08 Motion in Time

CHAPTER 8

Motion in Time

The starting point for an examination of the nature of motion in time is a recognition of the status of unit speed as the natural datum, the zero level of physical activity. We are able to deal with speeds measured from some arbitrary zero in our everyday life because these are not primary quantities; they are merely speed differences. For example, where the speed limit is 50 miles per hour, this does not mean that an automobile is prohibited from moving at any faster rate. It merely means that the difference between the speed of the vehicle and the speed of the portion of the earth’s surface over which the vehicle is traveling must not exceed 50 miles per hour. The car and the earth’s surface are jointly moving at higher speeds in several different directions, but these are of no concern to us for ordinary purposes. We deal only with the differences, and the datum from which measurement is made has no special significance.

In current practice we regard a greater rate of change in vehicle location relative to the local frame of reference as being the result of a greater speed, that quantity being measured from zero. We could equally well measure from some arbitrary non-zero level, as we do in the common systems of temperature measurement, or we could even measure the inverse of speed from some selected datum level, and attribute the greater rate of change of position to less “inverse speed,” In dealing with the basic phenomena of the universe, however, we are dealing with absolute speeds, not merely speed differences, and for this purpose it is necessary to recognize that the datum level of the natural system of reference is unity, not zero.

Since motion exists only in units, according to the postulates that define a universe of motion, and each unit of motion consists of one unit of space in association with one unit of time, all motion takes place at unit speed, from the standpoint of the individual units. This speed may, however, be either positive or negative, and by a sequence of reversals of the progression of either time or space, while the other component continues progressing unidirectionally, an effective scalar speed of 1/n, or n/1, is produced. In Chapter 4 we considered the case in which the vectorial direction of the motion reversed at each end of a one-unit path, the result being a vibrational motion. Alternatively, the vectorial direction may reverse in unison with the scalar direction. In this case space (or time) progresses one unit in the context of a fixed reference system while time (or space) progresses n units. Here the result is a translatory motion at a speed of 1/n (or n/1) units.

The scalar situation is the same in both cases. A regular pattern of reversals results in a space-time ratio of 1/n or n/1. In the example shown in the tabulation in Chapter 4, where the space-time ratio is 1/3, there is a one-unit inward motion followed by an outward unit and a second inward unit. The net inward motion in the three-unit sequence is one unit. A continuous succession of similar 3-unit sequences then follows. As indicated in the accompanying tabulation, the scalar direction

DIRECTION
Number
Unit
Vibratory Translation
Scalar Vectorial Scalar Vectorial
1 inward right inward forward
2 outward left outward backward
3 inward right inward forward
4 inward left inward forward
5 outward right outward backward
6 inward left inward forward

of the last unit of each sequence is inward. (A sequence involving an even number of n alternates n - 1 and n + 1. For instance, instead of two four-unit sequences, in which the last unit of each sequence would be outward, there is a three-unit sequence and a five-unit sequence.) The scalar direction of the first unit of each new sequence is also inward. Thus there is no reversal of scalar direction at the point where the new sequence begins. In the vibrational situation the vectorial direction continues the regular succession of reversals even at the points where the scalar direction does not reverse, but in the translational situation the reversals of vectorial direction conform to those of the scalar direction. Consequently, the path of vibration remains in a fixed location in the dimension of the oscillation, whereas the path of translation moves forward at the scalar space-time ratio 1/n (or n/1). This is the pattern followed by certain scalar motions that will be discussed later and by all vectorial motions: motions of material units and aggregates.

When the progression within a unit of motion reaches the end of the unit it either reverses or does not reverse. There is no intermediate possibility. It follows that what appears to be a continuous unidirectional motion at speed 1/n is, in fact, an intermittent motion in which space progresses at the normal rate of one unit of space per unit of time for a fraction 1/n of the total number of space units involved, and has a net resultant of zero, in the context of the fixed reference system, during the remainder of the motion.

