Chapter IV
Some Basic Relations
The next objective will be to evaluate some of the properties of the different elements as they are defined by the principles derived from the Fundamental Postulates. On beginning this task, one of the first things we will encounter is the fact that in many instances the variations in these properties will take place entirely within a single unit of space. For example, two atoms may be separated by t units of time. Since space and time are reciprocally related this separation in time is equivalent to a separation of 1/t units of space. If the time t increases the equivalent space decreases and the two atoms in effect move closer together. Such motion, however, differs in many respects from motion which involves actual units of space in association with actual units of time.
In view of the important differences in various physical phenomena which similarly depend on whether the magnitudes involved are above or below the unit levels we may conveniently regard these unit levels as dividing the universe into several general regions. At one extreme there is a region in which space remains at the minimum value, unity, and all variability is in time. This we will call the time region. Next we have a region in which one or more units of space are associated with a greater number of units of time. This region where the space-time ratio or velocity is below unity will be called the time-space region. A similar region on the other side of the neutral axis where the space-time ratio is greater than unity is the space-time region, and at the other extreme we have the space region where time remains at unity and all variability is in space.
We have found that the normal space-time progression involves n units of space for each n units of time, which means that the velocity of the progression, is always n/n or unity. In the time region space cannot progress but time does progress in the usual manner and since time and space are reciprocally related the progression of time t results in a progression of equivalent space 1/t. The velocity of the progression in this region is equivalent space 1/t divided by time t, or 1/t2.
In the time-space region the velocity corresponding to unit space and time t is 1/t. From the foregoing we find that in the time region it is 1/t2 . The time region velocity and all quantities derived therefrom, which means all of the physical phenomena of the region, are therefore second power expressions of the corresponding time-space region quantities. This is an important principle that must be taken into account in any relationship involving both regions. The intra-region relations may be equivalent; that is, the expression a = bc is the mathematical equivalent of the expression a2 = b2c2 . But if we measure the quantity a2 in the units applicable to a (the time-space region units), it is essential that the equation be written in the correct regional terms: a2 = bc. This principle is one of major significance because our measuring processes normally give us time-space region values.
Looking next at the direction of the motion, we note that in the time-space region the progression tends to move objects apart in space. Where motion is unimpeded the separation increases by n units of space in n units of time. In the time region the progression which increases time has the effect of decreasing equivalent space, since the space equivalent of time n is 1/n. This means that in the time region the space-time progression tends to move objects to positions which in effect are closer together.
If we appraise this situation in the usual manner, taking the mathematical zero as our datum, it appears inconsistent. We find the progression of space-time acting in a certain direction in one region and in the opposite direction in another region; a seemingly contradictory behavior. But the apparent conflict is only the result of using the wrong datum. It has already been pointed out that the true zero level of the physical universe is unity, not the mathematical zero. If we take unity as our datum the inconsistency disappears. We now find that space-time always progresses in the same direction: away from unity.
If two objects are initially separated by more than unit space they will move away from each other (outward from unity) under the influence of the space-time progression. If they are initially separated by the equivalent of less than unit space; that is, by t units of time, the space-time progression will take place in the same natural direction—away from unity—but in this case the result will be to move the objects toward each other, since outward from unity in this instance is toward zero.
The rotational motion of the atoms of matter necessarily opposes the space-time progression, for reasons previously explained, and it always acts in the direction toward unity. The resulting translational motion (gravitation) therefore causes the atoms to approach each other in the time-space region, but in the time region where unity lies in the opposite direction the gravitational motion increases the separation between the atoms.
Although it is quite apparent from the discussion thus far that both the space-time progression and the opposing motion due to the atomic rotation are always in existence, even if the resultant is no motion at all, it is convenient for many purposes to consider this resultant as having been brought about by a conflict of two forces tending to cause motion in opposite directions. We define force as that which will cause motion if not prevented from doing so by other forces, and we define the magnitude of the force as the product of mass and acceleration.
