# CHAPTER 6

# The Reciprocal Relation

Inasmuch as the fundamental postulates define a universe composed entirely of units of motion, and define space and time in terms of that motion, these postulates preclude space and time from having any significance other than that which they have in motion, and at the same time require that they *always* have that significance; that is, throughout the universe space and time are reciprocally related.

This general reciprocal relation that necessarily exists in a universe composed entirely of motion has a far-reaching and decisive effect on physical structures and processes. In recognition of its crucial role, the name “Reciprocal” has been applied to the system of theory based on the “motion” concept of the nature of the universe. The reason for calling it a “system of theory” rather than merely a “theory” is that its *subdivisions* are coextensive with other physical *theories*. One of these subdivisions covers the same ground as relativity, another parallels the nuclear theory of the atom, still another deals with the same physical area as the kinetic theory, and so on. It is appropriate, therefore, to call these subdivisions “theories,” and to refer to the entire new theoretical structure as the Reciprocal System of theory, even though it is actually a single fully integrated entity.

The reciprocal postulate provides a good example of the manner in which a change in the basic concept of the nature of the universe alters the way in which we apprehend specific physical phenomena. In the context of a universe of matter existing in a space-time framework, the idea of space as the reciprocal of time is simply preposterous, too absurd to be given serious consideration. Most of those who encounter the idea of “the reciprocal of space” for the first time find it wholly inconceivable. But these persons are not taking the postulates of the new theory at their face value, and recognizing that the assertion that “space is an aspect of motion” means exactly what it says. They are accustomed to regarding space as some kind of a container, and they are interpreting this assertion as if it said that “container space is an aspect of motion,” thus inserting *their own* concept of space into a statement which rejects all such previous ideas and defines a *new and different* concept. The result of mixing such incongruous and conflicting concepts cannot be otherwise than meaningless.

When the new ideas are viewed in the proper context, the strangeness disappears. In a universe in which everything that exists is a form of motion, and the magnitude of that motion, measured as speed or velocity, is the only significant physical quantity, the existence of the reciprocal relation is practically self-evident. Motion is defined as the relation of space to time. Its mathematical expression is the quotient of the two quantities. An increase in space therefore has exactly the same effect on the speed, the mathematical measure of the motion, as a decrease in time, and vice versa. In comparing one airplane with another, it makes no difference whether we say that plane A travels twice as far in the same time, or that it travels a certain distance in half the time.

Inasmuch as the postulates deal with space and time in precisely the same manner, aside from the reciprocal relation between the two, the behavior characteristics of the two entities, or *properties,* as they are called, are identical. This statement may seem incredible on first sight, as space and time manifest themselves to our observation in very different guises. We know time only as a progression, a continual moving forward, whereas space appears to us as an entity that “stays put.” But when we subject the apparent differences to a critical examination, they fail to stand up under the scrutiny.

The most conspicuous property of space is that it is three-dimensional. On the other hand, it is generally believed that the observational evidence shows time to be one-dimensional. We have a subjective impression of a unidirectional “flow” of time from the past, to the present, and on into the future. The mathematical representation of time in the equations of motion seems to confirm this view, inasmuch as the quantity *t* in v = *s/t* and related equations is scalar, not vectorial, as v and s are, or can be.

Notwithstanding its general and unquestioning acceptance, this conclusion as to the one-dimensionality of time is entirely unjustified. The point that is being overlooked is that “direction,” in the context of the physical processes which are represented by vectorial equations in present-day physics, always means “direction in space.” In the equation *v = s/t,* for example, the spatial displacement s is a vector quantity because it has a direction in *space*. It follows that the velocity v also has a direction in space, and thus what we have here is a *space velocity equation*. In this equation the term *t is* necessarily scalar *because it has no direction in space.*

It is quite true that this result would automatically follow if time were one-dimensional, but the one-dimensionality is by no means a necessary condition. Quite the contrary, time is scalar in this space velocity equation (and in all of the other familiar vectorial equations of modern physics; equations that are vectorial because they involve direction in space) *irrespective of its dimensions,* because no matter how many dimensions it may have, time has no direction in *space*. If time is multi-dimensional, as our theoretical development finds it to be, then it has a property that corresponds to the spatial property that we call “direction.” But whatever we call this temporal property, whether we call it “direction in time,” as we are doing for reasons previously explained, or give it some altogether different name, it is a temporal property, not a spatial property, and it does not give time any direction in space. Regardless of its dimensions, time cannot be a vector quantity in any equation such as those of present-day physics, in which the property, which qualifies a quantity as vectorial, is that of having a direction in space.

