## XIII

“The history of theoretical physics,” says Jeans, “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.”^{47} The Special Theory of Relativity adds another page of the same kind of history. In essence this theory is simply a mathematical system of compensating for the error introduced into relations at high velocities by the failure to recognize the existence of coordinate time. It can easily be seen that a *generally applicable *mathematical scheme of this kind is impossible, since no conceivable mathematical system can accurately describe a three-dimensional relationship in one-dimensional terms. If we limit the field of application to unidirectional transnational velocities, however, we are dealing with only one of the three dimensions of coordinate time, and in this special situation a mathematical compensation for the conceptual error is possible. The Einstein theory is an ingenious and carefully developed system of this nature and in relatively simple applications it gives consistently accurate results, but because of its inherent limitations it quickly runs into complications as soon as it attempts to go beyond these simple applications.

In spite of its shortcomings, Relativity Theory has been the accepted physical doctrine in its field for more than a half century and consequently any new theory that is presented must indicate where and to what extent it differs from Relativity, as a matter of facilitating tile task of grasping the content of the new theoretical structure, if for no other reason. A general picture of this situation can be obtained by considering first, the postulates on which Relativity is based, and second, the experimental and observational evidence which is offered in support of the theory.

Relativity theory utilizes many principles and relations drawn from the general body of physical knowledge. Except in the one case of the basic concept of the nature of time, there is no point of conflict in this area, and a discussion of these items will not be necessary in the present connection. In addition, the theory sets up four postulates of its own. Two of these appear in the Special Theory: (1) a denial of the existence of absolute velocity, and (2) the constant velocity of light. The General Theory adds two more: (3) the equivalence of gravitational and inertial mass, and (4) the so-called postulate of “covariance.”

The constant velocity of light is a necessary and direct consequence of the Fundamental Postulates of this present work. The same is true of the equivalence of gravitational and inertial mass. The Reciprocal System is therefore in full agreement with the Relativity Theory so far as postulates (2) and (3) are concerned. Furthermore, there is no definite disagreement concerning postulate (4). It is true that the implications of the Reciprocal System tend to reinforce the opinions of those who doubt whether there is any actual meaning in this postulate. Bridgman claims that “any assumed law of nature whatever can be expressed in covariant form,”^{48} and states that this was admitted by Einstein in 1918. Bondi brings out the same point very forcibly: “Thus whereas the special principle has physical content, the general principle is void of all physical meaning and is merely a mathematical challenge, a challenge to the ingenuity of mathematicians to find the same form for the laws of nature in all systems of coordinates, however different the content of these laws might be. Given sufficient high ingenuity on the part of the mathematician the principle places no restriction whatever oil the laws.”^{49} But in any event, whatever physical significance the postulate does have in the Relativity Theory, if any, is equally valid in the Reciprocal System; indeed, the problem of maintaining the “form-invariance’ is less acute in the new system as the theory itself and the mathematical expression thereof are both much simpler.

The area of disagreement between the Reciprocal System and the Relativity Theory, so far as the basic assumptions are concerned, therefore reduces to the difference in the respective concepts of the nature of time, and to a question as to the validity of postulate (1): the denial of the existence of absolute motion.

In setting up his Special Theory of 1905, Einstein took as his starting point the experimentally established constancy of the velocity of light, and expressed this as a postulate: a law of nature which we must accept, however strange and illogical it may seem in the light of preexisting thought. This immediately led to a direct conflict with the definition of velocity as *s/t*, space traversed divided by elapsed time, but Einstein found that the discrepancy, in the special case of uniform translational velocities, is a function of the velocity. He therefore deduced that the correct *mathematical *results could be obtained by denying the existence of absolute space and time and eliminating the discrepancy through an appropriate modification of the numerical values of the space and time terms.

This First Postulate which denies the existence of absolute motion, together with absolute space and time magnitudes, and thereby gives the Relativity Theory its name, is a purely negative proposition. It is sometimes stated in a positive manner; that is, the assertion is made that motion is relative rather than that it is not absolute. But absolute motion is also relative to any reference system which we may wish to set up, and the relativity assertion is therefore meaningless unless it is intended to signify that motion is relative * only, *which is just another way of saying that it is not absolute. As a negative proposition, this postulate cannot contribute anything positive to the theoretical structure. Herbert Dingle says flatly, “The principle of relativity in itself tells us nothing whatever about anything…”^{50} It merely serves the purpose of evading the contradiction which would otherwise exist, and it is somewhat analogous to the legal principle that prevents a wife from testifying against her husband. This prohibition does not strengthen the husband’s affirmative case in the least, but it may be extremely important in blocking the opposition. So it is with the First Postulate. When this postulate has eliminated the necessity of conforming to the space and time magnitudes determined by measurement, the way is left clear for the mathematical manipulation that is necessary in order to force an agreement with the constant velocity of light.

The First Postulate of Relativity is incompatible with the Reciprocal System described herein, as all physical magnitudes in this system are absolute, but as long as the only accomplishment of the postulate is to evade a contradiction, the new system arrives at exactly the same result without the postulate, since no contradiction develops in this system; the absolute space and time magnitudes on the new basis are in full agreement with the observed constant velocity of light.

We thus find that the Reciprocal System is in agreement with three of the four basic postulates of the Relativity Theory and it achieves the same results without the remaining postulate that Relativity Theory does with it. It is therefore appropriate to conclude that if these four postulates correctly represent the content of the theory, as Tolman and many others contend, any results which can * legitimately *be derived from the Relativity Theory will also be reached by the Reciprocal System. Other conclusions which result from attempts to apply the mathematical methods of Relativity to areas outside of their scope of applicability (that is, to areas in which the error due to the use of clock time rather than total time is not a function of the velocity alone) or which result from inferences drawn from the language in which the postulates are expressed will not be in agreement with the results obtained from the new system.

