As brought out in the preceding chapter, the development of a purely deductive theory of the physical universe has enabled reversing the customary procedure in scientific investigation. Instead of deriving mathematical relations applicable to the phenomena under consideration, and then looking for an explanation of the mathematics, we are now able, by deduction from very general premises, to derive a theory that is conceptually correct, and then look for an accurate mathematical representation of the theory. This is a much more efficient procedure, for reasons that were previously explained, but it does not necessarily follow that completing the task by solving the mathematical problems will be free from difficulty. In some cases, the search for the correct mathematical statement will require expenditure of a great deal of time and effort. During the course of this extended investigation there will be some defects in the mathematical “models” that are being used, just as there are defects in the conceptual models that are utilized in current practice.

The original development of the theory of the universe of motion, prior to the first publication is 1959, answered a number of the physical questions with which conventional physical science was (and still is) unable to deal. Atoms and sub-atomic particles were identified as combinations of scalar motions, and gravitation was identified as the inward translational manifestation of these motions. Electric charges were identified as one-dimensional motions of an oscillating character superimposed on the basic motion combinations, with similar translational (scalar) resultants. The basic forces were identified as the force aspects of these basic motions.

These identifications answered the questions as to how the forces are produced and the nature of the originating entities. They also answered the problem of explaining how the gravitational and electrostatic effects are (apparently) transmitted, and accounting for the instantaneous nature of the apparent transmission. Like many of the other answers to long-standing problems that have emerged from the development of this theory, the answer to the transmission problem took an unexpected form. From the postulates of the theory we deduce that each mass and charge follows its own course, and the apparent transmission is merely a result of the fact that the motion is scalar, and therefore has either an inward scalar direction, carrying all objects of these classes toward each other, or an outward scalar direction, carrying all such objects away from each other. There is no transmission of the effects, and hence no transmission time is involved.

As might be expected, the answers to most of these problems were initially incomplete, and the history of the theory since 1959 has been that of a progressive increase in understanding in all physical areas, during which one after another of the remaining issues has been clarified. In some cases, such as the atomic rotation, the mathematical aspects of the problems presented no particular difficulty, and the points at issue were conceptual. In other instances, the difficulties were primarily concerned with accounting for the mathematical forms of the theoretical relations and their numerical values.

The most troublesome problem of the latter kind has been that of the force equations. The force between electric charges can be calculated by means of the *Coulomb equation, *F = QQ’/d^{2}, which states that, when expressed in appropriate units, the force is equal to the product of the (apparently) interacting charges divided by the square of the distance between them. Aside from the numerical coefficients, this Coulomb equation is identical with the equation for the gravitational force that was previously discussed, and, as we will see later, with the equation for the magnetostatic force as well.

Unfortunately, these force relations occupy a dead end from a theoretical standpoint. Most basic physical relations have the status of points of departure from which more or less elaborate systems of consequences can be built up step by step. Correlation of these consequences with each other and with experience then serves either to validate the theoretical conclusions or to identify whatever errors or inadequacies may exist. No such networks of connections have been identified for these force equations, and this significant investigative aid has not been available to those who have approached the subjects theoretically. The lack of an explanation has not been as conspicuous in the case of the electric force, as the Coulomb equation, which expresses the magnitude of this force, is stated in terms of quantities derived from the equation itself, but there is an embarrassing lack of theoretical understanding of the basis for the relation that is expressed mathematically in the gravitational equation. Without such an understanding the physicists have been unable to tie this equation into the general structure of physical theory. As expressed in one physics textbook, “Newton’s law of universal gravitation is not a defining equation, and cannot be derived from defining equations. It represents an *observed relationship.” *

The problems involved in application of the theory of the universe of motion to the gravitational relations were no less formidable, and the early results of this application were far from satisfactory. Ordinarily, the results of incomplete investigations of this kind have not been included in the published material. The opportunities for publication of the findings of this investigation have been severely limited, and the material released for publication has therefore been confined, in general, to those results that have been established as correct both mathematically and conceptually, within the limits to which the investigation has been carried. If the gravitational motion and force were matters of an ordinary degree of importance, the best policy probably would have been to put the unsatisfactory results aside for the time being, and to wait for further developments in related areas to clarify the general situation enough to make further progress in the gravitational area possible. But because of the fundamental nature of the gravitational relations it has been necessary to make extensive use of them, in whatever form they might happen to be, as the theoretical investigation progressed. The previous publications, including the first volume of this present series, have therefore contained some of the tentative, and only partially correct, results of the earlier studies. However, much new light is being thrown on the subject matter by the continuing advances that are being made in related areas, and the status of the gravitational theory is consequently being updated in each new publication.

