Part Three
The Theory
IX
As pointed out in Part One, any explanation of a primary
physical phenomenon must necessarily rest upon assumptions or postulates
of some kind, and the ultimate in physical explanation is reached when
these assumptions refer to simple inherent properties of the universe,
the existence of which can be independently confirmed. The origin and
development of the postulates on which this work is based, and the nature
of the confirmatory evidence that is available were described in a previous
publication The Structure of the Physical Universe^{39}
and the full text will not be repeated here, but it may be helpful to
show briefly how the principal conceptual innovation involved in these
postulates can be derived from very elementary considerations.
Let us assume that we are undertaking a study of basic
physical relationships. Both past experience and theoretical considerations
indicate that it is sound practice to begin with the most fundamental
relation of this kind and then to build the superstructure of theories
and principles on this foundation. There is, of course, no definite signpost
to indicate just what this most fundamental relationship is, but few would
gainsay the statement that if we wish to select the two most basic entities
in the universe, the most likely candidates are space and time. The logical
starting point for the study is therefore an investigation to determine
the general relation between space and time.
At this point it is interesting to note that although
we arrive immediately and almost inevitably at this conclusion that we
should begin our study by examining the relation between space and time,
this question that should logically take first place in such a project
has never heretofore had any consideration as a part of the original
development of any basic physical theory. Newton never even realized
that there was any such relation, and in his system space and time are
completely independent. Einstein ultimately picked up and utilized Minkowski’s
hypothesis of a fourdimensional continuum with three space dimensions
and one time dimension, but this hypothesis played no part in his original
formulation; in fact, Minkowski did not even publish it until several
years after Einstein’s 1905 paper. The procedure suggested by an elementary
analysis of the situation therefore represents a new avenue of approach
to the problem. This, in itself, augurs well for the undertaking. The
odds against accomplishing anything significant by following previous
routes in areas which have had as much attention as this one are tremendous,
but the availability of a new approach to the problem makes the situation
very much more favorable.
In order to utilize this new line of attack to the best
advantage, it will be advisable to consider first the general situation
in which we are examining the relation of any quantity y to any
other quantity x. Let us illustrate this situation by the diagram,
Fig.1, in which the values of x and y are represented in
the usual coordinate manner. In the general situation there will be a
known region, which we will represent by the area to the left of the vertical
line aa, and an unknown region, which will be represented by the
area to the right of this line. The first step, obviously, is to determine
the relation existing in the known region, which we will represent by
the line OP. The problem then reduces to a question of determining the
corresponding relation existing in the unknown region.
Fig. 1
Since the relations in this region are, by definition,
not capable of being determined in any direct manner by the means now
at our command, our procedure must be to assume some relationship,
then develop the consequences of this relationship, select those of the
consequences which fall into the known region, and finally compare these
particular consequences with the corresponding known facts. If we find
an agreement, this verifies the assumption to a degree that depends on
the number and variety of the correlation’s available; if there is
a disagreement, the assumption is invalid and we must discard it. The
problem therefore becomes a matter of deciding what assumption should
be tried first.
Inasmuch as the relation in the unknown region is necessarily
unknown, it could be almost anything. When we consider this general situation,
however, without the distracting influences that normally interfere with
a clear view of any particular physical situation, it is obvious
that there is one possible assumption that is inherently far superior
to all others: an assumption which has so much greater probability of
being a true representation of the physical facts that we are never justified
in even considering any other possibility until we have given the consequences
of this assumption a complete and thorough examination. This greatly superior
assumption is, of course, the hypothesis that the same relation which
prevails in the known region also holds good in the unknown region; that
is, it is an extrapolation from the known to the unknown.
Our analysis of the general situation along these lines
thus tells us that the first move in investigating the relation between
space and time in the universe as a whole should be to test the consequences
of extrapolating the relation that we find existing in the known region
of the universe. In this known region the relation between space and time
is motion, and in motion space and time are reciprocally related. The
analysis thus indicates that we should postulate a general reciprocal
relation between space and time effective throughout the universe.
This reciprocal postulate is the keystone of the new
system of thought of which the gravitational theory herein described constitutes
an integral part, and in order to place it in the proper perspective it
should be emphasized that this is essentially the only conceptual
innovation introduced into physical theory by this new system. It is true
that a great many novel ideas, some of them surprising, perhaps even startling,
emerge from the development of the consequences of this one basic innovation,
but this is simply a result of the fact that this one new concept is introduced
at the very base of the theoretical structure and it therefore has some
kind of an effect on almost every part of that structure. In recognition
of this major role which the reciprocal postulate plays in the system
as a whole, this system will be designated as the Reciprocal System
for the purpose of convenient reference in the subsequent pages. The
word “system” is used rather than “theory” because
the full development of the consequences of the postulates on which it
is based leads to a whole network of physical relationships, each of which
is comparable in scope to the gravitational theory which is the primary
subject of the present discussion. This new development is not merely
a theory but an interconnected system of theories.
It is evident that the reciprocal postulate necessitates
the further assumption that space and time have the same dimensions, since
quantities of different dimensions cannot stand in a reciprocal relation
to each other. We can recognize three dimensions of space, and the simplest
assumption that is consistent with both the reciprocal postulate and these
observed properties of space is that both space and time are threedimensional.
Equally necessary in order to permit a reciprocal relationship of anything
other than a purely formal character is the limitation of space and time
to discrete units. Neither of these additional assumptions involves any
great departure from current scientific thought. The possible existence
of dimensional time is a frequent subject of speculation in theoretical
circles, and the continual extension of the property of discreteness to
more and more physical phenomena, first to matter, then to electricity,
then to radiant energy, then (somewhat tentatively) to magnetism, makes
the further extension of the same concept to the basic entities, space
and time, practically inevitable in the long run, irrespective of the
requirements of this present work.