If the speed is 1/n—one unit of space per n units of time—space progresses only one unit instead of the n units it would progress unidirectionally. The result of motion at the 1/n speed is therefore to cause a change of spatial position relative to the location that would have been reached at the normal rate of progression. Motion at less than unit speed, then, is motion in space. This is a well-known fact. But because of the uncritical acceptance of Einstein’s dictum that speeds in excess of that of light are impossible, and a failure to recognize the reciprocal relation between space and time, it has not heretofore been realized that the inverse of this kind of motion is also a physical reality. Where the speed is n/1, there is a reversal of the time component that results in a change of position in time relative to that which would take place at the normal rate of time progression, the elapsed time registered on a clock. Motion at speeds greater than unity is therefore motion in time.

The existence of motion in time is one of the most significant consequences of the status of the physical universe as a universe of motion. Conventional physical science, which recognizes only motion in space, has been able to deal reasonably well with those phenomena that involve spatial motion only. But it has not been able to clarify the physical fundamentals, a task for which an understanding of the role of time is essential, and it is encountering a growing number of problems as observation and experiment are extended into the areas where motion in time is an important factor. Furthermore, the number and scope of these problems has been greatly increased by the use of zero speed, rather than unit speed, as the reference datum for measurement purposes. While motion at speeds of 1/n (speeds less than unity) is motion in space only, when viewed relative to the natural (moving) reference system, it is motion in both space and time relative to the conventional systems that utilize the zero datum.

It should be understood that the motions we are now discussing are independent motions (physical phenomena), not the fictitious motion introduced by the use of a stationary reference system. The term “progression,” is here being utilized merely to emphasize the continuing nature of these motions, and their space and time aspects. During the one unit of motion (progression) at the normal unit speed that occurs periodically when the average speed is 1/n, the spatial component of this motion, which is an inherent property of the motion independent of the progression of the natural reference system, is accompanied by a similar progression of time that is likewise independent of the progression of the reference system, the time aspect of which is measured by a clock. Thus, during every unit of clock time, the independent motion at speed 1/n involves a change of position in three-dimensional time amounting to 1/n units.

As brought out in the preliminary discussion of this subject in Chapter 6, the value of n at the speeds of our ordinary experience is so large that the quantity 1/n is negligible, and the clock time can be taken as equivalent to the total time involved in motion. At higher speeds, however, the value of 1/n becomes significant, and the total time involved in motion at these high speeds includes this additional component. It is this heretofore unrecognized time component that is responsible for the discrepancies that present-day science tries to handle by means of fudge factors.

In the two-photon case considered in Chapter 7, the value of 1/n is 1/1 for both photons. A unit of the motion of photon X involves one unit of space and one unit of time. The time involved in this unit of motion (the time OX) can be measured by means of the registration on a clock, which is merely the temporal equivalent of a yardstick. The same clock can also be used to measure the time magnitude involved in the motion of photon Y (the time OY), but this use of the same temporal “yardstick” does not mean that the time interval OY through which Y moves is the same interval through which X moves, the interval OX, any more than using the same yardstick to measure the space traversed by Y would make it the same space that is traversed by X. The truth is that at the end of one unit of the time involved in the progression of the natural reference system (also measured by a clock), X and Y are separated by two units of total time (the time OX and the time OY), as well as by two units of space (distance). The relative speed is the increase in spatial separation, two units, divided by the increase in temporal separation, two units, or 2/2 = 1.

If an object with a lower speed v is substituted for one of the photons, so that the separation in space at the end of one unit of clock time is 1 + v instead of 2, the separation in time is also 1 + v and the relative speed is (1+ v)/(1 + v) = 1. Any process that measures the true speed rather than the space traversed during a given interval of standard clock time (the time of the progression of the natural reference system) thus arrives at unity for the speed of light irrespective of the system of reference.

When the correct time magnitudes are introduced into the equations of motion there is no longer any need for fudge factors. The measured coordinate differences and the measured constant speed of light are then fully compatible, and there is no need to deprive the spatial coordinates of their “metrical meaning.” Unfortunately, however, no means of measuring total time, except in certain special applications, are available at present. Perhaps some feasible method of measurement may be developed in the future, but in the meantime it will be necessary to continue on the present basis of applying a correction to the clock registration, in those areas where this is feasible. Under these circumstances we can consider that we are using correction factors instead of fudge factors. There is no longer an unexplained discrepancy that needs to be fudged out of existence. What we now find is that our calculations involve a time component that we are unable to measure. In lieu of the measurements that we are unable to make, we find it possible, in certain special cases, to apply correction factors that compensate for the difference between clock time and total time.

