# Chapter III

# Where We Stand: Mathematically

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.—Sir James Jeans

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One of the important functions of history is to enable us to learn from the experiences of the past, so that we do not have to repeat all of the mistakes of our ancestors. The bit of history cited by Jeans in the foregoing quotation is fully capable of performing this historical function, if we heed its message, as it points very clearly and distinctly to an important limitation on the power of mathematics in the realm of physical science; viz., *mathematical agreement is no guarantee of conceptual validity.*

What this means is that if we devise a theoretical explanation for 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 empirically 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. This may seem to be harsh doctrine. Perhaps it is. Certainly it makes the construction of a valid theory a much more difficult task than it would be if mathematical accuracy were sufficient, But nature does not go out of her way to make things easy for the theorist.

Actually, we do not even need the historical record to point the way to this conclusion. Elementary mathematical considerations would tell us the same thing. It is obvious that most mathematical expressions can be subjected to different operations that accomplish results, which are mathematically equivalent. For example, if we start with the equation x= y/z, we may (1) multiply z by a, (2) divide y by a, or (3) multiply y by n and z by m, maintaining the relationship m/n = a, and in all three cases the effect on the quantity x is exactly the same. The term x becomes x/a. In this example it is obvious that if we merely know that some kind of a change has occurred in the right hand side of the equation, and that this has caused x to become x/a, we cannot say which of the three possibilities that have been enumerated was the one that actually occurred. In fact, the range of possibilities is infinite, since there are an infinite number of combinations of m and n which have the quotient a. If anyone had the hardihood to contend that a decrease from x to x/a is positive proof that z has increased to az, we would simply laugh at him. Such a thing *could* be true, of course, but it is only one possibility out of many, and claiming that the observed decrease in x *is proof* of a corresponding increase in z is simply ridiculous.

Yet this is just exactly what the scientific community has permitted Einstein to get away with. If, instead of x = y/z, we write a = F/m, we have one of the alternate expressions of Newton’s Second Law of Motion. Experiments on high speed electrons by Kaufmann, Bucherer, and others showed that at these high speeds the observed values of the acceleration a fell below the level computed from the measured values of F and m, following a pattern which indicated that it would reach zero at the velocity of light. Einstein then decided that this was due to an increase in the mass at these high velocities. At this point he should have been told by his scientific colleagues that this variable mass hypothesis was only one of a number of mathematically equivalent possible explanations of the observed phenomenon, and that neither the hypothesis of mass increase nor any of the others could be accepted on more than a very tentative basis pending the accumulation of further evidence. But this is not the way that modern science operates. Einstein’s assumption was enthusiastically accepted without further ado, and since that time the original experiments that his explanation was designed to fit, together with subsequent results of the same nature in the particle accelerators, have been regarded as proof of the validity of the hypothesis: a flagrant example of circular reasoning.

The fact is that both Einstein’s choice of one particular explanation from among the several available, and the acquiescence of the scientific community in his choice, were based on preferences of a wholly unscientific nature. Since the particles whose acceleration was being measured in the experiments were given their velocity by electrical means, it was assumed that either mass or electric charge must vary, and variability of mass seemed intuitively more likely than variability of charge. This is, of course, sufficient justification for a *tentative* hypothesis of mass variation, but it provides no semblance of justification for talk of “proof.” Furthermore, in their haste to line up behind Einstein, the physicists have overlooked the fact that the electric charge does not enter the equation of motion directly, and hence even if mass is constant, this does not necessarily mean that charge must be variable; all that is required is that the *force* exerted by that charge varies with the velocity. The findings of the present investigation are that the charge does remain constant, but that there is no such thing as a constant force. What is now presumed to be a constant force is, in reality, a phenomenon whose magnitude decreases with the velocity of the object to which it is applied, following the inverse of the mathematical relation which is now customarily applied to the mass.

