14 Limits



One of the most significant features of the physical universe, as it emerges from the development of the consequences of the postulates that define the universe of motion, is the existence of limits. Everywhere we turn we find ourselves confronted with some kind of limit—a gravitational limit, a mass limit, an age limit, and so on and on. These limits exist because the postulates define a universe that is finite, with effective magnitudes that extend, not from zero, but from unit motion; that is, unit speed or unit energy. Since the deviations from this datum are finite, neither infinity nor zero is ever reached (except in a mathematical sense where the difference between two existing quantities of equal magnitude enters into some physical situation).

Many of the errors in present-day scientific theory owe their existence to a lack of recognition of the reality of these limits. Some particularly far-reaching conclusions of an erroneous nature that are pertinent at this stage of our investigation have been drawn from the Second Law of Thermodynamics. This law has been stated in a number of different ways. One of the simplest makes use of the physical quantity known as entropy, which is essentially a measure of the unavailability of energy for doing work. The statement of the Second Law on this basis is that the entropy of the universe continually increases. In the absence of recognition of any limits applicable to this process, the conclusion has been reached that the universe is on the way to becoming a featureless uniformity in which no significant action will take place. As expressed by Marshall Walker, “Apparently the universe is ’running down,’ and in the remote future it will consist of a disordered cold soup of matter dispersed throughout space at a uniform temperature of a few degrees above absolute zero.”163 Many writers dispense with qualifying words such as “apparently,” and express this point of view in uncompromising terms. Paul Davies, for example, puts it in this manner:

Unless, therefore, our whole understanding of matter and energy is totally misconceived, the inevitability of the end of the world is written in the laws of nature.164

James Jeans, writing half a century earlier, was already firmly convinced, and makes this equally positive assertion:

The energy is still there, but it has lost all capacity for change… We are left with a dead, although possibly a warm, universe—a “heat death.” Such is the teaching of modern thermodynamics. There is no reason for doubting or challenging it, and indeed it is so fully confirmed by the whole of our terrestrial experience, that it would be difficult to find any point at which it could be open to attack.165

But in that earlier day, when the “heat death” idea was new and still controversial, Jeans found it necessary to explain the reasoning by which this conclusion was reached (something his modern successors usually do not bother with), and in the course of this explanation he tells us this:

Thus the main physical process of the universe consists in the energy of exceedingly high availability which is bottled up in atomic and nuclear structures being transformed into heat-energy at the lowest level of availability.166

As we have seen in the preceding pages, this is not the “main physical process of the universe.” It is merely one of the subsidiary processes, an eddy in the main stream. The primary process of the material sector of the universe does not begin with energy of high intensity “bottled up in atomic structures”; it ends in that form, as the result of a long period of aggregation under the influence of gravitation. This high intensity state is one of the limits of the primary physical process. The highly dispersed state from which the aggregation started is the other. These limits could be compared to the high and low points in the travel of a pendulum. At its highest. point the pendulum is motionless, but it is subject to gravitation. The gravitational force pulls it down to its low limit, but in so doing imparts a motion to it, and this motion then takes the pendulum back to the level from which it started. Similarly, the new matter in the material sector is widely dispersed and motionless. Gravitational forces pull the dispersed units inward to a limiting concentration, but in so doing impart a system of motions to the matter. These motions initiate a chain of events that ultimately brings the matter back into the same dispersed and motionless condition from which it started.

This definition of the fundamental action of the universe as a cyclic process carries with it the existence of limits to the various subsidiary quantities. This present chapter will be addressed to an examination of some of the most significant of these limits.

The Reciprocal System of theory deals exclusively with units of motion, and thus it is quantitative from the start. As noted in the earlier volumes, the quantitative development goes hand in hand with the qualitative development as the theory is extended into additional areas and into more detail. Thus far in the present volume, however, the treatment of the subject matter has been almost entirely qualitative. There are two reasons for this. The first is that the objects with which astronomy deals are aggregates of the same kind of matter that was discussed in the previous volumes, differing only in that the range of sizes of the aggregates, and the range of conditions to which these aggregates are subject, are much greater than in the situations previously considered. By and large, therefore, the quantitative relations applicable to the astronomical aggregates are the same relations that were developed in the previous volumes. Thus one reason why there was not much quantitative discussion in the preceding pages is that most of the pertinent quantitative relations had already been covered in the earlier volumes.