A full explanation of the derivation of these correction factors, the Lorentz equations, is available in the scientific literature, and will not be repeated here. This conforms with a general policy that will be followed throughout this work. As explained in Chapter 1, most existing physical theories have been constructed by building up from empirical foundations. The Reciprocal System of theory is constructed in the opposite manner. While the empirically based theories start with the observed details and work toward the general principles, the Reciprocal System starts with a set of general postulates and works toward the details. At some point each of the branches of the theoretical development will meet the corresponding element of empirical theory. Where this occurs in the course of the present work, and there is agreement, as there is in the case of the Lorentz equations, the task of this presentation is complete. No purpose would be served by duplicating material that is already available in full detail.

Most of the other well-established relationships of physical science are similarly incorporated into the new system of theory, with or without minor modifications, as the development of the theoretical structure proceeds, not because of the weight of observational evidence supporting these relations, or because anyone happens to approve of them, or because they have previously been accepted by the scientific world, but because the conclusions expressed by these relations are the same conclusions that are reached by development of the new theoretical system. After such a relation has thus been taken into the system, it is, of course, part of the system, and can be used in the same manner as any other part of the theoretical structure.

The existence of speeds greater than unity (the speed of light), the speeds that result in change of position in time, conflicts with current scientific opinion, which accepts Einstein’s conclusion that the speed of light is an absolute limit that cannot be exceeded. Our development shows, however, that at one point where Einstein had to make an arbitrary choice between alternatives, he made the wrong choice, and the speed limitation was introduced through this error. It does not exist in fact.

Like the special theory of relativity, the theory from which the speed limitation is derived is an attempt to provide an explanation for an empirical observation. According to Newton’s second law of motion, which can be expressed as a= F/m, if a constant force is applied to the acceleration of a constant mass it should produce an acceleration that is also constant. But a series of experiments showed that where a presumably constant electrical force is applied to a light particle, such as an electron, in such a manner that very high speeds are produced, the acceleration does not remain constant, but decreases at a rate which indicates that it would reach zero at the speed of light. The true relation, according to the experimental results, is not Newton’s law, a = F/m, but a = -\/1—(v /c ) F/m. In the system of notation used in this work, which utilizes natural, rather than arbitrary, units of measurement, the speed of light, designated as c in current practice, is unity, and the variable speed (or velocity), v, is expressed in terms of this natural unit. On this basis the empirically derived equation becomes a = F/m.

There is nothing in the data derived from experiment to tell us the meaning of the term 1 - v2 in this expression; whether the force decreases at higher speeds, or the mass increases, or whether the velocity term represents the effect of some factor not related to either force or mass. Einstein apparently considered only the first two of these alternatives. While it is difficult to reconstruct the pattern of his thinking, it appears that he assumed that the effective force would decrease only if the electric charges that produced the force decreased in magnitude. Since all electric charges are alike, so far as we know, Whereas the primary mass concentrations seem to be extremely variable, he chose the mass alternative as being the most likely, and assumed for purposes of his theory that the mass increases with the velocity at the rate indicated by the experiments. On this basis, the mass becomes infinite at the speed of light.

The results obtained from development of the consequences of the postulates of the Reciprocal System now show that Einstein guessed wrong. The new information developed theoretically (which will be discussed in detail later) reveals that an electric charge is inherently incapable of producing a speed in excess of unity, and that the decrease in the acceleration at high speeds is actually due to a decrease in the force exerted by the charges, not to any change in the magnitude of either the mass or the charge.

As explained earlier, force is merely a concept by which we visualize the resultant of oppositely directed motions as a conflict of tendencies to cause motion rather than as a conflict of the motions themselves. This method of approach facilitates mathematical treatment of the subject, and is unquestionably a convenience, but whenever a physical situation is represented by some derived concept of this kind there is always a hazard that the correspondence may not be complete, and that the conclusions reached through the medium of the derived concept may therefore be in error. This is what has happened in the case we are now considering.