At the time of the first studies, the most obvious need was a clarification of the dimensions of the equations. Overall dimensional consistency is something that has never been attained by conventional physics. In some areas, such as mechanics, the currently recognized relations are dimensionally consistent, but in many other areas the dimensional confusion is so widespread that it has led to the previously mentioned conclusion that a rational system of dimensions for all physical quantities is impossible.

The present standard practice is to cover up the discrepancies by assigning dimensions to the numerical constants in the equations. Thus the gravitational constant is asserted to have the dimensions dyne-cm^{2}/gram^{2}. Obviously, this expedient is illegitimate. Whatever dimensions enter into physical expressions are properties of the physical entities that are involved, not properties of numbers. Dimensions are excluded from numbers by definition. Wherever, as in the gravitational case, an equation cannot be balanced without assigning dimensions to a numerical constant, this is prima facie evidence that there is something wrong in the understanding on which the dimensional assignments are based. Either the dimensions assigned to the physical quantities in the equation are incorrect, or the so-called “numerical constant” is actually the magnitude of an unrecognized physical property. Both types of dimensional errors have been encountered in our examination of current thought in the areas covered by our investigation.

One of the powerful analytical tools made available by the theory of the universe of motion is the ability to reduce all physical quantities to terms of space and time only. In order to be correct, an equation must have a space-time balance; that is, both sides of the equation must reduce to the same space-time expression. Another useful analytical tool derived from this theory is the principle of equivalence of units. This principle asserts that, inasmuch as the basic quantities, in all cases, are units of motion, there are no inherent numerical constants in the mathematical equations that represent physical relations, other than what we may call structural constants—values that have definite physical meanings, as, for instance, the number of active dimensions in one of the participating quantities. It follows that if the quantities involved in a valid physical equation are all expressed in natural units, or the equivalent in the units of another measurement system, the equation is in balance numerically, and no numerical constant is required.

The gravitational equation, in its usual form, fails by a wide margin to meet the test of dimensional consistency, but the general nature of the modifications that have to be made in the dimensional assignments was identified quite early in the investigation. The 1959 publication dealt with this dimensional problem, pointing out the need to reduce the distance term and one of the mass terms to a dimensionless status; that is, to recognize that they are merely ratios. It also emphasized the fact that an acceleration term must be introduced into the equation for dimensional consistency, and showed that this term represents the inherent acceleration of gravitating objects, which is unity, and therefore not perceptible in empirical measurements. Application of the principle of equivalence of natural units was attempted, without much success, but the tentative results of this study included a derivation of the gravitational constant.

By the time *Nothing But Motion *was published twenty years later, the lack of a fully satisfactory interpretation of the gravitational equation had become somewhat embarrassing. Furthermore, the validity of the original derivation of the gravitational constant was challenged by some of the author’s associates, and the evidence in its favor was not sufficient to meet that challenge effectively. It was therefore decided to abandon that interpretation, and to look for a new explanation to take its place. In retrospect it will have to be admitted that this 1979 revision was not a well-conceived attack on the problem. It was essentially an attempt to find a mathematical (or at least numerical) solution where the logical development of the theory had met an obstacle. This is the same policy that, as pointed out in Chapter 13, has brought conventional theory up against so many blank walls, and it has turned out to be equally unproductive in the present case. It became increasingly evident that some further study was necessary.

