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.
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
In the early
stages of this investigation the scalar nature of spacetime 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
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.
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 travelling 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.
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.
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 adhoc 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.
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.