In this connection, it should be noted that there is no observational evidence to support the First Postulate, the only one of the basic postulates of the Relativity Theory with which the Reciprocal System is in conflict, whereas there *is *evidence that tends to contradict it. It is true that all attempts to measure a translatory velocity relative to a hypothetical ether or to the general framework of space have failed, but this does not necessarily mean that absolute motion does not exist; it is equally consistent with the hypothesis that such motion exists but our present facilities are incapable of detecting it. Heisenberg views the situation in this manner: “This is sometimes stated by saying that the idea of absolute space has been abandoned. But such a statement has to be accepted with great caution… The equations of motion for material bodies or fields still take a different form in a ‘normal’ system of reference from another one which rotates or is in a nonuniform motion with respect to the ‘normal’ one.”^{51} Bergmann expresses the same thought: “It. appears as if general relativity contained within itself the seeds of its own conceptual destruction, because we can construct ’preferred’ coordinate systems.”^{52}

The most that can legitimately be contended, therefore, is that the experimental evidence leaves the question of the existence of absolute translational motion open. As indicated in the foregoing statements, absolute rotational motion *can *be detected. It is possible, for instance, to determine that the galaxies are rotating and even to get a rough idea as to the magnitude of the rotational velocity in each case simply by looking at them. All attempts that have been made to reconcile these facts with the First Postulate have been extremely awkward and far-fetched. In the words of Eddington, “We see at once that a relativity theory of translation is on a different footing from a relativity theory of rotation. The duty of the former is to explain facts; the duty of the latter is to explain away facts.”^{53}

The inapplicability of the First Postulate to rotational motion not only invalidates its claim to the status of a * general *physical principle; hut also creates a strong doubt as to its applicability to translational motion. The existence of absolute rotational motion and “preferred coordinate systems” strongly suggests that absolute translational motion also exists, since rotational and translational motion are interconvertible, and it is rather difficult to accept the idea that absolute motion can be converted to motion which has no absolute magnitude. It is, in fact, just this point that has made the proponents of the Relativity Theory so desperate in their attempt to “explain away” the physical evidence of absolute rotation.

The findings of this present work make these considerations somewhat academic, however, as these findings not only eliminate the *reason for questioning *the existence of absolute motion, but also indicated that absolute translational motion *can * be detected and measured. We cannot regard our usual measurements relative to the earth as absolute, since we know that the earth revolves around the sun; we cannot regard measurements relative to the sun or the solar system as absolute, since we know that the sun takes part in the rotation of the galaxy; we cannot regard measurements relative to the galactic center or to the galaxy as absolute, since we know that the galaxies have random motions of their own and are also receding from each other. Up to now there has also been the possibility that the galaxies are in motion as component parts of some larger unit. An extensive development of the consequences of the Fundamental Postulates of the Reciprocal System indicates, how ever, that the galaxy constitutes the maximum aggregation of matter, the so-called clusters of galaxies being merely temporary associations of no major significance which will ultimately disappear either by dispersion or by agglomeration. An absolute system of reference can therefore be obtained by correcting the galactic positions for the effects of the recession and the random movements of the individual galaxies. Such a system constitutes the general spatial framework of our physical universe and since we know only one universe, motion relative to this general framework is absolute motion. The principal problem involved in establishing this absolute system of reference lies in evaluating the random movements, as the correction for the recession due to the space-time progression is a straightforward operation, but available statistical methods should he adequate for this purpose.

## XIV

The ultimate test of the validity of any physical theory is agreement with the results of observation, and measurement. An actual *proof *of validity is, however, extremely difficult to achieve. In order to constitute a proof, the correlation between theory and observation must be comprehensive enough to reduce the possibility of a hidden conflict somewhere in the system to the point where it is negligible and, as previously pointed out, this means that there must be an exact correspondence in a large number of cases throughout the area affected, without exception, and without the use of contrived methods of evading contradictions or inconsistencies.

Because of the extraordinary difficulties that stand in the way of achieving a valid proof, it is necessary to relax the standards to some extent for practical purposes and to accept on a provisional basis a great many laws and principles which are far from qualifying as established truths under any reasonably rigid standards. Strictly speaking, such principles should be recognized as merely tentative, and in referring to them the words “We think…” should be used rather than “We know…” but the human mind is reluctant to admit ignorance and there is a very general tendency to regard today’s best guess as the equivalent of established fact. The individual who is firmly convinced of the truth of the currently accepted doctrines in his field is inclined to take at face value anything which tends to reinforce the position to which he is committed, but to be very critical of anything to the contrary. As a result, the meaning of the word “proof” is badly distorted in current usage.

One very common practice, for instance, is to present evidence in favor of some particular portion of a theory as proof of the validity of the theory as a whole. A great deal of publicity has recently been given to some new “proofs” of the Relativity Theory that have been made possible by modern technological developments. One of the most significant of the recent experiments of this kind was carried out at Columbia University by C. H. Townes and associates; another at Harvard University by R. V. Pound and G. A. Rebka, Jr. The results of this work were immediately reported in the news journals of the scientific world with headlines such as “Year-Long Tests Confirm Einstein’s Theory,” followed by unequivocal statements that these tests “have confirmed… that Einstein’s special theory of relativity is correct.”^{54} But when we look behind the facade to see just what was actually accomplished, we find that the Columbia group simply produced some additional verification of *one of the postulates *of the Relativity Theory: the constant velocity of light, whereas the Harvard investigators verified *another of the postulates: *the equivalence of gravitational and inertial mass.

To the extent that this additional evidence constitutes proof of anything, it is proof of these two postulates, not of the Relativity Theory as a whole. Furthermore, these particular postulates are not actually assumptions at all; they are experimental facts whose validity most physicists were willing to concede *before *Einstein incorporated them into his Relativity Theory. When we get down to bedrock, therefore, we find that these widely publicized “proofs” of the Relativity Theory simply verify knowledge that existed before the theory was born; they tell us nothing about the new ideas that Einstein put into the theory.

An even more serious threat to the integrity of scientific knowledge is the widespread tendency to accept evidence, which is *consistent with *a theory as *proof of *that theory. The situation with respect to the First Postulate of Relativity has already been discussed. Terrestrial experiments of many kinds have failed to disclose any effects that can be attributed to absolute translational motion of the earth. This is consistent with the hypothesis that no such absolute motion exists, to be sure, but by no stretch of the imagination can it qualify as proof, since the contrary hypothesis that such motion exists but cannot be detected by available methods is equally consistent with the facts. But Relativity is the currently fashionable doctrine, and it is standard practice to accept contentions favorable to the orthodox doctrine at their face value hence, as Herbert Dingle states the case, “A theoretical demonstration that the theory contains no internal contradictions—that it *could * be right—has frequently been regarded as a proof that it *is * right.”^{55}

Another striking illustration of the current trend is furnished by the concept of an increase in mass at high velocities. Everywhere we turn, we find references to the “proof” of this relation derived from experiment, and to our “knowledge” of the rate of increase of the mass. Here again the evidence is clearly consistent with the popular hypothesis, but even a very casual consideration is sufficient to show that this evidence is equally consistent with any one of several other explanations, and hence is no proof of any of them. The almost universal habit of treating this evidence as proof of the hypothesis of an increase in mass is scientifically indefensible.