If the assumption that a force applied to the acceleration of a mass remains constant in the absence of any external influences is viewed only from the standpoint of the force concept, it appears entirely logical. It seems quite reasonable that a tendency to cause motion would remain constant unless subjected to some kind of a modifier. But when we look at the situation in its true light as a combination of motions, rather than through the medium of an artificial representation by means of the force concept, it is immediately apparent that there is no such thing as a constant force. Any force must decrease as the speed of the motion from which it originates is approached. The progression of the natural reference system, for instance, is motion at unit speed. It therefore exerts unit force. If the force—that is, the effect—of the progression is applied to overcoming a resistance to motion (the inertia of a mass) it will ultimately bring the mass up to the speed of the progression itself: unit speed. But a tendency to impart unit speed to an object that is already moving at high speed is not equivalent to a tendency to impart unit speed to a body at rest. In the limiting condition, where an object is already moving at unit speed, the force due to the progression of the reference system has no effect at all, and its magnitude is zero.

Thus, the full effect of any force is attained only when the force is exerted on a body at rest, and the effective component in application to an object in motion is a function of the difference between the speed of that object and the speed that manifests itself as a force. The specific form of the mathematical function, ~ rather than merely 1-v, is related to some of the properties of compound motions that will be discussed later. Ordinary terrestrial speeds are so low that the corresponding reduction in the effective force is negligible, and at these speeds forces can be considered constant. As the speed of the moving object increases, the effective force decreases, approaching a limit of zero when the object is moving at the speed corresponding to the applied force—unity in the case of the progression of the natural reference system. As we will find in a later stage of the development, an electric charge is inherently a motion at unit speed, like the gravitational motion and the progression of the natural reference system, and it, too, exerts zero force on an object moving at unit speed.

As an analogy, we may consider the case of a container full of water, which is started spinning rapidly. The movement of the container walls exerts a force tending to give the liquid a rotational motion, and under the influence of this force the water gradually acquires a rotational speed. But as that speed approaches the speed of the container the effect of the “constant” force drops off, and the container speed constitutes a limit beyond which the water speed cannot be raised by this means. The force vanishes, we may say. But the fact that we cannot accelerate the liquid any farther by this means does not bar us from giving it a higher speed in some other way. The limitation is on the capability of the process, not on the speed at which the water can rotate.

The mathematics of the equation of motion applicable to the acceleration phenomenon remain the same in the Reciprocal System as in Einstein’s theory. It makes no difference mathematically whether the mass is increased by a given amount, or the effective force is decreased by the same amount. The effect on the observed quantity, the acceleration, is identical. The wealth of experimental evidence that demonstrates the validity of these mathematics therefore confirms the results derived from the Reciprocal System to exactly the same degree that it confirms Einstein’s theory. All that this evidence does in either case is to show that the theory is mathematically correct.

But mathematical validity is only one of the requirements that a theory must meet in order to be a correct representation of the physical facts. It must also be conceptually valid; that is, the meaning attached to the mathematical terms and relations must be correct. One of the significant aspects of Einstein’s theory of acceleration at high speeds is that it explains nothing; it merely makes assertions. Einstein gives us an ex cathedra pronouncement to the effect that the velocity terms represents an increase in the mass, without any attempt at an explanation as to why the mass increases with the velocity, why this hypothetical mass increment does not alter the structure of the moving atom or particle, as any other mass increment does, why the velocity term has this particular mathematical form, or why there should be a speed limitation of any kind.

Of course, this lack of a conceptual background is a general characteristic of the basic theories of present-day physics, the “free inventions of the human mind,” as Einstein described them, and the theory of mass increase is not unusual in this respect. But the arbitrary character of the theory contrasts sharply with the full explanation provided by the Reciprocal System. This new system of theory produces simple and logical answers for all questions, similar to those enumerated above, that arise in connection with the explanation that it supplies. Furthermore, one of these is, in any respect, ad hoc. All are derived entirely by education from the assumptions as to the nature of space and time that constitute the basic premises of the new theoretical system.

Both the Reciprocal System and Einstein’s theory recognize that there; a limit of some kind at unit speed. Einstein says that this is a limit on the magnitude of speed, because on the basis of his theory the mass reaches infinity at unit speed, and it is impossible to accelerate an infinite mass. The Reciprocal System, on the other hand, says that the limit is on the capability of the process. A speed in excess of unity cannot be produced by electromagnetic means. This does not preclude acceleration to higher speeds by other processes, such as the sudden release fo large quantities of energy in explosive events, and according to this new theoretical viewpoint there is no definite limit to speed magnitudes. In deed, the general reciprocal relation between space and time requires that speeds in excess of unity be just as plentiful, and cover just as wide a range, in the universe as a whole, as speeds less than unity. The apparent predominance of low-speed phenomena is merely a result of observing the universe from a location far over on the low-speed side of the neutral axis.