This introduces a new concept, that of mass, and in order to fit the force system into its proper position in the theoretical universe which we are developing from the Fundamental Postulates we must identify mass with the corresponding quantity in the velocity system; that is, we must reduce it to space-time terms. For this purpose we identify mass as the reciprocal of three-dimensional velocity. The correlation in this case is not as obvious as it has been in most of the identifications previously made, but this relation is inherent in the concept of force as it has been derived in the preceding paragraph and its validity will be demonstrated in the course of the subsequent discussion. In terms of space and time, mass may now be expressed as t3/s3. Force, which was defined as the product of mass and acceleration, becomes t3/s3 * s/t2 = t/s2. Acceleration and force are therefore analogous quantities, their space-time expressions having the same form with the space and time terms interchanged.
Before going on to a further consideration of force it will be desirable to point out that the space-time expression for energy or work, which is the product of force and distance, is t/s2 * s = t/s. This is the reciprocal of velocity s/t. Energy, therefore, is the reciprocal of velocity. When one-dimensional motion is not restrained by opposing motion (force) it manifests itself as velocity; when it is so restrained it manifests itself as potential energy. Kinetic energy is merely a measure of the energy equivalent of the velocity of a mass and it reduces to the same space-time terms as potential energy, since
½mv2 = ½t3/s3 x s2/t2 = ½ t/s
On the basis explained in the foregoing paragraphs we may treat gravitation as a force rather than a velocity. The gravitational force resulting from the rotation of an atom is equal to the mass corresponding to that rotation multiplied by unit acceleration. In this connection it will be desirable to state a general principle which we will call the Principle of Equivalence:
If a quantity a is expressed in terms of quantities x, y, z, etc., by means of the relationships derived from the Fundamental Postulates, and the quantities x, y, z, etc., are each given unit value, then the value of quantity a is also unity.
This is merely an expression of the obvious results of performing mathematical operations with all terms equal to unity, but it will not always be obvious in application to physical situations and we will find the principle useful in the subsequent development. In particular, it enables us to recognize the natural unit in cases where the usual measurement unit is arbitrary and the natural unit is not clearly identified physically. In the present instance it is evident that one unit of mass exerts one unit of force under unit conditions; that is, under such conditions that all of the factors x, y, z, etc., which enter into the determination of the gravitational force have unit value. This requirement of unit conditions is a very important point in all applications of the Principle of Equivalence. We cannot merely deduce from the general force expression F = ma that one unit of mass exerts one unit of gravitational force, as this general expression does not take into account all of the factors which affect the gravitational situation. In order to utilize the general equation for this specific purpose we must identify the special features that are involved and introduce them into the mathematical expression in such a manner that the resultant force is unity when each factor is likewise unity.
The first of these factors which should be considered is a consequence of the essential nature of force. As has been explained, force is merely a concept by means of 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 great convenience, but whenever a physical situation is represented by a derived concept of this kind there is always a hazard that the correspondence may not be complete and that conclusions reached through the medium of the derived concept may therefore be in error. A serious error of this kind has been introduced into the currently accepted theories concerning masses moving at high velocities.
The basic error in this case is the assumption that a force applied to the acceleration of a mass remains constant irrespective of the velocity of the mass. If we look at this assumption only from the standpoint of the force concept it appears entirely logical. Force is a tendency to cause motion and it seems quite reasonable that this tendency could remain constant. 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. The space-time progression, for instance, tends to cause objects to acquire unit velocity, and hence we say that it exerts unit force. But it is obvious that a tendency to impart unit velocity to an object which is already at a high velocity is not equivalent to a tendency to impart unit velocity to a body at rest. In the limiting condition, when the mass already has unit velocity, the force of the space-time progression (the tendency to cause unit velocity) has no effect at all, and its magnitude is zero.
It is evident that the full effect of any force is only attained when the force is exerted on a body at rest, and that the effective force component in application to an object in motion is a function of the difference in velocities. Ordinary terrestrial velocities are so low that the corresponding reduction in effective force is negligible and at these velocities forces can be considered constant. Experiments indicate, however, that acceleration decreases rapidly at very high velocities and approaches a limit of zero as the velocity of the mass approaches unity. Relativity theory explains the experimental results by the assumption that mass increases with velocity and becomes infinite at unit velocity (the velocity of light). In the theoretical universe being developed from the Fundamental Postulates this explanation is not acceptable as mass is constant, but the same results are produced by the fact that force is a function of the difference in velocities and drops to zero when the velocity of the mass reaches unity. In mathematical terms, the limiting zero value of a in the expression a = F/m (which is the fact determined by experiment) is not due to an infinite value of m but to a zero value of F.