The existing confusion in this area is no doubt due, at least in part, to the fact that the terms “dimension” and “dimensional” are currently used with two different meanings. We speak of space as three-dimensional, and we also speak of a cube as three-dimensional. In the first of these expressions we mean that space has a certain property that we designate as dimensionality, and that the magnitude applying to this property is three. In other words, our statement means that there are three dimensions *of* space. But when we say that a cube is three-dimensional, the significance of the statement is quite different. Here we do not mean that there are three dimensions of “cubism,” or whatever we may call it. We mean that the cube exists in space and extends into three dimensions of that space.

There is a rather general tendency to interpret any postulate of multi-dimensional time in this latter significance; that is, to take it as meaning that *time* extends into n dimensions of *space,* or some kind of a quasi-space. But this is a concept that makes little sense under any conditions, and it certainly is not the meaning of the term “three-dimensional time” as used in this work. When we here speak of time as three-dimensional we will be employing the term in the same significance as when we speak of space as three-dimensional; that is, we mean that time has a property, which we call dimensionality, and the magnitude of that property is three. Here, again, we mean that there are three dimensions *of* the property in question: three dimensions *of time.*

There is nothing in the role which time plays in the equations of motion in space to indicate specifically that it has more than one dimension. But a careful consideration along the lines indicated in the foregoing paragraphs does show that the present-day assumption that we *know* time to be one-dimensional is completely unfounded. Thus there is no empirical evidence that is inconsistent with the assertion of the Reciprocal System that time is three-dimensional.

Perhaps it might be well to point out that the additional dimensions of time have no metaphysical significance. The postulates of a universe of motion define a purely physical universe, and all of the entities and phenomena of that universe, as determined by a development of the necessary consequences of the postulates, are purely physical. The three dimensions of time have the same physical significance as the three dimensions of space.

As soon as we take into account the effect of gravitation on the motion of material aggregates, the second of the observed differences, the *progression* of time, which contrasts sharply with the apparent immobility of extension space, is likewise seen to be a consequence of the conditions of observation, rather than an indication of any actual dissimilarity. The behavior of those objects that are partially free from the gravitational attraction of our galaxy, the very distant galaxies, shows conclusively that the immobility of extension space, as we observe it, is not a reflection of an inherent property of space in general, but is a result of the fact that in the region accessible to detailed observation gravitation moves objects toward each other, offsetting the effects of the outward progression. The pattern of the recession of the distant galaxies demonstrates that when the gravitational effect is eliminated there is a progression of space similar to the observed progression of time. Just as “now” continually moves forward relative to any initial point in the temporal reference system, so “here” in the absence of gravitation, continually moves forward relative to any initial point in the spatial reference system.

Little additional information about either space or time is available from empirical sources. The only items on which there is general agreement are that space is homogeneous and isotropic, and that time progresses uniformly. Other properties that are sometimes attributed to either time or space are merely assumptions or hypotheses. Infinite extent or infinite divisibility, for instance, are hypothetical, not the results of observation. Likewise, the assertions as to spatial and temporal properties that are made in the relativity theories are, as Einstein says, “free inventions of the human mind,” not items that have been derived from experience.

In testing the validity of the conclusion that all properties of *either* space or time are properties of *both* space and time, such assumptions and hypotheses must be disregarded, since it is only conflicts with definitely established facts that are conclusive. The significance of a conflict with a questionable assertion cannot be other than questionable. “Homogeneous” with respect to space is equivalent to “uniform” with respect to time, and because the observations thus far available tell us nothing at all about the dimensions of time, there is nothing in these observations that is inconsistent with the assertion that time, like space, is isotropic. In spite of the general belief, among scientists and laymen alike, that there is a great difference between space and time, any critical examination along the foregoing lines shows that the apparent differences are not real, and that there is actually no observational evidence that is inconsistent with the theoretical conclusion that the properties of space and of time are identical.