These three postulates constitute the definition of spacetime
as formulated on the basis of the considerations discussed in the preceding
paragraphs. Together with the further assumption that spacetime as thus
defined is the sole constituent of the physical universe, they
can be combined into one comprehensive postulate which may be expressed
as follows:
First fundamental Postulate: The physical
universe is composed entirely of one component, spacetime, existing
in three dimensions, in discrete units, and in two reciprocal forms,
space and time.
In addition to this First Postulate which defines the
physical nature of the universe, it will be necessary to make some further
assumptions as to its mathematical behavior, in order that we may utilize
mathematical processes in developing the consequences of the First Postulate.
Until comparatively recently the validity of the mathematical relations
which will be assumed in this work was generally considered axiomatic,
but it has since been discovered that other more complex and unconventional
relations are also theoretically possible, and although the existence
of any physical realities corresponding to these unorthodox mathematics
has never been definitely verified, these recent inventions are widely
employed in presentday physical theory. In setting up a new theory, however,
it is obviously advisable to return to the simpler and more manageable
concepts of earlier days, unless and until this policy encounters obstacles.
We therefore have the
Second Fundamental Postulate: The
physical universe conforms to the relations of ordinary commutative
mathematics, its magnitudes are absolute and its geometry is Euclidean.
It was demonstrated in the previous work that these postulates
are sufficient in themselves, without the aid of any supplementary or
subsidiary assumptions, to define a complete theoretical universe, but
for present purposes we will confine the discussion to those aspects of
the theoretical universe which are relevant to the subject of gravitation
and we will limit the development of the consequences of the postulates
to those items which are required for a complete understanding of the
gravitational relations.
On examination of the Fundamental Postulates it is immediately
apparent that they require a progression of spacetime identical
with that which was assumed as a basis for the development in Part
Two. Let us consider some location A in spacetime. When one more unit
of time has elapsed this location has progressed to A + 1 in time. According
to the First Postulate the one unit of time is equivalent to one unit
of space, since the postulate specifies that space and time are reciprocal,
hence this location also progresses to A + 1 in space.
It is also evident that the reciprocal postulate requires
something beyond this equivalence of the single unit of time and the single
unit of space. If this were the extent of the relationship we would postulate
that time and space are equivalent; not that they are reciprocal. In order
to make the relation reciprocal there must be certain conditions, under
which associations of n units of one component exist, in which case the
sense of the postulate is that these n units of the one component are
equivalent to 1/n units of the other.
Next we will want to know how these associations originate;
that is, how it is possible to modify the 1 to 1 ratio of space to time
which exists in the general spacetime progression. It is evident that
such a modification cannot take place in spacetime itself, as space and
time are equal in any unit of spacetime and are therefore equal in any
number of units or any succession of units. The modification must be accomplished
by some alteration in the factors affecting space or time individually,
and in order to permit such an alteration there must be a difference between
space (or time) individually and space (or time) as a component of spacetime:
the capacity in which it participates in the spacetime progression. The
only such difference for which where is any provision in the postulates
is a difference in direction, and we therefore arrive at the conclusion
that spacetime as such is scalar and that direction is a property of
space and time individually.
On this basis, if we replace an individual unit of one
component by a multiple unit so that this multiple unit of the one component
is associated with a single unit of the other, the direction of the progression
of the multiple component must reverse at the end of each unit. Inasmuch
as spacetime is scalar, this reversal of space direction or time direction
means nothing from a spacetime standpoint and the regular rate of 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
involves 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 a multiple
unit replaces the normal unidirectional spacetime progression with a
progression which merely oscillates back and forth over the same path.
But when the translatory motion in one dimension of space is eliminated,
the oscillating unit is confined to a single space unit, and this unit
of space then progresses in the normal manner in another dimension, carrying
the oscillating unit with it. When viewed from the standpoint of a reference
system that does not progress, the combination of an oscillating progression
in one dimension and a unidirectional progression in a dimension perpendicular
to that of the oscillation takes the form of a sine curve.
Obviously this feature of the theoretical universe defined
by the Fundamental Postulates can be identified as radiation. Each
oscillating unit is a photon, and the spacetime ratio of the oscillation
is the frequency of the radiation. Since spacetime is scalar the
actual spatial 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 defined by the
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.
Another possible motion of the oscillating photon is
a rotation. Let us consider next the factors involved in rotational motion
of these units. Rotation differs from translation only in direction and
this difference has no meaning from a spacetime standpoint since spacetime
is scalar. Rotation at unit velocity is therefore indistinguishable from
the normal spacetime progression: that is, from the physical standpoint
it is essentially the equivalent of no rotation at all. In order to produce
any physical effects there must be what we will call a displacement:
a deviation from unity. The deviation is necessarily upward, as fractional
units do not exist, and the magnitude of any rotational motion of the
photons is therefore greater than that of the spacetime progression.
A second necessary characteristic of the rotational motion
of the photons is that its direction must be opposite to that of the spacetime
progression, because any added displacement in the positive direction
would result in a directional reversal and would produce a vibration rather
than a rotation, as previously explained. This means that when the photon
acquires a rotation it travels back along the line of the spacetime progression,
and since this retrograde motion is greater than that of progression (at
unit distance) these rotating units are reversing the pattern of free
spacetime and are moving inward toward each other, either in space or
in time, depending on the direction of the displacement.
