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 spacetime ratio or velocity
is below unity will be called the timespace region. A similar
region on the other side of the neutral axis where the spacetime ratio
is greater than unity is the spacetime 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 spacetime 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 timespace
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 timespace region quantities. This
is an important principle that must be taken into account in any relationship
involving both regions. The intraregion 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 timespace
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 timespace
region values.
Looking next
at the direction of the motion, we note that in the timespace 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
spacetime 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 spacetime
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 spacetime 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 spacetime
progression. If they are initially separated by the equivalent of less
than unit space; that is, by t units of time, the spacetime progression
will take place in the same natural directionaway from unitybut
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 spacetime 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 timespace 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 spacetime 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 spacetime
terms. For this purpose we identify mass as the reciprocal of threedimensional
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 spacetime 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 spacetime 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 onedimensional 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 spacetime terms as potential energy, since
½mv²
= 1/2t³/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 spacetime 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 spacetime
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 spacetime progression, which has no fixed
direction. It is true that a spacetime 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 spacetime 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 spacetime;
their motion is oppositely directed and hence they are continually passing
from one spacetime unit to another. As we have seen, the direction of
the spacetime progression with reference to a fixed system of coordinates
is indeterminate. Each time the atom enters a new unit of spacetime its
direction of motion with reference to a stationary coordinate system therefore
alters to oppose the direction of the spacetime 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 timespace 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 therefor start our correlation
with the natural unit of velocity equal to
2.9979 x 10^{10}
cm/sec.
Another wellestablished
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 x 10^{15} 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
halfcycle 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 X 10^{15}
halfcycles 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 x 10^{15}
seconds.
We may now multiply
this figure by the natural unit of velocity, 2.9979 x 10^{10}
cm/sec, to obtain the natural unit of space,
.4459 x 10^{5}
cm.
Here we have
the explanation of our distorted view of the spacetime 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 X 10^{10}
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 closeup 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 threedimensional. 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 closeup
vantage point. At this distance we are not even able to recognize more
than one dimension.
From our position
in spacetime 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 threedimensional velocity) is equal
to the cube of unit time divided by the cube of unit space, which gives
us 3.7115 x 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 centimetersecond
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
X 10^{32} by the appropriate gravitational constant. In the cgs
system this constant has the value 6.670 x 10^{8} and unit mass
becomes 0.5565 x 10^{24} grams. This is approximately onethird
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 x 10^{10}
cm/sec / 0.1521 X 10^{15} sec = 1.97 x 10^{26} cm/sec²
The natural
unit of force: unit time divided by the square of unit space and
by the gravitational constant.
0.1521 x 10^{15}
sec / ((0.4559 x 10^{5} cm) x 6.670 x 10^{8}) = 109.7
dynes
The natural
unit of energy: unit time divided by unit space and by the gravitational
constant.
0.1521 x 10^{15}
sec / (0.4559 x 10^{5} cm x 6.670 x 10^{8}) = 5.0 x
10^{4} ergs