Inasmuch as the gravitational equation will not normally be used in application at high velocities we will take this velocity situation into account for the present by limiting the application of the equation to low velocities, rather than introducing the necessary terms to make it generally applicable. There are two other factors, however, which will affect the normal application of the equation. Although the gravitational force of each unit of mass has an absolute value we will observe this force only in conjunction with the gravitational force of another mass and to use the Principle of Equivalence we must specify that this, be unit force. Likewise we must specify that the two interacting masses be separated by unit distance, since we will find that the gravitational force is also a function of the distance. With these two additions we may then say that unit mass exerts unit force against unit force at unit distance.
It follows that m units of mass exert m units of force on unit force at unit distance, and we may further conclude that m units of mass will exert mm’a units of force on m’a units of force at unit distance. It should be noted, however, that m’a is merely a ratio; it is m’a units of force divided by one unit of force and it has no physical dimensions. Therefore when we multiply the original expression ma or t/s2 by m’a we merely change the numerical value; we do not change the dimensions.
Since force is merely an aspect of motion it would seem on first consideration that no variation with distance should exist, as our usual concept of a velocity v in a direction AB is a magnitude which is not affected in any way by the distance between A and B. In the case of gravitation, however, the rotational velocity opposes the space-time progression, which has no fixed direction. It is true that a space-time unit which once starts in a given direction will continue in that direction indefinitely unless acted upon by an outside agency, simply through lack of any mechanism of its own which can cause a change. Radiation, for instance, which remains in the same space-time unit in which it originates, continues on unidirectionally as long as it remains undisturbed.
The rotating atoms, on the other hand, are not moving with the units of space-time; their motion is oppositely directed and hence they are continually passing from one space-time unit to another. As we have seen, the direction of the space-time progression with reference to a fixed system of coordinates is indeterminate. Each time the atom enters a new unit of space-time its direction of motion with reference to a stationary coordinate system therefore alters to oppose the direction of the space-time progression applicable to this particular unit. The probability principles require this motion to be distributed equally in all directions in the long run; hence the acceleration toward any specific area at a distance s from the rotating atom depends on the relationship of that area to the total area of the spherical surface of radius s. Since we have found that unit mass exerts unit force at unit distance, the force at distance s is inversely proportional to the ratio of areas; that is, inversely proportional to s2. Again we must take note of the fact that we are dealing with a pure ratio, s2 units of area divided by 12 units of area, and the introduction of this distance factor does not alter the dimensions of the original force equation F = ma.
The complete expression for gravitational force in the time-space region is then
| F units of force = (m units of mass * 1 unit of acceleration x m’a) / s2 |
(1) |
where m’a and s2 are pure numbers (ratios). With this understanding as to the nature of the magnitudes involved, we may simplify the equation for the purposes of numerical calculation by eliminating the terms which always have unit value.
| F = mm’/s2 |
(2) |
The derivation of this equation assumes that the various quantities are expressed in natural units. In order to use it in terms of conventional units we must therefore ascertain the relationship between each of the conventional units and the corresponding natural unit. This again involves a process of identification. For each of the fundamental quantities we must select some physical magnitude which we can identify in terms of natural units. The ratio between the values found for this particular quantity in the two systems is the conversion coefficient which is required for converting values from one system to the other. Since this ratio between the two systems is a constant for any specific property it can be derived from any quantity for which the value can be obtained in both systems. As a practical matter, however, it is desirable whenever possible to ascertain the conventional measurement corresponding to unit value in the natural system, since in most cases this unit quantity is readily identified and has been accurately measured in the conventional systems.
For example, the velocity of light in a vacuum obviously corresponds to unit velocity on the basis of the derivation of theory in the foregoing pages. This velocity has been measured very accurately and we therefore start our correlation with the natural unit of velocity equal to
2.9979×1010 cm/sec.