This brings up an issue that has been the subject of some comment. It is our contention that the many thousands of correlations between the observations and the consequences of the postulates of the theory of the universe of motion have established that this theory is a true and accurate representation of the actual physical universe. The skeptics then want to know how we can arrive at wrong conclusions in some cases, if we are applying a correct theory; why a conclusion reached in the first volume of a series had to be modified even before the second volume was published. The answer, as explained in many of our previous publications, is that while the theory is capable of producing the right answers, if properly applied, it does not necessarily follow that those who are attempting to apply it properly will always be successful in so doing. As stated earlier, an attempt has been made to confine the published material to firmly established items, aside from a few that are specifically identified as somewhat speculative, but nevertheless, some of the conclusions that have been published have subsequently been found to be incomplete, and in a few instances, incorrect.

There is no reason to be apologetic about these few errors and omissions. Present-day physical theory has been in the process of development for centuries, during which a myriad of conclusions that have been reached with respect to details of the theory (or theories) have subsequently had to be abandoned as incorrect. In comparison with this experience, the error rate in the development of the theory of the universe of motion is fantastically low. This is no accident. Inasmuch as all conclusions in all areas are derived deductively from the same set of basic premises, consistency of the interrelations between phenomena, the basic requirement for conceptual validity, is achieved automatically. Those cases in which the developers of the theory are having some trouble merely emphasize the easy and natural way in which solutions for most of the previously unresolved fundamental problems of physical science have emerged from the theoretical development.

The review of the gravitational situation that was recently undertaken was able to take advantage of some very significant advances that have been made in our understanding of the details of the universe of motion—that is, in the consequences of the postulates—in the years that have elapsed since publication of Volume I in 1979. Chief among these is the clarification of the nature and properties of scalar motion, discussed in Chapter 12, and covered in more detail in * The Neglected Facts of Science.* The improvement in understanding of this type of motion has thrown a great deal of new light on the force relations. It is now clear that the differences between the basic types of forces that were recognized from the start of the investigation as dimensional in nature are differences in the number of *scalar * dimensions involved, rather than geometric dimensions of space. This provides simple explanations for several of the issues that had been matters of concern in the earlier stages of the theoretical development.

The significant conceptual change here is in the nature of the relation between motion and its representation in the reference system. In previous physical thought motion was regarded as a change of position in a specifically defined physical space (Newton) or space-time (Einstein) during a specific physical time. This physical space and time thus constitute a background, or container. Changes of position due to motion relative to the spatial background are assumed to be capable of representation by vectors (or tensors of higher rank). In the theory of the universe of motion, on the other hand, space and time have physical existence only as the reciprocally related components of motion, and the three-dimensional space of our ordinary experience is merely a reference system, not a physical container. Furthermore, the development of the details of the theory in the preceding pages of this and the earlier volume shows that the spatio-temporal reference system which combines the three-dimensional spatial frame of reference with the time magnitudes registered on a clock, in incapable of representing the full range of existing motions. Some motions cannot be represented in their true character. Others cannot be represented in this reference system at all.

The deficiency of the reference system with which we are particularly concerned at this time is its inability to represent multi-dimensional scalar motion. This inability of the reference system to represent more than one scalar dimension of motion explains why the forces exerted by charges and masses are all one-dimensional, irrespective of the number of scalar dimensions applicable to the inherent motion of the charge or mass. Only one of these scalar dimensions is coincident with the dimension of the reference system, and the motion in this dimension is therefore the only one that can be represented in the reference system. As indicated earlier, this limitation on the capacity of the reference system is the reason for the great disparity in magnitude between the basic forces. The *total * magnitudes of the electric and gravitational forces are actually the same, but only the motion in the dimension of the reference system is effective. In our gravitationally bound system, the dimensional ratio (in cgs units) is 3×10^{10}. Thus the electric force, which is one-dimensional, and therefore fully effective, is relatively strong. The gravitational force actually has the same total strength, but it is distributed over three scalar dimensions, only one of which coincides with the dimension of the reference system. The * effective *gravitational force is therefore weaker than the effective electrostatic force by the factor 9×10^{20}.