The other reason for the limited amount of quantitative development is that, as noted in Chapter 1, the quantities of interest in astronomy are largely applicable to individual objects, whereas our present concern is with classes of astronomical objects and the general evolutionary patterns of those objects, rather than with individuals. Furthermore, even where quantitative relations are relevant to the subject matter under consideration, and have not been previously derived, it has appeared advisable, in most cases, to delay discussing them until after the general aspects of the astronomical universe, as seen in the context of the theory of the universe of motion, were examined and placed in their proper relations with each other. To the extent that this examination has not yet been carried out, these items will be discussed at appropriate points in the pages that follow. This chapter will take up the question of the quantitative limits applicable to some of the phenomena already examined from the qualitative standpoint in the earlier pages.

Chief among these, because of its fundamental nature, is the gravitational limit. On the basis of the general principles developed in Volume I, the gravitational limit of a mass is the distance at which the inward gravitational motion of another mass toward the mass under consideration is equal to its outward motion due to the progression of the natural reference system relative to our stationary system of reference. If the unit gravitational motion acted directly against the unit outward speed imparted to the mass unit by the outward progression of the reference system, the gravitational limit for unit mass would be at one natural unit of space, by reason of the relation between the natural units explained in Volume I. But the gravitational effect is distributed over all of the many dimensional variations of both the rotational and translational motion, and its effective magnitude is reduced by this dimensional distribution. As we saw in the earlier volumes, the rotational distribution extends over 128 units in each dimension. Since motion in space involves three dimensions of space and one dimension of time, the total rotational distribution is (128)4. Additionally there is a translational distribution over the eight units that we have previously identified as the linear maximum. The total of both distributions amounts to 8 × (128)4 = 2.1475×109. What this means is that the gravitational motion is distributed over 2.1475×109 units, only one of which is acting against the outward progression of the natural reference system in the line of the progression. The effective component of the gravitational force (motion) is thus reduced by this ratio, the rotational ratio, as we will call it.

Inasmuch as the one-dimensional analog of this rotational ratio, the interregional ratio, includes an additional component amounting to 2/9 of the 128 rotational units, increasing the ratio to 156.444, there may be some question as to why the rotational ratio does not contain a similar extra component. The explanation is that the atomic rotation is a rotation of a linear vibration. The total atomic motion is therefore distributed among the vibrational units (2/9 of 128) as well as among the 128 rotational units. But the 8-unit translational distribution included in the rotational ratio covers all of the possible linear motion, including the basic vibrational motions that is being rotated. Thus no additional term is required in the rotational ratio.

Within the gravitational limit the effective gravitational motion (or force) is inversely proportional to the square of the distance. Without the distribution to the multiple units, the equilibrium equation under unit conditions would be m/d02 = 1; that is, the gravitational force exerted by the natural unit of mass at the natural unit of distance would be in equilibrium with the unit force of the progression of the natural reference system. The distribution of the gravitational force reduces its effective value by the rotational ratio. The effective equilibrium is then.

4.65661×10-10 m/d02 = 1


Solving for d0 the gravitational limit. we obtain

d0 = 2.15792×10-5 m½


To convert this equation from natural to conventional units we divide the coefficient by the number of natural units of distance per light year, 2.0752x 10,23 and by the square root of the number of grams per natural unit of mass, 1.65979x 10.-14 In terms of grams and light years, equation 14-2 then becomes.

d0 = 8.0714×10-17 m½ light years


The mass of the sun has been calculated as 2×1033 grams. Applying the coefficient of equation 14-3, we find that the gravitational limit of the sun is 3.61 light-years. This is consistent with the observed separations. The nearest star system, Alpha Centauri, is 4.3 light years distant, and the average separation of the stars in the vicinity of the sun is estimated to be somewhat less than 2 parsecs, or 6.5 light years. Sirius, the nearest star larger than the sun, has its gravitational limit at 5.3 light years, and the sun, 8.7 light years away, is well outside this limit.

