CHAPTER 4

Speed Limits

At this point it will again be advisable to emphasize the purely ,factual nature of the development in this work. Perhaps this may seem to be unnecessary repetition, but many of the conclusions reached in the preceding pages are in conflict with currently accepted theories and concepts—products of human thought—and the general tendency will no doubt be to take it for granted that the new conclusions are similar products. On this basis, the issue presented to the reader would be the relative merits of the two lines of thought. But this is not the situation. This volume deals exclusively with factual material. It describes a type of motion that is known to exist, but has not heretofore been examined in detail. With the benefit of this more complete information it then identifies some known phenomena, the true nature of which has heretofore been unknown, as aspects of this scalar type of motion. All this is purely a matter of recognizing existing features of the physical world. No theories or assumptions are involved.

Once the fact that scalar motion exists is recognized, the determination of its properties is a straightforward operation, and the results thereof are equally factual. They do not depend, in any way, on any physical theory, or point of view. As brought out in Chapter 2, one of the significant properties of this type of motion is that, unlike vectorial motion, it is not restricted to one dimension. In a three-dimensional universe, scalar motion can take place coincidentally in all three dimensions.

The relevance of the foregoing comments in the present connection is a consequence of the nature of our next objective. We are now ready to take another step in the development of the properties of scalar motion, and the results of this extension of knowledge will again conflict with conclusions that have been reached from current theories. Scientists are understandably reluctant to abandon theories of long standing if this can possibly be avoided. It is important, therefore, to realize that we are not confronting the accepted theories with other theories, we are confronting these current theories with some newly established facts.

Of course, it is always painful to find that some idea or theory to which we have long been committed is wrong, and it is particularly distressing when the idea or theory is one that has been successfully defended against strong attacks in the past. The situation that will be discussed in this chapter is one of this nature, but the blow will be cushioned to some extent, as the rejection of the prevailing ideas is not total. We do not find that the theory currently accepted is wrong; we merely find that it claims too much. It has its field of applicability, but that field is considerably narrower than has heretofore been believed.

The question that we will now address is what, if any, limitations exist on speed magnitudes. The prevailing opinion is that the speed of light is an absolute maximum that cannot be exceeded. This opinion is based (1) on experiments, (2) on a theoretical analysis by Einstein, and (3) on the absence of any observation accepted as evidence of greater speeds.

The experiments, originally carried out by Bücherer and Kaufmann, and repeated by many other investigators, involved accelerating electrons and other particles to high speeds by electrical means. It was found that where the applied electric charge is held constant, the acceleration does not remain constant, as Newton’s Second Law of Motion, a = F/m, seems to require. Instead it is found to decrease as a function of the speed at a rate indicating that it would reach zero at the speed of light. The conclusion that was drawn from this experiment is that it is impossible to accelerate a physical object to a speed greater than that of light.

On first consideration, this conclusion appears to be justified, and it has not hitherto been successfully challenged, but the jump from the particular case to the general principle has been too precipitous. The electrons and other particles employed in the experiments can probably be taken as representative of matter in general, but there is certainly no adequate justification for assuming that the limitations applying to electrical processes are actually applicable to physical processes in general. What the experiments demonstrate, therefore, is not that it is impossible to accelerate physical objects to speeds in excess of that of light, but that it is impossible to do so by electrical means. Inasmuch as we have found, in the preceding pages, that electrical processes are confined to the one dimension of motion that can be represented in the spatial reference system, the results of this present investigation are consistent with this more limited conclusion. They do not, however, preclude acceleration to higher speeds by some other process, such as, for example, the sudden release of large quantities of energy by a violent explosion.

Turning now to the current theoretical view of the situation, Newton’s Second Law of Motion, F = ma, or a = F/ m, which is the form that enters into the present discussion, is a definition, and therefore independent of the physical circumstances. It follows that the observed decrease in acceleration at high speeds must be due either to a decrease in the force, F, or to an increase in the mass, m, or both. There is nothing in the experimental situation to indicate which of these alternatives is the one that actually occurs, so when Einstein formulated his theory of high speed motion he had to make what was, in essence, a blind choice. However, charge is known to exist only in units of a uniform size, and therefore has a somewhat limited degree of variability, while mass is much more variable. For this reason a variation in the mass at high speeds appeared to be the more likely alternative, and it is the one that Einstein selected.