It should be noted, however, that the difference in the number of effective scalar dimensions has this effect on the relative magnitude of the forces only because it is applied to the very large value of the unit of speed, the relation between the sizes of the units in which we measure space and time. This, in turn, is a consequence of our position in a gravitationally bound system that is moving inward in space at a high speed, opposing the spatial component of the outward progression of the natural reference system. The *net* motion of the gravitating system in space is relatively small, while the motion in time proceeds at the full speed of the progression. Thus we experience a small change in space coincidentally with a very large change in time. We assign values to the units of these quantities that reflect the manner in which we experience them, and on this basis we have defined a unit of time (in the cgs system) that is 3×10^{10} times as large as our unit of space. Our unit of speed is then 3×10^{10} space units (centimeters) per unit of time (second).

As can be seen from the foregoing, the magnitude that we assign to the unit of speed, the speed of light, customarily represented by the symbol *c*, is not an inherent property of the universe (although the magnitude of the speed itself is). The general range within which this value will fall is determined by our position in a system of gravitating objects, and the specific value within these limits is assigned arbitrarily. Any change in the unit of either space or time that is not counterbalanced by an equivalent change in the other alters the value of c, in our measurement system, and the relation between the magnitudes of the electric and gravitational forces, c^{2}, is changed accordingly. (The electric force is usually asserted to be 10^{39} or 10^{40} times as strong as the gravitational force, but this figure is based on a set of erroneous assumptions.)

The further clarification of the mutual nature of scalar motion accomplished in the most recent studies has also thrown a very significant additional light on the force situation. As brought out in Chapter 12, it is now evident that a scalar motion AB cannot be distinguished, in the absence of a fixed coupling to the reference system, from a scalar motion BA. This means that in considering the mutual gravitational motion of two masses we are dealing with only one motion, the representation of which in the reference system depends on external factors.

On this basis, the expression mm’ in the gravitational equation is not a product of two masses, but the product of one mass and the *number* of units of mass in the interacting object. Likewise, the distance term, s^{2}, is a pure number, the ratio of s^{2} units to 1^{2} unit. Thus the only dimensional quantity that appears in the equation, aside from the resultant force, is one of the mass terms. This result of the current study confirms the original finding reported in the 1959 publication. It likewise confirms the earlier finding that another dimensional term, a unit of acceleration, must be inserted into the equation to produce a dimensional balance. Force in general is the product of mass and acceleration. It follows that the expression for any * particular* force must reduce to F = ma when all dimensions are properly assigned. The existence of the acceleration term is not apparent without a theoretical analysis because the gravitational acceleration is unity, and therefore has no effect on the numerical result.

The difficulties that have previously been experienced in applying the principle of the equivalence of natural units to the gravitational equation are now seen to have been due to an inadequate understanding of the manner in which the dimensionless terms in the equation should be treated when the statement of the unit equivalence is formulated. We now recognize that these terms vanish if they are given unit value in the system of measurement in which the values of the dimensionless terms are stated, unless some structural factor is specifically applicable. However, the use of an arbitrary mass unit in the conventional measurement systems introduces a complication, as it means that two different systems of units are actually being used. As we saw in the discussion of physical fundamentals in Volume I, all physical quantities, including mass, can be expressed in terms of units of space and time only. It follows that when an arbitrary unit is used for the measurement of mass, we are expressing the mass and the acceleration in different measurement systems. This is equivalent to introducing a numerical factor into whatever physical relations may be involved: the ratio between the sizes of the respective units.

Introduction of this factor does not affect the numerical balance of an equation as long as both sides of the equation contain the same number of mass terms, but in the gravitational equation F = kmm’/d^{2}, there are two mass terms on one side of the equation, while the force, the lone term on the other side, contains only one mass term (F = ma). In order to balance the equation numerically, a correction factor must be applied to convert the extra mass term to the units applicable to space and time. The ratio of the natural space-time unit of mass to the arbitrary mass unit is the required correction factor. Together with whatever structural factors are applicable to the equation, it constitutes the *gravitational constant.*

The ratio of the natural unit of mass in the cgs system to the arbitrary unit, the gram, was evaluated in Volume I as 2.236055×10^{-8}. It was also noted in that earlier volume that the factor 3 (evidently representing the number of effective dimensions) enters into the relation between the gravitational constant and the natural unit of mass. The gravitational constant is then 3 × 2.236055×10^{-8} = 6.708165×10^{-8} (with a small adjustment that will be considered shortly).

