“To attempt a definite statement as to the meaning of so fundamental and underlying a notion as that of time is a task from which even philosophy may shrink,”^{1} says Richard Tolman in his classic treatise on Relativity. But the “notion” of time is basic in every field of science. In legal documents we often see the expression “Time is the essence of this contract.” It is no less the essence of physical theory—without the symbol *t* and all that it stands for, there would be little left in physical science. In order to make a definite and meaningful statement about *any* physical phenomenon it is therefore necessary to define the concept to which the name “time” is to be attached. This definition may not actually be expressed—indeed it is seldom expressed except in such basic works as Tolman‘s—but in any work that lays claim to scientific accuracy, the exact meaning of this concept *must* be specified, implicitly if not explicitly. Those who use the concept without defining it are not evading this requirement; they are simply accepting a definition set up by someone who has preceded them. How then does science meet this serious challenge at the very base of its theoretical structure: the absolute necessity of a precise and unequivocal definition of an entity that is so difficult to grasp that the mere thought of trying to understand it appalls the scientist? Tolman tells us frankly how he and his colleagues have met this issue:

“we shall assume without examination the unidirectional, one-valued, one-dimensional character of the time continuum.”

^{2}

Physical science justifiably prides itself on the “rigor” of its treatment of the subject matter which it covers: precise definitions, clear-cut distinctions, careful and critical development of theory by exact logical and mathematical processes. But when we examine the foundations of this work, we find that the entire structure of carefully developed theory rests upon nothing more substantial than three items which are “assumed without examination.” Scientific precision has here taken the form of precise formulation of pure assumptions: the most unreliable of all instruments of thought. Unfortunately, precision is no substitute for validity; an assumption is no less uncertain and speculative because it is expressed in definite and exact language. As matters now stand we have not grasped the essence; we see it only through a thick veil of uncertainty. And without the solid foundation which only a clear understanding of the true properties of time can give us, all of our vaunted logical and mathematical precision is spurious; indeed, if the premises are false, the more precise the logical development the more certain we are to arrive at the wrong conclusions. The physicist who fills pages of the Physical Review with complex mathematical calculations may be giving us a development that, in itself, is faultless, but if any of the properties of time that have been “assumed without examination” are not valid, then he is introducing some kind of an error every time he uses the symbol t and, in spite of its impeccable outward appearance, the work as a whole may be completely wrong.

If physical science had been uniformly successful in building up a consistent, integrated structure of theory, fully capable of meeting all demands upon it, this serious defect in the underpinnings of the structure could well be viewed with equanimity, on the ground that the assumptions are justified by the results thereof. It is admitted on all sides, however, that in spite of the spectacular successes that have been achieved in many areas, physical science is still far from having a comprehensive and satisfactory basic theory. In fact, many scientists have given up in despair, and no longer consider the construction of such a theory to be within the range of possibility. C.N. Yang, for instance, was quoted in a recent news item as “expressing some doubts about the ability of the human brain in general and his in particular to accomplish this task,”^{3} and Henry Margenau admits that “To the outsider the conclusions reached by a modern physicist seem almost like a declaration of the bankruptcy of science.”^{4}

In the light of this situation it would seem that science has now reached the point where it can no longer avoid facing the issue as to just what the properties of time actually are. Of course, we have no positive knowledge that errors in the assumptions regarding these properties are responsible for, or have contributed to, the failure to construct a satisfactory basic physical theory, but where the best efforts of the most competent investigators over a long period of years have failed to produce the expected results, it is certainly much more likely that the fault lies in basic premises that have been assumed arbitrarily and “without examination” than in any lack of “ability of the human brain” to apply logical and mathematical processes to these premises. A thorough and painstaking examination of the validity of the assumptions that have been made concerning the properties of time is therefore very much in order.

The question then arises as to how this issue can be approached. The scientific profession has hitherto believed that there is no alternative to the use of pure assumptions of the kind listed by Tolman, but the investigations which I have carried out have disclosed that it is possible to apply a much more reliable process to this problem, and thereby to arrive at some different conclusions as to the properties of space and time which eliminate most, if not all, of the basic difficulties that physical science now faces. This new approach substitutes a process of *extrapolation* for the arbitrary assumptions heretofore utilized. It is true that extrapolation is also, in a sense, a process of assumption, but the extrapolation assumption, the assumption that the situation or relation existing in the known region also exists in the unknown region, is inherently vastly superior to *any* other assumption that can be made, with a far greater probability of being a true representation of the physical facts, and in any case where extrapolation is possible, it is obviously sound policy to give the consequences of such an extrapolation a complete and thorough examination before anything else is even considered.

As a general proposition, the superiority of this approach is not open to serious question, but a *direct* extrapolation does not appear feasible in this case, as we have no positive knowledge as to what the properties of space and time actually are *anywhere*, and consequently there is no adequate base from which to extrapolate. All previous investigators have therefore relied upon assumptions—some related to our rather vague general impressions of space and time, others wholly conjectural—not because they preferred to do so, but because they felt that they had no option. The method which I have employed to overcome the existing difficulties is to approach the question indirectly, beginning with an examination of the relation between space and time. This relationship is one that has never been adequately explored heretofore. In the days of Newton, its existence was not recognized at all, the two entities being regarded as completely independent. Since then there has been a growing realization that they are not independent and that basically we must deal with space-time, not with space and time individually. Thus far, however, it does not appear to have been suspected that the existing concepts of the fundamental *nature* of space and time may be in error—that time, for instance, may be actually something other than a “unidirectional, one-valued, one-dimensional continuum” —and the hypotheses that have been advanced as to the character of the space-time relation, such as Minkowski‘s concept of a four-dimensional continuum, have retained these basic assumptions and thus have simply plied speculation upon conjecture.

