The basic postulate of the Reciprocal System of theory asserts the existence of motion. In itself, without qualification, this would permit the existence of any conceivable kind of motion, but the additional assumptions included in the postulates act as limitations on the types of motion that are possible. The net result of the basic postulates plus the limitations is therefore to assert the existence of any kind of motion that is not excluded by the limiting assumptions. We may express this point concisely by saying that in the theoretical universe of motion anything that can exist does exist. The further fact that these permissible theoretical phenomena coincide item by item with the observed phenomena of the actual physical universe is something that will have to be demonstrated step by step as the development proceeds.
Some objections have been raised to the foregoing conclusion that what can exist does exist, on the ground that actuality does not necessarily follow from possibility. But no one is contending that actual existence is a necessary consequence of possible existence, as a general proposition. What is contended is that this is true, for special reasons, in the physical universe. Philosophers explain this as being the result of a “principle of nature.” David Hawkins, for instance, tells us that “the principle of plenitude…says that all things possible in nature are actualized.”35 What the present development has done is to explain why nature follows such a principle. Our finding is that the basic physical entities are scalar motions, and that the existence of different observable entities and phenomena is due to the fact that these motions necessarily assume specific directions when they appear in the context of a three-dimensional frame of reference. Inasmuch as the directions are determined by chance, there is a finite probability corresponding to every possible direction, and thus every possibility becomes an actuality. It should be noted that this is exactly the same principle that was applied in Chapter 3 to explain why an expanding sphere of radiation emanates from each radiation source (a conclusion that is not challenged by anyone). In this case, too, scalar motions exist, each of which takes one of certain permissible directions (limited by the translational character of the motions), and these motions are distributed over all of the directions.
Inasmuch as it has been postulated that motion, as defined earlier, is the sole constituent of the physical universe, we are committed to the proposition that every physical entity or phenomenon is a manifestation of motion. The determination of what entities, phenomena, and processes can exist in the theoretical universe therefore reduces to a matter of ascertaining what kinds of motion and combinations of motions can exist in such a universe, and what changes can take place in these motions. Similarly, in relating the theoretical universe to the observed physical universe, the question as to what any observed entity or phenomenon is never arises. We always know what it is. It is a motion, a combination of motions, or a relation between motions. The only question that is ever at issue is what kinds of motions are involved.
There has been a sharp difference of opinion among those interested in the philosophical aspects of science as to whether the process of enlarging scientific understanding is “discovery” or “invention.” This is related to the question as to the origin of the fundamental principles of science that was discussed in Chapter 1, but it is a broader issue that applies to all scientific knowledge, and involves the inherent nature of that knowledge. The specific point at issue is clearly stated by R. B. Lindsay in these words:
Application of the term “discovery” implies that there is an external world “out there” wholly independent of the observer and with built-in regularities and laws waiting to be uncovered and revealed. They have always been there and presumably always will be; our task is by diligent search to find out what they are. On the other hand, the term “invention” implies that the physicist uses not only his observations but his imaginative powers to construct points of view that identify with experience.36
The “discovery” concept, says Lindsay, implies that the acquisition of scientific knowledge is cumulative, and that ultimately our understanding of the physical world should be essentially complete. On the contrary, the “point of view of invention means that the process of creating new experience and the construction of new ideas to cope with this experience go hand in hand.” On this basis, “the whole activity is open-ended” ; there is no place for the idea of completeness.
The Reciprocal System now provides a definitive answer to this question. It not only establishes scientific investigation as a process of discovery, but reduces that discovery to deduction and verification of the deductions. All of the information that is necessary in order to arrive at a full description of any theoretically possible entity or phenomenon is implicit in the postulates. A full development of the consequences of the postulates therefore defines a complete theoretical universe.