To apply the principle of equivalence of natural units to the gravitational equation, the dimensionless quantities m’ and d^{2} are given unit value in terms of the conventional measurement systems, so that they vanish from the equation. The dimensional terms, the mass term m and the acceleration term inserted into the equation, are then stated in the appropriate *natural *units, 1.6197×10^{-24} grams and 1.971473×10^{26} cm/sec^{2}, respectively. The natural unit of force derived from these values is 3.27223×10^{2} dynes.

The values thus derived exceed the measured gravitational constant and the previously determined value of unit force by the factor 1.00524. Since it is unlikely that there is an error of this magnitude in the measurements, it seems evident that there is another, quite small, structural factor involved in the gravitational relation. This is not at all surprising, as we have found in the earlier studies in other areas that the primary mass values entering into physical relations are often subject to modification because of secondary mass effects. The ratio of the unit of secondary mass to the unit of primary mass is 1.00639. The remaining uncertainty in the gravitational values is thus within the range of the secondary mass effects, and will probably be accounted for when a comprehensive study of the secondary mass situation is carried out.

A rather ironic result of the new findings with respect to the gravitational constant, as described in the foregoing paragraphs, is that they have taken us back almost to where we were in 1959. The repudiation of the 1959 result in the 1979 publication as a consequence of the criticism levied against it is now seen to have been a mistake. In the light of the additional information now available it appears that the shortcoming of the original results was not that they were wrong, but that they were incomplete and not adequately supported with explanations and confirmatory evidence, and were therefore vulnerable to attack. The more recent work has provided the support that was originally lacking.

Clarification of the gravitational force equation is not only important in itself, but has a further significance in that it opens the door to an understanding of the general nature of all of the primary force equations. Each of these equations is an expression representing the magnitude of the force (apparently) exerted by one originating entity (mass or charge) on another of the same, or equivalent, kind, at a specified distance. All take the same general form as the gravitational equation, F = kmm’/d^{2}.

With the benefit of the information developed in the earlier pages of this chapter, we may now generalize the equation by replacing m with X, which will stand for any distributed scalar motion with the dimensions (t/s)^{n}, and introducing a term Y with the value 1/s × (s/t)^{n-1}. The primary force equation is then F = kXY (X’/d^{2}).

Since only one dimension of an n-dimensional scalar motion is effective in the space of the conventional reference system, the effective space-time dimensions of the motion participating in the force equation are t/s. By definition, force has the dimensions t/s^{2}. The function of the term Y in the primary force equation is to reduce (t/s)^{n} to t/s and to introduce the term 1/s that is necessary to convert t/s to t/s^{2}. In the case of the gravitational equation, this involves multiplying by s^{2}/t^{2} × 1/s = s/t^{2}. These are the dimensions of acceleration. In the Coulomb equation the correction factor Y is merely 1/s.

The term X’/d^{2} is a combination of two ratios, and has unit value in the unit statement of the equation. The numerical constant k is also unity if all quantities are expressed in units that are consistent with the units in which the space and time magnitudes entering into the equation are measured. Where one or more of these quantities are expressed in units of another kind, the difference in the size of the units appears as the value of the numerical constant k. In the gravitational case, for example, the gravitational constant reflects the result of expressing mass in terms of a special unit (grams in the cgs system) rather than in sec^{3}/cm^{3}.

In essence, all that the force equations do is to reduce the scalar motions (mass, charge, etc.) to their effective one-dimensional values, introduce the 1/s term that relates the motion to the corresponding force, and correct for any inconsistencies in the units that are employed. It is somewhat of an anticlimax to arrive at such a simple explanation after years of exploring much more complicated hypotheses, but the simplicity of this result is consistent with the general nature of the findings in the basic areas of other physical fields. There are many complex phenomena in nature, to be sure, but throughout the development of the details of the universe of motion we have found that the fundamental relations are quite simple.