For present purposes we will consider only those photons
which are moving inward in space. Like the photons that are moving transitionally,
the rotating photons of this type which exist in the theoretical universe
defined by the Fundamental Postulates are readily correlated with observed
physical entities. With the exception of a few that are dimensionally
incomplete, they can be identified as atoms. Collectively the atoms constitute
matter, the inward motion resulting from the rotational velocity
is gravitation, and the incomplete atoms are subatomic particles.
At this point, then, it is clear that the two gravitational
assumptions of Part Two are necessary and direct consequences of the two
Fundamental Postulates; that is, the postulates lead directly to a progression
of spacetime and to an inherent motion of the atoms of matter in the
direction opposite to the progression. All of the conclusions and relations
derived from the gravitational assumptions in Part Two can therefore be
incorporated en bloc into the theoretical system that is here developed
from the Fundamental Postulates.
X
A very significant point about the theory outlined in
the foregoing paragraphs is that the same feature of the theory, which
leads to the existence of matter–rotation of the oscillating photons
in the direction opposite to the spacetime progression–also, causes
matter to gravitate. This is, of course, a major step toward simplification
of the basic physical relationships. It brings within the scope of one
general theory two important items, which have hitherto required completely
separate treatment. But this is by no means the full extent of the unification
of the theoretical structure that has been accomplished. The developments
in Part Two, which are also part of the new theoretical structure, since
the assumptions of Part Two have now been shown to be necessary consequences
of the Fundamental Postulates of the Reciprocal System, go on to derive
from the same initial premises the major characteristics of
gravitation, the instantaneous action, the absence of a medium and the
impossibility of modification, and in addition they explain the principal
deviations from what we may consider the “normal” pattern
of gravitational action in two of the three regions in which such deviations
occur. The somewhat abbreviated view of the situation given by the two
gravitational assumptions of Part Two was not capable of explaining the
deviations in the third of these regions, the region of atomic and molecular
interaction, but a more complete development of the consequences of the
Fundamental Postulates brings the relationships in this region within
the scope of the gravitational theory.
Ordinarily the effect of the spacetime progression is
to move physical objects farther apart. Any two objects which are initially
separated by x units of spacetime will be separated by x + n
units of space after n additional units of time have elapsed,
as each unit of time is equivalent to a unit of space. However, if there
is a spacetime displacement of such a nature that the two objects are
initially separated by x units of time in association with one unit of
space (that is, by the equivalent of less than one unit of space) the
result is quite different. In this case the progression takes place entirely
within one unit of space and consequently space remains constant at one
unit while time is free and progresses in the normal manner. In view of
the reciprocal relation between space and time, the increase in time due
to the unilateral progression is equivalent to a decrease in space, hence
the effect of the progression on two objects initially separated by the
equivalent of less than one unit of space is to decrease the equivalent
space separation.
At first glance it appears inconsistent for the same
spacetime progression to move objects farther apart in one region and
to move them closer together in another region. As emphasized in the previous
publication, however, the seeming inconsistency is due to the use of the
wrong datum in evaluating the situation. Because of the equivalence of
the unit of space and the unit of time, the initial point of all physical
activity is at unity, not at the mathematical zero. When we recognize
this fact, the apparent inconsistency disappears. Now the progression
always proceeds in the same natural direction: away from unity.
Above unit distance, away from unity is outward; below unit distance it
is inward.
Inasmuch as gravitation, by reason of its inherent nature,
always acts in the direction opposite to that of the progression, a similar
reversal occurs in the gravitational direction. Above unit distance, the
gravitational motion is inward toward unity. Below unit distance it acts
in the same natural direction, toward unity, but in this case toward unity
is outward. Gravitation therefore exerts a force of attraction between
two masses which are initially more than one space unit apart, but it
exerts a force of repulsion between two masses which are initially less
than the equivalent of one space unit apart.
With the benefit of this information, the nature of the
interatomic force equilibrium now becomes clear. In the region outside
unit distance there can be no equilibrium, as any motion resulting from
an unbalanced force accentuates the unbalance. If the inwarddirected
gravitational force exceeds all outwarddirected forces, for example,
an inward movement takes place, which strengthens the already dominant
gravitational force. In the region inside unit distance, on the other
hand, any movement due to an unbalanced force reduces the unbalance and
tends toward equilibrium. Here an excess gravitational force causes an
outward movement, which weakens that force, and ultimately reduces it
to equality with the constant force of the spacetime progression, whereupon
the motion ceases and the two objects take up equilibrium positions.
This explanation accounts for the existence of cohesion
in solids and liquids and thus extends the application of gravitational
theory to another major physical field for which a completely separate
theoretical structure has hitherto been necessary. With this addition,
the scope of the theory now includes the entire range of space intervals
from the shortest interatomic distance to the separation between the most
distant galaxies.
Throughout this entire immense region, the concept of
a progression of spacetime outward from unity provides the additional
effect of a general nature, which is necessary in order to account
for various phenomena that have hitherto resisted explanation. Physicists
are understandably very reluctant to accept any such idea. It is quite
distasteful to be compelled to admit that an unrecognized force of general
applicability can still exist after the many centuries that they have
devoted to intensive study of physical relations, and consequently they
are strongly inclined to close their eyes to the fact that many existing
situations are wholly inexplicable unless such a force is effective. But
occasionally an admission is encountered in the literature. Gold and Hoyle,
for example, tell us candidly, “Attempts to explain both the expansion
of the universe and the condensation of galaxies must be very largely
contradictory so long as gravitation is the only force field under consideration.
