As brought out in Chapter 3, the “space” of our ordinary experience, extension space, as we have called it, is simply a reference system, and it has no real physical significance. But the *relationships* that are *represented* in this reference system do have physical meaning. For example, if the distance between object A and object *B* in extension space is x, then if A moves a distance x in the direction *AB* while B remains stationary with respect to the reference system, the two objects will come in contact. The contact has observable physical results, and the fact that it occurs at the coordinate position reached by object *A* after a movement defined in terms of the coordinates from a specific initial position in the coordinate system demonstrates that the relation represented by the difference between coordinates has a definite physical meaning.

Einstein calls this a “metrical” meaning; that is, a connection between the coordinate differences and “measurable lengths and times.” To most of those who have not made any critical study of the logical basis of so-called “modern physics” it probably seems obvious that this kind of a meaning exists, and it is safe to say that comparatively few of those who now accept Einstein’s relativity theory because it is the orthodox doctrine in its field realize that his theory denies the existence of such a meaning. But any analysis of the logical structure of the theory will show that this is true, and Einstein’s own statement on the subject, previously quoted, leaves no doubt on this score.

This is a prime example of a strange feature of the present situation in science. The members of the scientific community have accepted the basic theories of “modern physics,” as correct, and are quick to do battle on their behalf if they are challenged, yet at the same time the majority are totally unwilling to accept some of the aspects of those theories that the *originators* of the theories claim are * essential* features of the theoretical structures. How many of the supporters of modern atomic theory, for example, are willing to accept Heisenberg’s assertion that atoms do not “exist objectively in the same sense as stones or trees exist”?^{40} Probably about as many as are willing to accept Einstein’s assertion that coordinate differences have no metrical meaning.

At any rate, the present general acceptance of the relativity theory as a whole, regardless of the widespread disagreement with some of its component parts, makes it advisable to point out just where the conclusions reached in this area by development of the consequences of the postulates of the Reciprocal System differ from the assertions of relativity theory. This chapter will therefore be devoted to a consideration of the status of the relativity concept, includes the extent to which the new findings are in agreement with it. Chapter 8 will then present the full explanation of motion at high speeds, as it is derived from the new theoretical development. It is worth noting in this connection that Einstein himself was aware of “the eternally problematical character” of his concepts, and in undertaking the critical examination of his theory in this chapter we are following his own recommendation, expressed in these words:

In the interests of science it is necessary over and over again to engage in the critique of these fundamental concepts, in order that they may not unconsciously rule us. This becomes evident especially in those situations involving development of ideas in which the consistent use of the traditional fundamental concepts leads us to paradoxes difficult to resolve.

^{41}

In spite of all of the confusion and controversy that have surrounded the subject, the factors that are involved are essentially simple, and they can be brought out clearly by consideration of a correspondingly simple situation, which, for convenient reference, we will call the “two-photon case.” Let us assume that a photon X originates at location O in a fixed reference system, and moves linearly in space at unit velocity, the velocity of light (as all photons do). In one unit of time it will have reached point x in the coordinate system, one spatial unit distant from 0. This is a simple matter of fact that results entirely from the behavior of photon X, and is totally independent of what may be done by or to any other object. Similarly, if another photon Y leaves point O simultaneously with X, and travels at the same velocity, but in the opposite direction, this photon will reach point y, one unit of space distant from O. at the end of one unit of elapsed time. This, too, is entirely a matter of the behavior of the moving photon Y. and is independent of what happens to photon X or to any other physical object. At the end of one unit of time, as currently measured, X and Y are thus separated by two units of space (distance) in the coordinate system of reference.

In current practice some repetitive physical process measures time. This process, or the device, in which it takes place, is called a * clock*. The progression of time thus measured is the standard time magnitude which, on the basis of current understanding, enters into physical relations. Speed, or velocity, the measure of motion, is defined as distance (space) per unit time. In terms of the accepted reference systems, this means distance between coordinate locations divided by clock registration. In the two-photon case, the increase in coordinate separation during the one unit of elapsed time is two units of space. The *relative* velocity of the two photons, determined in the standard manner, is then two natural units; that is, twice the velocity of light, the velocity at which each of the two objects is moving.