As will be seen in the pages that follow, the physical processes of the universe include a continuing series of interchanges between vectorial motions and scalar motions. In all of these interchanges causality is maintained; no motions of either type occur except as a result of previously existing motions. The concept of events occurring without cause, which enters into some of the interpretations of the theories included in the current structure of physics, is therefore foreign to the Reciprocal System. But the universe of motion is not deterministic in the strict Laplacian sense, because the directions of the motions are continually being redetermined by chance processes. The description of the physical universe that emerges from development of the consequences of the postulates of the Reciprocal System therefore identifies the general classes of entities and phenomena that exist in the universe, and the relations between them, rather than specifying the exact result of every interaction, as a similarly complete theory would do if it were deterministic.
In beginning our examination of these physical entities and phenomena, the first point to be noted is that the postulates require the existence of real units of motion, units that are similar to the units of motion involved in the progression of the natural reference system, except that they actually exist, rather than being fictitious results of relating motion to an arbitrary reference system. These independent units of motion, as we will call them, are superimposed on the moving reference background in much the same manner as that in which matter is supposed to exist in the basic space of previous physical theory. Since they are units of the same kind, however, these independent units are interrelated with the units of the background motion, rather than being separate and distinct from it, in the manner in which matter is presumed to be distinct from the space-time background in the theories based on the “matter” concept. As we will see shortly, some of the independent motions have components that are coincident with the background motion, and these components are not effective from the physical standpoint; that is, their effective physical magnitude is zero.
A point of considerable significance is that while the postulates permit the existence of these independent motions, and, on the basis of the principle previously stated, they must therefore exist in the universe of motion defined by the postulates, those postulates do not provide any mechanism for originating independent motions. It follows that the independent motions now existing either originated coincidentally with the universe itself, or else were originated subsequently by some outside agency. Likewise, the postulates provide no mechanism for terminating the existence of these independent motions. Consequently, the number of effective units of such motion now existing can neither be increased nor diminished by any process within the physical system.
This inability to alter the existing number of effective units of independent motion is the basis for what we may call the general conservation law, and the various subsidiary conservation laws applying to specific physical phenomena. It suggests, but does not necessarily require, a limitation of the independent units of motion to a finite number. The issue as to the finiteness of the universe does not enter into any of the phenomena that will be examined in this present volume, but it will come up in connection with some of the subjects that will be taken up later, and it will be given further consideration then.
The Reciprocal System of theory deals only with the physical universe as it now exists, and reaches no conclusions as to how that universe came into being, nor as to its ultimate fate. The theoretical system is therefore completely neutral on the question of creation. It is compatible with either the hypothesis of creation by some agency, or the hypothesis that the universe has always existed. Continuous creation of matter by action of the physical mechanism itself, as postulated by the Steady State theory of cosmology is ruled out, and there is nothing in that mechanism that will allow the universe to arrive at any kind of termination of its own accord. The question of creation or termination by action of an outside agency is beyond the scope of the theoretical development.
Turning now to the question as to what kinds of motion are possible at the basic level, we note that scalar magnitudes may be either positive (outward, as represented in a spatial reference system), or negative (inward). But as we observe motion in the context of a fixed reference system, the outward progression of the natural reference system is always present, and thus every motion includes a one-unit outward component. The discrete unit postulate prevents any addition to an effective unit, and independent outward motion is therefore impossible. All dependent motion must have net inward or negative magnitude. Furthermore, it must be continuous and uniform at this stage of the development, because no mechanism is yet available whereby discontinuity or variability can be produced.
Since the outward progression always exists, independent continuous negative motion is not possible by itself, but it can exist in combination with the ever-present outward progression. The result of such a combination of unit negative and unit positive motion is zero motion relative to a stationary coordinate system. Another possibility is simple harmonic motion, in which the scalar direction of movement reverses at each end of a unit of space, or time. In such motion, each unit of space is associated with a unit of time, as in unidirectional translational motion, but in the context of a stationary three-dimensional spatial reference system the motion oscillates back and forth over a single unit of space (or time) for a certain period of time (or space).
