gage in order to simplify the expr. (76) for the gravitational Lagrangian, which could be phrased in the physical manner: “the system of coordinates where Grav-vector and metric do not interact”.

The most general form linear first order expression for the gage has this expression:

(85)

If we apply “gage” relation to the underlined terms in expr. (84) we will reduce to only 8 terms:

(86)

This “interaction” terms could be vanished, if we add to the 5 gage -con- stants 3 more constants associated with transitioning to new (“shifted”) variable accordingly to the following linear expressions:

(87)

Applying the gage (85) and the “shift” (87) we will transfer the expr. (81) into a expression where has this form:

(88)

Demanding that vanishes―each figure bracket should be zero―pro- duces 8 equations for 8 unknown constants (thru and, ,), which yield this solutions for the gage’s and for the “shift”’s thru the constants’s. From eq. for (line 1) we get explicit value for the. From eq. for the (line 4) we get an explicit value for. The other 6 equations splits on 2 groups of 3 each―one for, , and the other for, ,

(89)

What important here is that these constants thru and thru are uniquely defined by the constants -s of the Lagrangian. After removing the interaction Lagrangian of metric and grav-field the gravitational Lagrangian can be written as:

(90)

The “double dash” in the reflects the fact that get additional terms due to the transition from to accordingly. The terms repre- sent the gage―see expr. (85). The equations of motion for the Grav-vector and are “standard” non-interactive vector equations with its source and “Einstein-like” equations for the metric field with its source.

(91)

For the gravitational vector we have:

(92)

I vacuum, written in components (t, x, y, z) or, these interconnected equations in the “rest” system coordinates will have this form:

(93)

The equation for obtained by variation of the Lagrangian (90) by leads to the equations:

(94)

And the “gage” equation is obtained by variation of with respect to Lagrange multiplier.

(95)

All constants in the above equations (93) and (94) are the linear combinations of the constants associated with the choice of the Lagrangian of the Total- Matter―see expr. (30)―and the -constants of “gage”.

Let us emphasize that the equations (93) and (94) are the first order approximation with respect to parameter (―Plank length and― atomic length) for the system with N number of particles. The functions and, which are small addition to Minkowski metric and constant time vector correspondingly, are defined by only number of particles N and by parameters of atomic scale: speed of light “c”, Plank’s constant, and particle mass (for proton).

In non-static case the equations of motions (93) for the could be, depending on the value of the -constants, either hyperbolic (when and) or elliptic (when and). The hyperbolic equations, like for Electromagnetic field, allow the field to exist (and propagate) on itself. On the other hand, the field governed he elliptic equations is not allowed independent existence of the time-oscillating field in vacuum. In elliptic equations the field will modify itself (thus will have time depending behavior), but it will quickly decay to zero any king of oscillating harmonics that it produces.

It is important to note that the possibility of elliptic equations are only due to the existence of unit vector or to existence of Gravitational vector (or).

It is always assumed that all equations of Nature are hyperbolic due to the fact that they are governed by Minkowski metric. That also means that equations for metric (as it is in General Relativity) must be hyperbolic. And thus must allow Gravitational waves (metric waves) similar to Electromagnetic waves [37] [38] . However, the fact that the LIGO project that ran for almost 50 years had not be able to detect the Grav-waves, might indicate that they don’t exist. In other words, the equations for both Grav-vector (or) and metric are not hyperbolic, but rather elliptical. If we consider a rotation of the Earth around the Sun (or Sun around the center of Galaxy), then from point of view of hyperbolic equations (even if we consider only General Relativity), the change of the metric (as the Earth moves so does the metric created by it) must produce waves (just like moving charge make Electromagnetic waves), which due to the “hyperbolicity” of equations runs away or “radiate” out taking with it some energy and thus forcing the orbit to decay and in the end for Earth (or Sun) to fall on the center of mass against which it rotates. However, if we assume that the equations for the Earth metric is governed by elliptical equations, we get totally different picture. The law of conservation of Energy-Momentum would state that flux on the surface due to the exponential decay in time is zero (no radiation). That means that the change of Total-Energy of Earth-Sun interaction is constant. And that means no degradation of the Earth orbit. The “radiation” from the star (or in our case from the Earth) is possible, but not by metric radiation”, but by emitting some Matter. If we choose our -constants based on the requirements to vanish “unwanted” terms―see expr. (53) and on―there is “50-50” chance that the equations for or turns out to be hyperbolically or elliptical. That means that we would need to impose additional requirement on them to be one way or the other. And if we choose it to be elliptical, the question is: should it be both or only one. The simple answer is both. It would seem to be logical that the pair and are tide together and thus have similar form of description― both elliptical.

Let us now discuss the fluxes and that defines the Grav-vector (or) and the metric accordingly. As we mentioned before the gravitational vector is a square-vector and thus has units. Its second derivative and thus the flux will have units, which is the same as energy-momentum tensor (measured in units). And it seems to be logical for the macroscopic system―with large number of particles to equate the flux with component of the tensor energy-momentum

(96)

The constant is unit-less constant with the value about (by order of magnitude) and of course might depend on the form of matter under consideration (like gas, liquid, etc.) and its parameters (like pressure and temperature). Similarly, we can use this approach (justification) for the treatment of metric flux. Per our definition of metric-correction the has the units. Its second derivative has units, which is the same as Energy-Momentum tensor measured in units. With this justification we can write:

(97)

which transfer the equation for into “Einstein form”. Of course, only symbolic sense. The are quite different from Einstein’s.

We need to point out that because we transformed the description of Gravitational field and metric into Minkowski flat space, we automatically obtain the law of conservations that are the consequences of Lagrangianian description. These laws are of “approximate” nature as the flat Minkowski approximation of the space.

