In beginning an examination of the consequences of the two Fundamental Postulates we note first that they involve a progression of space-time which is similar to the progression of time as it is ordinarily visualized. Let us consider a location A somewhere in space-time. During the next unit of time this location progresses to A + 1 in time and since one unit of time, on the basis of the First Fundamental Postulate, is equivalent to one unit of space the location also progresses to A + 1 in space. When n units of time have elapsed the location has progressed to A + n both in space and in time.
It should be emphasized that this statement does not refer to some object that might happen to occupy the location A; it refers to the location itself. If the hypothetical object has no independent motion of its own it will also be found at location A + n after n units of time, but this does not involve any motion of the object. It remains stationary at the same location in space-time but the location itself moves.
We thus arrive at a concept of the physical universe as being characterized by a continuous process of expansion. Although this idea of the fundamental nature of space-time is new and unfamiliar it should not be difficult to visualize since it is merely an extension of the universally recognized progression of time, and it is also entirely in harmony with the large-scale picture of the universe which has been reached through astronomical observations. As will be brought out in the subsequent discussion, the expansion of the universe deduced by the astronomers from the motions of the distant galaxies is a direct consequence of the progression of space-time itself.
Now let us consider some further implications of the postulate that space and time are reciprocally related. As already noted, this means that each individual unit of space is equivalent to an individual unit of time, but if this were the full extent of the relationship there would be no physical phenomena at all, since each unit would be exactly equivalent to all other units and the entire universe would be one vast domain of perfect uniformity in which nothing could ever happen. It is apparent that no physical phenomena can exist except as a result of a divergence from this one to one correspondence: a displacement of space-time from the unit ratio. The space-time ratio of unity therefore constitutes the initial level of all physical activity, the datum from which all phenomena extend.
This is a principle of great significance. In the subsequent development we will find that throughout the physical world relationships are simplified and seemingly contradictory facts are brought into harmony when we take unity as our datum rather than zero. We may, in fact, regard unity rather than the mathematical zero as the true physical zero.
The space-time displacements which are necessary for the existence of physical phenomena originate because the reciprocal postulate involves something more than the equivalence of the individual units. If this were the extent of the relationship we would postulate that space and time are equivalent, not that they are reciprocal. The reciprocal postulate includes the further requirement that under certain conditions associations of n units of one component must exist and that under those conditions the n units of this kind are equivalent to 1/n units of the other component.
We are then led to inquire how it can be possible for n units of space or time to act as an association when each of the individual units in this association is required to Progress uniformly with a unit of the opposite kind as an integral part of the general space-time progression. A detailed consideration of this point discloses that it requires the existence of a difference between space (or time) as a constituent of space-time and space (or time) as a separate entity. The only such difference permitted by the Fundamental Postulates is a difference in direction; hence we arrive at the conclusion that space-time is scalar and that direction is a property of space and time individually.
In the early stages of this investigation the scalar nature of space-time was embodied in an additional postulate. Further study indicated that it was a necessary consequence of the previous assumptions, as indicated in the preceding paragraph, and it was therefore eliminated from the list of postulates. However, if there is any question as to the logic involved in deriving this conclusion from the First Postulate the additional postulate can be restored and the number of basic physical assumptions will be increased from four to five. This comment is being made to clarify the point that the status of this principle has no bearing on the validity of the subsequent development of theory. The scalar nature of space-time is a part of the system; the only question at issue is whether or not it needs to be expressed as an additional postulate.
From the foregoing it is apparent that where n units of one component replace a single unit in association with one unit of the other kind in a linear progression, the direction of the multiple component must reverse at each end of the single unit of the opposite variety. Since space-time is scalar the reversal of direction is meaningless from the space-time standpoint and the uniform progression, one unit of space per unit of time, continues just as if there were no reversals. From the standpoint of space and time individually the progression has involved n units of one kind but only one of the other, the latter being traversed repeatedly in opposite directions. It is not necessary to assume any special mechanism for the reversal of direction. In order to meet the requirements of the First Postulate the multiple units must exist, and they can only exist by means of the directional reversals. It follows that these reversals are required by the Postulate itself.