Equally common is the thoroughly unsound practice of accepting a hypothesis as an established truth simply because it happens to be the best explanation available at the moment, even if the confirmatory evidence is wholly inadequate This is the situation which exists today with reference to the Relativity Theory as a whole. In order to verify this statement, we will next proceed to summarize the evidence in favor of the theory and then to analyze the extent to which this evidence is sufficient to justify the two assertions that are now commonly made on behalf of Einstein’s ideas: (1) that tile Relativity Theory is superior to the Newtonian system, and (2) that the Relativity Theory is a correct representation of the physical relationships.

The following are the points that are generally advanced in support of the theory:

- It reduces to Newton’s system at low velocities and hence is in agreement with all of the great mass of experimental and observational data that supports Newton’s generalizations.
- It is in agreement with the observed fact that the velocity of light is constant and independent of the reference system.
- It accounts for the advance of the perihelion of Mercury.
- It predicts the interconvertibility of mass and energy and furnishes the correct mathematical expression for this conversion.
- It supplies an explanation for the deviation from the Newtonian relation F =
*ma*that is observed at high velocities.

Einstein proposed two other tests of the theory: bending of a light ray in passing a massive body and a shift of atomic spectra toward the red in a strong gravitational field. The first observations made after the original publication of the theory were interpreted as confirming these theoretical predictions, and for many years these correlations were accepted as proofs of the validity of the theory. More recently, however, skepticism has been growing, and what may be regarded as the “official” opinion at present is that the status of both of these tests is doubtful. A statement by H. P. Robertson at a conference on “Experimental Tests of Theories of Relativity” held at Stanford University in July 1961 contains the following conclusions, as reported by L. I. Schiff: “the deflection of light by the sun has not been measured with great precision,” “the red shift follows from more elementary considerations and is not really a test of general relativity,” and “only the precession of the perihelion of the orbit of the planet Mercury provides an accurate test of Einstein’s theory.”^{56} In view of the existing uncertainties, we are not justified in taking the deflection of light and the gravitational red shift into consideration in the present analysis.

If we examine the five points listed in the foregoing tabulation from the standpoint of their relevance to the question as to the relative merits of Newton’s and Einstein’s theories, it is apparent that the verdict is definitely in favor of Einstein. Newton’s theory gives the wrong answer in the case of item 2, the constant velocity of light and also item 5, the decrease in acceleration at high velocities, and it provides no explanation of the observed facts concerning item 3, the advance of the perihelion of Mercury. It is likewise silent on item 4. “This latter cannot be counted against Newton, as this subject is not specifically within the scope of his theories, but the ability to cover a larger field is a point in favor of Einstein’s theory. On the other side of the picture, Newton can claim no offsetting advantage over Einstein because of item 1, which means that all of the positive evidence in favor of Newton’s system is equally favorable to the Relativity Theory. This statement should perhaps be qualified to some extent, as the ability of Einstein’s theory (the General Theory, in particular) to achieve the same results as Newton’s system cannot be definitely checked in the more complex applications because the mathematics become too difficult to handle by any means now available. Mc Vittie comments on this point as follows: “But whether it (General Relativity) can also embrace the phenomena associated with the idea of rotation—the tides, for example—which presented little difficulty to Newtonian theory, is still an unsolved problem.”^{57}

The early history of the Relativity Theory is commonly portrayed, in present-day writings, as a contest between Einstein and Newton in which the points outlined in the foregoing paragraphs were gradually recognized by the scientific profession, so that the decision ultimately went to Einstein. Actually, however, Michelson and Morley destroyed the validity of Newton’s Laws as physical principles of general application in conjunction with existing ideas of space and time in 1887, not by Einstein, who first published his theory in 1905. As soon as the authenticity of the results of the Michelson-Morley experiment was conceded, the generality of Newton’s Laws was automatically invalidated, even though it took some years to overcome the reluctance of the scientific profession to accept this distasteful fact. There was now a direct conflict between the Newtonian concept of motion and the experimentally verified constancy of the velocity of light. Something in the fabric of physical theory obviously had to be altered, and the issue facing the scientific community was essentially a question as to what this should be. Not recognizing the incomplete nature of the existing ideas of time, tile scientists of this era selected the concept of absolute magnitudes of space, time and motion as the item to be sacrificed. Fitzgerald first advanced the idea of a contraction of space in the direction of motion, Lorentz enlarged and improved the concept, and finally Einstein put the whole development on a firm mathematical and theoretical footing.

Between 1887 and 1905 there was no general theory of motion that could even *claim *validity. Thus the Relativity Theory did not attain its present position by triumph over an opposing idea; it simply filled a conceptual vacuum, and its general acceptance in spite of its many weaknesses is due primarily to this fact. Because of these numerous and serious weaknesses powerful voices were raised against it in its youth. P. W. Bridgman, for example, once predicted (referring particularly to the General Theory) that “the arguments which have led up to the theory and the whole state of mind of most physicists with regard to it may some day become one of the puzzles of history.”^{26} Paul R. Heyl was equally critical. “Here,” he says, speaking of rotational motion, “the relativity concept shows plainly its nature: a hollow mathematical shell, with no real content; useful as far as it fits the facts, useless where it does not.”^{58} But the battle was won by default. Bridgman, Heyl and their fellow critics were ultimately silenced because they had no alternative to offer; they could only attack the shortcomings of the theory itself and, as the politicians so aptly put it, “You cannot beat something with nothing.”

But when we turn to the second issue, the question as to whether the Relativity Theory is a correct representation of the actual physical relationships, all of the deficiencies and shortcomings pointed out by these early critics to no avail once more become pertinent. Here the theory has much more difficult requirements to meet; it must be so strongly supported that the *possibility *of an error of any consequence in the structure of the theory is negligible. As previously stated, this means that it must agree with the observed and measured facts in a large number of individual applications throughout the area affected, without exception, and without the use of contrived methods of evading inconsistencies or contradictions. Obviously the theory does not even begin to meet these requirements. Aside from the fact that it claims to incorporate Newton’s low velocity relations, which are firmly established, the Relativity Theory can point to only a very few instances of agreement with the established facts, and in the most important of these, the situation with respect to the constant velocity of light, the agreement has been reached only by means of one of those evasive devices which vitiate any attempt at proof.