It is evident that such a distribution of a very large number of objects in space, where the minimum separation is two-thirds of the average, requires some kind of a barrier on the low side; it cannot be the result of pure chance. The results of this present investigation show that the reason why the stars of the solar neighborhood do not approach each other closer than about four light years is that they can not do so. This finding automatically invalidates all theories that call for star systems making contact or a close approach (as contemplated in some theories of the formation of planetary systems), and all theories that call for the passage of one aggregate of stars through another (such as the currently accepted theory of “elongated rectilinear orbits” of the globular clusters).

Furthermore, these results show that the isolation of the individual star system is permanent. These systems will remain separated by the same tremendous distances because each star, or star system, or pre-stellar cloud continually pulls in the material within its gravitational range, and this prevents the accumulation of enough matter to form another star in this volume of space. The immense region within the gravitational limit of each star is reserved for that star alone.

The interstellar distance calculated from the number of stars per unit volume is less in the interiors of the globular clusters and the central regions of the galaxies. But since the three-dimensional region of space extends only to the gravitational limit, the reduction in volumetric dimensions beyond this limit, due to the gravitational effect of the aggregate as a whole, is in equivalent space rather than in actual space. It therefore does not alter the spatial relation of a star to the gravitational limits of its neighbors.

Galactic masses are usually expressed in terms of a unit equal to the solar mass. Since we have already evaluated the gravitational limit of the sun, we may express equation 14-3, for application to the galaxies, in the convenient form.

d0 = 3.61 (m/ms)½ light years


This relation enables us to verify the conclusions reached in Chapter 2 with respect to the process of cannibalism by which the giant spheroidal galaxies have reached their present sizes. As noted in that earlier discussion, the development of theory indicates that galaxies as large as the Milky Way are pulling in not only a large amount of diffuse material, but also individual stars, globular clusters, and small galaxies. The Magellanic Clouds were identified as galaxies in the process of being captured. In order for such a capture to take place, the smaller unit must be within the gravitational limit of the larger. Let us see, then, just what distances are involved.

Estimates of the mass of the Galaxy range from 1011 to 5×1011 solar masses. If we accept an intermediate value, 3×1011 for present purposes, equation 14-4 indicates a gravitational limit of about two million light years. On this basis, the Magellanic Clouds are well on their way to capture by the Galaxy.

It was also brought out in the previous discussion that the irregular structures of the Magellanic Clouds are due to distortion of the original spiral or elliptical structures by differences in the gravitational forces acting on different parts of the Clouds. The diameters of the Clouds are approximately 20,000 and 30,000 light years respectively. These distances are obviously large enough in proportion to the distance from the Galaxy to give rise to significant differences between the forces acting on the near sides of the Clouds and those acting on the far sides.

Both of these findings with respect to the Magellanic Clouds can be generalized. We can say that any galaxies within a distance of about two million light years of a galaxy as large as the Milky Way are in the process of being pulled in toward the galaxy and will eventually be captured. We can also say that any galaxy on the way to capture will experience structural distortion in the last stages of its approach.

The calculations that support the theoretical conclusions as to what is now happening to the globular clusters and small galaxies in the vicinity of larger aggregates can be extended to confirm the further conclusions concerning what is going to happen in the future as the evolutionary development of the galaxies proceeds. In the original discussion of the aggregation process it was pointed out that the possibility of growth by capture depends on the location of the gravitational limit. Consolidation of two star systems cannot take place, because each such system is outside the gravitational limits of all others, and these limits can be extended only at a relatively slow rate, since there is nothing in interstellar space subject to capture other than diffuse matter and a few small objects such as comets. On the other hand, the theoretical consideration of the galactic situation in Chapter 2 showed that the globular clusters and early type galaxies are inside the gravitational limits of their immediate neighbors because of the nature of the process by which they were formed. Consolidation of these objects into larger and larger aggregates therefore proceeds until the greater part of the mass in each large region of space is gathered into one giant spheroidal galaxy.