In 1887, an experiment by Michelson and Morley compared the velocity of light traveling over round trip paths in different directions relative to the direction of the earth’s motion. The investigators found no difference in the velocities, although the accuracy of the experiment was far greater than would be required to reveal the expected difference had it been present. This experiment, together with others, which have confirmed the original findings, makes it necessary to conclude that the velocity of light in a vacuum is constant irrespective of the reference system. The determination of velocity in the standard manner, dividing distance traveled by elapsed time, therefore arrives at the wrong answer at high velocities.

As expressed by Capek, the initial impact of this discovery was “shattering.” It seemed to undermine the whole structure of theoretical knowledge that had been erected by centuries of effort. The following statement by Sir James Jeans, written only a few decades after the event, shows what a blow it was to the physicists of that day:

For more than two centuries this system of laws (Newton’s) was believed to give a perfectly consistent and exact description of the processes of nature. Then, as the nineteenth century was approaching its close, certain experiments, commencing with the famous Michelson-Morley experiment, showed that the whole scheme was meaningless and self-contradictory.

^{42}

After a quarter of a century of confusion, Albert Einstein published his special theory of relativity, which proposed a theoretical explanation of the discrepancy. Contradictions and uncertainties have surrounded this theory from its inception, and there has been continued controversy over its interpretation in specific applications, and over the nature and adequacy of the various explanations that have been offered in attempts to resolve the “paradoxes” and other inconsistencies. But the *mathematical* successes of the theory have been impressive, and even though the mathematics antedated the theory, and are not uniquely connected with it, these mathematical successes, in conjunction with the absence of any serious competitor, and the strong desire of the physicists to have *something* to work with, have been sufficient to secure general acceptance.

Now that a new theory has appeared, however, the defects in the relativity theory acquire a new significance, as the arguments which justify using a theory in spite of contradictions and inconsistencies if it is the only thing that is available are no longer valid when a new theory free from such defects makes its appearance. In making the more rigorous appraisal of the theory that is now required, it should be recognized at the outset that a theory is not valid unless it is correct both mathematically and conceptually. Mathematical evidence alone is not sufficient, as *mathematical agreement is no guarantee of conceptual validity.*

What this means is that if we devise a theoretical explanation of a certain physical phenomenon, and then formulate a mathematical expression to represent the relations pictured by the theory, or do the same thing in reverse manner, first formulating the mathematical expression on an empirical basis, and then finding an explanation that fits it, the mere fact that this mathematical expression yields results that agree with the corresponding experimental values does not assure us that the theoretical explanation is correct, even if the agreement is complete and exact. As a matter of principle, this statement is not even open to question, yet in a surprisingly large number of instances in current practice, including the relativity theory, mathematical agreement is accepted as complete proof.

Most of the defects of the relativity theory as a *conceptual* scheme have been explored in depth in the literature. A comprehensive review of the situation at this time is therefore unnecessary, but it will be appropriate to examine one of the long-standing “paradoxes” which is sufficient in its self to prove that the theory is conceptually incorrect. Naturally, the adherents of the theory have done their best to “resolve” the paradox, and save the theory, and in their desperate efforts they have managed to muddy the waters to such an extent that the conclusive nature of the case against the theory is not generally recognized.

The significance of this kind of a discrepancy lies in the fact that when a theory makes certain assertions of a general nature, if any * one* case can be found where these assertions are not valid, this invalidates the generality of the assertions, and thus invalidates the theory as a whole. The inconsistency of this nature that we will consider here is what is known as the “clock paradox.” It is frequently confused with the “twin paradox,” in which one of a set of twins stays home while the other goes on a long journey at a very high speed. According to the theory, time progresses more slowly for the traveling twin, and he returns home still a young man, while his brother has reached old age. The clock paradox, which replaces the twins with two identical clocks, is somewhat simpler, as it evades the question as to the relation between clock registration and physical processes.