At first glance, it might appear that the reversals of scalar direction at each end of the basic unit are inadmissible in view of the absence of any mechanism for accomplishing a reversal. However, the changes of scalar direction in simple harmonic motion are actually continuous and uniform, as can be seen from the fact that such motion is a projection of circular motion on a diameter. The net effective speed varies continuously and uniformly from +1 at the midpoint of the forward movement to zero at the positive end of the path of motion, and then to -1 at the midpoint of the reverse movement and zero at the negative end of the path. The continuity and uniformity requirements are met both by a continuous, uniform change of direction, and by a continuous, uniform change of magnitude.
As pointed out earlier, the theoretical structure that we are developing from the fundamental postulates is a description of what can exist in the theoretical universe of motion defined by those postulates. The question as to whether a certain feature of this theoretical universe does or does not correspond to anything in the actual physical universe is a separate issue that is explored in a subsequent step in the project, to be started shortly, in which the theoretical universe is compared item by item with the observed universe. At the moment, therefore, we are not concerned with whether or not simple harmonic motion does exist in the actual physical universe, why it exists, if it does, or how it manifests itself. All that we need to know for present purposes is that inasmuch as this kind of motion is continuous, and is not excluded by the postulates, it is one of the kinds of motion that exists in the theoretical universe of motion, under the most basic conditions.
Under these conditions simple harmonic motion is confined to individual units. When the motion has progressed for one full unit, the discrete unit postulate specifies that a boundary exists. There is no discontinuity, but at this boundary one unit terminates and another begins. Whatever processes may have been under way in the first unit cannot carry over into the next. They cannot be divided between two totally independent units. Consequently, a continuous change from positive to negative can occur only within one unit of either space or time.
As explained in Chapter 3, motion, as herein defined, is a continuous process—a progression—not a succession of jumps. There is progression even within the units, simply because these are units of progression, or scalar motion. For this reason, specific points within the unit—the midpoint, for example—can be identified, even though they do not exist independently. The same is true of the chain used as an analogy in the preceding discussion. Although the chain exists only in discrete units, or links, we can distinguish various portions of a link. For instance, if we utilize the chain as a means of measurement, we can measure 10½ links, even though half of a link would not qualify as part of the chain. Because of this capability of identifying the different portions of the unit, we see the vibrating unit as following a definite path.
In defining this path we will need to give some detailed consideration to the matter of direction. In the first edition the word “direction” was used in four different senses. Exception was taken to this practice by a number of readers, who suggested that it would be helpful if “direction” were employed with only one significance, and different names were attached to the other three concepts. When considered purely from a technical standpoint, the earlier terminology is not open to legitimate criticism, as using words in more than one sense is unavoidable in the English language. However, anything that can be done to facilitate understanding of the presentation is worth serious consideration. Unfortunately, there is no suitable substitute for “direction” in most of these applications.
Some of the objections to the previous terminology were based on the ground that scalar quantities, by definition, have no direction, and that using the term “direction” in application to these quantities, as well as to vectorial quantities, is contradictory and leads to confusion. There is merit in this objection, to be sure, in any application where we deal with scalar quantities merely as positive and negative magnitudes. But as soon as we view the scalar motions in the context of a fixed spatial reference system, and begin talking about “outward” and “inward,” as we must do in this work, we are referring not to the scalar magnitudes themselves, but to the representation of these magnitudes in a stationary spatial reference system, a representation that is necessarily directional. The use of directional language in this case therefore appears to be unavoidable.
There are likewise some compelling reasons for continuing to use the term “direction in time” in application to the temporal property analogous to the spatial property of direction. We could, of course, coin a new word for this purpose, and there would no doubt be certain advantages in so doing. But there are also some very definite advantages to be gained by utilizing the word “direction” with reference to time as well as with reference to space. Because of the symmetry of space and time, the property of time that corresponds to the familiar property of space that we call “direction” has exactly the same characteristics as the spatial property, and by using the term “direction in time,” or “temporal direction,” as a name for this property we convey an immediate understanding of its nature and characteristics that would otherwise require a great deal of discussion and explanation. All that is then necessary is to keep in mind that although direction in time is like direction in space, it is not direction in space.