This explanation advanced by the Reciprocal System produces exactly the same mathematical results in application to the behavior of high speed particles as Einstein’s theory of an increase in mass. It would therefore be equally entitled to claim that the results of experience “prove” its validity. But, of course, they do no such thing, either for the Reciprocal System or for Einstein. Mathematical agreement proves nothing but mathematical validity. It does not prove conceptual validity; it merely establishes the fact that this particular conceptual explanation *could* be correct, and it leaves open the possibility that the correct explanation is contained in some other hypothesis that is mathematically equivalent to the one in question. There may well be many such.

Sherwin, for instance, tells us that while there are practical advantages in treating this phenomenon as an increase in mass,

there is an alternative and more exact way of thinking about the inertial properties of a moving particle…. The increased mass is a sort of artifact, which results from the “distorted” measurements of space and time that are the heart of the theory of relativity…. Rather than think of the inertial mass as increasing because of its velocity, we could instead think of the particle as possessing a constant rest mass but note that, because of unavoidable effects on the measurement of space and time, the observed deflection of the particle produced by a given impact decreases as its velocity increases.

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As Sherwin says, there is no doubt about the experimental fact: “Moving particles are harder to accelerate than the same particles when nearly at rest.” But in spite of the confident and categorical assertions that emanate so freely from modern physicists, this experimental fact does *not* prove that mass increases with velocity. There are at least four explanations that have already been suggested: (1) Einstein’s hypothesis of an increase in mass, (2) the unwelcome, but nevertheless equally plausible hypothesis of a decrease in the charge (where the applied force is electrical, as it has been in the experiments), (3) the possibility that the observed effect is a result of factors affecting the measurement, as suggested in the preceding paragraph, and (4) the conclusion of the Reciprocal System that the effect of a presumably constant force—electrical, magnetic, or gravitational—actually decreases with the velocity of the object to which it is applied. The mathematical “proof” which is supposed to establish the validity of Einstein’s hypothesis applies with equal force to any one of the other three explanations, and it may well be that there are still others as yet unknown which are equally valid mathematically.

This brings us to the issue as to what further steps are necessary in order to establish conceptual validity after mathematical agreement has been obtained. How can we arrive at a definite conclusion as to which of the four hypotheses as to the reason for the decrease in acceleration at high velocities (if any of them) is valid? Here again we can find an answer in a closer analysis of the mathematical situation. Returning to the behavior of high velocity particles, we note that in application to this problem the equation a = F/m contains two unknowns. The acceleration is measured, but the magnitudes of F and m are known only at rest; neither can be measured at high velocity. Mathematical theory tells us that we cannot solve a single equation with two unknowns. We can select values for the unknowns which will make the equation mathematically valid, but these will not be the *correct *values, unless by accident. If the equation represents some physical situation or other meaningful relation, correct values do exist, but in order to determine these values we must have a * set of simultaneous equations.*

The same principle applies to such cases as the one now under consideration. We cannot determine the true reason for the decrease in acceleration at high velocities by a study of this phenomenon alone; we must have at least one independent but related phenomenon which can be utilized together with the behavior of high velocity particles in a manner similar to the solving of a set of simultaneous equations. Usually it will take an even larger set of auxiliary relationships in this kind of a situation than would be expected on the basis of the number of unknowns that appear to be involved, because these physical relations do not have the exact parallelism that characterizes simultaneous mathematical equations, and furthermore, it is not always easy to determine in advance just how many unknowns enter into the situation.

Invoking the aid of auxiliary relations in an attempt to prove the conceptual validity of a hypothesis can work in reverse and furnish a disproof rather than a proof; indeed, this is the usual result, as the great majority of all theories that are proposed are conceptually wrong. The few that survive represent only a very small fraction of the number originally submitted for consideration. But here again the current tendency is to relax the standards for the benefit of the popular theories of the moment, and to ignore or “explain away” contradictions and discrepancies, particularly if they appear in subordinate or collateral applications of the theory in question.