Accepting the expr. (96) and (97) we will acquire 2 more laws of conservations that are the consequence of the of the law of conservation of Energy-Momen- tum:

(98)

In case of static (independent of time) spherical symmetry the Equation (94) for the metric, defines as, in vacuum reduce themselves to only two obvious equations as a function of distance:

(99)

where the constants and are defined by the tensor (i.e.). These are exactly the equations and solutions of Einstein General relativity, where and. The only difference we have is that in our derivation we also have selected a special “preferred” system coordinates defined by the gage Equation (95), which reduces to this relation:

(100)

From which follows that and, which also leads to the linear relation between, and. And if we want that this “rest” system of coordinates were the conformely-euclidian we need to request that

It worth mentioning that the Landau gage applied to spherical-symmetrical metric yields this relation:

which is neither Schwardchild nor conformely-euclidian representation (to be precise the deviation from Minkowski metric). The later of course is used in the calculation of two standard GR tests―bending of light and drift of Mercury orbit.

The solution for the grav-vector has also an obvious form:

(101)

Combining expr. (99) and (100) we get the final form solution for the “single bar” Grav-vector:

(102)

Per our assumption that all particle have gravitational field which is accumulated to a very large magnitude, the constant must be positive. Or more appropriate it should be the same sign for all baryons. It is a condition on the functions that are the core of the “source” flux, which we will address in Section 9―Mass-Matter. Or it could be viewed as one more condition on unknown constants -s―see expr. (12)―as well as -s, expr. (30).

One of the most important subject that we have not addressed yet is the subject of motion of the point-mass in gravitational field. Accordingly Einstein’s GR this is governed by the Einstein’s “geodesic” postulate which states the point- mass body moves along geodesic line of the curved metric.

It need to be pointed out that this postulate cannot be taken as one of fundamental principles of Physics, but only as an approximation for two reasons. First, the motion of the body should be derived from the Lagrangian of the physical description. In other words, this postulate is not necessary. In second, it by its nature applies to the “point-mass” physical configuration. It is totally looses its meaning when we consider a field description of some physical entity―like quantum mechanical description of the electron or as in our case a field description of the Matter (Tensor-Potential).

In our case the correct approach is to derive the movement of point-mass from the Lagrangian―expr. (74)―of the Matter, which includes the interaction between Matter and Gravitational fields (and). And based on the results of this derivations we might have a situation that mass-point depends on both the metric and the Grav-vector and thus on both the metric and the Grav-vector. If we assume that Einstein’s geodesic postulate is true we must impose an additional requirements on -constant to vanish the dependence of mass-point trajectory on the vector.

In this regard it is worthwhile to point out that described above procedure of “decoupling” the Grav-vector and the metric―where we “shifted” the Grav-vector and left metric unchanged per expr. (87)―is not unique. One can consider the “decoupling” of and by “shifting” only the metric:

(103)

which in coordinate form has this form:

(104)

If in the expression (87) the Grav-vector is “shifted” while the metric is unchanged, in the expression above the metric is “shifted” by three parameters while Grav-vector is unchanged. This means that if we require that Einstein’s geodesic postulate holds, we get different conditions on -con- stants.

In fact both approaches―“shifting or “shifting―could be combined in more general “decoupling” approach if we consider transitioning to new variables and accordingly to the following linear expressions:

(105)

with total 12 -s and -s constants. These expressions, of course, must be reversible―that is we should be able to express and thru the variables and. This can always be done as long as the -constants satisfy some (“non-equal zero” type) conditions.

Writing the expr. (105) above for the spacial components of the metric tensor and we get.

Writing expr. (105) for the metric components “” and we get this system of equations:

(106)

The second “non-equal zero” condition of reversibility comes from contracting the the first line expr. (105) by and the second line of expr. (105) by and separately by:

(107)

In order to reverse the relations the determinant of it should not be zero――where:

(108)

If we substitute the expressions (105) and expr. (85) into the expr. (76) we again get quadratic with respect to and Lagrangian. The part that contains quadratic with respect to terms will have only four constants. The last part of the Lagrangian that is quadratic with respect to will depend on 8 constants. And the middle part――will include all 12 -s constants.

If we add to this the 5 -constants of the gage (85), we will have total 17 constants that we can choose to vanish and to simplify and Lagrangians. With 7 constants needed for vanishing and 4 possible constant available to simplify we have 6 constants available for us to simplify. We can use these constants to yield the dependence mass-point movement along geodesic lines only.

When written in the form (90) the equations for are independent of the Grav-vector and thus have exactly the Einstein’s form and do not contain “dark matter”, not even a gravitational constant (as it is in linearized Einstein’s equations).

There are however the differences between this set of equations for the correction of metric and Einstein’s one.

a) The Lagrangian for the metric correction even though quadratic with respect to in form has different constants as compared to GR.

b) It most likely will have a second derivative of by time which it does not have in the GR.

c) It is only true in the “special” Minkowski flat system coordinates, that is defined by the gage (85) and associated with it equations on the functions.

d) We know that it cannot be used for the “large” systems where its own gravitational field is compared to the outside Grav-field. And any attempts to use these equations for the description of Universe, probably Galaxies or near event horizon should be considered as unjustified.

In our approach of description, we started from the curved space and by a way of linearizion came up to a flat Minkowski space and description. If we knew the procedure how to do it, we could at this step derive the point-mass approximation directly from the form of Lagrangian of the Matter and thus the trajectory of the “point mass” test particle.

We now can make one more step further (sort of step in reverse direction) and ask ourselves a question: could all the description of point-mass particle behavior can be absorbed into 10 functions that would represent the curvature of the space. In other words, can we replace partial derivatives inside the Lagrangian with a covariant derivatives associated with some metric and add the Einstein equations to complete the description? Or phrase it differently, can we “hide” Lagrangian (including Grav-vector) in the some effective metric and Einstein’s equations of GR?

If this program is possible, it would lead us directly to the Berkenstein [39] idea of effective space metric, which is the basis of his TeVeS theory.