The use of principles of impotence or other evasive devices has become such a commonplace feature of present-day physical theory that their true character has been to a large extent obscured. What these devices actually accomplish is to dispose of a contradiction or inconsistency by postulating that this discrepancy shall not be counted as a discrepancy. It is, of course, possible that the device may be entirely legitimate; the universe may actually be constructed in some such weird manner. But there is no way in which we can determine whether or not it is legitimate in any particular case, and hence the use of such a device precludes any possibility of proof, irrespective of whether or not the contentions are well-founded. If we have to utilize a device of this kind to arrive at the truth, then we can never be certain that it is the truth. As Einstein himself has pointed out, “For it is often, perhaps even always, possible to adhere to a general theoretical foundation by securing the adaptation of the theory to the facts by means of artificial additional assumptions.”^{59}

Either of these deficiencies, the lack of adequate observational confirmation or the use of the unsupported assumption of the “rubber yardstick,” is sufficient to stamp the Relativity Theory as unproved, hence our second question, the question as to whether the theory is a correct representation of the physical facts, will have to receive an inconclusive answer for the moment. There are no instances where the theory is definitely in conflict with established facts, aside from such bearing as the existence of absolute rotation may have on the issue, but the theory becomes more and more vague as it passes from general principles to details, and there is ample justification for a suspicion that it is only the concealment thus provided that prevents recognition of many conflicts with the physical facts. Einstein tacitly admits this when he speaks of “…the ever widening logical gap between the basic concepts and laws on one side and the consequences to be correlated with our experience on the other—a gap which widens progressively with the developing unification of the logical structure.”^{60}

## XV

The development of the Reciprocal System now confronts the Relativity Theory with the kind of an acid test which it has never had to meet before: a direct item by item comparison with a new theoretical structure that agrees with the facts of observation and experiment in an easy and natural way, without the use of evasive devices such as the First Postulate of Special Relativity or vague and obscure “artificial additional assumptions” of the kind that are employed so freely in the development of the General Relativity Theory. The presentation in this volume advances two contentions on behalf of the Reciprocal System similar to those, which have previously been offered on behalf of the Relativity Theory. These are (1) that the Reciprocal System is superior to the Relativity Theory, and (2) that this system is a correct representation of the physical facts. The second contention includes the first and it would actually be sufficient to establish this point alone, but it will help to clarify several significant issues if the two questions are considered separately.

In beginning an analysis of these two questions, let us first summarize the position of the Reciprocal System with respect to the five points raised in support of the Relativity

- This system also reduces to Newton’s system at low velocities.
- It is also in agreement with the observed constant velocity of light.
- It furnishes a different, but equally accurate, explanation of the advance of the perihelion of Mercury.
- It arrives at the same mathematical expression for the interconversion of mass and energy.
- It furnishes a different, but equally consistent, explanation of the deviation from the relation

F =*ma*at high velocities.

From this tabulation it can be seen that the points which led to the triumph of Relativity over Newton’s system are not available as arguments in the contest with the Reciprocal System. On the contrary, unless it can be shown that Relativity Theory furnishes a * better *explanation in one or more of those cases where the two theories arrive at the same results by different routes, the Relativity Theory has no argument at all to support a contention that it is superior to the Reciprocal System. Let us therefore examine the differences between the two theories in these particular areas.

So far as the agreement with Newton’s Laws at low velocities is concerned, the Reciprocal System is in much the better position. The adherents of the Relativity Theory claim that the equations of this theory give the same results as Newton’s Laws at low velocities but, as pointed out earlier, this claim cannot be vitrified in any other than the very simplest applications, as the mathematics of the theory are too complicated to be workable elsewhere. The Reciprocal System does not merely give *the same results as *Newton’s Laws; at low velocities the equations of this system are Newton’s expressions, hence there cannot be any question *as *to the kind of results which this system produces in the low velocity field.

The comparison in the field of motion at high velocities is also very definitely favorable to the Reciprocal System, as Relativity Theory is confronted with a contradiction between the constant velocity of light and the definition of velocity which the theory utilizes: a contradiction that is removed only by the use of an arbitrary assumption of a wholly unsupported nature. In the Reciprocal System, on the other hand, no such contradiction exists and no evasive assumption is required. The constant velocity of light emerges easily and naturally from the development of the basic postulates of this system.

Closely connected with this question of the constant velocity of light is the advance of the perihelion of Mercury. It has been known since the time of Leverrier that the orbit of this planet is constantly moving ahead of the position calculated on the basis of Newton’s Laws, the unexplained increment being almost twenty miles per revolution or something over 40 seconds of arc per century. According to the Reciprocal System, this is merely another effect of the same factors that are responsible for the negative result of the Michelson-Morley experiment. As long as the orbital velocity is low, the difference between clock time and total time is negligible, but the velocity of Mercury is great enough to introduce an appreciable amount of coordinate time and during this added time the planet travels through an additional distance.

Einstein’s mass-energy equation E = *mc*^{2} is entirely in accord with the relations derived from the Reciprocal System. In the previous publication mass was identified as the reciprocal of three-dimensional velocity, *t ^{3}/s^{3}*, and energy as the reciprocal of one-dimensional velocity,

*t/s.*When reduced to the space-time terms of the new system, the mass-energy equation becomes

*t/s = t*As this is a valid equality, the equation E =

^{3}/s^{3}× s^{2}/t^{2}*mc*

^{2}hold good in the Reciprocal System just as it does in the Relativity Theory.