Another well-established value is that of unit frequency, which has been determined from a study of the characteristics of radiation. It is known as Rydberg’s fundamental frequency and has the value 3.2880×1015 cycles per second. In this measurement the cycle per second has been taken as the unit on the assumption that frequency is a function of time only. From the explanation previously given it is apparent that frequency is a velocity, a ratio of space to time, and consequently the natural unit of frequency is one unit of space divided by one unit of time. This is the equivalent of one half-cycle per unit of time rather than one full cycle, as a full cycle involves one unit in each direction. For our purposes the measured value of the Rydberg frequency should therefore be expressed as 6.576×1015 half-cycles per second.
Expressing the frequency, which is actually a velocity, in terms of reciprocal time in this manner is equivalent to using the natural unit of space in combination with the cgs unit of time as the cgs unit of frequency. In other words, omitting consideration of the space term in selecting the unit of measurement has the same effect as giving it unit value. The natural unit of time in cgs terms is therefore the reciprocal of the Rydberg frequency or
0.1521×10-15 seconds.
We may now multiply this figure by the natural unit of velocity, 2.9979×1010 cm/sec, to obtain the natural unit of space,
0.4459×10-5 cm.
Here we have the explanation of our distorted view of the space-time relations: the reason why space seems so much more real and understandable to us than time. Because the retrograde motion of gravitating matter neutralizes the progression of space in our sector of the universe while the progression of time continues unchecked, we are dealing with relatively large time magnitudes and relatively small space magnitudes.
The common units of space and time are not directly comparable as they were set up independently without any idea that there is a definite relationship between the two phenomena, but their practical utility depends on their being of the same order of magnitude with respect to human sensations. just because they are designed to be useful the centimeter and the second or any similar pair of practical units of space and time are approximately equal from the human standpoint; that is, the are about equally distant from the threshold of sensation. But the second, the unit of time which to us is of the same order of magnitude as the centimeter, is actually 3×1010 times as large. No wonder time seems elusive and mysterious to us when it goes by so fast that we experience in one second the time equivalent of 186,000 miles. This enormous difference in magnitudes is obviously one of the principal reasons why we fail to credit time with the properties that we distinguish so readily in space.
We have here a difference comparable to looking at a forest first from a distance of a few yards and then from an airplane several miles up above it. From the close-up viewpoint we are able to distinguish the details: the kind of trees, their sizes, spacing, etc. Furthermore, it is quite apparent that the forest is three-dimensional. On the other hand we learn nothing at all about the extent or shape of the wooded area. From the plane the latter information can be readily ascertained but we can obtain no information regarding those details which were so easily observed from the close-up vantage point. At this distance we are not even able to recognize more than one dimension.
From our position in space-time where only a relatively small amount of space is within our field of view we are able to observe such features as the multiple dimensions, but the space progression is difficult to detect and we catch a glimpse of it only with the aid of our largest telescopes. Our view of time is so extended that we can recognize the large scale feature, the progression, but we cannot identify any of the details that we see in space.
The natural unit of mass (the reciprocal of three-dimensional velocity) is equal to the cube of unit time divided by the cube of unit space, which gives us 3.7115×10-32 sec3/cm3. However, the relationship between mass and the two basic quantities, space and time, has not heretofore been recognized and mass has been taken as another fundamental quantity for which an arbitrary unit has been established. The ratio of the centimeter-second unit of mass to this arbitrary unit can be obtained from measurements of the force of gravity and is known as the gravitational constant. To obtain the natural unit of mass in conventional terms we divide 3.7115×10-32 by the appropriate gravitational constant. In the cgs system this constant has the value 6.670×10-8 and unit mass becomes 0.5565×10-24 grams. This is approximately one-third of the mass of the smallest unit of matter, the hydrogen atom. The exact relation will be developed later.
From the basic conversion ratios similar relations can be computed for the derived units. Among those which we will find useful are the following:
The natural unit of acceleration: unit velocity divided by unit time.
2.9979×1010 cm/sec / 0.1521 X 10-15 sec = 1.97×1026 cm/sec2
The natural unit of force: unit time divided by the square of unit space and by the gravitational constant.
0.1521×10-15 sec / ((0.4559×10-5 cm) × 6.670×10-8) = 109.7 dynes
The natural unit of energy: unit time divided by unit space and by the gravitational constant.
0.1521×10-15 sec / (0.4559×10-5 cm x 6.670×10-8) = 5.0×10-4 ergs