In this case, we being in flat Minkowski space, would be in no way able to distinguish whether such curved space is real―that is equal to―or not. Such determination could only be possible, if we consider a larger system where linear approximation of a weak gravitation is not applicable or where mass-point approximation is clearly is not the case.

In principal one can ask even a more general question: can not only for point-mass but for any Matter describe by the Lagrangian of this theory (expr. (1)) be equally replaced by a Einstein Lagrangian density

with some effective curved metric

space. It’s quite possible that due to the “linear” description of the gravitation, such replacement is always exist. In such case the Einstein equations are always right for any weak gravitational field, except for the fact the it produces not real, but “effective” curvature of the space.

8. Electromagnetic and Other Vector Fields

In this section we deal with the Lagrangian of the Matter and particular with the Lagrangians for the three vector fields, and. The Lagrangian of the Matter which as we showed earlier can be written as―see expr. (59):

(109)

where is a linear function of the tensor (or and) and the vectors, and.

The tensor is a tensor of Matter, which is Total-Matter without the Gravitational field―no and no curvature tensor. In the expr. above we consider all variables (tensors) as a “bar” tensors―or tensors on top of Minkowski space. This means that all its derivatives as just partial derivatives and manipulation of indeces is done with Minkowski metric.

The identification of the Electromagnetic field is not a straight forward procedure. Potentially any of the three vector fields, and could be chosen as one. The problem here is not only to choose the vector field that leads to Maxwell equations (or to equations the closest to Maxwell’s), but also to be sure that the remaining two vectors fields are in some sense unique and not just a repetition of the vector field that we had identified with Electromagnetic one. The major difference between the vector fields comes from the fact that today in physics there is only one (apart from gravitation) long-distance field―the Electromagnetic field. That means that the other 2 vector fields―out of total 3, and―must be short-distance. The short-distance here means that the “time-component” of that vector does not have asymptotic at infinity, but rather.

We begin with defining the expression for Total-Matter tensor by choosing -constants as:

(110)

The tensor of Total-Matter as a function of―or its symmetrical and its anti-symmetrical parts―has this expression:

(111)

and “” represents a covariant derivative “using”. Our next step is to rewrite the expression above in a form where instead of we use the tensor Potential:

(112)

(113)

In the expression above we kept the label on the symmetrical part of the Tensor-Potential so to remind us that it includes the vector inside, which is not a part of the anti-symmetrical part of the Tensor-Potential. And in the final step we need to write the tensor of Total-Matter as a function of sub-fields (or and), , , and (we added a symbol to indicate that the vector always comes as anti-symmetric tensor).

The symmetrical part of the Tensor-Potential we has this form:

(114)

and for anti-symmetrical part of of the Tensor-Potential we have:

(115)

The tensor that comes with a factor comes from quadratic expression for:

(116)

The tensor is linear with respect to any sub-field. Thus for example for the field is defined by this expression:

(117)

The above expressions can be somewhat simplified. This could be beneficial (and in fact mandatory) if we would be caring out the exact calculations. But for our purposes what is important is the fact that is a function of sub-field. Similarly we can calculate the expression for the tensor.

However, if we are to write the expression for, we must point out that unlike for the sub-fields and where it is defined by symmetrical part of the Tensor-Potential, the will be defined by both symmetric and anti- symmetrical part of the Tensor-Potential. The anti-symmetric dependence

comes from term representing covariant derivatives. The term (that come with factor in open form has this expr. (*need bar*):

(118)

Using (112) and (116) we get this expression for the tensor:

(119)

where we added square brackets with the labels “sym” and “a-sym” to indicate where those terms came from. The exact expression of thru is not that important at this point, except for the fact it is linear with respect to and it does not depend on value of Grav-vector G, but only on its unit vector.

For the other two vector fields―and―the and are defined only by anti-symmetrical part of the Tensor-Potential and will depend on the values of -constants. It is not difficult to see that for our choice of - constants―expr. (112) and (112)―.

However for the in general it’s not the case. From (112) the expression for has this form:

(120)

If we substitute in the expr. above the exact form of, we get this result for the tensor:

(121)

And if we choose the constants the. This is exactly the reason why we made this choice of―so the contains no vectro field.

Our next step is to write the tensor of Matter as a function of all sub- fields. Because the tensor of Matter is quadratic function with respect to the Tensor-Potential when written thru sub-fields it can be split on two groups: sum of sub-fields’ Matters and their interaction (labeled):

(122)

First we note that because of our choice of, the tensor of Matter for has no quadratic terms and because it contains the first derivatives only in “Maxwellian” form:

(123)

and because thru all interaction tensors are not present.

The situation is somewhat different for the sub-field (we use *-symbol to indicate the fact that vector field always comes with a fully-anti-symmetrical tensor.

Because of our choice of the constants, the tensor of Matter for the vector contains the first derivatives only in Maxwellian form:

(124)

And because per our choice of and all quadratic terms for vanish.

We can now calculate the “interaction” terms, and.

The interaction term comes only from the terms proportional to and of the expr. (112):

(125)

Form which follows this expression for

(126)

due to the fact that is fully symmetrical and traceless.

Unlike for the interaction of vector field with the vector or do not vanish.

For interaction terms of vector and come from the same expression (112). However now instead of we need to use. Form this follows this expression for:

(127)

and for any choice of, except for, the interaction terms do exist. Unfortunately, we cannot use due to the fact that expr, (25) would not be reversible and thus the determination of vectors and from the Tensor-Potential would not be uniquely determined.