This theoretical finding implies that most of the galaxies included in what is known as the Local Group—our Milky Way, the Andromeda Galaxy, M 33, the newly discovered Maffei galaxies, and a considerable number of smaller aggregates—will eventually combine. For an evaluation of the likelihood of such an outcome, let us look at the gravitational limits. We have found that if we take the mean of present estimates of the size of our galaxy, its gravitational limit is about two million light years. But if we take the highest of the previously quoted values, this limit becomes 2½ million light years. It is generally believed that the Andromeda Galaxy, M 31, about two million light years distant, is somewhat larger than our own. Thus, if the higher estimate of the mass of the Milky Way is correct, we are already within the gravitational limit of M 31.

If the actual masses are smaller than these estimates indicate, this conclusion is premature. But M 31 is growing. Intergalactic space contains a multitude of material aggregates—globular clusters, unconsolidated dust and gas clouds of globular cluster size, dwarf galaxies, and stray stars—all of which are subject to capture by the giant spirals, along with quantities of diffuse matter. If the Milky Way is not yet within the gravitational limit of M 31, it certainly will be within that limit when the capture process is a little farther along. In the meantime, the total mass of the various aggregates and diffuse matter between the major galaxies helps to hold the Local Group together while those galaxies continue their cannibalism. The calculation of the gravitational limits of the local galaxies thus confirms the theoretical conclusion from more general premises that the eventual result of the evolutionary process will be a giant spheroidal galaxy containing most of the mass of the Local Group.

It is even possible that a larger, hitherto unrecognized, member of the group might swallow both M 31 and the Milky Way galaxy. A very large volume of space in our immediate vicinity is hidden from our view by the dense portions of our own galaxy, and we do not know what is behind this barrier. The two Maffei galaxies were just recently discovered on the fringes of this unobservable zone, and there are reports indicating the existence of two more in a “heavily obscured region in Cygnus.”18 One is said to be a “bright elliptical.” If this is correct, the elliptical galaxy could be the dominant member of the Local Group.

The conclusion as to the ultimate consolidation of most of the mass of the Local Group conflicts with current astronomical opinion, as expressed by Gorenstein and Tucker, who assert flatly that “the probability of our galaxy’s ever colliding with the Andromeda galaxy is close to zero.”167 But that opinion is a relic of the traditional view of the galaxies as objects that originated in essentially their present forms in the early stages of the existence of the universe, and are moving randomly, a view that, although it is still orthodox doctrine, is gradually giving way as more and more evidence of galactic collisions and cannibalism accumulates.

Another critical magnitude in which we are interested is the limiting distance beyond which all galaxies recede at the full speed of light. The importance of this distance lies in the fact that its relation to the speed of light determines the rate of increase in the recession speed relative to the distance, the Hubble constant, as the astronomers call it. This is a place where the theory of the universe of motion arrives at conclusions that are altogether different from those that the astronomers have reached from their consideration of the only piece of observational information that has been available to them, the existence of a galactic recession at a speed that is proportional to the distance. On this slender factual basis, they have erected an elaborate framework of assumption and inference that serves as the orthodox theory of the large-scale action of the universe. The cosmological aspects of this theory will be discussed in the final chapters of this volume. Our concern at present is with the recession speed. In the absence of any evidence to the contrary, the astronomers have assumed that the Hubble constant is a fixed characteristic of the physical universe, and they further assume that the increase in the recession speed with increasing distance continues indefinitely, approaching the speed of light asymptotically.

While these two assumptions seemed reasonable in the light of the information available to their originators, the development of a theoretical understanding of the recession process now shows that both of them are wrong. The recession is not peculiar to the galaxies. It is a general property of the universe, a relation between the reference system that we utilize and the reference system to which natural phenomena actually conform, and it applies not only to the galaxies, but to all material objects, and also to non-material entities, such as the photons. The outward motion of the photons at the speed of light is the same phenomenon as the recession of the galaxies, differing only in that the galactic recession is slowed by the opposing gravitational forces, whereas the photons are not subject to any significant amount of gravitational retardation, except in the immediate vicinity of very large masses. The Hubble constant“ is not, as currently assumed, a basic property of the physical universe. Like the gravitational limit, it is a property of each individual mass aggregate. In application to the galactic recession this so-called constant is a function of the total galactic mass, exclusive of the outlying globular clusters and halo stars that are still in free fall.