In the usual statement of the paradox, it is assumed that a clock *B is* accelerated relative to another identical clock A, and that subsequently, after a period of time at a constant relative velocity, the acceleration is reversed, and the clocks return to their original locations. According to the principles of special relativity, clock *B*, the moving clock, has been running more slowly than clock A, the stationary clock, and hence the time interval registered by *B is* less than that registered by A. But the special theory also tells us that we cannot distinguish between the motion of clock *B* relative to clock A and the motion of clock A relative to clock *B*. Thus it is equally correct to say that A is the moving clock and *B is* the stationary clock, in which case the interval registered by clock A is less than that registered by clock *B*. Each clock therefore registers both more and less than the other.

Here we have a situation in which a straightforward application of the special relativity theory leads to a conclusion that is manifestly absurd. This paradox, which stands squarely in the way of any claim that relativity theory is conceptually valid, has never been resolved except by means which contradict the basic assumptions of the relativity theory itself. Richard Schlegel brings this fact out very clearly in a discussion of the paradox in his book *Time and the Physical World*. “Acceptance of a preferred coordinate system” is necessary in order to resolve the contradiction, he points out, but “such an assumption brings a profound modification to special relativity theory; for the assumption contradicts the principle that between any two relatively moving systems the effects of motion are the same, from either system to the other.”^{43} G. J. Whitrow summarizes the situation in this way: “The crucial argument of those who support Einstein (in the clock paradox controversy) automatically undermines Einstein’s own position.”^{44} The theory based primarily on the postulate that all motion is relative contains an internal contradiction which cannot be removed except by some argument relying on the assumption that *some* motion is not relative.

All of the efforts that have been made by the professional relativists to explain away this paradox depend, directly or indirectly, on abandoning the general applicability of the relativity principle, and identifying the acceleration of clock *B* as something *more* than an acceleration relative to clock A. Moller, for example, tells us that the acceleration of clock *B is* “relative to the fixed stars.”^{45} Authors such as Tolman, who speaks of the “lack of symmetry between the treatment given to the clock A, which was at no time subjected to any force, and that given to clock *B* which was subjected to…forces…when the relative motion of the clocks was changed,”^{46} are simply saying the same thing in a more roundabout way. But if motion is *purely relative,* as the special theory contends, then a force applied to clock *B cannot* produce anything more than a relative motion—it cannot produce a kind of motion that does not exist—and the effect on clock A must therefore be the same as that on clock *B*. Introduction of a preferred coordinate system such as that defined by the average positions of the fixed stars gets around this difficulty, but only at the cost of destroying the foundations of the theory, as the special theory is built on the postulate that no such preferred coordinate system exists.

The impossibility of resolving the contradiction inherent in the clock paradox by appeal to acceleration can be demonstrated in yet another way, as the acceleration can be eliminated without altering the contradiction that constitutes the paradox. No exhaustive search has been made to ascertain whether this streamlined version, which we may call the “simplified clock paradox” has been given any consideration previously, but at any rate it does not appear in the most accessible discussions of the subject. This is quite surprising, as it is a rather obvious way of tightening the paradox to the point where there is little, if any, room for an attempt at evasion. In this simplified clock paradox we will merely assume that the two clocks are in uniform motion relative to each other. The question as to how this motion originated does not enter into the situation. Perhaps they have always been in relative motion. Or, if they were accelerated, they may have been accelerated equally. At any rate, for purposes of the inquiry, we are dealing only with the clocks in uniform relative motion. But here again, we encounter the same paradox. According to the relativity theory, each clock can be regarded either as stationary, in which case it is the faster, or as moving, in which case it is the slower. Again each clock registers both more and less than the other.

There are those who claim that the paradox has been resolved experimentally. In the published report of one recent experiment bearing on the subject the flat assertion is made that “These results provide an unambiguous empirical resolution of the famous clock paradox.”^{47} This claim is, in itself, a good illustration of the lack of precision in current thinking in this area, as the clock paradox is a *logical* contradiction. It refers to a specific situation in which a strict application of the special theory results in an absurdity. Obviously, a *logical* inconsistency cannot be “resolved” by *empirical* means. What the investigators have accomplished in this instance is simply to provide a further verification of some of the *mathematical* aspects of the theory, which play no part in the clock paradox.

This one clearly established logical inconsistency is sufficient in itself, even without the many items of evidence available for corroboration, to show that the special theory of relativity is incorrect in at least some significant segment of its conceptual aspects. It may be a useful theory; it may be a “good” theory from some viewpoint; it may indeed have been the best theory available prior to the development of the Reciprocal System, but this inconsistency demonstrates conclusively that it is not the *correct* theory.