For interaction terms for vector and come from two places. First, from the “” covariant derivatives (112), which would be proportional to the constant where tensor comes thru its con-torsion form. And secondly, from the quadratic terms associated with constants―see expr. (1)― which is due to relation is fully anti-symmetrical. Using (12) and (124) we have:

(128)

The expr. above could be simplified: some terms vanish, some could be combined together. The actual expression of (128) is not that important (at this moment), but what is important here is that the interaction of vector and tensor does exist for all parameters, and. Combining the results of (124), (127) and (128) we get these expressions for the tensors written in symbolic form:

(129)

If we now calculate the expression for the tensor of Matter associated with the vector we get all possible terms, which symbolically can be written as:

(130)

Important to note that in expr. above does not have “Max” lable attache to it, implying that derivatives of include all possible terms of―for example, or. Combining the results of (123), (129) and (130) we get these expressions for the tensors, and the Lagrangian of the Matter L:

(131)

where we used square brackets with index “H”, “B” and “E” to indicate the sub- fields’ Lagrangians.

At this point we can transition to the “bar”―variables and to flat Minkowski space―and, so the expr. (131) above takes this form:

(132)

Let us first consider Lagrangian for the vector. It is not difficult to see that in this case the Lagrangian (132) for just the vector takes this form:

(133)

where is an anti-symmetric 2-index tensor and is a function of the tensors, and. The antisymmetry of is due to the antisymmetry of tensor. Or reinstating “summation” and the -s:

(134)

In the index representation has only 3 invariants and can be written in the following manner:

(135)

where is a 1-index (vector) flux of the field.

The constants thru are linear combinations of the -constants of expr. (132) and -constants of (132). If we impose one more requirement on the these constants such that, the Lagrangian for the sub-field takes Maxwellian form―and thus could be a basis for equating the sub-field with electromagnetic field. However, there is a problem. This Maxwellian (in form) Lagrangian has one peculiar property, which has to do with the form of the flux. Because this flux is the “diversion” of the anti-symmetrical tensor it cannot represent the electrically charged particle. That

is, it can not have a solution where vector potential has behavior

with respect to distance from the source. Indeed, if such solution existed, then the constant should be found as an integral of the source written in spherical coordinates:

(136)

But since the flux is has the property 1 this integral is zero as it follows from these calculations.

(137)

This will be true for any “well” localized behavior of the flux―that is the flux that has asymptotics faster that. In other words, the static solution for may not have a distance asymptotic, but it can have the asymptotic of higher order―like, corresponding to a short range interaction similar to a dipole (thus the letter “D” for this vector field).

It needs to be mentioned here that the idea of representing the flux of a charged particle as a diversions of anti-symmetrical tensor had been put froward by Gustav Mie some 100 years ago. W. Pauli in his “Theory of Relativity” [40] has analyze it in some detailed and as an end result showed that it has significant problems―albeit for the different reasons.

In vacuum the Lagrangian is identical to Maxwell one and the D-field has the same property as electromagnetic field. So in a language of quantum mechanics we can call it D-photon.

Everything that we said above about the sub-field is in fact true even if we remove the requirement. Since Lagrangian (135) (with) depends on anti-symmetrical tensor only, the law of conservation of of the flux still holds.

Also, we can “eliminate” the whole factor if we introduce “scaled time” and a new vector. In these new variables the Lagrangian of the sub-field (or now) has this Maxwell form:

(138)

This can be easily seen if we introduce instead 4-dimensional tensor two 3-dimensional vectors―“electrical” and “magnetic” This allows us to rewrite the Lagrangian in this form:

(139)

And if we introduce new vector and “scaled time”, we can rewrite the Lagrangian in Maxwell form:

(140)

It needs to be mentioned that the static solution of equations for sub-field that come for the Lagrangian (132) (where) is identical to the Maxwell static equations. This might make the requirement not needed.

We now consider Lagrangian for the vector―or―since it always comes with fully-antisymmetric Levi-Chivita tensor (in flat Minkowski space). It is not difficult to see (see expr. (124)) that the Lagrangian for the sub-field has this form:

(141)

The square of fully anti-symmetric tensor (in flat Minkowski space) can be expressed thru Minkowski metric and thus in terms with “double star”, the could be dropped. So the final expression for the Lagrangian of the sub-field has this form:

(142)

This Lagrangian differs from the Maxwell’s one by a) having an extra invariant in the first symbolic term and b) by the presence

of the underlined symbolic terms. As in the case of the sub-field we require that the (although as we will discuss it in few paragraphs later it might not be an absolute must) the -constants (and the constants, and) must satisfy a condition so that the extra term vanishes. In addition we also must require that the -constants (and still undefined constants,) must satisfy a condition so that all invariants of double underlined sym-

bolic term, which consist of only three invariants, and, which consist of only x invariants, vanish.

(143)

This requirement steams from the necessity to avoid non-physical situation in the equations for the vector. Indeed, if such term exists, then in equations for (or) it will correspond to the “source” that is only function of Electromagnetic field. And since (or) is localized (has asymptotics faster than) it would require that should be localized as well, which of course is not the case because it contradicts to the long-distance property of Electromagnetic filed. Strictly speaking, the underlined terms don’t have to vanish, but they must not contradict to the “short” distance of either or at least in static solution. But for now we assume double underlined terms must vanished.

In order for the Lagrangianin―expr. (142)―to have the Maxwell’s form and (thus to represent Electromagnetic field), we must demand that the single underlined term vanishes as well, although as we will discuss it in few paragraphs below this might not be an absolute must―and in fact it might better represent the physical reality.

If we drop all the unwanted terms we get this expression for the:

(144)

In the expression above we split the Electromagnetic flux on two components:

first, , that was derived from the term proportional to and the

second, that was derived from the term. There is a significant difference between these two fluxes. The first (see discussion for sub-field) always corresponds to the zero total charge while the second can have non-zero total charge.

If we extend this description to a elementary particles such as electron or proton―which in our case are represented by short-distance (or localized) functions (or and), we must satisfy one more condition, which reflects an experimental fact that the electric charge of a such particles is always. This requirement must be treated as a condition on the constants of integrations for either field or mass-matter tensors and (or).