The current literature of science makes no reference, for instance, to the very obvious fact that Einstein’s postulate of an increase in mass at high velocities, the one we have just been discussing, is wholly inconsistent with his explanation of the conversion of mass to energy. Mass cannot be an *accompaniment of* kinetic energy, an entity that increases as the kinetic energy increases, as required by the aspect of Einstein’s theories that explains the behavior of particles in the accelerators, and also an entity that can be *converted to* kinetic energy, one that decreases as the kinetic energy increases, as required by the aspect of Einstein’s theories that explains the atomic bomb. Both of these aspects of the theories are *mathematically* correct, so far as we are able to determine at present, but they are mutually contradictory from the conceptual standpoint, and therefore at least one of them is *conceptually* wrong. The overwhelming mass of evidence *in favor* of the hypothesis that mass and energy are interconvertible is thus equally overwhelming evidence *against* the hypothesis that mass increases with velocity.

Since the objective of this chapter is to survey the mathematical knowledge of space and time that is available to supplement the conceptual knowledge described in Chapter II, this discussion of the hypothesis of variable mass may seem somewhat off the main subject, but a clear understanding of the difference between mathematical knowledge and conceptual knowledge is necessary before we can proceed with our survey, and this variable mass hypothesis provides a particularly good example of the nature of the difference and of the way in which modern science is confusing the two categories. The essential point here is that no matter how definitely the validity of a mathematical relation may be established, this does not in any way confirm the validity of a particular *interpretation* of that relation.

The modern development which is of most particular concern to us in this chapter, because it relates to the inherent nature of space and time, is the Special Theory of Relativity, and the principal relevance of the foregoing discussion of the variable mass hypothesis lies in the fact that this hypothesis, which is itself wholly unsubstantiated and actually in conflict with observed facts, is commonly regarded as one of the main supports of the Special Theory. Furthermore, the logical status of the Special Theory is similar to that of the variable mass hypothesis, in that it is also a well-established mathematical relation, upon which has been placed a conceptual interpretation that is totally unverified and, in reality, has no claim to special merit other than that it is the current favorite among the many possible explanations of the mathematical results.

The existence of alternative explanations is well known and incontestable. Hesse, for instance, tells us that “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.”^{54} Hutten is equally explicit: “This (the Special Theory) is a prime example of modern scientific method, and it illustrates that facts alone do not make a theory, as is often believed; but that an experiment may be interpreted in many different ways.”^{55} What the physicists have actually done is to examine the alternative explanations, to the extent that they were able to visualize them, and to arrive at the conclusion that the Special Theory is the best of these alternatives. “The principle of relativity was not accepted without a thorough-going analysis of all conceivable alternatives,”^{56} reports Sherwin.

Unfortunately, the Special Theory is not being presented to us today in its true aspect as the current choice among a number of possible explanations of the mathematical results; it is being presented as the *correct *explanation. “The conceptions of space and time as given by relativity theory are, even today, accepted as final,”^{57}we are told, in spite of the general recognition of the fact that there are other alternatives that are equally well substantiated observationally. Furthermore, the prevailing opinion, as expressed by Sherwin, that “all conceivable alternatives” have been examined is clearly in error. No one is ever in a position to say that there are no other conceivable alternatives. The most that he can legitimately assert is that no other alternatives have yet been thought of, which puts the situation in an entirely different light, as it eliminates any possible justification for the assumption that “the best we have” is equivalent to “correct.“

In this present work a new alternative to the Special Theory has been produced. This immediately and automatically destroys the contention that no more alternatives exist, and it cuts the ground out from under the argument that has been the principal support of the Relativity Theory, the argument expressed by Sherwin that it is the best of “all conceivable alternatives.” With the knowledge, then, that the Special Theory is only one of many, perhaps very many, possible explanations of the mathematical facts in this case, and that it must justify itself by comparison with observation and experiment in the same manner as any other physical hypothesis, let us briefly review the situation as it now stands.