Actually, it should not be difficult to get away from the habit of always interpreting “direction” as meaning “direction in space” when we are dealing with motion. We already recognize that there is no spatial connotation attached to the term when it is used elsewhere. We habitually use “direction” and directional terms of one kind or another in speaking of scalar quantities, or even in connection with items that cannot be expressed in physical imagery at all. We speak of wages and prices as moving in the same direction, temperature as going up or down, a change in the direction of our thinking, and so on. Here we realize that we are using the word “direction” without any spatial significance. There should be no serious obstacle in the way of a similar conception of the meaning of “direction in time.”
In this edition the term “direction” will not be used in referring to the deviations upward or downward from unit speed. In the other senses in which the term was originally used it seems essential to continue utilizing directional language, but as an alternative to the further limitations on the use of the term “direction” that have been suggested we will use qualifying adjectives wherever the meaning of the term is not obvious from the context.
On this basis vectorial direction is a specific direction that can be fully represented in a stationary coordinate system. Scalar direction is simply outward or inward, the spatial representation of positive or negative scalar magnitudes respectively. Wherever the term “direction” is used without qualification it will refer to vectorial direction. If there is any question as to whether the direction (scalar or vectorial) under consideration is a direction in space or a direction in time, this information will also be given.
Vectorial motion is motion with an inherent vectorial direction. Scalar motion is inward or outward motion that has no inherent vectorial direction, but is given a direction by the factors involved in its relation to the reference system. This imputed vectorial direction is independent of the scalar direction except to the extent that the same factors may, in some instances, affect both. As an analogy, we may consider a motor car. The motion of this car has a direction in three-dimensional space, a vectorial direction, while at the same time it has a scalar direction, in that it is moving either forward or backward. As a general proposition, the vectorial direction of this vehicle is independent of its scalar direction. The car can run forward in any vectorial direction, or backward in any direction.
If the car is symmetrically constructed so that the front and back are indistinguishable, we cannot tell from direct observation whether it is moving forward or backward. The same is true in the case of the simple scalar motions. For example, we will find in the pages that follow that the scalar direction of a falling object is inward, whereas the scalar direction of a beam of light is outward. If the two happen to traverse the same path in the same vectorial direction, as they may very well do, there is nothing observable that will distinguish between the inward and outward motion. In the usual situation the scalar direction of an observed motion must be determined from collateral information independently of the observed vectorial direction.
The magnitude of a simple harmonic motion, like that of any other motion, is a speed, the relation between the number of units of space and the number of units of time participating in the motion. The basic relation, one unit of space per unit of time, always remains the same, but by reason of the directional reversals, which result in traversing the same unit of one component repeatedly, the speed of a simple harmonic motion, as it appears in a fixed reference system, is 1/x (or x/1). This means that each advance of one unit in space (or time) is followed by a series of reversals of scalar direction that increase the number of units of time (or space) to x, before another advance in space (or time) takes place. At this point the scalar direction remains constant for one unit, after which another series of reversals takes place.
Ordinarily the vectorial direction reverses in unison with the scalar direction, but each end of a unit is the reference point for the position of the next unit in the reference system, and conformity with the scalar reversals is therefore not mandatory. Consequently, in order to maintain continuity in the relation of the vectorial motion to the fixed reference system the vectorial direction continues the regular reversals at the points where the scalar motion advances to a new unit of space (or time). The relation between the scalar and vectorial directions is illustrated in the following tabulation, which represents two sections of a 1/3 simple harmonic motion. The vectorial directions are expressed in terms of the way the movement would appear from some point not in the line of motion.
The simple harmonic motion thus remains permanently in a fixed position in the dimension of motion, as seen in the context of a stationary reference system; that is, it is an oscillatory, or vibratory, motion. An alternative to this pattern of reversals will be discussed in Chapter 8.
Like all other absolute locations, the absolute location occupied by the vibrating unit, the unit of simple harmonic motion, is carried outward by the progression of the natural reference system, and since the linear motion of the vibrating unit has no component in the dimensions perpendicular to the line of oscillation, the outward progression at unit speed takes place in one of these free dimensions. Inasmuch as this outward progression is continuous within the unit as well as from one unit of the reference frame to the next, the combination of a vibratory motion and a linear motion perpendicular to the line of vibration results in a path which has the form of a sine curve.