One of the reasons why Einstein’s assertion as to the existence of limiting speed was so readily accepted is an alleged absence of any observational evidence of speeds in excess of that of light. Our new theoretical development indicates, however, that there is actually no lack of evidence. The difficulty is that the scientific community currently holds a mistaken belief as to the nature of the change of position that is produced by such a motion. We observe that a motion at a speed less than that of light causes a change of location in space, the rate of change varying with the speed (or velocity, if the motion is other then linear). It is currently taken for granted that a speed in excess of that of light would result in a still greater rate of change of spatial location, and the absence of any clearly authenticated evidence of such higher rates of location change is interpreted as proof of the existence of a speed limitation. But in a universe of motion an increment of speed above unity (the speed of light) does not cause a change of location in space. In such a universe there is complete symmetry between space and time, and since unit speed is the neutral level, the excess speed above unity causes a change of location in three-dimensional time rather than in three-dimensional space.

From this it can be seen that the search for “tachyons” , hypothetical particles that move with a spatial velocity greater than unity, will continue o be fruitless. Speeds above unity cannot be detected by measurements if the rate of change of coordinate positions in space. We can detect them only by means of a direct speed measurement, or by some collateral effects. There are many observable effects of the required nature, but their status as evidence of speeds greater than that of light is denied by present-day physicists on the ground that it conflicts with Einstein’s assumption of an increase in mass at high speeds. In other words, the observations are required to conform to the theory, rather than requiring the theory to meet the standard test of science: conformity with observation and measurement.

The current treatment of the abnormal redshifts of the quasars is a glaring example of this unscientific distortion of the observations to fit the theory. We have adequate grounds to conclude that these are Doppler shifts, and are due to the speeds at which these objects are receding from the earth. Until very recently there was no problem in this connection. There was general agreement as to the nature of the redshifts, and as to the existence of a linear relation between the redshift and the speed. This happy state of affairs was ended when quasars were found with redshifts exceeding 1.00. On the basis of the previously accepted theory, a 1.00 redshift indicates a recession speed equal to the speed of light. The newly discovered redshifts in the range above 1.00 therefore constitute a direct measurement of quasar motions at speeds greater than that of light.

But the present-day scientific community is unwilling to challenge Einstein, even on the basis of direct evidence, so the mathematics of the special theory of relativity have been invoked as a means of saving the speed limitation. No consideration seems to have been given to the fact that the situation to which the mathematical relations of special relativity apply does not exist in the case of the Doppler shift. As brought out in Chapter 7, and as Einstein has explained very clearly in his works, the Lorentz equations, which express those mathematics, are designed to reconcile the results of direct measurements of speed, as in the Michelson-Morley experiment, with the measured changes of coordinate position in a spatial reference system. As everyone, including Einstein, has recognized, it is the direct speed measurement that arrives at the correct numerical magnitude. (Indeed, Einstein postulated the validity of the speed measurement as a basic principle of nature.) Like the result of the Michelson-Morley experiment, the Doppler shift is a direct measurement, simply a counting operation, and it is not in any way connected with a measurement of spatial coordinates. Thus there is no excuse for applying the relativity mathematics to the redshift measurements.

Inasmuch as the “time dilatation” aspect of the Lorentz equations is being applied to some other phenomena that do not seem to have any connection with spatial coordinates, it may be desirable to anticipate the subsequent development of theory to the extent of stating that the discussion in Chapter 15 will show that those “dilatation” phenomena that appear to involve time only, such as the extended lifetime of fast-moving unstable particles, are, in fact, consequences of the variation of the relation between coordinate spatial location (location in the fixed reference system) and absolute spatial location (location in the natural moving system) with the speed of the objects occupying these locations. The Doppler effect, on the other hand, is independent of the spatial reference system.