In order to reduce the Lagrangian (132) to Maxwell’s we requested that the terms vanish. However what we try to show is that because of “Normalization” procedure such requirement is not necessary. Even with the terms present this Lagrangian could serve as generalized Lagrangian for the Electromagnetic field.

In standard Maxwell equations the total electrical charge of the system is an integral of the flux over all 3D-space:

(145)

Now if we consider Lagrangian (132) with, then due to the fact that (and as we will show few paragraphs below too) is highly localized it might not influence much on description of the sub-field outside the mass-matter, except for adding some constant to a total charge value:

(146)

which could be scaled down or totally absorbed by the scaling procedure.

To illustrate this point let us consider a toy model that corresponds to a case where Electromagnetic field has only zero (time) component and functions (or and) depend only on the distance (―spherical symmetry) and could be expressed by a piece-meal functions: that is a constant for the distances smaller than the size of the particle―and zero for the distances.

In that case the equations for (which we will labeled as) will have this form:

(147)

where and are constants, which approximate terms and correspondingly, and is the size of the particle. The solution for―which is regular everywhere inside the “particle”―as a function of distance has this form:

(148)

where is an arbitrary constant of integration. In our “toy model” we must choose the constant in such a way that the charge of particle (or −1). The presence of term in the Lagrangian for field will modify the behavior of the inside the mass-matter (particle) but―due to the Normalization procedure―will not change its asymptotics. In other words, the “Normalization” procedure guarantees that the asymptotic of Electromagnetic potential (in our toy model―) outside the particle is always. And as long as the asymptotics at infinity is still, the Maxwell description of the Electromagnetic field of a macro system (in statistical sense) could be recovered by introducing Dirac’s -function as a single particle’s flux.

(149)

Such -function approximation is possible if the “single” particles are far enough from each other (as compared to their size) so we can use the approximation, which might be harder and harder to achieve as the speed of “single” particles increases toward the speed of light.

By maintaining the term in Lagrangian for the vector we in fact postulating that the standard Maxwell equations for the Electromagnetic field is an approximation of more complex equations (132) derived thru Affine Unification.

The other point that needs to be made is related to the law of charge conservation that is part of the Maxwell equation. However, by itself the law of conservation is not enough, simply because it does not forbid for example for electron to be split on two halves―each with a charge −1/2 as long as the total charge is still −1. What makes this impossible is the statement that electron is not splittable. On the other hand if electron is not splittable, the law of conservation of its charge is automatically holds. So we come to the conclusion that the law of charge conservation is more a property of electron localization and it “normalization” procedure than the outcome of equations of motion for the Electromagnetic field.

We can now consider the Lagrangian for the vector. In symbolic writing it has quite different form all due to the fact that is a function of the vector (we will ignore for now vectors and):

(150)

where the first 3 terms in expression for correspond to the vacuum, the single-underlined terms could be written as a flux and the double-underlined terms represent non-linear interaction between vector and the mass-matter.

We need to be reminded that the invariants are formed not only with Minkowski metric, but also with a pair of unit vectors.

In vacuum the Lagrangian represented by the first three terms of (150) consists of 10 invariants―3 for invariants and 7 for:

(151)

It is important to point out that in expr. above the terms are not of “Minkowski type”, i.e. anti-symmetric, but of general type―that is includes both anti-symmetrical and symmetrical parts of the tensor. This will lead a situation when the Lagrangian for vector in its square derivatives will contain all 6 possible terms, very much similar to the gravitation vector―see expr. (83). One of the problems here is that unlike for Electromagnetic field (which had been thoroughly studied over last 100 year), we don’t have similar knowledge about non-linear fields, like. So we can only limit ourselves to more or less general statements.

Since the Electromagnetic vector is the only (apart from Gravitational vector) long-distance filed, the sub-field must be (in vacuum) a short distance field as well.

The vacuum equation of motion has this form:

(152)

It is not difficult to see that if we are to look for the long-distance solution (,-constant), we get a quadratic equation for the constant. If we assume that this equation has no roots, we will automatically have the property that vector cannot be a long-distance vector.

But even if such solution did exit its value is of order magnitude of the -constants that define the equation (152)―or magnitude of atomic size. The non-linearity prevent the value of vector to grow.

If we consider the question of propagation thru vacuum, we will have two possibilities, that reflect the structure of equation (152) for. The first possibility is when the equation (152) are of hyperbolic type―that is to say that the second derivative by time and the second derivative by spacial coordinates are of apposite sign,. In this case the propagation of -vector in vacuum should be possible, because for small amplitudes of the non-linear terms could be ignored. The non-linearity, however, would be much pronounced for low frequency/high amplitude waves (―in units 1/cm).

That means that once emitted this B-particle will split on several smaller (in terms of magnitude) but higher frequency (harmonics) particle that would travel thru the space as the zero mass particle, very much similar to the regular photons. And because the contains terms that include the unit vector (just like in case the Gravitational vector) the vacuum speed for such particles might differ from speed of light by some factor around 1 (from 0.1 to 10)―and thus be even larger than the speed of light―but probably not by several factors of magnitude.

In the second possibility that corresponds to the situation when equation (152) are of elliptical type―that is to say that the second derivative by time and the second derivative by spacial coordinates are of the same sign,. In this case the propagation is not possible at all.

It seems to be contradictory that in the Minkowski space the equations for the some form of Matter (in this case) were not hyperbolic. However, this is quite possible due to the existence of unit vector. In addition to contractions (in forming the invariants) using metric, there will be terms where contraction is done using a pair of unit vectors. It is these terms will create some terms that will change the sign in front of from +1 to −1 and thus change the type of equations from hyperbolic to elliptic.

It is interesting to point out that in vacuum there is some interactions between Electromagnetic field and either field or. But because in vacuum comes only in the form of Maxwell (anti-symmetric) tensor, the interaction for the vector will have a scattering effect (although with some energy loss). However, for the vector, the Electromagnetic field will have an effect of a source, and thus dragging some along with it.