Whether space and time are absolute or relational—that is, whether they are prior to or determined by events—has been a subject of debate ever since the earliest days of thinking about scientific subjects, but until the closing years of the nineteenth century it was taken for granted that the *magnitudes* of these entities and of their quotient, velocity, were absolute in the sense that accurate measurements would yield identical values irrespective of the conditions under which the measurements were made. Such a conclusion does not preclude the measurement of relative velocities, but it means that the magnitudes of these relative velocities are themselves absolute, and as stated in the textbooks, “The velocity of body A relative to body C is the vector sum of the velocity of body A relative to body B and the velocity of body B relative to body C.“

In 1887 the Michelson-Morley experiment dropped a bombshell into physical theory by demonstrating that the foregoing rule as to the composition of relative velocities does not apply to the velocity of light, and that the measured value of this velocity is constant irrespective of the reference system. By this time, nearly a hundred years later, the sensational impact of the findings of this experiment is beginning to grow somewhat dim, and there is an increasing tendency to minimize its importance in the development of present-day physical theory, even to the extent, in some cases, of contending that Einstein was not appreciably influenced by the experiment in formulating his theories. But Einstein himself says that the results of the experiment created “one of the most dramatic situations in the history of science.”^{58} These results were not only totally unexpected, but they caught the scientific world completely unprepared to offer any plausible explanation.

An explanation of sorts was finally devised by Fitzgerald, who postulated that the length of an object in motion contracts in the direction of motion just enough to account for the observed discrepancy. Some theoretical backing for this postulate was subsequently supplied by Lorentz, who attributed the contraction to the effect of motion on the cohesive forces between the molecules, but there was a very definite *ad hoc* flavor about the whole situation, which left scientists very uneasy.

At this point Einstein came on the scene with what is now known as the Special Theory of Relativity. Instead of attributing the contraction to physical changes in the moving objects, he took the drastic step of abandoning the concept of absolute magnitudes of space and time, and postulating that the observed deviations from Newton’s velocity relations are due to variability in these space and time magnitudes. According to this viewpoint, intervals of space and time are not fixed but vary with the relative velocity of the objects concerned.

Mathematically, both the Fitzgerald-Lorentz contraction hypothesis and the Special Theory of Relativity are correct. Both are specifically devised in such a manner that their formulation leads to a set of mathematical expressions—the Lorentz transformations—which were obtained empirically and are, for that reason, necessarily in agreement with the facts of observation. The first question involved in the present examination of the Special Theory: Is the theory mathematically correct? can thus be answered in the affirmative. It is mathematically correct because it was specifically designed to fit the results of the experiments.

However, this does not automatically give us the answer to the next question: Is the theory conceptually correct? If it could be established that this theory is the *only* possible explanation of the mathematical relations, then the theory would necessarily be conceptually correct as well but, as already pointed out, this is, from a practical standpoint, impossible. There is, to be sure, a tendency on the part of the theorists to conclude that when a problem has been under intensive study by the best minds in the scientific profession for a long period of time and no more alternatives have been discovered, this shows that there are no more alternatives, but this assumes an omniscience on the part of the investigators which the record certainly does not justify.

As brought out earlier in the discussion, such a conceptual situation involving more than one unknown can be resolved only in the same way that a mathematical problem with n unknowns can be solved; that is, by the equivalent of the mathematical device of a set of simultaneous equations. In other words, we cannot establish the conceptual validity of a theory by showing that it explains the phenomenon that it was designed to fit, even if the agreement is complete and exact. In order to prove conceptual validity we must have at least two, preferably several, independent lines of evidence converging to the same theoretical conclusions. This requirement obviously cannot be met by the Special Theory. Even the study of non-uniform motion, which is the phenomenon most nearly related to the uniform motion treated by the Special Theory, has led to conclusions, which are different from, and in some respects inconsistent with, the assertions of the Special Theory. As Bergmann says, the General Theory “discards, in a sense, the conceptual framework of its predecessor.”^{59}