The way in which motion in time manifests itself to observation depends on the nature of the phenomenon in which it is observed. Large redshifts are confined to high speed astronomical objects, and a detailed examination of the effect of motion in time on the Doppler shift will be deferred to Volume II, where it will be relevant to the explanation of the quasars. At this time we will take a look at another of the observable effects of motion in time that is not currently recognized as such by the scientific community: its effect in distorting the scale of the spatial reference system.

It was emphasized in Chapter 3 that the conventional spatial reference systems are not capable of representing more than one variable—space—and that because there are two basic variables—space and time—in the physical universe we are able to use the spatial reference systems only on the basis of an assumption that the rate of change of time remains constant. We further saw, earlier in this present chapter that at all speeds of unity or less time does, in fact, progress at a constant rate, and all variability is in space. It follows that if the correct values of the total time are used in all applications, the conventional spatial coordinate systems are capable of accurately representing all motions at speeds of 1/n. But the scale of the spatial coordinate system is related to the rate of change of time, and the accuracy of the coordinate representation depends on the absence of any change in time other than the continuing progression at the normal rate of registration on a clock. At speeds in excess of unity, space is the entity that progresses at the fixed normal rate, and time is variable. Consequently, the excess speed above unity distorts the spatial coordinate system.

In a spatial reference system the coordinate difference between two points A and B represents the space traversed by any object moving from A to B at the reference speed. If that reference speed is changed, the distance corresponding to the coordinate difference AB is changed accordingly. This is true irrespective of the nature of the process utilized for measurement of the distance. It might be assumed, for instance, that by using something similar to a yardstick, which compares space directly with space, the measurement of the coordinate distance would be independent of the reference speed. But this is not correct, as the length of the yardstick, the distance between its two ends, is related to the reference speed in the same manner as the distance between any other two points. If the coordinate difference between A and B is x when the reference speed has the normal unit value, then it becomes 2x if the reference speed is doubled. Thus, if we want to represent motions at twice the speed of light in one of the standard spatial coordinate systems that assume time to be progressing normally, all distances involved in these motions must be reduced by one half. Any other speed greater than unity requires a corresponding modification of the distance scale.

The existence of motion at greater-than-unit speeds has no direct relevance to the familiar phenomena of everyday life, but it is important in all of the less accessible areas, those that we have called the far-out regions. Most of the consequences that apply in the realm of the very large, the astronomical domain, have no significance in relation to the subjects being discussed at this early stage of the theoretical development, but the general nature of the effects produced by greater-than-unit speeds is most clearly illustrated by those astronomical phenomena in which such speeds can be observed on a major scale. A brief examination of a typical high speed astronomical object will therefore help to clarify the factors involved in the high speed situation.

In the preceding pages we deduced from theoretical premises that speeds in excess of the speed of light can be produced by processes that involve large concentrations of energy, such as explosions. Further theoretical development (in Volume II) will show that both stars and galaxies do, in fact, undergo explosions at certain specific stages of their existence. The explosion of a star is energetic enough to accelerate some portions of the stellar mass to speeds above unity, while other portions acquire speeds below this level. The low speed material is thrown off into space in the form of an expanding cloud of debris in which the particles of matter retain their normal dimensions but are separated by an increasing amount of empty space. The high speed material is similarly ejected in the form of an expanding cloud, but because of the distortion of the scale of the reference system by the greater-than-unit speeds, the distances between the particles decrease rather than increase. To emphasize the analogy with the cloud of material expanding into space, we may say that the particles expanding into time are separated by an increasing amount of empty time.

The expansion in each case takes place from the situation that existed at the time of the explosion, not from some arbitrary zero datum. The star was originally stationary, or moving at low speed, in the conventional spatial reference system, and was stationary in time in the moving system of reference defined by a clock. As a result of the explosion, the matter ejected at low speeds moves outward in space and remains in the original condition in time. The matter ejected at high speeds moves outward in time and remains in its original condition in space. Since we see only the spatial result of all motions, we see the low speed material in its true form as an expanding cloud, whereas we see the high speed material as an object remaining stationary in the original spatial location.

Because of the empty space that is introduced between the particles of the outward—moving explosion product, the diameter of the expanding cloud is considerably larger than that of the original star. The empty time introduced between the particles of the inward—moving explosion product conforms to the general reciprocal relation, and inverts this result. The observed aggregate, a white dwarf star, is also an expanding object, but its expansion into time is equivalent to a contraction in space, and as we see it in its spatial aspect, its diameter is substantially less than that of the original star. It thus appears to observation as an object of very high density.