But this agreement as to the mathematical *form * of the relationship does not signify agreement as to the *meaning * of the equation accepted l y both systems. Einstein claims that a body at rest possesses a quantity of energy equivalent to its mass, and that kinetic energy of motion likewise corresponds to an equivalent amount of mass. A body in motion therefore acquires an additional mass, which “varies with changes in its energy” and “becomes infinite when q (the velocity) approaches 1, the velocity of light.”^{61} “According to the theory of relativity,” Einstein says, “there is no essential distinction between mass and energy. Energy has mass and mass represents energy.”^{62}

The Reciprocal System is in direct conflict with this * interpretation *of the equation. From the Fundamental Postulates of this system we find that energy is a one-dimensional displacement of space-time, whereas mass is a three-dimensional displacement (rotational). Under appropriate conditions the dimensions of the displacement can be altered, hence mass is convertible to energy and vice versa. The displacement can exist *either *as mass or as energy (that is, either in three dimensions or in one dimension) but obviously *not as both simultaneously. *Mass is not *associated with *energy; it *is convertible to *energy, and the mass-energy equation merely indicates the relation between the magnitudes involved *when and if *the conversion takes place. Energy is mass only if it is converted to mass, and when such a conversion takes place so that a quantity of mass makes its appearance, the equivalent quantity of kinetic energy ceases to exist.

As Bridgman has pointed out, many of Einstein’s conclusions have been accepted without adequate critical scrutiny, and this mass-energy relation definitely falls in this category. If this relationship is examined from the standpoint of logic, it is apparent that Einstein’s contentions are internally inconsistent and must eventually fall of their own weight, irrespective of what any other theory may say. Mass cannot be something that is associated with energy (and therefore increases as the energy increases) and at the same time something that is convertible to energy (and therefore decreases as the energy increases). But this obvious conceptual contradiction is one of the things that Relativity Theory expects us to accept. If “mass and energy, are only different expressions for the same thing,”^{61} as Einstein declares, then we cannot have a conversion of one to the other; we cannot convert anything into itself. But such a conversion clearly *does *take place. An atomic explosion, for example, is not a mere alteration in terminology or a conceptual reorientation; it is an actual physical event, and hence Einstein’s viewpoint cannot be correct. It does not meet the requirements of elementary logic.

It is generally believed that the hypothesis of an increase in mass accompanying increased velocity is firmly established by experiment, and scientific literature is full of positive statements to that effect: statements which emanate not only from rank and file physicists, but also from the most eminent leaders of science. Louis de Broglie states unequivocally, “…the variation of mass with velocity deduced by Einstein… is verified daily by observation of the motion of the high-speed particles of which nuclear physics currently makes such extensive use.”^{63} Planck was equally positive: “The theory of relativist mechanics was verified by experiment in the case of rapidly moving electrons, for this experiment showed that mass is not independent of velocity,”^{64} and Eddington tells us flatly, “…the mass depends on the velocity—a fact unknown in Newton’s day.”^{65}

Yet, oddly enough, while a host of scientific authorities of the highest rank are thus proclaiming that the postulated increase of mass with velocity has been proved by experiments with high-velocity electrons and verified by the successful use of the theory in the design and construction of the particle accelerators, almost every elementary physics textbook admits, explicitly or tacitly, that this hypothesis of an increase in mass is only an arbitrary selection from among several possible explanations of the observed facts. Richard Schlegel even manages to put both points of view into the same sentence. “Even before Einstein’s first paper on special relativity,” he tells us, “W. Kaufmann had observed an increase in the mass of electrons moving with very high velocities—or more precisely he had found a decrease in the e/m ratio of electrons, where e is the electron charge, a magnitude unaffected by relative velocity.”^{66} Still more precisely, we may say that Schlegel’s final comment about the effect of velocity on the magnitude of the charge is pure assumption. The truth is that the experiments with high velocity particles and the experience with the particle accelerators merely show that if a specific force is applied to a specific mass, the acceleration decreases at high velocities, following a pattern which indicates that it will reach zero at the velocity of light. II we are to maintain the relation *a* = *F/m * it then necessarily follows that either the mass increases or the force decreases, or both. Certainly the hypothesis of an increase in mass is * consistent *with the observed facts, but this is by no means the equivalent of the proof that is claimed. The door is wide open for ally alternative explanation which calls for a decrease in the effective force: either a decrease in the magnitude of the entity responsible for the force (an electric charge, in the usual case) or a reduction in the effective component of the force. The latter is the explanation that we obtain from the Reciprocal System.

In this system mass is absolute in magnitude, and it therefore remains constant irrespective of velocity. Here, however, force is *not *constant. Force, according to the principles of the Reciprocal System, is simply a special way of looking at motion. If we assume a velocity *v _{1}* acting in a certain direction and then superimpose an equal velocity

*v*acting in the opposite direction, the net velocity is

_{2}*v*= 0. In describing this situation we may take the stand that both velocities actually exist and that the null result is due to the fact that one cancels the other, or alternatively, we may say that there is a force F

_{1}-v_{2}_{1}tending to produce velocity

*v*and an oppositely directed force F

_{1}_{2}tending to produce velocity

*v*, but that, since the resultant of the two forces is zero, no motion takes place.

_{2}It is clear from the development in Part Three that the motions actually do exist and that the concept of force is simply an artificial way of looking at the situation. This does not mean that there is anything inherently wrong in the use of such a concept. If there is an element of convenience in utilizing an artificial contrivance of this kind, as there certainly is in this particular case, it is perfectly legitimate to take advantage of this more convenient mode of expression, providing that the concept is recognized for what it really is, and its limitations are taken into account. But if this true status is not recognized and the limitations are ignored, it is inevitable that this artificial contrivance will lead us astray sooner or later.

From the standpoint of the force concept itself, the idea of a *constant *force seems entirely logical, and up to now the existence of forces of constant magnitude has not been questioned. On the basis of the explanation of the nature of force given in the preceding paragraphs, however, there can be no such thing as a constant force. The space-time progression, for instance, tends to cause objects to acquire unit velocity and we therefore say that it exerts unit force. But a tendency to impart unit velocity to a mass, which is already at a high velocity, is not equivalent to a tendency to impart unit velocity to a body at rest. The *effective *force is a function of the difference between the velocities, and the full effect of any force is only attained when that force is exerted on a body at rest. As velocity increases the velocity difference decreases and hence the effective force also decreases. In the limiting condition, when tile mass already has unit velocity, the force (the tendency to cause unit velocity) has no effect at all and the effective force component is zero. The acceleration is then also zero, as the experimental results indicate.