Because of the dimensional relationship between the oscillation and the linear progression, there is a corresponding relation between the vectorial directions of these two components of the total motion, as seen in the context of a stationary reference system, but this relation is fixed only between these two components. The position of the plane of vibration in the stationary spatial system of reference is determined by chance, or by the characteristics of the originating object.
Although the basic one-to-one space-time ratio is maintained in the simple harmonic motion, and the only change that takes place is from positive to negative and vice versa, the net effect, from the standpoint of a fixed system of reference is to confine one component—either space or time—to a single unit, while the other component extends to n units. The motion can thus be measured in terms of the number of oscillations per unit of time, a frequency, although it is apparent from the foregoing explanation that it is actually a speed. The conventional measurement in terms of frequency is possible only because the magnitude of the space (or time) term remains constant at unity.
Here, in this oscillating unit, the first manifestation of independent motion (that is, motion that is separate and distinct from the outward motion of the natural reference system) that has emerged from the theoretical development, is the first physical object. In the motion of this object we also have the first instance of “something moving.” Up to this time we have been considering only the basic motions, relations between space and time that do not involve movement of any “thing.” Experience in presenting the theory to college audiences has indicated that many persons are unable to conceive of the existence of motion without something moving, and are inclined to argue that this is impossible. It should be realized, however, that we are definitely committed to this concept just as soon as we postulate a universe composed entirely of motion. In such a universe, “things” are combinations of motions, and motion is thus logically prior to “things.”
The concept of a universe of motion is generally conceded to be reasonable and rational. The long list of prominent and not-so-prominent scientists and philosophers who have essayed to explore the implications of such a concept is sufficient confirmation of this point. It follows that unless some definite and positive conflict with reason or experience is encountered, the necessary consequences of that concept must also be presumed to be reasonable and rational, even though some of them may conflict with long-standing beliefs of some kind.
There is no mathematical obstacle to this unfamiliar type of motion. We have defined motion, for purposes of a theory of a universe of motion, by means of the relation expressed in the equation of motion: v = s/t. This equation does not require the existence of any moving object. Even where the motion actually is motion of something, that “something,” does not enter into any of the terms of the equation, the mathematical representation of the motion. The only purpose that it serves is to identify the particular motion under consideration. But identification is also possible where there is nothing moving. If, for example, we say that the motion we are talking about is the motion of atom A, we are identifying a particular motion, and distinguishing it from all other motions, but if we refer to the motion which constitutes atom A, we are identifying this motion (or combination of motions) on an equally definite basis, even though this is not motion of anything.
A careful consideration of the points brought out in the foregoing discussion will make it clear that the objections that have been raised to the concept of motion without anything moving are not based on logical grounds. They stem from the fact that the idea of simple motion of this kind, merely a relation between space and time, is new and unfamiliar. None of us likes to discard familiar ideas of long standing and replace them with something new and different, but this is part of the price that we pay for progress.
This will be an appropriate time to emphasize that combinations or other modifications of existing motions can only be accomplished by adding or removing units of motion. As stated in Chapter 2, neither space nor time exists independently. Each exists only in association with the other as motion. Consequently, a speed 1/a cannot be changed to a speed 1/b by adding b-a units of time. Such a change can only be accomplished by superimposing a new motion on the motion that is to be altered.
In carrying out the two different operations that were involved in the investigation from which the results reported herein were derived, it would have been possible to perform them separately; first developing the theoretical structure as far as circumstances would permit, and then comparing this structure with the observed features of the physical universe. In practice, however, it was more convenient to identify the various theoretical features with the corresponding physical features as the work progressed, so that the correlations would serve as a running check on the accuracy of the theoretical conclusions. Furthermore, this policy eliminated the need for the separate system of terminology that otherwise would have been required for referring to the various features of the theoretical universe during the process of the theoretical development.