The circumstances surrounding scientific developments tend to be forgotten in the course of time, and it is quite generally accepted these days that Einstein must have had some reliable basis for selecting mass as the variable quantity. An examination of the older textbooks will show that this was not the understanding closer to Einstein’s own time. The word “if” figures prominently in the explanations given in these older texts, as in this quotation from one of them: “If this decrease is interpreted as an increase of mass with speed, charge being constant…”⁴⁸

The reason for this quite cautious attitude toward the assumption was a general realization at the time that too little was known about the nature of electric charges to justify a firm decision in favor of the variable mass alternative. The findings reported in this work now show that this caution was amply justified. We can now see that it is not the charge that enters into the acceleration equation; it is the force aspect of that charge (motion). A constant charge is a constant motion, not a constant force. The existence of the motion results in the existence of a force, a property of the motion, but there is no legitimate basis for assuming that the force aspect of a constant motion is necessarily constant. On the contrary, it seems rather evident that the ability of a motion to cause another motion is limited by its own magnitude.

The mathematical expression of Einstein’s theory, stated in terms of the variable mass concept, has been thoroughly tested, and is undoubtedly correct. Unfortunately, this validation of the mathematical aspects of the theory has been generally accepted as a validation of the theory as a whole, including the conceptual interpretation that Einstein gave to it. Acceptance of mathematical validity as complete proof is an unsound practice that is all too prevalent in present-day science. All complete physical theories consist of a mathematical statement, and a conceptual statement, essentially an interpretation of the mathematics. Validation of the mathematics does not in any way guarantee the validity of the interpretation; it merely identifies this interpretation as one of those that could be correct.

It is much more difficult to validate the interpretation than to validate the mathematics. As soon as it is shown that the mathematics are in full agreement with the observed facts, the mathematical task is complete. Any other mathematical statement that is also in full agreement with the facts is necessarily equivalent to the first, and in mathematics equivalent statements are merely alternate ways of saying the same thing. On the other hand, two different interpretations of the same mathematics are not equivalent. The prevailing tendency to accept the first one that comes along, without any rigorous inquiry into its authenticity, has therefore been a serious obstacle to scientific progress. As expressed by Jeans in an oft quoted statement:

The history of theoretical physics is a record of the clothing of mathematical formulae which were right, or very nearly right, with physical interpretations which were often very badly wrong.⁴⁹

The situation that we are now examining is a good example of the kind of thing that Jeans was talking about. Einstein’s theory of high speed motion (that is, his mathematical expression and his interpretation thereof) is accepted as having been “confirmed by a large number of experiments,” and it is currently part of the dogma of conventional physics. The truth is, however, that those experiments, no matter how great their number may have been, or how conclusive their results, have confirmed only the mathematical aspects of the theory. The point that now needs to be recognized is that the speed limitation does not come from these confirmed mathematics; it comes from the untested interpretation.

If Einstein’s assumption that the mass varies with the speed is valid, then the mass of a moving object reaches infinity at the speed of light. A greater speed is thus impossible. But this is only one of the possible interpretations of the mathematics, and neither Einstein nor anyone else has produced any tangible evidence to support this interpretation. New “tests of Einstein’s theory” are continually being reported, but they are all tests of the mathematics of the theory, not tests of the theory.

The findings of the scalar motion investigation agree with the mathematical expression of this theory of Einstein’s, as they must do, since physical facts do not disagree with other physical facts, but they indicate that he made the wrong guess when he chose mass as the variable quantity in the acceleration equation. It is a decrease in the effective force that accounts for the decrease in acceleration at high speeds, not an increase in the mass. An interesting point in this connection is that there is a universal law that bars the mass alternative, and would have prevented this wrong choice, but unfortunately it has not been accepted to any significant degree by science, even though it plays an important role in many other branches of knowledge. This law, the law of diminishing returns, bars infinities—actually it is one expression of the principle that there are no infinities in nature—and it is just as applicable to the acceleration equation as to the many situations in the fields, such as economics, where it is officially recognized. This law tells us that the ratio of the incremental output of a physical process to the incremental input does not remain constant indefinitely, but eventually decreases, and ultimately reaches zero. On the basis of this law, the effective force at high speed is not the force measured at low speed, but a quantity that decreases with increasing speed.