9. Mass-Matter

One the advantages of the theory based on Affine Unification is that it allows us to view the equations for tensor and (or for complex and its complex conjugate) as equations for the atomic matter (mass-matter)―such as electron, proton, etc.―or at least their classical description or/and approximation. First we need to emphasize that in weak gravitational field (as in our Solar system) these equations are the 3-index equations (with appropriate symmetry). on top of flat Minkowski space with all covariant derivatives replaced by partial derivatives.

Also important to point out that without any specific requirements it would be too unrealistic to expect that the Lagrangian of the Matter does not contain unity vector.

And because the equations are derived by means of variation of a Lagrangian, they will contained all the “law of conservations”, such as conservation of energy-momentum tensor―associated with it. Those “laws” are not totally universal, but only true as approximation in the “weak” gravitational fields. But for an atomic particles, the “correction” associated with non-Minkowski space (or with curvature of space) is about for 2 protons.

Besides tensor these equations will (or may) contain also vector fields, and, equations for which we have already considered in the previous sections. We know for instance that if we consider a charged particle (like electron or proton) the electromagnetic field must be present. It is not quite clear what role the other fields serve and in which case their presence in the equations could be ignore. Or may be never.

In addition to the fact that these equations―when we consider elementary mass-particles―are written in flat Minkowski space they (the equations) contain no functions associated with the space curvature or grav-vector.

They also are highly non-linear―that is the Lagrangian contains terms proportional to third and even forth power of tensor, such as and. This non-linearity, serves exactly the same purpose as a postulate of quantum mechanics, which states that no two fermion particles can occupy the same space. Or in other words, the super-position of solutions is not a solution. The non-linearity also produces the “localization” of the mass-particle. If the localization is unstable, the mass-particle will eventually disappear passing its energy to other fields, such as vector fields, and.

There are two type of “stable” solutions that should be consider. The first one―the obvious one―is a stationary solution, which does not contain time coordinate. The second is the oscillating type of solution, in which the time is present only under some periodic function―like or) along with associated with it frequency. The second is of course more general solution and does include the first type as a particular case corresponding to. It is not clear―since we can not simply switch to a Fourier representation―if such solutions (with) do in fact exist for a highly non-linear equations. And if they do, do they produce the discreet states (or quantization) similar to Quantum Mechanics.

It is not the goal of this paper to give a full investigation of the non-linear solutions for the mass-matter tensor. But instead to look at some simple features of such solutions.

First we need to point that the equations of must be properly symmetrisized accordingly the symmetry of the tensor or. For example, if we consider in the Lagrangian for Mass-Matter a term:

(153)

The equations of motion for the tensor will have this form:

(154)

But since has the following linear symmetries―is fully symmetric and―so should the equations. This lead to the Equation (154) to take this form:

(155)

Likewise if we take a Lagrangian term that contains unit vector it too must be properly symmetrisized:

(156)

or in Minkowski flat coordinates:

(157)

9.1. Localization

One of the questions that we would like to consider in this section is asymptotic behavior of the (or and) on large distances or possibility of localized solutions. We must postulate that to be truly localized these solutions (or dependence vs. distance) must decay with the distance faster than. More realistically like or even faster. It is not difficult to see that at infinity the nonlinear terms in the equations for will vanish (for example, if, then, while and) transferring the equations of motion for the tensor to a linear system of equations similar to the equations of QM. This in its turn may produce the “basis” for transitioning (or coupling) this Affine Unification description to linear QM description of elementary particles.

Typically, the equations of motion of second order (second derivatives)―such as for vector potential for electromagnetic field―produce static solutions with an asymptotic. But because the large number of independent functions that describe the tensor, there is a possibility for a static solution with much stronger than behavior at infinity. We can demonstrate it on a particular case, where. We don’t know for certain if such assumption is physical―it might be that every mass-matter must have both tensors and present― in which case this considerations should be viewed as a toy model. However the conclusion that large number of independent functions in can lead to very much localized solutions is still holds.

In case of spherical symmetry is described by 6 independent functions――from which the components of the tensor can be easily deduced using the invariant form:, which in case spherical symmetry could be written as:

(158)

From this immediate follows expressions for the 8 non-zero components of the tensor

(159)

The condition (in Minkowski flat space) in curve-linear coordinates lead to these equations:

(160)

In general the equations written in spherical coordinates has a form of linear system of 4 equations for 4 functions―, , ,:

(161)

where the symbol represent a derivative by distance― and are constants with indeces “p” corresponding to the row and “q” corresponding to the the term within the row. The -s constants are linear combinations of the constants -s of the expr. (1), which in their turn depend on the constants -s of the Lagrangian 1.

Asymptotically at large distances the functions, , and should have behavior, where K is a constant. Assuming that all of then have the same “n” (but different K―, , and) and substituting it in the set of above we get a system of 4 linear equations for, , ,.

(162)

The system of equations for the constants has a non-zero solution(s) if the determinant of the eq. (162) is zero. This will lead to a 8-power equations for the parameter “n”, which solely depends on the set of constants or ultimately on the set of constants -s. Our postulate of “localization” thus requires that the constants (and thus the constants -s) were such that the solution of (162) had a root. We can look at this problem in slightly different way. We can set n to 2 (n = 2) and view the Equation (162) as requirement for the (and thus the constants -s). Similarly, we could choose (or any other value greater that 1). Another word, by adding one (or more) condition on parameters -s (or by choosing different forms of Lagrangians) we can set the of “n” to any number. The problem here is that we really don’t know what number(s) we should pick. So we are left with 2 options: a) set no conditions for “n” and hope that -s themselves will provide a proper value, or b) derive “n” from other mathematical (or physical) considerations and use it to impose additional conditions on parameters -s.