The independent line or lines of evidence needed to verify the conclusions of the Special Theory would normally come from an application of principles previously established in related physical fields, but it should be realized that up to the time of this present investigation no one has ever applied principles developed elsewhere in physical science, in their original forms and without *ad hoc* modification, to the solution of this particular problem. On the contrary, the solutions thus far proposed have done just the opposite; they have repudiated principles developed in other areas and previously regarded as firmly established. The *ad hoc *character of the Fitzgerald contraction hypothesis (which is in disfavor with the scientific community and therefore fair game for criticism) is freely condemned. Capek reports, “Today this explanation is usually presented as a perfect specimen of an *ad hoc* hypothesis, artificially postulated in order to save the appearances.”^{60} But Einstein’s solution is equally *ad hoc* and open to exactly the same criticism. It, too, is a hypothesis artificially postulated to meet the requirements of this particular situation, and while it is mathematically correct, as it must be, since it was deliberately designed to fit the mathematical results already available, this does not in any way assure us that the theory is conceptually correct.

It is therefore clear that the conceptual validity of the Special Theory has not been established, but for present purposes we will want to go one step farther and ask: Are there any reasons for believing that its validity will *never* be established; that is, are there any definite items of evidence to show that it is * incorrect?* And if we examine the situation carefully and critically, without the partisan bias in favor of the theory that colors the viewpoint of the modern physicist, we must necessarily answer: Yes, there is such evidence.

Before a theory can even undertake to pass the crucial test of comparison with experience, it must be self-consistent. It must agree with itself before there is any object in trying to determine whether it agrees with observation. But the Special Theory is not self-consistent. One serious internal contradiction is revealed by the clock paradox. In the statement of this paradox we assume 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 motion of clock B relative to clock A and 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 time 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 Theory leads to a conclusion that is manifestly absurd. As was emphasized in *Beyond Newton*, this paradox, which stands squarely in the way of any claim that the Relativity Theory is physically 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.”^{61} 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.”^{62} The theory based primarily on the postulate that *all* motion is relative contains an internal contradiction which cannot be removed except by some argument based on the assumption that *some* motion is *not* relative.

All of the efforts that have been made by the professional relativists to resolve 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,”^{63} and those 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,”^{64} clothing 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 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, since the Special Theory is built on the postulate that no such preferred coordinate system exists.

This one inconsistency is sufficient in itself to show that the Special Theory is not conceptually correct, in spite of the irreproachable character of its mathematical credentials. It may be a useful theory; it may be a “good” theory; it may, indeed, be the best theory available at the moment (aside from the development in the subsequent pages of this volume); but this evidence shows that it is not the *correct* theory. However, we do not have to rely on this one inconsistency alone; there are other contradictions. One that is closely allied with the clock paradox is the existence of absolute acceleration.

It is true, as the relativists contend, that we cannot determine, without recourse to some external criterion, whether a system is at rest or in uniform translational motion, and this is the principal piece of experimental evidence advanced in support of the relativity postulate. But this evidence only shows that if we deliberately restrict ourselves to the information that we can gain from observation of uniform translational motion within the moving system itself, there is nothing that contradicts the relativity postulate. There is no sound reason, however, why we should bar the use of external criteria as an aid in determining the true facts, and these criteria tell us definitely that absolute motion—that is, motion with reference to the general framework of the universe—does exist. Furthermore, just as soon as we go beyond uniform translational motion we can determine, without reference to external criteria, whether or not the system is undergoing absolute acceleration. Since absolute acceleration is a change in the magnitude of absolute motion, this necessarily involves the existence of absolute motion.

The passengers in a space vehicle, for instance, will have no physical evidence of translational motion within the vehicle after they attain a uniform velocity, but they will be acutely conscious of acceleration during the period just after the rocket is fired, and they will be very aware of the situation if, for any reason, the vehicle begins to rotate. The relativists have never been able to incorporate such accelerations into their system other than by calling them “accelerations relative to the fixed stars” which is merely a euphemism for absolute acceleration.