As brought out in Chapter 4, deviations from unit speed, the basic one-to-one space-time ratio, are accomplished by means of reversals of the direction of the progression of either space or time. As a result of these reversals, one component traverses the same path in the reference system repeatedly, while the other component continues progressing unidirectionally in the normal manner. Thus the deviation from the normal rate of progression may take place *either* in space or in time, but not in *both* coincidentally. The space-time ratio, or speed, is either 1/n (less than unity, the speed of light,) or n/1 (greater than unity). Inasmuch as everything physical in a universe of motion is a motion—that is, a relation between space and time, measured as speed—and, as we have just seen, the properties of space and those of time are identical, aside from the reciprocal relationship, it follows that every physical entity or phenomenon has a reciprocal. There exists another entity or phenomenon that is an exact duplicate, except that space and time are interchanged.

For example, let us consider an object rotating with speed 1/n and moving translationally with speed 1/n. The reciprocal relation tells us that there must necessarily exist, somewhere in the universe, an object identical in all respects, except that its rotational and translational speeds are both n/1 instead of 1/n. In addition to the complete inversions, there are also structures of an intermediate type in which one or more components of a complex combination of motions are inverted, while the remaining components are unchanged. In the example under consideration, the translational speed may become n/1 while the rotational speed remains at 1/n, or vice versa. Once the normal (1/n) combination has been identified, it follows that both the completely inverted (n/1) combination and the various intermediate structures exist in the appropriate environment. The general nature of that environment in each case is also indicated, inasmuch as change of position in time cannot be represented in—a spatial reference system, and each of these speed combinations has some special characteristics when viewed in relation to the conventional reference systems. The various physical entities and phenomena that involve motion of these several inverse types will be examined at appropriate points in the pages that follow. The essential point that needs to be recognized at this time, because of its relevance to the subject matter now under consideration, is the *existence* of inverse forms of all of the normal (1/n) motions and combinations of motions.

This is a far-reaching discovery of great significance. In fact the new and more accurate picture of the physical universe that is derived from the “motion” concept differs from previous ideas mainly by reason of the widening of our horizons that results from recognition of the inverse phenomena. Our direct physical contacts are limited to phenomena of the same type as those that enter into our own physical makeup: the direct phenomena, we may call them, although the distinction between direct and inverse is merely a matter of the way in which we see them, not anything that is inherent in the phenomena themselves. In recent years the development of powerful and sophisticated instruments has enabled us to penetrate areas that are far beyond the range of our unaided senses, and in these new areas the relatively simple and understandable relations that govern events within our normal experience are no longer valid. Newton’s laws of motion, which are so dependable in everyday life, break down in application to motion at speeds approaching that of light; events at the atomic level resist all attempts at explanation by means of established physical principles, and so on.

The scientific reaction to this state of affairs has been to conclude that the relatively simple and straightforward physical laws that have been found to apply to events within our ordinary experience are not universally valid, but are merely approximations to some more complex relations of general applicability. The simplicity of Newton’s laws of motion, for instance, is explained on the ground that some of the terms of the more complicated general law are reduced to negligible values at low velocities, and may therefore be disregarded in application to the phenomena of everyday life. Development of the consequences of the postulates of the Reciprocal System arrives at a totally different answer. We find that the inverse phenomena that necessarily exist in a universe of motion play no significant role in the events of our everyday experience, but as we extend our observations into the realms of the very large, the very small, and the very fast, we move into the range in which these inverse phenomena replace or modify those which we, from our particular position in the universe, regard as the direct phenomena.

On this basis, the difficulties that have been experienced in attempting to use the established physical laws and relations of the world of ordinary experience in the far-out regions are very simply explained. These laws and relations apply specifically to the world of immediate sense perception, phenomena of the direct space-time orientation, and they fail in application to any situation in which the events under consideration involve phenomena of the inverse type in any significant degree. They do not fail because they are wrong, or because they are incomplete; they fail because they are misapplied. No law—physical or otherwise—can be expected to produce the correct results in an area to which it has no relevance. The inverse phenomena are governed by laws distinct from, although related to, those of the direct phenomena, and where those phenomena exist they can be understood and successfully handled only by using the laws and relations of the inverse sector.