For if the expansive kinetic energy of matter is adequate to give universal
expansion against the gravitational field it is adequate to prevent local
condensation under gravity, and vice versa. That is why, essentially,
the formation of galaxies is passed over with little comment fin most
systems of cosmology.^{40}
The analogous problem of the globular clusters is “passed
over” with even less comment. Examination of one after another of
the available textbooks and dissertations on astronomy gives us no indication
that there is any difficulty in accounting for the existence of these
numerous and conspicuous objects. But if we search diligently through
the astronomical journals we will find a few papers in which attempts
have been made to analyze this problem, and almost invariably these papers
begin with an admission such as the following from E. FinlayFreundlich: “All attempts to explain the existence of isolated globular star
clusters in the vicinity of the galaxy have hitherto failed.”^{41}
Here again the difficulty arises from the fact that at
least two forces are required in order to explain the existing
situation, whereas the astronomers have only one force to work with. “Their
structure must be determined solely by the gravitational field set up
by the stars which constitute such a cluster,” says Freundlich, and
he goes on to admit that on this basis, “The main problem presented
by the globular star clusters is their very existence as finite systems...” In view of the absence of any evidence of rapid rotation, the only theory
which the astronomers have been able to produce is that the behavior of
the stars in the cluster is analogous to that of the molecules in a gas. “But the analogy is useless,” says R. v. d. R. Woolley, the
Astronomer Royal, “unless collisions among the stars are sufficient
to set up equipartition of energy, and not very valuable unless the mean
free path for stellar encounters is small compared with the dimensions
of the stellar aggregation considered. Calculation shows that the mean
free path is probably large compared with the effective diameter of a
cluster, so that in clusters the analogy with a gas is an idea that cannot
be pushed very far.”^{42}
All that we actually know about these clusters indicates
that they are very stable structures and have probably existed in approximately
their present forms for billions of years. This implies that they are
in a state of equilibrium: a condition that existing theory cannot explain.
Both Woolley and Freundlich concede that an isothermal equilibrium similar
to that, which would be the final result in a gaseous system, would involve
dispersion of the cluster. “The isothermal gas sphere’ is not a finite
object,” Wooliey admits. Those astronomers who attempt to approach
this problem are thus forced either to assume that the clusters are not
in equilibrium. In the face of the observational evidence indicating that
they are among the most stable and permanent of all astronomical objects,
or else to modify the gas relations in an arbitrary and ad hoc manner,
which destroys the cogency of the argument in favor of using the gas analogy.
Resort to such tactics is not necessary in the theoretical
universe of the Reciprocal System, as this system provides the additional
force that is necessary for equilibrium. Here each individual star (or
multiple star system) is outside the gravitational limits of its neighbors
and hence is moving away from them because of the spacetime progression.
At the same time, however, the gravitational effect of the cluster as
a whole is pulling the individual stars in toward the center of the cluster.
The net effect is equilibrium between these opposing motions (or forces).
When we turn to the interatomic situation, we find essentially
the same thing. Here again the gravitational force and the postulated
electrical force of attraction between the atoms are not sufficient to
account for the equilibrium that actually exists, but the complete inadequacy
of the currently accepted theories in this particular respect is simply
ignored by all but a very few of the physicists. One of the few authors
that has recognized and conceded the true nature of this problem is Karl
Darrow, who examined the whole situation in 1942 in an article entitled
Forces and Atoms.^{43} Darrow points out that both gravitation
and the electrical force of cohesion (the existence of which he assumes,
in accordance with current theory) are forces of attraction, and in order
to produce an equilibrium there must be a third force: an “antagonist” to the attractive forces. “This essential and powerful force has
no name of its own,” Darrow explains, “This is because it is
usually described in words not conveying directly the notion of force.” By this means the physicist “manages to avoid the question.”
Darrow concludes his discussion of the problem with the
comment, “This combination of a shortrange attraction with a repulsion
still shorter in range cries out for explanation. Could one but somehow
reduce it all to inversesquare forces, one would be more contented.” This comment is particularly apropos at the moment, since what the Reciprocal
System has accomplished is essentially what Darrow envisions: a reduction
of the whole problem to equilibrium between an inversesquare force and
a force of constant magnitude. In this system the constant spacetime
progression provides the inward force that is responsible for cohesion,
while the gravitational force, reversed at the unit level, is the “antagonist” that accounts for the equilibrium.
The validity of this definition of the interatomic force
system can be verified by an examination of the response of the system
to external forces impressed upon it; that is, by a study of solid and
liquid compressibility’s. A comprehensive study of this kind has been
carried out in connection with the investigation on which this present
work is based, and it is planned to include a summary of the results in
a new and more complete edition of The Structure of the Physical Universe
to be published in the near future. This data will show that the theoretical
compressions derived from the equations of the Reciprocal System are in
agreement with the measured values, within the probable experimental error,
over the entire range of the experimental work, up to 100,000 atm.. static
pressure and to several million atmospheres by the recently developed
shock wave techniques.
XI
In our ordinary experience space and time appear to be
altogether different in character. Space is the kind of a phenomenon that
we intuitively feel that we can understand: a wellbehaved entity with
a comfortable sort of permanence that makes observation and measurement
relatively simple. It is true that there are some basic philosophical
questions concerning its ultimate nature that have been controversial
issues ever since man first began to speculate about such subjects, but
nevertheless space can be broadly classified as one of the more familiar
features of our physical universe.
On the other hand, time has always been mysterious and
elusive. It undoubtedly exists; there is certainly something that distinguishes
the present from the past and from the future, and there is certainly
some physical meaning attached to the symbol t that enters into so many
of the mathematical expressions that we; use for the purpose of expressing
physical relationships. But when we attempt to be more specific and to
develop a more tangible concept to replace these rather hazy ideas, we
encounter some extraordinary difficulties. We have not even been able
to devise any direct measurement of time; the best we can do is to select
some type of periodic motion and to assume that successive coincidences
of identifiable spatial points connected with this motion distinguish
intervals of time.