Much the same considerations apply to the presentation of the results, and we will therefore identify each theoretical feature as it emerges from the development, and will refer to it by the name that is customarily applied to the corresponding physical feature. It should be emphasized, however, that this hand in hand method of presentation is purely an aid to understanding. It does not alter the fact that the theoretical universe is being developed entirely by deduction from the postulates. No empirical information is being introduced into the theoretical structure at any point. All of the theoretical features are purely theoretical, with no empirical content whatever. The agreement between theory and observation that we will find as we go along is not a result of basing the theoretical conclusions on appropriate empirical premises; it comes about because the theoretical system is a true and accurate representation of the actual physical situation.
Identification of the theoretical unit of simple harmonic motion that we have been considering presents no problem. It is obvious that each of these units is a photon. The process of emission and movement of the photons is radiation. The space-time ratio of the vibration is the frequency of the radiation, and the unit speed of the outward progression is the speed of radiation, more familiarly known as the speed of light.
When considered merely as vibrating units, there is no distinction between one photon and another except in the speed of vibration, or frequency. The unit level, where speed 1/n changes to n/l cannot be identified in any directly observable way. We will find, however, that there is a significant difference between the manner in which the photons of vibrational speed 1/n enter into combinations of motions and the corresponding behavior of photons of vibrational speed greater than unity. This difference will be examined in detail in the chapters that follow.
One of the things that we can expect a correct theory of the structure of the universe to do is to clear up the discrepancies and “paradoxes” of previously existing scientific thought. Here, in the explanation of the nature of radiation that emerges from the development of theory, we find this expectation dramatically fulfilled. In conventional thinking the concepts of “wave” and “particle” are mutually exclusive, and the empirical discovery that radiation acts in some respects as a wave phenomenon, and in other respects as an assembly of particles has confronted physical science with a very disturbing paradox. Almost at the outset of our development of the consequences of the postulates that define a universe of motion we find that in such a universe there is a very simple explanation. The photon acts as a particle in emission and absorption because it has the distinctive feature of a particle: it is a discrete unit. In transmission it behaves as a wave because the combination of its own inherent vibratory motion with the translatory motion of the progression of the natural reference system causes it to follow a wavelike path. In this case the problem that seemed impossible to solve while radiation was looked upon as a single entity loses all of its difficult features as soon as it is recognized as a combination of two different things.
Another difficult problem with respect to radiation has been to explain how it can be propagated through space without some kind of a medium. This problem has never been solved other than by what was described by R. H. Dicke as a “semantic trick” ; that is, assuming, entirely ad hoc, that space has the properties of a medium.
One suspects that, with empty space having so many properties, all that had been accomplished in destroying the ether was a semantic trick. The ether had been renamed the vacuum.5
Einstein did not challenge this conclusion expressed by Dicke. On the contrary “he freely admitted” not only that his theory still employed a medium, but also that this medium is indistinguishable, other than semantically, from the “ether” of previous theories. The following statements from his works are typical:
We may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether.37
We shall say: our space has the physical property of transmitting waves, and so omit the use of a word (ether) we have decided to avoid.38
Thus the relativity theory does not resolve the problem. There is no evidence to support Einstein’s assumption that space has the properties of a medium, or that it has any physical properties at all. The fact that no method of propagating radiation through space without a medium has ever been conceived is therefore still unreconciled with the absence of any evidence of the existence of a medium. In the theoretical universe of the Reciprocal System the problem does not arise, since the photon remains in the same absolute location in which it originates, as any object without independent motion must do. With respect to the natural reference system it does not move at all, and the movement that is observed in the context of a stationary reference system is a movement of the natural reference system relative to the stationary system, not a movement of the photon itself.
In both the propagation question and the wave-particle issue the resolution of the problem is accomplished in the same manner. Instead of explaining why the seemingly complicated phenomena are complex and perplexing, the Reciprocal System of theory removes the complexity and reduces the phenomena to simple terms. As other long-standing problems are examined in the course of the subsequent development we will find that this conceptual simplicity is a general characteristic of the new theoretical structure.