The question then arises: In the face of these facts, why are present-day scientists so thoroughly convinced of the validity of the special theory? Why do front-rank scientists make categorical assertions such as the following from Heisenberg?

The theory…has meanwhile become an axiomatic foundation of all modern physics, confirmed by a large number of experiments. It has become a permanent property of exact science just as has classical mechanics or the theory of heat.

^{48}

The answer to our question can be extracted from this quotation. “The theory,” says Heisenberg, has been “confirmed by a large number of experiments.” But these experiments have confirmed only the * mathematical* aspects of the theory. They tell us only that special relativity is mathematically correct, and that it therefore *could* be valid. The almost indecent haste to proclaim the validity of theories on the strength of mathematical confirmation alone is one of the excesses of modern scientific practice which, like the over-indulgence in ad hoc assumptions, has covered up the errors introduced by the concept of a universe of matter, and has prevented recognition of the need for a basic change.

Like any other theory, special relativity cannot be confirmed as a theory unless its conceptual aspects are validated. Indeed, the conceptual aspects are the theory itself, as the mathematics, which are embodied in the Lorentz equations, were in existence before Einstein formulated the theory. However, establishment of conceptual validity is much more difficult than confirmation of mathematical validity, and it is virtually impossible in a limited field such as that covered by relativity because there is too much opportunity for alternatives that are mathematically equivalent. It is attainable only where collateral information is available from many sources so that the alternatives can be excluded.

Furthermore, consideration of the *known* alternatives is not conclusive. There is a general tendency to assume that where no satisfactory alternatives have thus far been found, there is no acceptable alternative. This gives rise to a great many erroneous assertions that are given credence because they are modeled after valid mathematical statements, and have a superficial air of authenticity. For example, let us consider the following two statements:

- As a mathematical problem there is virtually only one possible solution (the Lorentz transformation) if the velocity of light is to be the same for all. (Sir George Thomson)
^{49} - There was and there is now no understanding of it (the Michelson Morley experiment) except through giving up the idea of absolute time and of absolute length and making the two interdependent concepts. (R. A. Millikan)
^{50}

The logical structure of both of these statements (including the implied assertions) is the same, and can be expressed as follows:

- A solution for the problem under consideration has been obtained.
- Long and intensive study has failed to produce any alternative solution.
- The original solution must therefore be correct.

In the case of statement A, this logic is irrefutable. It would, in fact, be valid even without any such search for alternatives. Since the original solution yields the correct answers, any other valid solution would necessarily have to be mathematically equivalent to the first, and from a mathematical standpoint equivalent statements are merely different ways of expressing the same thing. As soon as we obtain *a* mathematically correct answer to a problem, we have *the* mathematically correct answer.

Statement B is an application of the same logic to a *conceptual* rather than a mathematical solution, but here the logic is completely invalid, as in this case alternative solutions are *different* solutions, not merely different ways of expressing the same solution. Finding *an* explanation which fits the observed facts does not, in this case, guarantee that we have *the* correct explanation. We must have additional confirmation from other sources before conceptual validity can be established.

Furthermore, the need for this additional evidence still exists as strongly as ever even if the theory in question is the *best* explanation that science has thus far been able to devise, as it is, or at least should be, obvious that we can never be sure that we have exhausted the possible alternatives. The theorists do not like to admit this. When they have devoted long years to the study and investigation of a problem, and the situation still remains as described by Millikan—that is, only one explanation judged to be reasonably acceptable has been found—there is a strong temptation to assume that no other possible explanation exists, and to regard the available theory as necessarily correct, even where, as in the case of the special theory of relativity, there may be specific evidence to the contrary. Otherwise, if they do not make such an assumption, they must admit, tacitly if not explicitly, that their abilities have thus far been unequal to the task of finding the alternatives. Few human beings, in or out of the scientific field, relish making this kind of an admission.