This explains the ability of the Reciprocal System of theory to deal successfully with the recently discovered phenomena of the far-out regions, which have been so resistant to previous theoretical treatment. It is now apparent that the unfamiliar and surprising aspects of these phenomena are not due to aspects of the normal physical relations that come into play only under extreme conditions, as previous theorists have assumed; they are due to the total or partial replacement of the phenomena of the direct type by the related, but different phenomena of the inverse type. In order to obtain the correct answers to problems in these remote areas, the unfamiliar phenomena that are involved must be viewed in their true light as the inverse of the phenomena of the directly observable region, not in the customary way as extensions of those direct phenomena into the regions under consideration. By identifying and utilizing this correct treatment the Reciprocal System is not only able to arrive at the right answers in the far-out areas, but to accomplish this task without disturbing the previously established laws and principles that apply to the phenomena of the direct type.

In order to keep the explanation of the basic elements of the theory as simple and understandable as possible, the previous discussion has been limited to what we have called the direct view of the universe, in which space is the more familiar of the two basic entities, and plays the leading role. At this time it is necessary to recognize that because of the *general* nature of the reciprocal relation between space and time every statement that has been made with respect to space in the preceding chapters is equally applicable to time in the appropriate context. As we have seen in the case of space and time individually, however, the way in which the inverse phenomenon manifests itself to our observation may be quite different from the way in which we see its direct counterpart.

Locations in time cannot be represented in a spatial reference system, but, with the same limitations that apply to the representation of spatial locations, they can be represented in a stationary three-dimensional temporal reference system analogous to the three-dimensional spatial reference system that we call extension space. Since neither space nor time exists independently, every physical entity (a motion or a combination of motions) occupies both a space location and a time location. The location as a whole, the location in the physical universe, we may say, can therefore be completely defined only in terms of two reference systems.

In the context of a stationary spatial reference system the motion of an absolute location, a location in the natural reference system, as indicated by observation of an object without independent motion, such as a photon or a galaxy at the observational limit, is linearly outward. Similarly, the motion of an absolute location with respect to a stationary temporal reference system is linearly outward in time. Inasmuch as the gravitational motion of ordinary matter is effective in space only, the atoms and particles of this matter, which are stationary with respect to the spatial reference system, or moving only at low velocities, remain in the same absolute locations in time indefinitely, unless subjected to some external force. Their motion in three-dimensional time is therefore linearly outward at unit speed, and the time location that we observe, the time registered on a clock, is not the location in any temporal reference system, but simply the *stage of progression*. Since the progression of the natural reference system proceeds at unit speed, always and everywhere, clock time, if properly measured, is the same everywhere. As we will see later in the development, the current hypotheses which require repudiation of the existence of absolute time and the concept of simultaneity of distant events are erroneous products of reasoning from premises in which clock time is incorrectly identified as time in general.

The best way to get a clear picture of the relation of clock time to time in general is to consider the analogous situation in space. Let us assume that a photon A is emitted from some material object X in the direction Y. This photon then travels at unit speed in a straight line XY which can be represented in the conventional fixed spatial reference system. The line of progression of time has the same relation to time in general (three-dimensional time) as the line XY has to space in general (three-dimensional space). It is a one-dimensional line of travel in a three-dimensional continuum; not something separate and distinct from that continuum, but a specific part of it.

Now let us further assume that we have a device whereby we can measure the rate of increase of the spatial distance XA, and let us call this device a “space clock” , Inasmuch as all photons travel at the same speed, this one space clock will suffice for the measurement of the distance traversed by *any* photon, irrespective of its location or direction of movement, as long as we are interested only in the scalar magnitude. But this measurement is valid only for objects such as photons, which travel at unit speed. If we introduce an object, which travels at some speed other than unity, the measurement that we get from the space clock will not correctly represent the space traversed by that object. Nor will the space clock registration be valid for the *relative* separation of moving objects, even if they are traveling at unit speed. In order to arrive at the true amount of space entering into such motions we must either measure that space individually, or we must apply an appropriate correction to the measurement by the space clock.

Because objects at rest in the stationary spatial reference system, or moving at low velocities with respect to it, are moving at unit speed relative to any stationary *temporal* reference system, a clock which measures the time progression in any one process provides an accurate measurement of the time elapsed in *any* low speed physical process, just as the space clock in our analogy measured the space traversed by *any* photon. Here, again, however, if an object moves at a speed, or a relative speed, differing from unity, so that its movement in time is not the same as that of the progression of the natural reference system, then the clock time does not correctly represent the actual time involved in the motion under consideration. As in the analogy, the true quantity, the net total time, must be obtained either by a separate measurement (which is usually impractical) or by determining the magnitude of the adjustment that must be applied to the clock time to convert it to total time.