The assumption that the speed of light is a limiting value, which the recession speed never quite reaches, is also invalidated by the theoretical findings. As we saw earlier in this chapter, the effect of the distribution of the gravitational motion over all of the rotational and translational units involved in the atomic rotation is to require 2.1475×109 mass (gravitational) units to counterbalance each unit of the outward motion of the natural reference system. The point at which this is accomplished is the gravitational limit. Here the net speed, relative to the conventional spatial reference system, is zero. Beyond the gravitational limit the reduced gravitational speed is only able to neutralize a part of the outward progression, and there is a net outward motion: the galactic recession.

The net outward speed increases with the distance, but it cannot continue increasing indefinitely. Eventually the attenuation of the gravitational motion by distance brings it down to the point where the remaining motion of each mass unit is sufficient only to cover the distribution over the dimensional units involved in direct one-dimensional contact between motion in space and motion in time. Less than this amount (a natural compound unit) does not exist. Beyond this point, therefore, the gravitational effect is eliminated entirely, and the recession takes place at the full speed of light. The limiting distance with which we are now concerned can thus be obtained by substituting the one-dimensional relation, 128 (1+2/9) = 156.44 (previously identified as the inter-regional ratio), for the rotational ratio in equation 14-1. The new equilibrium force equation is then

1/156.44 × m/d1= 1


Again, solving for the distance, which in this case we are calling d1, we have

d1 = m½/12.55


which can be expressed in terms of solar masses as

d1 = 13350 (m/ms)½ light years


If we again take the intermediate estimate of the mass of the Galaxy that was used earlier, 3×1011 solar masses, and apply equation 14-7, we find that the limiting distance, d1, is 7.3×109 light years. Disregarding the relatively short distance between the Galaxy and its gravitational limit, we may now calculate the distance from our galaxy to any other galaxy of the same or smaller mass by converting the redshift in the spectrum of that galaxy to natural units (traction of the speed of light). and multiplying by 7.3×109 light years, or 2.24×109 parsecs. This is equivalent to a value of 134 km/sec per million parsecs for the Hubble constant.

This calculated value for the Hubble constant does not apply to the recession of a galaxy larger than our own, as the effective gravitational force that defines the limiting distance is the force exerted by the larger of the two aggregates. The mass of the smaller aggregate is immaterial from this standpoint. It is true that the controlling mass exerts a greater gravitational force on a large mass than on a smaller one, but the opposing effect of the progression of the natural reference system is subject to the same proportionate increase, and the equilibrium point remains the same. The astronomers estimates of the value of the Hubble constant are based largely on observations of the more massive galaxies, the ones that are most easily observed. The masses of these giant galaxies are quite uncertain, and the estimates vary widely, but as a rough approximation we may take the mass of a galaxy of maximum size to be ten times that of the Milky Way galaxy. Substituting this mass for that used in the previous calculation, we obtain a Hubble constant of 42 km/see per million parsecs.

The value currently accepted by most astronomers is between 50 and 60 km/sec. Before 1952 the accepted value was 540. By the time the first edition of this work was published in 1959 it was down to about 150. Subsequent revisions have brought it down to the present 50 or 60. These latest results are consistent with the theoretically calculated values within the accuracy of the galactic mass determination.

Emission of radiation from the rotating atoms of matter is also subject to a dimensional distribution, but radiation is a much simpler process than the gravitational interaction, and the distribution is correspondingly more limited. As noted in Volume II, where the dimensional distribution effective in gravitation was discussed, the theoretical conclusions with respect to the dimensional distribution of the primary motions are still somewhat tentative, although the satisfactory agreement with observation gives them a significant amount of support. It appears from these findings that the radiation distribution is confined to the basic 128 rotational positions in one dimension.

The application of this distribution with which we are now concerned is its effect on the magnitude of the increase in the wavelength of radiation (the redshift) due to the outward progression of the reference system. Since all physical entities are subject to this progression, it has no effect on ordinary physical phenomena, but it does alter the neutral point, the boundary between motion in space and motion in time. The outward progression in space relative to the location from which we are making our observations shifts the boundary in the direction of longer wavelengths. Observers in the cosmic sector (if there are any) see a similar shift in the direction of the shorter wavelengths.