Here, then, is the reason why the serious shortcomings of the special theory are currently looked upon so charitably. Nothing more acceptable has been available (although there are alternatives to Einstein’s interpretation of the Lorentz equations that are equally consistent with the available information), and the physicists are not willing to concede that they could have overlooked the correct answer. But the facts are clear. No new valid *conceptual* information has been added to the previously existing body of knowledge by the special theory. It is nothing more than an erroneous hypothesis: a conspicuous addition to the historical record cited by Jeans:

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.

^{51}

”As an emergency measure,” say Toulmin and Goodfield, “physicists have resorted to mathematical fudges of an arbitrary kind.”^{52} Here is the truth of the matter. The Lorentz equations are simply fudge factors: mathematical devices for reconciling discordant results. In the two-photon case that we are considering, if the speed of light is constant irrespective of the reference system, as established empirically by the Michelson-Morley experiment, then the speed of photon X relative to photon Y is unity. But when this speed is measured in the standard way (assuming that this might be physically possible), dividing the coordinate distance xy by the elapsed clock time, the relative speed is two natural units (2c in the conventional system of units) rather than one unit. Here, then, is a glaring discrepancy. Two different measurements of what is apparently the same thing, the relative speed, give us altogether different results.

Both the nature of the problem and the nature of the mathematical answer provided by the Lorentz equations can be brought out clearly by consideration of a simple analogy. Let us assume a situation in which the property of direction exists, but is not recognized. Then let us assume that two independent methods are available for measuring motion, one of which measures the speed, and the other measures the rate at which the distance from a specified reference point is changing. In the absence of any recognition of the existence of direction, it will be presumed that both methods measure the same quantity, and the difference between the results will constitute an unexpected and unexplained discrepancy, similar to that brought to light by the Michelson-Morley experiment.

An analogy is not an accurate representation. If it were, it would not be an analogy. But to the extent that the analogy parallels the phenomenon under consideration it provides an insight into aspects of the phenomenon that cannot, in many cases, be directly apprehended. In the circumstances of the analogy, it is evident that a fudge factor applicable to the general situation is impossible, but that under some special conditions, such as uniform linear motion following a course at a constant angle to the line of reference, the mathematical relation between the two measurements is constant. A fudge factor embodying this constant relation, the cosine of the angle of deviation, would therefore bring the discordant measurements into mathematical coincidence.

It is also evident that we can apply the fudge factor anywhere in the mathematical relation. We can say that measurement 1 understates the true magnitude by this amount, or that measurement 2 overstates it by the same amount, or we can divide the discrepancy between the two in some proportion, or we can say that there is some unknown factor that affects one and not the other. Any of these explanations *is mathematically* correct, and if a theory based on any one of them is proposed, it will be “confirmed” by experiment in the same manner that special relativity and many other products of present-day physics are currently being “confirmed.” But only the last alternative listed *is conceptually* correct. This is the only one that describes the situation as it actually exists.

When we compare these results of the assumptions made for purposes of the analogy with the observed physical situation in high speed motion we find a complete correspondence. Here, too, mathematical coincidence can be attained by a set of fudge factors, the Lorentz equations, in *a special set of circumstances only*. As in the analogy, such fudge factors are applicable only where the motion is constant both in speed and in direction. They apply only to uniform translational motion. This close parallel between the observed physical situation and the analogy strongly suggests that the underlying cause of the measurement discrepancy is the same in both cases; that in the physical universe, as well as under the circumstances assumed for purposes of the analogy, one of the factors that enters into the measurement of the magnitudes involved has not been taken into consideration.

This is exactly the answer to the problem that emerges from the development of the Reciprocal System of theory. We find from this theory that the conventional stationary three-dimensional spatial frame of reference correctly represents locations in extension space, and that, contrary to Einstein’s assertion, the distance between coordinates in this reference system correctly represents the spatial magnitudes entering into the equations of motion. However, this theoretical development also reveals that time magnitudes in general can only be represented by a similar three-dimensional frame of reference, and that the time registered on a clock is merely the one-dimensional path of the time progression in this three-dimensional reference frame.