9.2. Normalization

There are some requirements that can not be achieved by proper choice of parameters -s and do relay on the constant of integration of each solution for. Among such requirements are the “normalization” requirements.

In the previous section we have discuss the normalization procedure associated with the law of “fixed minimum charge”―such as proton charge to be + 1 and never less or more. This means that for any solution that we identify with a proton, we must get its charge or asymptotic at infinity as a function of distance equal to.

The other requirement could be called “Einstein law” which states that energy of particle is.

It is not difficult to see that because all equations are derived from unified Affine description, one of the constant of integration is a scale of coordinate. The units of is 1/cm. If is a solution, then for any constant the

is a solution too. The tensor energy-momentum derived for

function using corresponding Lagrangian and bring expressed in the units has the same units. The Einstein law that total energy of a particle (say proton) is can be written as:

(163)

Another words, the Einstein law is a one particular normalization procedure. From physics point of view this normalization does not explain why electron’s mass is 1800 times less than proton’s one. Or―using electron characteristic length―it is 1800 times greater than of proton’s. It simply assign a proper value. In order to get the factor 1800 we need one more condition that would effectively compared the length between themselves. Perhaps some common asymptotic at infinity. For example if we postulate that at infinity has the same asymptotic, where K is a fixed constant. That will produce a relation between the characteristic length (and, which of course must be 1836, which is additional condition for the -s constants of the right Lagrangian.

9.3. Particle, Anti-Particle

It seems logical, considering the frame work of the Eddington Affine derivation, to assume that if a some form of the tensor potential represents a particle, then (minus) represents an anti-particle. However, if we consider (in symbolic writing) the Lagrangian for tensor potential (see expr. (109))

(164)

we can see that because of the term, if is a solution is (in general) not. So the more accurate definition of anti-particle should be: transfers to and the coordinate transfers to. And since is also a solution then annihilation of particles is possible, assuming of course that all other laws of conservations (energy, etc) are preserved.

This immediately could be apply to any mass-matter (particle) (ignoring for now all the vector field, ,). If particle is described by some solution then the solution describes anti-particle.

From that point of view both linear photons that we discussed in Section 8 in vacuum are identical to their anti-photons due to the fact that Lagrangian of these particles is quadratic with respect to their first derivatives.

It is not difficult to see―see expr. (142) that such procedure correspond to the rule: if particle is changed to anti-particle, the electrical charge (current) changes its sign.

However, if we switch from matter (particle) to anti-matter (anti-particle) expression for Energy-Momentum for the Matter

does not change. Similarly, the unit vec-

tor does not change. This is a consequence of the fact that the gravitation vector is a square-vector:, where is deduced directly from the tensor potential. With the change of and the coordinates the vector changes sign, but the square-vector remains unchanged.

In conclusion it needs to be pointed out that we take as a description of a particle (mass-matter) based primarily on its form of 3-index tensor. It is a possibility that this tensor could be “reduced” to simpler forms―such for example as 6 spinors. In that case one might want to associate the simple forms with the “basic particles” and construct the atomic particles from them.

9.4. Elliptic vs. Hyperbolic Equations

We have pointed out before (see Section 7 and Section 8) that existence of Unit vector allows for the equations of Gravitational field and metric to be elliptical. This possibility even more important for the mass-matter fields.

The difficulties of establishing gravitational waves (which in fact should have been an easy task taking into account the sophistication of measuring technologies) might be taken as a proof of the fact that Gravitational field and/or metric described by elliptical equations.

However we almost certain that the mass-matter should be described by elliptical equations. The reason behind it is its localization and its stability. If the equations were elliptic, then any statistical fluctuation or any collision of 2 mass-particles would create deviation in the form of the localized particle which would in time degrade to zero. But if the equations are hyperbolic, the deviations are the composition of “running waves” that can live on its own and “run” away (propagate in space) from the particle, taking with it some energy. This, of course would make the particle to degrade and eventually disappear. But some particles―like electron and proton do not disappear. And the only way to assure it is to postulate that the equations that describe mass-field are elliptical.

9.5. Moving System of Coordinates

Since equations for mass-matter (proton, electron, etc) are written in tensorial form, the solution in “rest” system coordinate, where Minkowski metric is and, could be used to obtain a solution in any moving with a constant speed system coordinate.

The equations that describe the mass-matter (electron, proton, etc.) particle contain unity vector and thus in general are not Lorentz invariant. That is to say that in “rest” system coordinate, where Minkowski metric is and, we have one form of Lagrangian (and thus equations), while writing it in moving―say in z direction (even with constant speed) we get different form simply because the unit vector in moving coordinate is going to have non-zero “z” component: .

Another words, the moving system of coordinate are not identical to the “rest” one. The solution for will be different and will depend on the speed factor both in amplitude and direction sense. This change might be imperceptible for small, but if we accelerate say ion of helium to the speed compared to the speed of light, we might expect that its atomic levels change or it might simply loose the last electron transferring itself into alpha-particle.

Interesting, that beta decay (weak interaction) of large atoms, has the value about which is comparable to the speed of our Sun moving around the Milky-Way Galaxy. So, we might suggest that Sun’s speed (or “non-rest” system of coordinates) is sufficient large (even though it’s only of speed of light) to cause large atoms (nuclei) to change its shape enough to cause the decay of those atoms.

This forces us to ask a following question: how does the localized solution for look if the “particle” moves with a constant speed. Let us consider a simple 2 dimensional Minkowski flat space toy model, which as we will show has all the elements that are associated with moving mass-matter. We will consider the following Lagrangian of function

(165)

which leads to the equation

(166)

In the “rest” system coordinate where the equation has this form:

(167)

which has static (time independent) solution

(168)

If the Equation (1) take elliptical form, with no “running wave” solutions―the solution in the form has real

(169)

Our next step is to find the solution for a mass-matter that moves with a constant speed v, which of course is a rotation in space that leaves metric unchanged. For that we need to switch to moving system coordinate associated with moving particle and write the Lagrangian and equations and find the static solution in that system coordinate. Since (169) written in tensorial form, switching to moving system coordinate is simple to rename coordinates to. There is though an exception―the vector which in moving system has both components depending on velocity v:

(170)

with the static solution:

(171)

And with (our case), we can see that the localization is remained on all velocity’s values. However as v approaches to speed of light the “particle” get stretched in spacial dimension.