In practical applications, such as the design of particle accelerators, for example, Einstein’s theory is used in the form of a mathematical equation, and his interpretation of the mathematics does not enter into the result. Consequently, those who use the theory are not particularly concerned as to whether the interpretation is correct or not, and it tends to be accepted without any critical consideration. This casual acceptance of the interpretation by the physicists has placed a roadblock in the way of gaining an understanding of phenomena in which speeds greater than that of light are involved. Since, as we have found, the decrease in acceleration is due to a reduction in the effective force of the electric charge, there is nothing in the mathematical relations that would prevent acceleration to higher speeds where means of applying greater forces are available. This conclusion, reached by correcting the interpretation of Einstein’s equation, without affecting the equation itself, is the same conclusion that we reached when we subjected the experimental results to a critical consideration. The mathematics of Einstein’s theory describe the process of acceleration by means of a one-dimensional (electric) force. They do not apply to the maximum possible acceleration by other means.

Now let us see how the information about scalar motion presented in the preceding pages fits in with these revised conclusions drawn from the acceleration experiments and Einstein’s mathematical development. There is nothing in the scalar motion development thus far that requires a speed limit, but neither is there anything that precludes the existence of such a limit. (The reason for its existence will be derived from some further properties of scalar motion that will be examined in the next chapter.) The previous findings are therefore consistent with the experimental evidence indicating a limit at the speed of light. It is evident, however, from what has been learned about scalar motion that this limit applies to the speed represented in the spatial reference system; that is, it is a one-dimensional spatial limit. Einstein’s theoretical conclusion that the speed of light cannot be exceeded will therefore have to be modified to assert that motion in space in the dimension of the reference system cannot take place at a speed greater than that of light.

Here is a conclusion that agrees with all of the positive evidence. To complete the picture we will also want to take a look at what is offered as negative evidence. The third line of argument currently offered in support of an absolute limit at the speed of light is the asserted absence of any evidence of greater speeds. As applied, however, this argument is meaningless, because anything that might appear to be evidence of speeds beyond that of light is immediately dismissed as unacceptable because it conflicts with Einstein’s theory. For instance, measurements that appear to indicate that some components of certain quasars are moving apart with speeds up to eight or ten times the speed of light are not accepted as authentic, even though the astronomers are becoming more and more confident of the validity of their measurements.

Aside from these controversial measurements, the significance of which will be considered later, after some further relevant information has been developed, most of the evidence of speeds in the higher ranges is in the form of effects that are not recognizable as products of greater-than-light speeds without the benefit of an understanding of the properties of scalar motion. Recognition of this evidence by adherents of conventional physical theory therefore could not be expected. But there is one type of actual measurement of speeds greater than the speed of light that should have been recognized in its true light. This is the Doppler shift of the radiation from the quasars.

From the manner in which this shift in the frequency of the incoming radiation is produced, it follows that the relative speed of the emitting object, in terms of the speed of light as unity, is simply the ratio of the shift in wavelength to the laboratory wavelength. There was no suggestion, prior to the discovery of the quasars that there might be any kind of a modification of this relation at high speeds. But when quasar redshifts above 1.00 were measured, indicating speeds in excess of the speed of light, the astronomers were unwilling to accept the fact that they were measuring speeds that Einstein called impossible, so they applied a mathematical factor to keep these speeds below the 1.00 level.

In two other cases, particle acceleration and the composition of velocities, it had been possible to bring the pre-Einstein physical relations into conformity with the values derived by direct measurement at high speeds by applying Einstein’s reduction factor

(1-v²/c²)^½. In the acceleration case, the magnitudes calculated from Newton’s Second Law of Motion exceed the speed of light at high speeds, whereas the direct measurement approaches a limit at that speed. The reduction factor is therefore applied to the calculated magnitudes to bring them into agreement with the direct measurements. In the composition of velocities, the magnitudes calculated from the relation of coordinate differences to clock time exceed the speed of light, whereas the direct measurements approach a limit at that speed. The reduction factor is therefore applied to the calculated magnitudes to bring them into agreement with the direct measurements. The Doppler shifts above 1.00 again confronted the physicists with a situation in which a speed greater than that of light was indicated. The same expedient was therefore employed to keep the indicated quasar speeds within Einstein’s limit.