The foregoing discussion shows that the Reciprocal System provides a consistent and logical explanation for each of the five points on which Relativity Theory rests its case. Since this is true, there is no scientific basis on which Relativity can claim any superiority. On the contrary, whatever advantage does exist favors the Reciprocal System, since this system does not have to utilize any principle of impotence such as the First Postulate of Relativity, nor does it contain any internal inconsistency such as giving mass both the status of something associated with energy and the status of something convertible to energy. Even when it meets the Relativity Theory on that theory’s own ground, therefore, the Reciprocal System makes a very favorable showing. But the facts brought out for this purpose represent only a minor portion of the evidence supporting the validity of the new system. Unlike the Relativity Theory, which can be checked against experiment and observation in only a few cases and which, as Einstein says, confronts us with an “ever widening gap” between the theoretical concepts and the facts of experience as the theory is extended into additional areas, the consequences of the Reciprocal System are clearly and sharply defined at all points, and they can be checked against experience in a multitude of applications, not only in the areas which Relativity Theory purports to cover, but also in many additional fields which have hitherto been considered totally unrelated to the gravitational phenomenon.

In presenting the case in favor of the broader contention that the gravitational theory derived from the Fundamental Postulates of the Reciprocal System is a correct representation of the physical facts, we will limit the discussion to gravitation, the specific subject under consideration in this present work, rather than dealing with the Reciprocal System as a whole. No attempt will be made, therefore, to present all of the immense volume of data confirming the validity of the system in general. Enough of these data were included in the previous publication *The Structure of the Physical Universe *to establish the solid factual status of the system, which may be described by the statement that the necessary consequences of the Fundamental Postulates of this system, without the aid of subsidiary or supplemental assumptions, and without the use of any contrived or artificial methods of evading contradictions, constitute a complete theoretical system of physical entities and relationships which is in agreement with observations and measurements in thousands of applications throughout the physical universe, and thus far has not been found inconsistent with the established facts in any instance. Just because of the validity of these postulates, and without the intervention of any other factor, radiation, matter, electrical and magnetic phenomena, and the other major features of the observed universe *must *exist in the theoretical universe, and the primary characteristics which these phenomena theoretically *must *have are identical with the characteristics of the corresponding observed phenomena.

In its general aspects, therefore, the Reciprocal System meets all of the requirements for proof of its validity; that is, it agrees with the observed and measured facts in a large number of individual cases throughout all of the general areas in which such facts are available, there are no known cases in which positively established facts are inconsistent with the theoretical conclusions, and no use has been made anywhere in the theoretical development of principles of impotence or other artificially contrived devices for evading such inconsistencies. It is true, of course, that the new system is in conflict with much of the currently accepted thought of the scientific profession, but in every case where such conflicts occur, it can be shown that the existing ideas, however firmly entrenched they may be, are not established facts; they are extrapolations of, interpretations of, or inferences drawn from these facts, or else they are pure assumptions not connected with the facts at all. If the issue is squarely faced, therefore, it must be conceded that the validity of the system in general has been established.

It does not follow, however, that every deduction, which may be made from the Fundamental Postulates of the Reciprocal System necessarily, participates in the proof of the validity of the system as a general proposition. As pointed out in Part One, once the validity of general principles of this kind has been established, it is possible to prove the validity of certain other conclusions by deductive methods; that is, by showing that these conclusions are necessary and unavoidable consequences of the principles already established, or of these principles taken together with certain known facts. The extent to which this kind of proof can be carried is somewhat limited; however, as one can rarely, if ever, be absolutely certain that a long line of reasoning is entirely free from error. Because of this situation the method of deductive proof is not ordinarily sufficient in itself; the purpose that it normally serves is reduce the number of correlation’s with established facts that are required in order to bring the possibility of a concealed error down to the vanishing point. A theoretical relation that is definitely in conflict with positively established facts is wrong regardless of its derivation, but w here there is no contradiction or inconsistency, a relation that is derived in a straightforward manner from principles whose validity has already been proved is obviously in a much better initial position than a relation that is purely hypothetical. What we have done here is to reduce the general question of the possible existence of some contradictory fact to the more limited question of the validity of the deductive process, and hence a smaller number of factual correlation’s is sufficient to reduce the probability of a hidden error to the neighborhood of zero.

The extent to which the deductive proof is effective in this respect naturally depends upon the length of the chain of reasoning involved in deducing the conclusion from the previously established principles. If the connection is immediate and direct, comparatively little factual corroboration should be required; the absence of any contradiction or inconsistency should be almost enough in itself under these circumstances. As the number and complexity of the steps in the process of deduction increases, the chance of a logical or mathematical error somewhere in the process likewise increases, and the need for more factual corroboration increases accordingly. In the limiting condition, where the deductive chain is extremely long and involved, the requirements for proof are essentially no different than in the case of a pure hypothesis.

On this basis, there is abundant proof of the validity of the gravitational theory presented herein. To begin with, this theory is an immediate and direct consequence of the Fundamental Postulates of the Reciprocal System, the validity of which, as has been stated, is confirmed by a great mass of evidence that meets all of the requirements of proof. In view of this status as a direct deduction from principles already established, a relatively small amount of factual corroboration should be adequate to complete the proof, and since everything that is actually known about gravitation is in full agreement with the theory, this should be sufficient for the purpose, even though it is true that the existing knowledge in this field is quite limited.

Furthermore, a very substantial amount of additional support is developed in fields, which have not hitherto been recognized as falling within the scope of gravitational theory. As the previous discussion has indicated, the new gravitational theory not only explains the origin of this phenomenon and the characteristics which it manifests in the generally recognized aspects of the gravitational action, but also goes on to provide explanations for other phenomena, such as the recession of the distant galaxies, the cohesion of solids, and the abnormal distances between the stars, which have heretofore been considered as totally unrelated to gravitation. The agreement between theory and established facts in these additional fields not only constitutes a major addition to the rather meager number of factual correlation’s which it is possible to obtain in the narrow field hitherto connected with gravitation, but also has another very significant aspect, in that such an extension of the field of application is a recognized indication of merit in a new theory of any kind. In this case the extension that has been achieved is extremely far-reaching and it adds substantial additional weight to the already strong case in favor of the new gravitational theory

The question as to just how far it is necessary to go before we can say that the remaining probability of the existence of a hidden error is essentially zero is a matter of opinion, but if the new gravitational theory does not actually qualify, it is certainly not far away from qualification. In any event, it is much closer to positive proof than most physical theories, and far superior in this respect to the current favorite, Einstein’s General Theory of Relativity, which has not achieved its present standing on its own merits, but because of the fact that nothing better has hitherto been available.