Inasmuch as the natural unit in vibrational motion is a half cycle, the cycle is a double unit. The wavelength corresponding to unit speed is therefore two natural units of distance, or 9.118×10-6 cm. The distribution over 128 positions increases the effective distance to 1.167×10-3 cm (11.67 microns). This, then, is the effective boundary between motion in space and motion in time, as observed in the material sector. On the high frequency (short wavelength) side of the boundary there is first the near infrared, from 1.167×10-3 cm to 7×10-5 cm, next the optical region from the infrared boundary to 4×10-5 cm, and finally the x-ray and gamma ray regions at the highest frequencies. Because of the reciprocal relation between space and time these high frequency regions are duplicated on the low frequency (long wavelength) side of the neutral level.

Inasmuch as the processes of the region below unit speed involve transfer of fractional units of speed—that is, units of energy—the frequencies of the normal radiation from these processes are on the energy side of the boundary. This is high frequency radiation. At speeds above unity, this situation is reversed. The physical processes at these speeds involve transfer of fractional amounts of energy, and the frequencies of the normal radiation are on the speed side of the unit boundary. In the regions accessible to our observation, these low frequency processes are less common than those in the high frequency range, and the instrumentation that has been developed for dealing with them is much less advanced. Consequently they are not as well known, and only two subdivisions are recognized. The far infrared corresponds to the near infrared and the optical ranges, while the radio range corresponds to the X-ray and gamma ray ranges.

The term “normal” in the foregoing paragraph refers to radiation in full units of the type appropriate to the speed of the emitting objects. For example, thermal radiation is a product of processes operating at speeds below unity (the speed of light). The full units produced at these speeds constitute frequencies on the upper side of the unit boundary. Fractional units do not exist, but adding or subtracting units of time can produce the equivalent of fractional changes in the amount of space. This enables extension of a portion of the frequency distribution of thermal radiation into the far infrared, below the unit level. In fact, if the radiating object is cool, it may radiate entirely in this lower range. But if this radiating object is hot enough to produce a substantial amount of radiation, the great bulk of this radiation is in the upper frequency ranges. Thus strong thermal radiation comes from matter in the speed range below unity. The same principle applies to radiation produced by any other processes of the low speed range. Conversely, strong radiation of the inverse type—far infrared and radio comes from matter in the upper speed ranges (above unity).

The existence of a sharp line of demarcation between the kind of objects that radiate in the near infrared and the kind that radiate in the far infrared is clearly recognized, even at this rather early stage of infrared astronomy, but the fact that there is an equally sharp distinction in the nature of the radiation from these objects has not yet been recognized by the astronomers. For example, Neugebauer and Becklin suggest that the observed strong radiation from some objects at 100 microns is “thermal radiation from dust heated by star light,”168 that is, it is essentially equivalent to the radiation from cool stars, although they also report that the objects which radiate strongly at two microns (in the near infrared) are altogether different from those that radiate at 20 or 100 microns (in the far infrared). The ten brightest sources at two microns, they report. are all stars: three supergiants, three giants. and four long-period variables—the same stars that are bright in the visible region On the other hand, none of the ten brightest sources at 20 microns is an ordinary star. They include the center of the Galaxy. several nebulae, and a number of objects whose nature is, as yet. not clearly understood. As the investigators say, “At present we lack the information needed to understand the sources unambiguously.”

Our findings show that what is needed is recognition of the existence of the unit boundary at 11.67 microns. Strong radiation in the far infrared, beyond 11.67 microns, comes from matter with speeds in the upper ranges, above the speed of light, not from relatively cool thermal sources like those that radiate weakly in the far infrared. As we will see in the pages that follow, strong infrared emission is one of the conspicuous features of the objects that we will identify as involving motion at upper range speeds: quasars, Seyfert galaxies, the cores of other large galaxies, exploding galaxies such as M 82, etc. The infrared radiation from the quasars is estimated to be 1000 times the radiation in the visible range.167 The association between infrared emission and radiation in the radio range (which we identify with upper range speeds) is another feature of these objects, which, like the infrared emission, is unexplained in current astronomical theory. The significance of the results of the surveys of the infrared sources within the Galaxy, such as the one reported by Neugebauer and Becklin, is that they demonstrate the existence of the line of demarcation between the far infrared of the upper range speeds and the near infrared of the speeds below unity.