The most striking and prominent feature of time, as we
observe it, is the continuous flow or progression. This is, in fact, just
about all that we know about time. The most prominent feature of
space is its extension in three dimensions. According to the Fundamental
Postulates of the Reciprocal System, however, space and time are absolutely
symmetrical and all of the properties now observed in either space or
time individually are actually applicable to both. On the basis of this
new viewpoint the great dissimilarity in the observed characteristics
of the two entities is not due to any real difference between the two,
but is a result of the gravitational motion of matter in the direction
opposite to that of the spacetime progression. This oppositely directed
motion in space cancels the effect of the space progression in the local
region and the results of the progression are visible only at the extreme
range of our giant telescopes. The existence of this phenomenon has therefore
remained unrecognized.
In our observations of time we recognize the progression
but not the threedimensional extension. A modification of the normal
spacetime ratio by substitution of an association of units for a single
unit of one of the components produces a motion either in space or in
time, but not in both. The inherent motion of matter is in space (because
it is the units whose motion is in space that we call matter) and so far
as matter is concerned, the progression of time continues as in free space.
Since the velocity of the progression is so high, 186,000 miles per second,
the differences in time location comparable to the differences, which
we observe in spatial location, are, in most instances, relatively minor,
and they are so overshadowed by the progression that their true nature
has not been perceived. Here again, there actually is a noticeable effect
under extreme conditions (motion at very high velocities) but it has not
hitherto been realized that this discrepancy is chargeable to a misconception
of the properties of time.
Fig.2
As an aid in visualizing this situation, let us consider
the motion of a distant galaxy. In the constellation Hydra there is a
faint galaxy which, according to the red shift in its spectrum, is receding
from us at a velocity of over 35,000 miles per second, onefifth of the
velocity of light. So far as we are able to determine t his galaxy is
moving directly away from our location and, except for the somewhat lower
velocity, the recession of this and other distant galaxies has exactly
the same characteristics as the progression of time: that is, it is a
scalar motion which always proceeds in the same direction: linearly outward.
According to the findings of this work, this galactic recession not only
appears similar to the progression of time; it actually is the
same kind of a phenomenon. It is the space equivalent of the time progression.
The only reason why the galactic velocity is lower than that of light
is that the Hydra galaxy, in spite of the enormous distance which separates
it from our location in space, is still close enough to be subject to
a small gravitational effect. At greater distances there undoubtedly are
galaxies which are receding from us at practically the full velocity of
light.
In Fig.2 the Hydra galaxy was at point A at time to.
During a time interval t the recession carries it from point A to point
B. The intervening distance AB is the space equivalent of a time interval
resulting from the constant progression of time, and since we refer to
the latter as clock time, we may utilize the same terminology and
call the distance AB the clock space. From our far distant location,
this movement from A to B in clock space is the only movement of the Hydra
galaxy that we can distinguish, but we know from observation of less distant
galaxies that these galactic aggregations also have random motions in
space, and we can therefore deduce that the Hydra galaxy will not actually
be found at point B when t units of clock time have elapsed; it will be
found at some other point C. The random motions of the galaxies are not
restricted to the dimension of the recession; that is, the distance BC,
which represents the random motion during time t is not necessarily a
prolongation of AB, but may have any direction in threedimensional space.
The total distance traveled by the Hydra galaxy during time t is therefore
the vector resultant of the clock distance AB and the distance BC due
to the random motion, the latter being a distance in the familiar threedimensional
space of our everyday experience. In order to distinguish this kind of
space from the clock space, we may call it coordinate space, since
we usually define it by means of some coordinate system.
Summarizing the foregoing, we may say that the total
space traversed by the Hydra galaxy in any specified interval of clock
time consists of two separate components: the clock space, which
is the distance covered by the galactic recession (the progression of
space) and the coordinate space, which is the distance covered
in threedimensional space by the random motion of the galaxy. Inasmuch
as there is no reason to believe that this particular galaxy is exceptional
or privileged in this respect, we may apply the same conclusion to all
galaxies, including our own. In the case of the far distant galaxies,
such as Hydra, we can detect only the motion in clock space; in observation
of our own or other galaxies of our local group the motion in clock space
is masked by the gravitational motion, and we see only the motion in coordinate
space. But it is clear that these are merely observational deficiencies;
the two separate components exist in all cases whether we can detect them
or not.
In view of the reciprocal relation between space and
time it is evident that exactly the same conclusions apply to time. The
total time interval in any physical situation includes not only the clock
time due to the constant time progression, a onedimensional movement
analogous to the galactic recession, but also another component, the coordinate
time, due to random movement in threedimensional time. Normally we
detect only the clock time because the coordinate time component is negligible
(relatively), but under extreme conditions, such as very high velocities,
the coordinate time may be quite significant. As in the analogous case
of the receding galaxy, the actual point c in time occupied by an object
at a particular stage of the progression is something other than the clock
time b, and since many values of c may correspond to the same value of
b, the true time interval cannot be expressed in terms of the clock system
of reference (that is, in clock time) except in certain special cases,
such as that in which b and c are practically coincident (which is true
at low velocities) or where the local motion follows a pattern of a restricted
type, such as uniform transnational velocity.
This is the basic error in all previous theories of motion.
All theories that have enjoyed any substantial degree of support have
assumed a onedimensional, onevalued time. “...we shall assume without
examination... the unidirectional, onevalued, onedimensional character
of the time continuum,”^{44}
says Tolman. But this is clock time, not the total time that actually
enters into physical processes.