The white dwarf is one member of a class of extremely compact astronomical objects discovered in recent years that is today challenging the basic principles of conventional physics. Some of these objects, such as the quasars are still without any plausible explanation. Others, including the white dwarfs, have been tied in to current physical theory by means of ad hoc assumptions, but since the assumptions made to explain each of these objects are not applicable to the others, the astronomers are supplied with a whole assortment of theories to explain the same phenomenon: extremely high densities. It is therefore significant that the explanation of the high density of the white dwarf stars derived from the postulates of the Reciprocal System of theory is applicable to all of the other compact objects. As will be shown in the detailed discussion, all of these extremely compact astronomical objects are explosion products, and their high density is in all cases due to the same cause: motion at speeds in excess of that of light.

This is only a very brief account of a complex phenomenon that will be examined in full detail later, but it is a striking illustration of how the inverse phenomena predicted by the reciprocal relation can always be found somewhere in the universe, even if they involve such seemingly bizarre concepts as empty time, or high speed motion of objects stationary in space.

Another place where the inability of the conventional spatial reference systems to represent changes in temporal location, other than by distortion of the spatial representation, prevents it from showing the physical situation in its true light is the region inside unit distance. Here the motion in time is not due to a speed greater than unity, but to the fact that, because of the discrete nature of the natural units, less than unit space (or time) does not exist. To illustrate just what is involved here, let us consider an atom A in motion toward another atom B. According to current ideas, atom A will continue to move in the direction AB until the atoms, or the force fields surrounding them, if such fields exist, are in contact. The postulates of the Reciprocal System specify, however, that space exists only in units. It follows that when atom A reaches point X one unit of space distant from B. it cannot move any closer to B in space. But it is free to change its position in time relative to the time location occupied by atom B. and since further movement in space is not possible, the momentum of the atom causes the motion to continue in the only way that is open to it.

The spatial reference system is incapable of representing any deviation of time from the normal rate of progression, and this added motion in time therefore distorts the spatial position of the moving atom A in the same manner as the speeds in excess of unity that we considered earlier. When the separation in time between the two atoms has increased to n units, space remaining unchanged (by means of continued reversals of direction), the equivalent spatial separation, the quantity that is determined by the conventional methods of measurement, is l /n units. Thus, while atom A cannot move to a position less than one unit of space distant from atom B. it can move to the equivalent of a closer position by moving outward in time. Because of this capability of motion in time in the region inside unit distance it is possible for the measured length, area, or volume of a physical object to be a fraction of a natural unit, even though the actual one, two, or three-dimensional space cannot be less than one unit in any case.

It was brought out in Chapter 6 that the atoms of a material aggregate, which are contiguous in space, are widely separated in time. Now we are examining a situation in which a change of position in the spatial coordinate system results from a separation in time, and we will want to know just where these time separations differ. The explanation is that the individual atoms of an aggregate such as a gas, in which the atoms are separated by more than unit distance, are also separated by various distances in time, but these atoms are all at the same stage of the time progression. The motion of these atoms meets the requirement for accurate representation in the conventional spatial coordinate systems; that is, it maintains the fixed time progression on which the reference system is based. On the other hand, the motion in time that takes place inside unit distance involves a deviation from the normal time progression.

A spatial analogy may be helpful in getting a clear view of this situation. Let us consider the individual units (stars) of a galaxy. Regardless of how widely these stars are separated, or how much they move around within the galaxy, they maintain their status as constituents of the galaxy because they are all receding at the same speed (the internal motions being negligible compared to the recession speed). They are at the same stage of the galactic recession. But if one of these stars acquires a spatial motion that modifies its recession speed significantly, it moves away from the galaxy, either temporarily or permanently. Thereafter, the position of this star can no longer be represented in a map of the galaxy, except by some special convention.