Because of the periodic reversal of direction the multiple unit of space or time replaces the normal unidirectional space-time progression with a progression which merely oscillates back and forth over the same path. But when the translatory motion in this dimension is eliminated there is nothing to prevent the oscillating unit from progressing in another dimension, and it therefore moves outward at the normal unit velocity in a direction perpendicular to the direction of vibration. When viewed from the standpoint of a reference system which remains stationary and does not participate in the space-time progression the resultant path of the oscillating progression takes the form of a sine curve.
It is now possible to make some identifications. The oscillating system which has been described will be identified as a photon. The process of emission and movement of these photons will be identified as radiation and the space-time ratio of the oscillation will be identified as the frequency of the radiation.
Since space-time is scalar the actual direction in which any photon will be emitted is indeterminate and where a large number of photons originate at the same location the probability principles whose validity was assumed as a part of the Second Fundamental Postulate require that they be distributed equally in all directions. We find then that the theoretical universe which we are developing from the Fundamental Postulates includes radiation consisting of photons traveling outward in all directions from various points of emission at a constant velocity of one unit of space per unit of time; that is at unit velocity.
At this point it is in order to call attention to the fact that even in this early stage of the development simple explanations are already emerging for items with which previous theories have experienced great difficulty. The dual nature of radiation which causes it to travel as a wave but to act as a particle in emission and absorption has been a controversial issue for decades, yet the foregoing explanation shows that the reasons for this behavior are actually very simple. The photon acts as a particle in emission or absorption because it is a single independent entity; it travels as a wave because the resultant of its own inherent motion and that of the space-time progression has the form of a wave.
Furthermore, it is clear that this wave motion requires no medium; no troublesome hypothetical ether needs to be brought into the picture. Nor is there any need to make the unwelcome and disturbing postulate of action at a distance. The photon, having no independent translatory motion, remains at the same space-time location permanently but it is carried along by the progression of space-time itself. It acts only upon objects which do not participate in the progression and are therefore encountered in the path of motion. The nature of these objects will be discussed shortly.
A simple explanation is also provided for the observed fact that the velocity of radiation remains constant regardless of the reference system. Let us consider two photons originating at the same point and traveling in opposite directions. Each moves one unit of space in one unit of time. When the first unit of motion is complete the photons are separated by two units of space, and in the Newtonian system the relative velocity is obtained by dividing the increase in separation, two units, by the elapsed time, one unit. The result is a relative velocity of two units. But experiments indicate that if this velocity were measured it would be found to be unit velocity, not two units. The Newtonian system therefore fails at these high velocities.
Einstein met this situation by adopting a hypothesis previously advanced by Fitzgerald and Lorentz in which it is assumed that distance is not an absolute magnitude but varies with the velocity of the reference system in such a manner as to keep the relative velocity of radiation constant. In the case under consideration the velocity equation s/t = v, which produces the incorrect result 2/1 = 2 in the Newtonian system, now becomes s/1 = 1. Here it is assumed that the distance, s, automatically takes whatever value is required in order to arrive at the observed constant value of the velocity, the latter being accepted as being fixed by a law of nature. The highly artificial character of this solution of the problem aroused strong opposition when it was first proposed but it has won general acceptance by default, no reasonable alternative having heretofore appeared to challenge it.
In the theoretical universe being developed from the Fundamental Postulates physical magnitudes are absolute, and the variability which relativity theory introduces into the measurement of distance cannot be accepted. In this system, however, there is no necessity for any ad hoc assumption of this kind to force agreement with the observed facts since the constant relative velocity of radiation is a natural and unavoidable consequence of the Postulates.
The controlling factor in this situation is the three-dimensional nature of time. In the particular example under consideration each photon moves one unit of space in one unit of time (the normal unit velocity of the space-time progression). Both Newton and Einstein accepted the unit of time applicable to photon B as the same unit of time which is applicable to photon A. But the Postulates of this work specify that each unit of space is equivalent to a unit of time and since the motion involves two different units of space the equivalent units of time are also two separate and distinct units. Therefore when the photons increase their separation by two units of space they also increase their separation by two units of time; that is it takes two units of time to move the photons apart two units in space. The relative velocity is then 2/2 = 1, which is completely in agreement with the observed facts.
This unit velocity relative to a photon moving in the opposite direction is identical with the velocity relative to a stationary object, and the same result is obtained for any intermediate velocity of the reference system. We therefore arrive at the general principle that the velocity of radiation in free space is independent of the reference system. Basically this is a necessary consequence of the status of unity as the true physical zero.