One of the factors that have helped to create the existing confusion with respect to the logical status of the Special Theory is the introduction of the concept of the ether. As originally conceived, this ether was supposed to be something filling all space and stationary in that space. If there were any such entity, motion with respect to the ether would measure motion with respect to absolute space, and such experiments as that of Michelson and Morley were directed toward a measurement of this kind. Since all such experiments have failed to yield a positive result, it has by this time been demonstrated, to a reasonable degree of certainty, that motion of the earth with respect to absolute space cannot be detected by measurement of motion with respect to a hypothetical ether. At this point the logical conclusion would have been that resort to *other means* to accomplish this purpose would be necessary, but in the meantime Einstein had come forward with his theory based on the assumption that absolute space and absolute motion do not exist. The scientific world was desperately anxious to have some kind of a plausible explanation of the results of the Michelson-Morley experiment, hence in order to accommodate the new theory and prevent its immediate destruction, the physicists took the position that they would refuse to concede the existence of absolute space and time unless this existence could be demonstrated by some means within the moving system itself. In effect, this ruled out any possible universal frame of reference other than an ether. As Arthur Beiser states the “official” position: “The absence of an ether, then, implies that there is no universal frame of reference, so that all motion exists solely relative to the person or instrument observing it.”^{65}

In this way the use of external criteria for the purpose of determining absolute motion was arbitrarily ruled illegal. This is a most astounding innovation in scientific procedure. We are quite familiar with such situations in jurisprudence. Many a lawbreaker has gone free because the rules of evidence would not permit the introduction of the testimony which would have convicted him. But bringing this kind of thing into science is not only unprecedented, but completely out of order. Nature recognizes no “scientific rules of evidence,, and there is no sound reason why science should tie its own hands. Since we cannot observe locations in space directly, we must identify them by means of something observable that is present therein, but there is no necessity that this be an ether, or anything resembling an ether. On the contrary, we are on much more solid ground if we utilize objects that we know actually exist, rather than something purely hypothetical. The obvious choice for a reference system is the spatial framework defined by means of the fixed stars.

Whether or not we regard this reference system as defining an “absolute space” is immaterial. Some *of* the professional relativists concede that it does. Moller, for instance, admits that “the fixed stars as a whole may be regarded as approximately at rest relative to the ’absolute space.’…”^{66} Eddington has made a similar concession. But in any event, these stars provide us with a universally applicable frame of reference, the kind of a “preferred coordinate system“, that Relativity claims does not exist, and whether or not we call it “absolute space” is merely a question of semantics. Motion relative to this universal frame of reference is exactly the same thing, from the scientific and practical standpoints, if not philosophically, as the “absolute motion” that the world of science knew before Relativity came on the scene. The frequency with which the relativists themselves call upon “motion relative to the fixed stars” to get out of tight corners is clear evidence of how necessary an absolute framework actually is, even to those whose basic theories rest upon a denial that any such thing exists. It is also highly significant that the astronomers, the scientific group whose work is the most directly affected by the principles of Relativity, carry out their calculations in callous disregard for those principles, just as if they had never heard of Einstein, even though, as loyal members of the scientific community, they may pay lip service to this phase of current scientific dogma. McVittie tells us explicitly:

In discussing stellar proper motion and radial velocities, astronomers tacitly assume that these represent the rates of change of local distances with respect to the absolute time of classical Newtonian mechanics. Moreover local distance is identified with the absolute distance of classical theory.

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In addition to the direct evidence against the conceptual validity of the Special Theory which is supplied by the clock paradox and the existence of absolute acceleration, the inability of the adherents of this theory to extend its field of applicability or to establish firm connections with other physical theories, in spite of the immense amount of effort devoted to the task is, in a sense, additional direct evidence against it, because non-uniform motion is so closely related to uniform motion that a conceptually valid theory applicable to the more limited field should be capable of extension to the general case without any serious difficulty. We must therefore conclude that the conceptual validity of the Special Theory is not only *unproved * but definitely* disproved.* The situation with reference to the General Theory will not be considered in this volume, as this phase of the Relativity ideas is outside the scope of the present discussion. It was, however, examined in detail in *Beyond Newton*.