Fig. 3
Newton’s Laws of Motion are based on the primitive
concepts of space and time: a threedimensional Euclidean space (coordinate
space) and a onedimensional time progressing uniformly and having the
same value at all points in space at each stage of the progression (clock
time). For two hundred years these laws met every test, with nothing more
than minor discrepancies which were not regarded very seriously. Then
in 1887 the MichelsonMorley experiment shattered the foundations of Newton’s
structure. Fig.3, adapted from Tolman,^{45}
shows the nature of the problem introduced by the results of this experiment.
Let us assume that a ray of light from a distant source S passes from
A to B and from A’ to B’ in two parallel systems. Then let us
assume that the systems AB and APB’ are in motion in opposite directions
as shown, and are in coincidence as the light ray passes A and A’.
Because of the motions of the respective systems, point B will have moved
to some point C closer to A by the time the light reaches it, whereas
B’ will have moved to some more distant point C’.
Yet if the results of the MichelsonMorley experiment
are to be believed, the velocity of the incoming ray at C is identical
with the velocity of the incoming ray at C’; that is, the velocity of
light is independent of the reference system. As Tolman expresses it,
the time required for the light to pass from A to C measures the same,
as the time required to pass from A ’to C’: a conclusion which, as he
says, is “in direct opposition to the requirements of socalled common
sense.”^{45}
This comment by Tolman shows very clearly just where
and how the thinking of the scientific profession was diverted into the
wrong channels. In reality the MichelsonMorley experiment does not
indicate that the time ac is equivalent to the time a’c’; it
merely shows that the velocity of light over the path AC is the
same as the velocity over the path A’C’. The further conclusion that the
two times are equivalent is not an experimental finding; it is an interpretation
of the experimental findings in the light of the currently popular
assumption as to the nature of time.
It is evident from the points brought out in the preceding
paragraphs that we do not need to abandon common sense to explain this
situation; all that we need to do is to get a broader view of time which
will encompass all of its properties, not just the progression. The correct
explanation of Tolman’s diagram is that points A and B are not only
separated by the coordinate distance AB; they are also separated by an
equal amount of coordinate time, since each unit of space, according to
the Fundamental Postulates, is equivalent to a unit of time. The movement
of point B to location C not only reduces the space separation between
the original location of A and the new location of B by the; amount of
coordinate space BC, but also reduces the time separation by bc, the same
amount of coordinate time. If the velocity of the system AB is relatively
low, as most velocities are in the world of our everyday experience, the
time bc is negligible in comparison with the time of the progression,
but if the velocity is great enough to make it necessary to take the distance
BC into account, then we must also take the equivalent time be into
account. The Newtonian concept of time in conjunction with the results
of the MichelsonMorley experiment leads to the relation
AC/t = A’C’/t, which is absurd, as Tolrman tells us. But when
we realize that the motion which reduces the distance from AB to AC also
reduces the time from ab to ac, the relation of the velocities in the
two systems becomes AC/ac = A’C’/a’c’ which is fully
in accord with both common sense and common mathematics.
The extreme condition is illustrated in Fig.4. Here two
light photons leave point O simultaneously and travel in opposite directions.
Photon a moves one unit of space OA in one unit of time. Photon b moves
one unit of space OB in one unit of time. (The system of space and time
units is immaterial. Irrespective of the conventional units employed,
we reduce them to the same proportionality by defining the velocity of
light as unit velocity.) According to Newton, the relative velocity of
the two photons is then 2/1 = 2, since the space separation at the end
of one unit of time is AB, or two space units. But the experimental results
show that the velocity of light is independent of the system of reference,
and that the relative velocity is actually one unit, not two. Newton’s
system therefore gives us the wrong answer.
Fig. 4
Einstein met the situation by abandoning the concept
of absolute space and time magnitudes and accepting the hypothesis, originally
advanced by Fitzgerald, that space contracts in the direction of motion.
On this basis the distance AB no longer has the value 2, as it does in
Newton’s system. The constant velocity of light is accepted as a
fundamental property of nature, and it is assumed that the distance AB
is automatically reduced to whatever value is necessary in order to make
the quotient s/t equal to this constant velocity, generally represented
by the symbol c or, as in the present discussion, defined as unit velocity.
In the Einstein system the equation of motion for Fig.4 becomes s/1 =
1, where s is arbitrary, or more generally s/t = 1, where both s and t
are arbitrary, as for some purposes, at least, it becomes necessary to
assume a dilatation of time as well as a contraction of space.
Einstein’s concept of space as a purely relative magnitude,
varying according to the location and velocity of the observer, is incompatible
with the somewhat intuitive ideas synthesized from everyday experience
and generally described by the term “common sense.” The scientists
of the early twentieth century were therefore very reluctant to accept
it, but Newton’s system, with its absolute time and space, had been invalidated
by the experimental demonstration of the constant velocity of light, and
since no plausible alternative was proposed (other than some variations
of the Relativity Theory itself) this concept of a “rubber yardstick” won acceptance by default. A factor that contributed greatly to this acceptance
was that the principal obstacle standing in its way, the prejudice against
violation of common sense principles, was undermined by the fact that
the experimental results appeared to be, as Tolman remarked in the statement
previously quoted, “in direct opposition to the requirements of socalled
common sense.” Obviously if the facts themselves are in conflict
with common sense, it is no longer consistent to demand that theory stay
within common sense boundaries.