In order to see the particle in the rest system coordinate all we need to do is replace with, which would describe a

static solution of the particle moving with a constant speed with respect to “rest” system coordinate. If we consider now the possibility of “running waves” solutions―that is the solutions in a form―we get this equation:

(172)

from which it’s easy to see that in our case for all the the expr. under the square root is negative implying that no “running waves” are possible. In other words, the particle moving with any speed will deform itself, but it will mountain it ellipticity of its equations. Of course this is a toy model and we should not put to much into it. It is quite possible that for complex nuclear assemblies (large number of baryons) the deformation due to the high speed could be “too much” for the assembly to stay together and we might expect that at some high velocity such assembly fall apart (decays).

10. Cosmology

A cosmological description of the Universe with a vector field as gravitational matter is both difficult and simple. The simplicity comes from its “statistical” uniformity. If we assume that Universe is uniform closed 3D-sphere with oscillating (or static) radius―and this is the only physical and philosophical right assumptions―then the physical point of view, the presence of Gravitational field allows this interpretation of Universe’ behavior. As Universe expands the average gravitational field decreases, the effective value of Newton’s gravitational constant increases, which increases gravitational pull of masses. Eventually, this pull will be strong enough to stop the expansion and reverse it to compression of the Universe. At the other end the shrinking of the size of Universe increases the average value of the gravitational field, effectively decreasing the Newton’s gravitational constant and thus decreasing gravitational pull of masses. The mass’ kinetic energy (the temperature of Universe) would be large enough to stop contraction of the Universe and reverse it back to expansion, thus leading to the eternal oscillation of the Universe.

The difficulties, on the other hand, lie in its mathematical model. First, since in cosmology we have to assume that gravitational field defined by all the particles of Universe is comparable to the Matter, we cannot use perturbation series (by parameter G), but must consider the exact Lagrangian. However, we don’t know what the exact Lagrangian is until we actually do the analysis that we described in Section 6―Lagrangian. That is we need to find what is the parameter “n” in expr. (12) and then to derive the actual values of all -s constants associated with that Lagrangian. Also we would need to resolve the issue with extra undetermined parameters of some extra invariants. In other words, we would need to show that all -s are uniquely defined. Second, even if we assume that Lagrangian is known, we would be required to perform “averaging” procedure over the functions describing the Matter. The assumptions that matter is uniformly distributed in the Universe, and the assumption that Universe’s metric could be approximated with closed Universe 3-dimensional sphere should significantly simplify the mathematical side of the problem. The averaging procedure might require of (or lead to) introduction, in addition to a “density”, some statistical functions as “pressure” and “temperature”, which in its turn would demand the laws of “state” for those functions, similar to statistical theory of gas. It is not clear what those “laws” are. Could we equate Universe with ideal “gas” or the interaction between Galaxies are strong enough that one must postulate (or derive) more realistic “laws” of non-ideal gas. Third, the Lagrangian for the vector field would contain not only square of the first derivatives of the Grav-field, but also higher powers of it. If we assume that in expr. (12) n = 4, then we will have. Perhaps it could be shown that these equations yield the results (or reduce to) similar to the ones obtained in the phenomenological theories of Berkenstein’s TeVeS [39] and and Milgrom’s MOND [41] .

The development of Physics―whether we realize it or not―is in large driven by it’s mathematical representation. It is based on Einstein General Relativity, that we arrive to such constructions as: Event Horizon, Black Holes, Open and Close Universe, Big Bang, Black Energy, etc. If this Eddington Unification theory with accumulating Gravitational field is correct, it is not clear which of these phenomena survive. Or what their mathematical description would be if they still exist. For example, we might learn that “black holes” that don’t allow light to escape do exist (which would not be a big surprise to Astronomers), but those “black hole” have no event horizon. In fact, if the “event horizon” did exist, would we see a ring of a bright glow around each black hole, due to the “stacked” light (from outsider point of view) of all the stars that fell into that “hole” over millions and millions year of its history?

There are more questions than answers. For example, we don’t know what the value of Gravitational field is as compared to the value of the field created by a Galaxy. Can it be that the Gravitational field between Milky Way (our Galaxy) and the Galaxy next to it is much smaller than the Grav-field in our Solar System created by our Galaxy. And if that is the case, then may be―even with hyperbolic equations for and―that is the reason why we cannot detect the gravitational waves that should come from other Galaxies. Perhaps they get reflected back just like water waves when they reach the area where level of water is very low. Or perhaps the waves don’t come to us because of the non-linearity effects in description of Grav-vector―both in terms of and. However, if proposed in this paper theory based on accumulative large Grav-field reflects the physical reality, it is quite realistic to assume that Einstein’s equations (with or without dark matter) can not be used for description of Universe (or Galaxies) and must be viewed only as a linear approximation applicable only for calculating correction of metric and/or Gravitational field (Dark Matter) in a system of week gravitational interactions.

In the end we would like to emphasize that in proposed theory the Gravitation is described only thru atomic parameters―i.e. as a measure of energy and mass, being the speed of light and being the atomic length―and the number of particle in the Universe. And because of that it is not difficult to see that the scale of the basic parameters of the Universe―the radius of Universe (measures in atomic length) and the Time-Scale of the Universe―are given by a simple relation with respect to the number of particles in the Universe N, where both of basic Universe parameters are proportional to a square root of the number of particles:, with.

Conflicts of Interest

The authors declare no conflicts of interest.

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