The success of this mathematical expression in the earlier applications, together with the preeminent status accorded to Einstein’s limitation on speed evidently conspired to prevent any critical consideration of the justification for applying the same mathematics to the Doppler shift, as it can easily be seen that the Doppler situation is altogether different from the other two. ln both of these other cases, the direct measurement is accepted as correct, and the adjustment factor is applied to the results computed by means of certain relations that hold good at low speeds to bring these calculated results into agreement with the direct measurements. In the Doppler situation there is nothing that needs to be adjusted to agree with the direct measurement. The only magnitude involved is the shift itself, and it is the direct measurement.

There is no valid reason for assuming that the Doppler shifts above 1.00 are anything other than direct measurements of speeds greater than the speed of light. It should be noted, however, that on the basis of the points brought out in the preceding discussion, the speed that can be represented in the spatial reference system, the speed that causes change of spatial position, is limited to the speed of light. The increment above this speed, corresponding to the increment of the Doppler shift above 1.00, is a scalar addition to the speed represented in the reference system. It appears in the Doppler shift because that shift measures the total magnitude of the speed, not the change of spatial position.

The difference between this and the gravitational situation is significant. The gravitational motion that is measured (as a force) takes place within the limits of the reference system. In this case, therefore, the effective magnitude is fully represented in the reference system. The gravitational motion in the other two scalar dimensions is not so represented, but it has no effect in the dimension of the reference system. On the other hand, a speed in excess of that of light in the dimension that is represented in the reference system is a physical magnitude in that dimension, and even though it cannot be represented by a difference in the spatial coordinates, it participates in any measurement of magnitudes, such as the Doppler shift, which is independent of coordinate differences.

This capability of addition of magnitudes in different speed ranges, independently of the limitations of the spatial reference system, is a general property of scalar magnitudes that has an important bearing on many physical phenomena. As noted earlier, scalar magnitudes cannot be combined in any way analogous to the addition of vectors, but any two scalar quantities in the same dimension are additive. Thus the Doppler shift due to motion in one dimension above unit speed (a scalar quantity) adds to the shift due to motion of the same object in the range below unity (another scalar quantity), which is in the same dimension because the motion in the higher speed range is an extension of the motion in the lower speed range.

Summarizing the foregoing discussion of the question as to the limitations on speed, the evidence shows that it is not possible to accelerate material objects to speeds in excess of that of light by means of electrical forces. We have found that the electric charge is a one-dimensional distributed scalar motion. The meaning of the experimental results therefore is that the speed of light is the limiting speed in one scalar dimension. The three scalar dimensions are independent, and there is nothing to distinguish one from another. It follows that the limiting speed in each dimension is the speed of light. Thus the limiting value of the total scalar speed of an object is 3c: three times the speed of light. Consequently, there are three speed ranges of scalar motion. One coincides with the range of speed of vectorial motion. Speeds in this range have magnitudes 1-x, where the speed of light is taken as unity, and x is some fraction thereof. If the scalar motion is two-dimensional, the speeds are 2-x, while if the motion is three-dimensional, they are 3-x. The reason for expressing the speeds in this particular manner will be explained in Chapter 6.

The concept of an absolute limit at the speed of light, as laid down by Einstein, is thus erroneous. His mathematics are correct, but they apply only to motion in one dimension, the dimension of the conventional spatial reference system. The new information derived from the investigation of scalar motion makes it evident that the general acceptance of Einstein’s conclusion as to the impossibility of speeds greater than that of light has been a monumental roadblock in the way of scientific progress, probably second only to Aristotle’s conception of the nature of motion, characterized by Alfred N. Whitehead as “a belief which had blocked the progress of physics for two thousand years.”⁵⁰

There is, indeed, a rather close parallelism between the two cases. Both of these serious errors were products of the outstanding scientists of their day: men with many notable achievements to their credit, who had attained such a standing in the scientific community that disagreement with their conclusions was, in effect, prohibited. Both of the conclusions now seen to be erroneous were supported by what originally seemed to be adequate empirical evidence. But both encountered increasing difficulties as physical understanding improved, and both ultimately reached the point where they were maintained as orthodox scientific doctrine on the strength of the authority of their originators, rather than on their own merits. This is generally recognized so far as Aristotle’s theory is concerned, where we have the benefit of the historical perspective. It is not so generally appreciated in Einstein’s case, but a critical examination of current scientific literature will reveal the remarkable degree to which his pronouncements are treated as incontestable dogma, with a standing superior to the empirical facts.