In the case of complex objects in which both upper range and normal speeds are strongly represented, the existence of a discontinuity is evident in the spectra. For instance, the IRAS observations show that “The spectrum of the Crab nebula ’breaks,’ or turns over in the far infrared,” leading to the conclusion that “something must happen in the infrared region that lies between the near infrared and radio bands.”350

The processes other than thermal that give rise to these various radiation frequencies, and their identification with the speeds of the emitting objects, will be discussed in Chapter 18.

As we saw in Chapter 6, the inverse process of adding energy accomplishes the increase in speed in the range below the unit level. Addition of n - 1 units of energy to zero speed (1 - 1/1) results in a speed of 1 - 1/n2 Obviously, this is a very inefficient method of increasing the speed, inasmuch as a large increment of energy produces only a very small increase in the speed. Furthermore, the maximum speed that can be attained by this means is limited to one unit (that is, the speed of light). But in spite of these highly unfavorable aspects, this is the way in which additions to the speed are made in the range between zero speed and unit speed, simply because there is no alternative. Fractional units of speed do not exist.

The subsequent pages of this work will be concerned mainly with phenomena that take place at speeds in excess of unity, and one of the arguments that will be advanced against the reality of such speeds by the adherents of orthodox physical thought will be that the amount of energy required to produce speeds of this magnitude is incredibly great. Indeed, such arguments are already being raised against suggestions that call for the ejection of galactic fragments at speeds that merely approach the speed of light. The answer to these objections is that the upper range speeds are not produced by the inefficient inverse process of adding energy; they are produced by the direct addition of units of speed, a much more efficient process.

To illustrate the difference, let us consider the result of adding energy or speed to the initial situation just mentioned, where the energy is unity and the net speed is 1 - 1/1 = 0. (Most speeds in the material sector are differences (1 - 1/n2) - (1 -1/m2), but it will be convenient to deal with the simpler situation). If we add two units to the energy component the net speed increases to 1 -1/9 = 0.889. On the other hand, if we add two units to the speed component the net speed increases to 3 - 1/1 = 2.000, at the threshold of the ultra high-speed range. The significance of these figures lies in the fact that in the universe of motion a unit of speed and a unit of energy are equivalent. It follows that an event that is capable of increasing the net speed of an object only as far as 0.889 by the process of adding energy is capable of increasing it to 2.000 by the process of adding speed, if speed is available in unit quantities. The conclusion that we reach from the theoretical development is that matter reaches its age and size limits under conditions that result in gigantic explosions in which speed is, in fact, released in unit quantities, and is available for acceleration of the explosion products to the upper range speeds. This means that intermediate and ultra high speeds are well within the capability of known processes.

It is probable that most readers who encounter this idea for the first time will see it as a strange and unprecedented addition to physical thought. But, in fact, the basic principle that is involved is the same one that governs a well known, and quite common, type of physical situation. The only thing new is that this principle has not heretofore been recognized as applying to the phenomenon now being discussed. The truth is that what we are here dealing with is simply a threshold effect, something that we meet frequently in physical theory and practice. The photoelectric effect is a good example. In order to eject electrons from cold metal, the frequency of the impinging radiation must be above a certain level, the threshold frequency. The result cannot be accomplished by increasing the total amount of low frequency energy. Ejection does not take place at all unless the energy is available in units of the required size. When units of this size are applied, even a small total energy is sufficient. The production of speeds in excess of the speed of light is governed by the same kind of a limitation. Speed units of the required magnitude must be available.

A number of other magnitudes that are significant in the quantitative description of the evolution of the contents of the universe are subject to calculation on the theoretical basis that has been provided, including such items as the average duration of the various evolutionary stages, maximum and minimum sizes of the various aggregates, and so on. Lack of time has prevented undertaking any systematic investigation of these subjects, but some results have been obtained as by-products of studies made for other purposes. These are mostly properties of objects moving at upper range speeds, and they can be discussed more conveniently in connection with the general examination of the phenomena of these upper speed ranges in the subsequent chapters.

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