The separations in time discussed in Chapter 6 are analogous to the separations in space within the galaxies. The material aggregates that we are now discussing retain their identities just as the galaxies do, because their individual components are progressing in time at the same rates. But just as individual stars may acquire spatial speeds which cause them to move away from the galaxies, so the individual atoms of the material aggregates may acquire motions in time which cause them to move away from the normal path of the time progression. Inside unit distance this deviation is temporary and quite limited in extent. In the white dwarf stars the deviations are more extensive, but still temporary. In the astronomical discussions in Volume II we will consider phenomena in which the magnitude of the deviation is sufficient to carry the aggregates that are involved completely out of the range of the spatial coordinate systems.

So far as the inter-atomic distance is concerned, it is not material whether this is an actual spatial separation or merely the equivalent of such a separation, but the fact that the movement of the atoms changes from a motion in space to a motion in time at the unit level has some important consequences from other standpoints. For instance, the spatial direction AB in which atom A was originally moving no longer has any significance now that the motion is taking place inside unit distance, inasmuch as the motion in time which replaces the previous motion in space has no spatial direction. It does have what we choose to call a direction in time, but this temporal direction has no relation at all to the spatial direction of the previous motion. No matter what the spatial direction of the motion of the atom may have been before unit distance was reached, the temporal direction of the motion after it makes the transition to motion in time is determined purely by chance.

Any kind of action originating in the region where all motion is in time is also subject to significant modifications if it reaches the unit boundary and enters the region of space motion. For example, the connection between motion in space and motion in time is scalar, again because there is no relation between direction in space and direction in time. Consequently, only one dimension of a two-dimensional or three-dimensional motion can be transmitted across the boundary. This point has an important bearing on some of the phenomena that will be discussed later.

Another significant fact is that the effective direction of the basic scalar motions, gravitation and the progression of the natural reference system, reverses at the unit level. Outside unit space the progression of the reference system carries all objects outward in space away from each other. Inside unit space only time can progress unidirectionally, and since an increase in time, with space remaining constant, is equivalent to a decrease in space, the progression of the reference system in this region, the time region, as we will call it, moves all objects to locations which, in effect, are closer together. The gravitational motion necessarily opposes the progression, and hence the direction of this motion also reverses at the unit boundary. As it is ordinarily observed in the region outside unit distance, gravitation is an inward motion, moving objects closer together. In the time region it acts in the outward direction, moving material objects farther apart.

On first consideration, it may seem illogical for the same force to act in opposite directions in different regions, but from the natural standpoint these are not different directions. As emphasized in Chapter 3, the natural datum is unity, not zero, and the progression of the natural reference system therefore always acts in the same natural direction: away from unity. In the region outside unit distance away from unity is also away from zero, but in the time region away from unity is toward zero. Gravitation likewise has the same natural direction in both regions: toward unity.

It is this reversal of coordinate direction at the unit level that enables the atoms to take up equilibrium positions and form solid and liquid aggregates. No such equilibrium can be established where the progression of the natural reference system is outward, because in this case the effect of any change in the distance between atoms resulting from an unbalance of forces is to accentuate the unbalance. If the inward-directed gravitational motion exceeds the outward-directed progression, a net inward motion takes place, making the gravitational motion still greater. Conversely, if the gravitational motion is the smaller, the resulting net motion is outward, which still further reduces the already inadequate gravitational motion. Under these conditions there can be no equilibrium.

In the time region, however, the effect of a change in relative position opposes the unbalanced force, which caused the change. If the gravitational motion (outward in this region) is the greater an outward net motion takes place, reducing the gravitational motion and ultimately bringing it into equality with the constant inward progression of the reference system. Similarly, if the progression is the greater, the net movement is inward, and this increases the gravitational motion until equilibrium is reached.

The equilibrium that must necessarily be established between the atoms of matter inside unit distance in a universe of motion obviously corresponds to the observed inter-atomic equilibrium that prevails in solids and, with certain modifications, in liquids. Here, then, is the explanation of solid and liquid cohesion that we derive from the Reciprocal System of theory, the first comprehensive and completely self-consistent theory of this phenomenon that has ever been formulated. The mere fact that it is far superior in all respects to the currently accepted electrical theory of matter is not, in itself, very significant, inasmuch as the electrical hypothesis is definitely one of the less successful segments of present-day physical theory, but a comparison of the two theories should nevertheless be of interest from the standpoint of demonstrating how great an advance the new theoretical system actually accomplishes in this particular field. A detailed comparison will therefore be presented later, after some further groundwork has been laid.