The gravitational situation has already been discussed. As von Laue admits in the statement that was quoted in Chapter 2, the repudiation of the results of observation “is a result solely of the theory of relativity.” The situation with respect to the Doppler shifts of the quasars, mentioned earlier in this chapter, is another instance where the experimental evidence has been reconstructed to agree with Einstein’s dictum. The true state of affairs in most other physical areas is obscured by the ad hoc assumptions that are made to “save” the theory, but the prevailing tendency to elevate Einstein’s conclusions to an unchallengeable status is clearly illustrated by the general readiness to throw logic and other basic philosophical considerations to the wolves whenever they stand in the way of his pronouncements. Hans Reichenbach, for example, tells us,

This discovery of a physicist (the relativity theory) has radical consequences for the theory of knowledge. It compels us to revise certain traditional conceptions that have played an important part in the history of philosophy.⁵¹

Kurt Güdel similarly sees far-reaching consequences following from Einstein’s interpretation of special relativity, even though it is well known that this is merely the current choice from among a number of equally possible explanations of the mathematical results. M. B. Hesse points this out in the following statement: “There are some other logical questions raised by the theory of relativity . . . because there are a number of alternative theories which all appear observationally equivalent.”⁵² On this slippery ground, Güdel finds “unequivocal proof.”

Following up the consequences (of the assertions of special relativity) one is led to conclusions about the nature of time which are very far reaching indeed. In short, it seems that one obtains an unequivocal proof for the view of those philosophers who… deny the objectivity of change.⁵³

Warren Weaver is ready to jettison logic to accommodate Einstein. He tells us that the close observer “finds that logic, so generally supposed to be infallible and unassailable, is, in fact, shaky and incomplete. He finds that the whole concept of objective truth is a will-o’-the-wisp.”⁵⁴ Now where does this remarkable conclusion come from’? A few pages later in the same work Weaver answers this question. “A major consequence of the developments in relativity and quantum theory over the past half century,” he says, has been the destruction of “both ultimate precision and ultimate objectivity,” and he goes on to assert that “presuppositions which have neither a factual nor a logical—analytical basis…enter into the structure of all theories and into the selection of the group of `facts’ to be dealt with.”⁵⁵

The revolutionary character of the apotheosis of the relativity theory in modern science cannot be fully appreciated unless it is realized that this logic that Weaver and his colleagues propose to sacrifice on the Einsteinian altar, along with the objective facts of gravitation, Doppler shifts, and other physical phenomena, is one of the basic pillars of the scientific structure. As expressed by F. S. C. Northrop:

In this third stage of inquiry, which permits the introduction of unobservable entities and relations in order to solve one’s problem, and which is called the stage of deductively formulated theory, the use of formal logic is a necessity. For it is only by recourse to formal logic that one can deduce consequences from one’s hypothesis concerning unobservable entities and relations and thereby put this hypothesis to an empirical and experimental test.⁵⁶

The basic reason for the similarity in the history of the two theories under consideration is that they are both products of invention, rather than of induction from factual premises. Aristotle was an observer, “a pure empiricist…exclusively inductive in his procedure,”⁵⁷ as described by Northrop. But the amount of empirical knowledge that had been accumulated up to his time was altogether inadequate for his purposes, and he found it necessary to resort to invention to fill in the gap. In his theory of motion, “the things that were in motion had to be accompanied by a mover all the time,”¹ and the “unseen hands” mentioned in Chapter 1, that “had to be in constant operation” to provide this service, were certainly inventions.

Einstein was definitely a protagonist of the “inventive” school of science. “The axiomatic basis of theoretical physics cannot be an inference from experience, but must be free invention,”⁵⁸ he tells us. Elaborating, in another connection, he further asserts:

The theoretical scientist is compelled in an increasing degree to be guided by purely mathematical, formal considerations in his search for a theory, because the physical experience of the experimenter cannot lift him into the regions of highest abstraction.⁵⁹

Notwithstanding Einstein’s brave words, physical science, in practice, resorts to invented principles only when and where inductive results are not available. In Aristotle’s day relatively few physical relationships of a general character had been definitely established, and invented principles predominated. By this time, however, the subsidiary laws and principles of physical science, including almost all of the relations utilized by the engineers, the practitioners in the application of science, have been derived inductively from empirical premises. Einstein’s theories and other products of scientific invention have gained their present ascendancy in the fundamental areas only because the previous system of inductive theory applicable to these areas, that generally associated with the name of Newton, was unable to keep pace with the progress of empirical discovery around the end of the nineteenth century.