As noted earlier, the reference point of a scalar motion, the point in the fixed reference system to which an object in the scalar motion system is coupled, may be in motion vectorially. The mass of this object is a measure of its three-dimensional distributed scalar motion, the inward gravitational motion. The vectorial motion is outward, and it order for it to take place, a portion of the inward gravitational motion must be overcome. The mass is thus also a measure of the magnitude of the resistance to vectorial motion, the *inertia * of the object. In the light of the points brought out in the preceding pages, it is evident that in these manifestations of mass we are looking at two aspects of the same thing, just as in the case of the rocket, where the quantity of acceleration imparted *by *the combustion products (the force) is the same as the quantity of acceleration imparted *to *the rocket.

This point was not recognized by the early investigators because they were not aware of the existence of motion in different scalar dimensions. It appeared to them that two different quantities were involved: a *gravitational mass *and an *inertial mass. *Very accurate measurements showed that these two masses are identical, a finding that the physics of that day could not explain. As one observer says, “Within the framework of classical physics there is no explanation. When attention was directed to the problem, it seemed like a complete mystery.”^{42} A step toward solution of the problem was taken by Einstein. In the absence of an understanding of scalar motion, he was not able to see that gravitation* is *a motion, but he formulated a “Principle of Equivalence,” in which he postulated that gravitation is *equivalent to *a motion. Since he viewed “motion” as synonymous with “vectorial motion,” the postulate meant that gravitation is equivalent to an accelerated frame of reference, and it is often expressed in these terms. But such an equivalence is inconsistent with Euclidean geometry. As explained by Tor Gerholm:

If acceleration and gravitation are equivalent, we must apparently also be able to imagine an acceleration field, a field formed by inertial forces. It is easy to realize that no matter how we try, we will never be able to get such a field to have the same shape as the gravitational field around the earth and other celestial bodies…If we want to save the equivalence principle…If we want to retain the identity between gravitational and inertial mass, then

we are forced to give up Euclidean geometry!Only by accepting a non-Euclidean metric will we be able to achieve a complete equivalence between the inertial field and the gravitational fields. This is the price we must pay.^{43}

Identification of gravitation as a distributed scalar motion has now thrown an entirely new light on the situation. Gravitation is an accelerated motion, but it is not geometrically equivalent to an accelerated frame of reference. Einstein’s attempt to reconcile these two phenomena by resort to non-Euclidean geometry is misdirected. Whatever mathematical results are obtained by the use of this expedient (actually not very many. As Paul Davies points out, “technical problems of a mathematical nature render all but the simplest systems hopelessly insoluble”^{44}) are not indicative of the true relations. The scalar gravitational motion of an object and any vectorial motion that it may possess are quite different in their nature and properties.

In the case of the propagation of radiation, the principal stumbling block for the ether theory was the contradictory nature of the properties that the hypothetical substance “ether” must possess in order to perform the functions that were assigned to it. Einstein’s solution was to replace the ether with another entity that was assumed to have no properties, other than an ability to transmit the radiation, an ability which he says we should “take for granted.”^{27}

Similarly, the obstacles to accounting for the observed results of the addition of velocities were the existence of absolute magnitudes and fixed spatial coordinate locations. Here, the answer was to deny the reality of absolute magnitudes, and as Einstein says, to “free oneself from the idea that co-ordinates must have an immediate metrical meaning.”^{45} Now we find that he deals with the gravitational problem in the same way, loosening the mathematical constraints, rather than looking for a conceptual error. He invents the “equivalent of motion.” a hypothetical something which has enough of the properties of motion to enable accounting for the mathematical results of gravitation (at least in principle) without having those properties of vectorial motion that are impossible to reconcile with the observed behavior of gravitating objects. In all of these cases, the development of the theory of the universe of motion has shown that the real reason for the existence of these problems was the lack of some essential information. In the case of the composition of velocities, the missing item was an understanding of motion in time. In the other two cases cited, the problems were consequences of the lack of recognition of the existence of scalar motion.

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