## XVI

In spite of the status of gravitation as the primary subject of this volume, the foregoing discussion of the Relativity Theory has been directed largely at the Special ’Theory since the General Theory, which actually deals with gravitation, is supposed to be an extension of the principles of the Special Theory to the wider field of nonuniform motion, and a full understanding of the true nature of the Special Theory gives us a better indication of the position of the General Theory than we can obtain by making a detailed analysis of the rather confused structure of the latter. It has been shown in the preceding pages that the Special Theory is simply a mathematical device which compensates for the error introduced into the relations of moving bodies by the failure to recognize the existence of coordinate time. The General Theory represents an attempt to extend this compensating mechanism into the field of nonuniform motion.

The Special Theory is mathematically correct even though it is expressed in terms of totally erroneous concepts, because its mathematical content is empirical and independent of the language in which it is described by the theory (in fact, this mathematical content antedates the theory itself). The validity of the empirical relations is dependent, however, on restricting their application to those cases where the error due to using clock time instead of total time is a definite function of the velocity. It is evident, therefore, that when the relations of the Special Theory are extended to rotational and other accelerated motion, where this error is normally *not *a specific function of the velocity, for reasons that have been detailed in the preceding discussion, the derived relations cannot be correct. Thus it is impossible for the General Theory to supply us with mathematical expressions which will serve the same purpose with respect to nonuniform motion that the equations of the special theory (the Lorentz transformations, etc.) do for uniform translational motion. No mathematical system, regardless of how complex and sophisticated it may be, can provide an accurate representation of the relations between quantities which, in truth, are not definitely related in any mathematical way. Any theory, which attempts to achieve this objective, must inevitably bog down in unworkable mathematical complexities and conceptual confusion, just as has actually happened in the case of the General Theory.

“In physics,” said Herbert Dangler twenty-five years ago, “the name of Relativity is notorious: if one claims to understand it he is looked at askance, and his subsequent statements are received with suspicion.”^{67} Another quarter of a century in which Relativity has had the field all to itself without serious competition has silenced most of the critics, but it has not lessened the real force of their criticism. The General Theory is just as full of inconsistencies and loose ends today as it was in that earlier era; it has made no appreciable advance in the interim. If we examine the two postulates, which Tolman tells us contain the essence of the General Theory, the reason for this sterility is evident. The objective of the theory, its originator asserts, is an extension of the findings of the Special Theory into the area of nonuniform motion, particularly accelerated motion. Now let us ask, just what does the Principle of Equivalence contribute toward this objective? The answer must be—nothing. This postulated principle merely asserts that gravitational mass and inertial mass are equivalent, and hence gravitation can be treated as the equivalent of an accelerated motion. The present work has no quarrel with this conclusion; on the contrary, it goes a step farther and says that gravitation *is *an accelerated motion. But this simply means that we have only one problem—accelerated motion; we do not have two problems—gravitation and accelerated motion—as had been thought previously. All that the equivalence postulate does is to establish this point; it makes no contribution whatever to the solution of the one problem which does exist.

When we turn to the second of the two postulates of General Relativity, the postulate of covariance, we encounter a very odd situation. It was pointed out originally by Kretschmann, emphasized by Bridgman and others, and admitted both by Tolman and by Einstein, that this postulate actually imposes no restriction on physical theory, yet we find some of the most far-reaching conclusions of General Relativity ostensibly based upon it. This strange situation is the subject of a very penetrating comment by Bridgman: “It must, I think, strike one on reflection as paradoxical to attempt to get information about nature from the requirement of covariance, for this is at bottom merely an attempt to get information about nature from an analysis of the language in terms of which we describe it, whereas the fundamental idea back of the argument as it is worked out in detail is that the sort of language with which we describe nature must be a matter of indifference.”^{48}

The two postulates, which are supposed to express the content of General Relativity thus, turn out to have no bearing on the primary problem of extending the application of Special Relativity to accelerated systems. “The astonishing thing about Einstein’s equations is that they appear to have come out of nothing,”^{68} says one observer. Even the status of the General Theory as an extension of the Special Theory is open to serious question. Peter G. Bergmann states categorically, “It is quite true that the general theory of relativity is not consistent with the special theory any more than the special theory is with Newton’s mechanics—each of these theories discards, in a sense, the conceptual framework of its predecessor.”^{69}

The question therefore arises, just how *does * General Relativity come to grips with this problem? Einstein himself supplies the answer to this question. He tells us that he had completed his analysis of the factors involved in gravitation and accelerated motion in general by 1908, and then goes on to say, “Why were another seven years required for the construction of the general theory of relativity? The main reason lies in the fact that it is not so easy to free oneself from the idea that co-ordinates must have an immediate metrical meaning.”^{70} He later defines this expression “a metrical meaning” as the existence of a specific relationship between differences of coordinates and measurable lengths and times.

Here we have the *real *essence of General Relativity. Special Relativity accomplished its objective of providing a mathematical correction for the conceptual error in the conventional view of time by abandoning the idea that the magnitudes of time and space intervals measured with respect to coordinate systems of reference have fixed values, and introducing a fictitious variability in these magnitudes. To meet the additional problem of accelerated motion, Einstein simply prescribed a bigger dose of the same medicine. It took him seven years to figure out where the additional flexibility could be introduced, but finally he created more latitude for numerical variation by depriving the coordinates themselves of any meaning so far as mensuration is concerned. As Moller sums up the new picture, “In accelerated systems of reference the spatial and temporal coordinates thus lose every physical significance; they simply represent a certain arbitrary, but unambiguous, numbering of the physical events.”^{71}

If the situation were one which *could *be reconciled with the conventional views of time by purely mathematical means, this strategy might have been successful (to the extent that constructing a theory that is mathematically right but conceptually wrong can be considered a success) just as Special Relativity is able to compensate for an error in its concept of the nature of time by introducing a counterbalancing error in its treatment of space. But since no general mathematical relationships of this kind exist once we get away from uniform translational velocity, General Relativity cannot expect to do anything in the field of motion beyond the little that has already been accomplished; that is, to provide complicated and rather vague solutions for certain very limited and mainly hypothetical problems.