With the benefit of the information that has been developed
in this work, however, it is clear that Einstein’s drastic step in abandoning
absolute space and time was neither necessary nor justifiable. Both Newton
and Einstein failed to recognize that there are two components of physical
time and both set up their theories on the assumption that we are dealing
only with clock time. On this assumption the time required by photon a
to travel from O to A is the same unit of time as the time required
by photon b in traveling from O to B. But the unit of distance is separate
and distance from the unit OA and since two separate units of distance
are traversed by the photons, the equivalence of the individual units
of space and time postulated in this work requires the corresponding units
of time to be separate and distinct. In other words, when the photons
are at points A and B respectively and are separated by two units of space
they are also separated by two units of time. The equation of motion is
then 2/2 = 1, which is completely in accord with the results of the experiments.
It is now apparent that Tolman misunderstood the nature
of the message, which the MichelsonMorley experiment was trying to convey
(as did Tolman’s colleagues, including Einstein. Tolman’s work is being
used for purposes of illustration merely because it is a particularly
clear presentation of the currently accepted viewpoint). The results of
this experiment do not, as Tolman asserts, require us to contradict the “requirements of common sense” and accept the time elapsed between
A’ and C’ as being the same as that between A and C. What these results
actually tell us, if we read their message correctly, is that the concept
of time which leads to this absurd conclusion, the concept that has hitherto
been universally accepted, is wrong.
This orthodox concept of time is based on a narrow view
which recognizes only one of its aspects, the progression, whereas the
preceding pages have shown that a theoretical analysis of the situation,
supported by observations of the motions of the distant galaxies and by
the observed properties of radiation, leads to the conclusion that time
also possesses all of the attributes that are recognized in space. When
it is thus realized that space and time are completely symmetrical, it
becomes apparent that all of the magnitudes applicable to time are commensurate
with the corresponding magnitudes applicable to space. The individual
locations are not necessarily coincident. In Fig.3. for example, the time
locations a, b and c, which now correspond to space locations A, B and
C respectively, may be changed to d, e and f as a result of the progression
of time, but the new time separation de then corresponding to the space
separation AB will still be equal to AB and also to ate. This simplifies
the problem of measurement of the time separation very materially, since
we can readily measure the coordinate space separation between any two
accessible objects, and this value, when expressed in appropriate units,
is also the coordinate time separation between these objects.
Both the complexities and the limitations of Einstein’s
Relativity Theory arise from the fact that he was unable to see the broad
picture and attempted to describe all physical events in terms of clock
time only. As indicated in the preceding discussion, a revision of the
basic concepts to take all of the properties of space and time
into account eliminates the necessity for any arbitrary manipulation either
of the mathematical relationships or the physical magnitudes. Everything
then falls into line easily and naturally without any kind of artificial
maneuvering or any conflict with common sense.
XII
One of the most significant features of the Reciprocal
System is that it is a purely theoretical construct. All of its elements
are sharply and positively defined because they are derived from pure
theory and have no empirical content (even though the theory itself was
derived by inductive reasoning from empirical premises). This system is
therefore completely untouched by most of the uncertainties and ambiguities
that have troubled conscientious critics of previous theories. Bridgman
points out; for example, that Relativity Theory is based in the first
instance on the assumption of the existence of some kind of a physical
framework defined by means of rigid measuring rods and clocks. But, as
he says, there is no “very articulate analysis” of what is meant
by “rigid,” and he continues, “The specification of what
is meant by a clock is usually even less articulate, and has been felt
to be a matter of much difficulty by a number of critics... in practice
it almost appears as though the only criterion for a clock is whether
it functions in the way that the equations demand that a clock function.”^{46}
Similar considerations apply to all of the concepts entering
into physical theory, and a recognition of this situation has led to the
emergence of the “operational” point of view, the adherents
of which contend that physical concepts should be defined solely in terms
of the operations by which they will be detected and measured. This is,
of course, impossible for such elementary concepts as those of clocks
and measuring rods, but once these basic elements have been defined, the
methods by which they are utilized in observing and measuring other more
complex entities can be used as bases for the physical definition of those
entities. Such reasoning, applied to the concept of simultaneity, played
a major role in Einstein’s development of the Theory of Relativity and
it has exerted a significant influence on modern physical theory as a
whole.
Questions of this kind do not arise in the Reciprocal
System. The universe developed from the Fundamental Postulates of this
system is purely theoretical and any concept such as that of a clock can
be specifically defined on a strictly theoretical basis. By means of the
same structure of theory the kind of a physical entity that can qualify
as a clock under this definition can also be specified unambiguously.
Up to this point it is not necessary to take the actual physical world
into consideration in any way, but once we have arrived at a conclusion
which, for example, might be that any body in uniform rotational motion
constitutes a clock, the next move is to examine the physical universe
to determine whether or not we can identify any body in uniform rotational
motion. If we can find such an entity, then we have a clock. In following
this procedure we have actually used, on a rigidly correct theoretical
basis, the practical criterion described by Bridgman; that is, we say
that a rotating body is a clock because it functions in the way that a
clock theoretically should function. If it functions exactly according
to the theoretical requirements, then it is an accurate clock.
This is the same procedure that we apply to all physical
phenomena when we utilize the Reciprocal System. We first determine what
sort of thing should exist in the theoretical universe and then we look
to see if we can find an entity in the observed physical universe, which
conforms to the theoretical description. For instance, we do not put gravitation
into our theoretical universe because we know that it exists in the real
physical universe. We put nothing into the theoretical universe
but the Fundamental Postulates, and gravitation is there only because
its existence is a necessary and unavoidable consequence of those postulates.
Similarly, the properties, which are attributed to gravitation in the
theoretical universe, do not depend in any way on the properties that
we find by observation of the physical phenomenon of gravitation; these
theoretical properties are also necessary consequences of the postulates.