The reason for this emergence of inventive theory only when there are gaps in the inductive structure is that the inventive theories are inherently wrong, in their conceptual aspects. This is an inevitable result of the circumstances under which they are able to gain acceptance. The scientific problems that are responsible for the existence of gaps in the structure of inductive theory do not continue to exist because of a lack of technical competence on the part of the scientists who are trying to solve them, or because the methods available for dealing with them are inadequate. The lack of success, where it exists, is due to the absence of some essential piece, or pieces, of information. If the necessary information is available, there is no need for invention; the correct theory can be derived by induction. Without the essential information it is not possible to construct the correct theory by any method.

The gravitational situation is a good example. Newton derived a mathematical expression for the gravitational effect. Subsequently it was found that the range of application of this expression was limited, and Einstein formulated a new expression that presumably has a more general applicability. Both of these were inductive products; that is, they were based on the mathematical aspects of the results of observation and measurement. Neither of the investigators was able to complete his theory by deriving an interpretation of his mathematics inductively. It can now be seen that the reason for this failure was the lack of recognition of the existence of distributed scalar motion. As long as the existence of this type of motion was unknown, the identification of the nature of the gravitational effect required for the inductive formulation of the correct gravitational theory was impossible. Newton, who was committed to the inductive approach, was therefore unable to devise any complete theory (mathematical statement and interpretation thereof). Without the essential item of information, Einstein was equally unable to formulate the correct theory, but on the basis of his contention that the source of basic physical principles must be “free inventions of the human mind,” he was at liberty to complete his theory by inventing an explanation to fit the mathematical expression that he had derived.

Whether or not an inventive theory of this kind serves any useful purpose during the time before the correct inductively derived theory becomes available is a debatable issue. So far as the particular phenomena to which the theory is directly applicable are concerned, the conceptual interpretation is essentially irrelevant. For practical purposes, the theory is applied mathematically, and it makes little, if any, difference whether the user understands the real significance of the mathematical operations. As Feynman observes, “Mathematicians…do not even need to know what they are talking about.”⁶⁰ The conceptual interpretation of the mathematics is important primarily because it is one of the essentials for an understanding of the relations between physical phenomena. While a wrong interpretation may occasionally stimulate a line of thought that leads in the right direction, it is much more likely to impede progress. The justification for the construction and use of inventive theories is therefore highly questionable.

It would appear that the main purpose served by inventing a theory is to enable the scientific community to avoid the painful necessity of admitting that they have no answer to an important problem. What the inventive scientist is able to do, when his inductive counterpart is stymied, is to construct a theory that is mathematically correct, and that meets some of the conceptual requirements. Until the correct theory appears (or even for a time thereafter, if the Establishment can maintain discipline), the inventive theory can stand its ground on the strength of the assertions (1) that it produces the correct mathematical results (often claimed to be complete verification), and (2) that the assertions of the theory have not been definitely disproved (something that is very difficult to accomplish because of the free use of ad hoc assumptions to avoid contradictions). The extent to which this preposterously inadequate amount of support is currently accepted as conclusive by a scientific community desperately anxious to have some kind of a theory in each fundamental area is graphically illustrated by the description of the prevailing attitude toward Einstein’s theories in the preceding paragraphs. However, the fiction can be maintained only for a limited time. Ultimately the inventive theories of Einstein and his school, like the inventive theories of Aristotle, will accumulate too many ad hoc modifications—too many epicycles, we may say—and they will have to give way to theories, derived inductively, that are both mathematically and conceptually correct.

Inasmuch as the presentation in this work is purely factual, it does not offer any new inductive theories to replace the inventive theories currently in vogue. It merely calls attention to a large number of hitherto undiscovered, unrecognized, or disregarded physical facts, all of which the theories of physics, inventive or inductive, as the case may be, will hereafter have to be prepared to deal with. From now on, the requirements for acceptance of theories will be substantially enlarged. No theory will be viable unless it incorporates an acceptable explanation of scalar motion and its consequences.