Strangely enough, the reputation of the General Theory of Relativity, which is essentially, a theory of motion (gravitational and other) rests primarily on items which are only indirectly connected with motion. If this theory had to rely on its achievements in the field of motion alone (that is, on what it has accomplished in extending the relations of Special Relativity to accelerated motion) it would be in a very sorry state. But other, somewhat incidental, conclusions derived from or suggested by the Relativity Theory have had a spectacular success, and the success in these collateral areas has had the effect of sidetracking any critical scrutiny either of the extent to which the theory has accomplished its primary objective or the extent to which these widely publicized collateral derivatives are legitimate products of the Relativity Theory. “Though this treatment gets over some difficulties,” says Sir George Thomson, “it does so at the cost of considerable violence to commonsense. It may be doubted if it would have received the full acceptance that it in fact has but for the remarkable applications to mechanics… leading to predictions concerning the identity of mass and energy which have been brilliantly verified in nuclear physics.”^{72}

When the originator of a new theory tells us that his theory leads to the (at that time) astonishing conclusion that mass and energy are equivalent and interconvertible, and subsequently the conversion of mass to energy is demonstrated in an awe-inspiring manner; when he also tells us that the mass of a body in motion increases with the velocity, becoming infinite at the velocity of light, and that this mass increase will decrease the acceleration of high speed particles subjected to constant forces, and subsequently it is found that particles traveling at high velocities behave in exactly the manner predicted, this practically closes the door to any attempt at a critical analysis of the theoretical background. Few investigators are willing to attack such a strongly entrenched position, and little attention is paid to those who do have the temerity to make the attempt.

As a result, no one seems to have given any consideration to the fact that, while each of these two conclusions alleged to have been derived from the Relativity Theory makes an impressive showing by itself, they are mutually contradictory, and if either one is valid, the other is necessarily wrong. At least *one * of these impressive successes of the theory is fictitious. Mass cannot be an *accompaniment *of energy, as demanded by the aspect of the theory that explains the operation of the particle accelerators, and also something that can be *converted into *energy, as demanded by the aspect of the theory that explains the atomic bomb. These two concepts are incompatible and it is obvious that a theory, which claims to have derived both results from the same basic source, is in error somewhere. A thorough examination of the whole development is therefore very much in order.

Unfortunately, such an examination encounters major obstacles. One of the principal items of this kind, a factor that has played an important part in preventing the emergence of any full-scale attempts at a critical analysis of the alleged achievements of the Relativity Theory is the extreme difficulty of getting at the real essence of the theory. The situation that faces anyone who attempts to find out where the conclusions of the theory actually come from has already been mentioned. The mathematical basis of the theory is equally elusive. In the words of H. Bondi, “The equations describing general relativity are, in all but the simplest applications, exceedingly complex and difficult to unravel.”^{3} When it is extremely difficult to determine just what is *in *a theory, it is an almost hopeless task to arrive at a critical judgment as to the legitimacy of conclusions, which the originator and his supporters claim to have obtained *out of *the theory.

However, now that the Reciprocal System has provided us with a complete and consistent theoretical structure that is in agreement with the observed facts at all points, it is possible to examine the General Theory in the light of this new information and to get a more intelligible picture of the status of the points at issue, as has been done in the preceding pages. The conclusions of the foregoing analysis may be summarized as follows:

*As to the general objective.*The real, even though unrecognized, purpose of the theory (as interpreted in the context of the new information now available) is to provide a mathematical means of correcting for the error introduced into calculations involving nonuniform motion by the failure to recognize the existence of coordinate time. However, the magnitude of the required correction, unlike that for uniform translatory motion, is not a specific function of the velocity, hence the primary objective of the General Theory is an impossible goal.*As to the postulates of the theory.*The Principle of Equivalence is fully in accord with the newly developed information; indeed, the Reciprocal System goes a step farther and asserts that gravitation is an accelerated motion, not merely the equivalent of an accelerated motion. The Principle of Covariance is also accepted by the new system, although the significance of this principle is minimized. Most of the conclusions purporting to be derived from it have actually been introduced into the General Theory*ad hoc.**As to the correlation’s with observation.*The equivalence of mass and energy (more properly the interconvertibility of mass and energy) deduced from the Relativity principles is verified by the Reciprocal System, but the hypothetical increase in mass accompanying an increase in velocity is inconsistent with the interconvertibility and is erroneous. The observed decrease in acceleration at high velocities is due to a decrease in the effective component of the presumably “constant” force, instead of an increase in mass. The advance of the perihelion of Mercury is a consequence of the same factors that are responsible for the negative result of the Michelson-Morley experiment and it is therefore related to the Special Theory, even though it involves nonuniform motion, rather than to the General Theory.*General conclusion.*Unlike the Special Theory, which is mathematically correct, even though conceptually wrong, the General Theory must be considered erroneous in all of its basic aspects. The tangible achievements that it can claim, such as the prediction of the interconvertibility of mass and energy, have only a very tenuous connection with the theory itself, and rest primarily on*ad hoc*assumptions*suggested by*the theory. The widespread acceptance of the General Theory is based primarily on the achievements of the Special Theory, which is definitely superior to Newton’s Laws of Motion in application to bodies moving at high velocities. The argument here is that if the Relativity principles are correct in application to uniform translatory motion, as the mathematical results seem to indicate, then they are probably also correct, and therefore superior to Newton’s system, in application to nonuniform motion. In view of the admitted lack of consistency between the General Theory and the Special Theory this reasoning is clearly invalid, but in any event the issue being examined in this work is not between the Relativity Theory and Newton’s system, but between the Relativity Theory and the newly developed Reciprocal System, and here the Relativity Theory is badly outperformed. The Reciprocal System is not only superior on an item by item basis in every instance where there is any significant difference between the too, but it also makes out a very good case when matched against the requirements for positive proof of its validity: something that the Relativity Theory cannot even approach.

The many glaring deficiencies and weaknesses that show up in the structure of the Relativity Theory as soon as it is subjected to a critical examination and judged on its own merits rather than merely on the basis of a comparison with Newton’s system, indicate very clearly wily Bridgman was troubled by “the whole state of mind of most physicists with regard to it.” We do not necessarily have to go along with his opinion that this will “some day become one of the puzzles of history,” however, as it is actually quite evident that this is simply another manifestation of the psychological trait which makes the scientist (in common with his fellow human beings in other pursuits) unwilling to admit ignorance and leads him to treat today’s best guess as the equivalent of an established fact, however weak and vulnerable that guess may be. Relativity has been, in reality, merely a makeshift: something to which the physicist could cling temporarily rather than drifting in a sea of uncertainty. It has survived only because of the lack of any serious competition, coupled with a general feeling that something is better than nothing: that even a poor theory is better than none at all.