Inasmuch as gravitation operates in space in our material sector of the universe, the progression of time continues unchecked, and the change of position in time represented by the clock registration is a component of the time magnitude of any motion. In everyday life, no other component of any consequence is present, and for most purposes the clock registration can be taken as a measurement of the total time involved in a motion. But where another significant component is present, we are confronted with the same kind of a situation that was portrayed by the analogy. In uniform translational motion the mathematical relation between the clock time and the total time is a constant function of the speed, and it is therefore possible to formulate a fudge factor that will take care of the discrepancy. In the general situation where there is no such constant relationship, this is not possible, and the Lorentz equations cannot be extended to motion in general. Correct results in the general situation can be obtained only if the true scalar magnitude of the time that is involved is substituted for clock time in the equations of motion.

This explanation should enable a clear understanding of the position of the Reciprocal System with respect to the validity of the Lorentz equations. Inasmuch as no method of measuring total time is currently available, there is a substantial amount of convenience in being able to arrive at the correct numerical results in certain applications by using a mathematical fudge factor. In so doing, we are making use of an incorrect magnitude that we are able to measure in lieu of the correct magnitude that we cannot measure. The Reciprocal System agrees that when we need to use fudge factors in this manner, the Lorentz equations are the correct fudge factors for the purpose. These equations simply accomplish a mathematical reconciliation of the equations of motion with the constant speed of light, and since this constant speed, which was accepted by Lorentz as an empirically established fact, *is deduced* from the postulates of the Reciprocal System, the mathematical treatment is based on the same premises in both cases, and necessarily arrives at the same results. To this extent, therefore, the new system of theory is in accord with current thinking.

As P. W. Bridgman once pointed out, many physicists regard “the content of the special theory of relativity as coextensive with the content of the Lorentz equations.”^{53} K. Feyerabend gives us a similar report:

It must be admitted, however, that Einstein’s original interpretation of the special theory of relativity. For them the theory of relativity is hardly ever used by contemporary physicists. For them the theory of relativity consists of two elements:

(1) The Lorentz transformations; and (2) mass-energy equivalence.^{54}

For those who share this view, the results obtained from the Reciprocal System of theory in this area make no change at all in the existing physical picture. These individuals should find it easy to accommodate themselves to the new viewpoint. Those who still take their stand with Einstein will have to face the fact that the new results show, just as the clock paradox does, that Einstein’s interpretation of the mathematics of high speed motion is incorrect. Indeed, the mere * appearance* of a new and different explanation of a rational character is a crushing blow to the relativity theory, as the case in its favor is argued very largely on the basis that there is no such alternative. As Einstein says, “if the velocity of light is the same in all C.S. (coordinate systems), then moving rods must change their length, moving clocks must change their rhythm…there is no other way.”^{55} The statement by Millikan quoted earlier is equally positive on this score.

The status of an assertion of this kind, a contention that there is no alternative to a given conclusion, is always precarious, because, unlike most propositions based on other grounds, which can be supported even in the face of some adverse evidence, this contention that there is no alternative is immediately and utterly demolished when an alternative is produced. Furthermore, the use of the “no alternative” argument constitutes a tacit admission that there is something dubious about the explanation that is being offered; something that would preclude its acceptance if there *were* any reasonable alternative.

In contribution, in the form of the special theory, can be accurately evaluated only if it is realized that this, too, is a fudge, a conceptual fudge, we might call it. As he explains in the statement that has been our principal target in this chapter, what he has done is to eliminate the “metrical meaning,” of spatial coordinates; that is, he takes care of the discrepancy between the two measurements by arbitrarily decreeing that one of them shall be disregarded. This may have served a certain purpose in the past by enabling the scientific community to avoid the embarrassment of having to admit inability to find any explanation for the high speed discrepancy, but the time has now come to look at the situation squarely and to recognize that the relativity concept is erroneous.

It is not always appreciated that the mathematical fudge accomplished by the use of the Lorentz equations works in both directions. If the velocity is not directly determined by the change in coordinate position during a given time interval, it follows that the change in coordinate position is not directly determined by the velocity. Recognition of this point will clear up any question as to a possible conflict between the conclusions of Chapter 5 and the constant speed of light.

In closing this discussion of the high speed problem, it is appropriate to point out that the identification of the missing factor in the motion equations, the additional time component that becomes significant at high speeds, does not merely provide a new and better explanation of the existing discrepancy. It *eliminates* that discrepancy, restoring the “metrical meaning” of the coordinate distances in a way that makes them entirely consistent with the constant speed of light.