The question now arises, If the foregoing conclusions are correct, and this can hardly be denied if the issues are squarely faced, 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.

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We can throw some light on these questions by considering 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)^{69} - 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)
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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.
- Hence the original solution must be correct.

In the case of statement A, this logic is irrefutable. It would, in fact, be valid even without any 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 simply different ways of expressing the same thing. The statements x = ab and x/a = b, for example, are not two different mathematical relations; they are simply two different ways of stating the same relation. Hence 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. “A physical theory consists of a formal calculus and an interpretation,” explains Hutten, “but the relation between calculus and interpretation is in fact not unique. A single calculus may be interpreted in terms of various concepts.”^{71} Merely finding an explanation, which fits the observed facts does not, in this case, guarantee that we have the correct explanation. As brought out previously, 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 plausible explanation has been found—there is a strong temptation to assume that no other possible explanation exists, and to regard the existing theory as necessarily correct, even though, as in the case of the Special Theory, 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 better is now available, and the physicists are not willing to concede that they might have overlooked the correct answer. But the facts are clear. No new *conceptual* information has been added to the previously existing body of knowledge concerning space and time by the Michelson-Morley experiment and the theoretical developments aimed at explaining the results of that experiment. The Special Theory of Relativity is nothing more than an erroneous hypothesis: a conspicuous addition to the historical record cited by Jeans; another example of a mathematical formula that is right, or very nearly right, clothed with a physical interpretation that is very badly wrong.

To many of those who, from their earliest contacts with physical science, have been taught that the Special Theory “belongs to the firm foundation of modern physics and cannot be disputed in our present situation,” as Heisenberg puts it, this conclusion may seem almost incredible, but the realization that the mathematical validity of the Lorentz transformations is no proof of the validity of Einstein’s *interpretation* of these mathematical results has been growing. “It must be admitted, however, that Einstein’s original interpretation of the special theory of relativity is hardly ever used by contemporary physicists,” says Feyerabend, “For them the theory of relativity consists of two elements: (1) the Lorentz transformations; and (2) massenergy equivalence.”^{72} Bridgman also comments on the tendency “to define the content of the special theory of relativity as coextensive with the content of the Lorentz equations,” and he points out that on this basis there is no “theory” of relativity:

Nothing explicit in the (Lorentz) equations themselves determines the nature of the physical application, but this has to be specified in some way apart from the equations. Not until we have specified the details of the physical application do we have the right to speak of the equations as part of a physical “theory.”

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The emotional reaction of most physicists to any suggestion that Einstein’s theory will have to be abandoned is largely due to a mistaken impression that the choice is between Einstein and pre-Einstein; that the proponent of change is advocating a retreat to positions that are now known to be untenable. But as long as we concede the validity of the Lorentz equations a return to pure Newtonian theory is impossible. The findings presented in this work do not suggest a retreat from Einstein to pre-Einstein; they suggest an advance from Einstein to post-Einstein. The Special Theory of Relativity is mathematically correct but conceptually wrong. What is needed is not to abandon the correct mathematical structure, but to discard Einstein’s erroneous interpretation of the mathematical results, an action that, as Feyerabend and Bridgman report in the foregoing statements, is already under way, and then to ascertain what new concepts are necessary in order to produce a theory that is *both* mathematically and conceptually correct.

The correct theory of space and time must therefore be based not only on the *conceptual* information summarized at the end of Chapter II, but also on the additional *mathematical* information about these two entities which has been obtained from the Michelson Morley experiment and the subsequent studies of the results of that experiment. This information may be expressed as follows:

The velocity of light is independent of the reference system.

Other velocities measured in the normal manner in one reference system can be expressed in terms of another reference system moving translationally at a constant speed relative to the first system by means of the mathematical relations known as the Lorentz transformations.