In application to motion in space, the total time, like the clock registration, is a scalar quantity. Some readers of the previous edition have found it difficult to accept the idea that time can be three-dimensional because this makes time a vector quantity, and presumably leads to situations in which we are called upon to divide one vector quantity by another. But such situations are non-existent. If we are dealing with spatial relations, time is scalar because it has no spatial direction. If we are dealing with temporal relations, space is scalar because it has no temporal direction. *Either* space or time can be vectorial in appropriate circumstances. However, as explained earlier in this chapter, the deviation from the normal scalar progression at unit speed may take place either in space or in time, but not in both coincidentally. Consequently, there is no physical situation in which both space and time are vectorial.

Similarly, scalar rotation and its gravitational (translational) effect take place *either* in space or in time, but not in both. If the speed of the rotation is less than unity, time continues progressing at the normal unit rate, but because of the directional changes during rotation space progresses only one unit while time is progressing n units. Thus the change in position relative to the natural unit datum, both in the rotation and in its gravitational effect, takes place in space. Conversely, if the speed of the rotation is greater than unity, the rotation and its gravitational effect take place in time.

An important result of the fact that rotation at greater-than-unit speeds produces an inward motion (gravitation) in time is that a rotational motion or combination of motions with a net speed greater than unity cannot exist in a spatial reference system for more than one (dimensionally variable) unit of time. As pointed out in Chapter 3, the spatial systems of reference, to which the human race is limited because it is subject to gravitation in space, are not capable of representing deviations from the normal rate of time progression. In certain special situations, to be considered later, in which the normal direction of vectorial motion is reversed, the change of position in time manifests itself as a distortion of the spatial position. Otherwise, an object moving normally with a speed greater than unity is coincident with the reference system for only one unit of time. During the next unit, while the spatial reference system is moving outward in time at the unit rate of the normal progression, gravitation is carrying the rotating unit inward in time. It therefore moves away from the reference system and disappears. This point will be very significant in our consideration of the high speed rotational systems in Chapter 15.

Recognition of the fact that each effective unit of rotational motion (mass) occupies a location in time as well as a location in space now enables us to determine the effect of mass concentration on the gravitational motion. Because of the continuation of the progression of time while gravitation is moving the atoms of matter inward in space, the aggregates of matter that are eventually formed in space consist of a large number of mass units that are contiguous in space, but widely dispersed in time. One of the results of this situation is that the magnitude of the gravitational motion (or force) is a function not only of the distance between objects, but also of the effective number of units of rotational motion, measured as mass, that each object possesses. This motion is distributed over all space-time directions, rather than merely over all space directions, and since an aggregate of n mass units occupies n time locations, the total number of space-time locations is also n, even though all mass units of each object are nearly coincident spatially. The total gravitational motion of any mass unit toward that aggregate is thus n times that toward a single mass unit at the same distance. It then follows that the gravitational motion (or force) is proportional to the product of the (apparently) interacting masses.

It can now be seen that the comments in Chapter 5, with respect to the apparent change of direction of the gravitational motions (or forces) when the apparently interacting masses change their relative positions are applicable to multi-unit aggregates as well as to the individual mass units considered in the original discussion. The gravitational motion always takes place toward all space-time locations whether or not those locations are occupied by objects that enable us to detect the motion.

A point that should be noted in this connection is that two objects are in effective contact if they occupy adjoining locations in *either* space or time, regardless of the extent of their separation in the other aspect of motion. This statement may seem to conflict with the empirical observation that contact can be made only if the two objects are in the same place at the same clock time. However, the inability to make contact when the objects reach a common spatial location in a fixed reference system at different clock times is not due to the lack of coincidence in time, but to the progression of space that takes place in connection with the progression of time which is registered by the clock. Because of this space progression, the location that has the same spatial coordinates in the stationary reference system is not the same spatial location that it was at an earlier time.

Scientific history shows that physical problems of long standing are usually the result of errors in the prevailing basic concepts, and that significant conceptual modifications are a prerequisite for their solution. We will find, as we proceed with the theoretical development, that the reciprocal relation between space and time which necessarily exists in a universe of motion is just the kind of a conceptual alteration that is needed to clear up the existing physical situation: one which makes drastic changes where such changes are required, but leaves the empirically determined relations of our everyday experience essentially untouched.