Of course, the theoretical development gives us no names;
it simply gives us a description of the various components of the theoretical
universe and, unless we wish to coin some new names for the theoretical
entities, which would be confusing and would serve no useful purpose,
we have to locate the physical entity corresponding to the theoretical
description in each case before we arrive at a name. The theory merely
tells us, for example, that a certain phenomenon exists in which oscillating
units, with various frequencies of oscillation, originate at different
points in space and travel away from these points in all directions at
a constant velocity. When we turn to the actual physical universe, we
find that a phenomenon with exactly the same characteristics also exists
there. We are then justified in concluding that this physical phenomenon,
which we call radiation, is the physical equivalent of the theoretical
phenomenon deduced from the Fundamental Postulates, and we therefore apply
the name “radiation” to both.
When we have thus identified a physical entity with its
theoretical equivalent, we can carry over into the physical field all
of the properties and relationships applying to the corresponding theoretical
entity, irrespective of whether or not the available observational data
are adequate for a physical verification of all of the theoretical conclusions.
In the case of gravitation, for instance, we can confidently assert that
the gravitational effect is instantaneous, because it is necessarily instantaneous
in the theoretical universe, even though this point is contested by most
theoretical physicists, including Einstein, not because of any evidence
to the contrary, but because it is inconsistent with their theories.
In asserting that the gravitational effect is instantaneous,
the Reciprocal System does not arrive at any new conclusion; it
merely takes a position on one side of a longstanding controversy, but
we are equally justified in going still farther and applying to the physical
universe other features of the theoretical universe defined by this system
which are completely new and in some cases totally foreign to current
thinking. The reversal of direction of gravitation at unit distance is
an item of this kind. Here is something that has never even been suspected,
and which does not fit in very well with orthodox lines of thought. It
will consequently meet with much resistance, but there is no logical or
factual basis on which it can be rejected, since it is not inconsistent
with any known fact, while on the affirmative side there is the very powerful
argument that the explanation of the cohesion of solids provided by this
gravitational reversal enables the interatomic distances in solids, and
the changes in these distances under pressure, to be accurately calculated
from pure theory.
This is the kind of a place where the immense advantage
of a theory that is both complete and correct makes itself
manifest. All that we know about solid and liquid cohesion form observation
and experiment is that there must be two forces involved in the interatomic
equilibrium: a force of attraction that holds the atoms together in the
two condensed states, and a force of repulsion that limits the closeness
of approach. (This is equally true whether or not the atoms are in contact.
If they are in contact there must still be a force within the atom resisting
deformation.) It does not appear likely that any more detailed information
about these forces will be obtained from observation without some fairly
specific clue as to what to look for, and in order to find such a clue
we must first formulate a theory of the interatomic force system.
Heretofore there has been no available source of such
a theory other than pure invention, and it is painfully obvious that the
theorists’ inventive capacity has been completely inadequate for this
task. There is no doubt as to the existence of the repulsive force. If
we apply external pressure to a solid or liquid aggregate, the atoms move
closer together, and as they do so the resistance to the compression increases
as a function of the displacement from the original equilibrium positions–up
to the experimental limits of several million atmospheres, at least–but,
as Darrow pointed out in the article previously quoted,93 the physicists
have not even attempted to construct a theory which would account
for anything other than the for ce of attraction. They have simply “managed
to avoid the question” of the “essential and powerful force” that plays the role of an “antagonist” to the attractive force.
A hypothesis which makes no attempt to account for more than one of the
two participants in an equilibrium certainly cannot claim to be an explanation
of the phenomenon in question, and presentday science therefore has nothing
that can be legitimately called a theory of the interatomic forces.
In such a case, where the information obtainable from
direct observation of the phenomenon itself is too meager to point the
way to an adequate theory, there is an alternative possibility: the theoretical
principles governing the situation may be deduced from relationships previously
established in collateral fields. This is where the necessity for a complete
and correct theory arises. An incomplete and approximate theory may be
of considerable value in the field to which it is directly applicable,
but it seldom accomplishes anything of any consequence outside of that
field. Even the gravitational theory based on the two assumptions of Part
Two, which is actually correct, as far as it goes gives us no help at
all with the interatomic problem. But when these two gravitational assumptions
are traced back to their source in the Fundamental Postulates of the Reciprocal
System, and the further consequences of these postulates are developed
in detail, we can obtain a complete and comprehensive picture of the interatomic
forces in all of their manifestations and the acute questions concerning
the nature of solid and liquid cohesion, including the identification
of both the attractive and the repulsive forces, are answered as a part
of the clarification of the general situation.
In the present state of physical knowledge it is not
at all likely that a hypothesis such as that of a “natural” direction of gravitation– always toward unity–would ever be formulated
ad hoc; there is nothing elsewhere in the physical universe to
suggest anything of this kind. On the other hand, when it develops that
this constant direction toward unity, which involves a reversal of the
gravitational force at unit distance, is a necessary and unavoidable consequence
of principles firmly established in other fields, and on applying this
conclusion to the cohesion problem we find that it accounts for the observed
facts both qualitatively and quantitatively, it is in order to conclude
that this hypothesis is a correct statement of the true physical situation.
Throughout the many fields, which have thus far been covered in developing
the details of the theoretical universe of the Reciprocal System, similar
cases have been encountered frequently. In each of these instances the
solution of a difficult problem of long standing was found to require
some conceptual innovation or reversal of a habitual trend of thinking
which by itself was almost inconceivable, yet proved to be fully in accord
with the known facts when a careful and critical examination was finally
undertaken because the need for such an innovation was established in
the course of the development of the Reciprocal System.