Instead of starting with arbitrary assumptions, the first move in the present investigation has been to extrapolate to the universe as a whole the relation between space and time which 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. This is not surmise or assumption, nor is its accuracy in any way open to doubt. It is positive knowledge from which we can extrapolate. Irrespective of the nature and properties of space and time individually, the method of extrapolation leads directly to the conclusion that we should postulate a *general reciprocal relation between space and time effective throughout the universe.*

Of course, *any* new viewpoint that conflicts with long-standing beliefs concerning space and time, no matter how firmly based it may be, will seem strange and hardly credible on first consideration, but nothing that we actually know about space or time is inconsistent with this reciprocal postulate. The truth is that we know very little about either of these entities. Time has always been mysterious and elusive, but even space, which seems so much ore understandable, has been a difficult problem for those who have sought to discover its true nature, and no general agreement on this score has ever been reached. To Aristotle, space was merely a *relationship* between physical objects; to Democritus and his fellow atomists it was a *container* in which such objects exist; to Einstein it was a *medium* connecting these objects. Certainly it cannot be claimed that there now exists any positive knowledge about the inherent nature of space to which a new theory must conform. On the contrary, the conclusion of this current investigation, which, in effect asserts that space is merely an *aspect* of *motion*, has a much greater a *prior* probability of being correct than any of its predecessors, since it has been reached by way of a more reliable process. Nevertheless, the proof of the pudding must be in the eating; that is, we must develop the consequences of the new concept and see whether they give us a more logical and consistent picture of physical relations than the currently accepted ideas.

It will not be possible in a short article of this kind to describe all of the results that have been obtained in the application of the reciprocal hypothesis to a wide variety of physical phenomena during the many years that this investigation has been under way, but the general nature of the results can be demonstrated by a typical example, and in the discussion that follows, the consequences of the reciprocal postulate will be developed far enough to produce an explanation of gravitation; something that no other physical theory has been able to do. The gravitational findings are particularly interesting because they not only demonstrate the ease with which this new development surmounts the difficulties that have stood in the way of progress in such areas as this, but also show why we get a distorted view of space and time from our everyday experience, and why most of the inferences as to the nature of these entities that we draw from such experience are erroneous and misleading.

No doubt many readers will be surprised at the assertion that gravitation still remains unexplained, as there is a very common misconception that Einstein‘s General Theory of Relativity supplies such an explanation. But, as Willem de Sitter has pointed out very clearly *no* hypothesis thus far advanced to *explain* gravitation “has ever had the least chance, they have all been failures.” Einstein‘s contribution, de Sitter says, is to make gravitation identical with inertia, and thus to put it in the category of “one of the fundamental facts of nature, which have to be accepted without explanation, like the axioms of geometry.”^{5} After fifty years, the inadequacy of this treatment is clearly apparent. As R. H. Dicke puts it, gravitation is still an “enigma,” and “It may well be the most fundamental and least understood of the interactions.”^{6} A recently published review of the proceedings of the First Soviet Gravitational Conference confirms this opinion with the following comments: “... the gathering seemed painfully perplexed with endless questions, nearly all of which remain unanswered.”^{7}

The crux of the gravitational problem is the dilemma which no previous theory has been able to avoid: the apparent necessity of postulating either action at a distance, which is philosophically unacceptable to most scientists, or propagation through a medium, which is completely lacking in observational support and is faced with seemingly insurmountable practical obstacles. After three hundred years in which it has been agreed that these are the only two possibilities, the new development based on the reciprocal postulate now produces a third alternative that has been completely overlooked by previous investigators: one in which gravitation acts in a perfectly natural and understandable way, instantaneously, without an intervening medium or a medium-like space, and in such a way that it cannot be screened off or modified in any way; all of which are exactly in accord with what our observations have indicated.

To begin the explanation of how these results were obtained, let us now return to the basic assumption of a reciprocal relation between space and time. It is evident that this assumption necessitates a further postulate 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 with the simplest assumption that is consistent with both the reciprocal postulate and the observed properties of space is that both space and time are three-dimensional. Limitation of both space and time to discrete units is also necessary in order to make the reciprocal postulate mathematically workable. Extrapolation of the relation between space and time that is observed in the phenomenon of motion thus leads directly to three conclusions about the properties of time and space which can replace the assumptions that the physicists have made “without examination” . Together with the further assumption that space-time as thus defined is the *sole* constituent of the physical universe, these can be combined into one comprehensive postulate as follows:

FIRST FUNDAMENTAL POSTULATE

The physical universe is composed entirely of one component, space-time (motion), 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, some further assumptions as to mathematical behavior will be necessary, and since this present development does not get into any difficulties of the kind that have forced modern physics to resort to the use of complex and abstruse mathematics, it will be possible to formulate the following simple postulate:

SECOND FUNDAMENTAL POSTULATE

The physical universe conforms to the relations of ordinary commutative mathematics, its magnitudes are absolute, and its geometry is Euclidean.

On examination of these two postulates, it is apparent that they require a progression of space-time similar to the progression of time as ordinarily visualized. Let us consider some location A in space-time. When one more unit of time has elapsed, this location has progressed to A+1 in time. Since one unit of time is equivalent to one unit of space, according to the First Postulate, this location has also progressed to A+1 in space. At the very outset, therefore, the new development confronts us with an important basic phenomenon which has not hitherto been recognized: a progression of space similar to the observed progression of time. We thus have an immediate opportunity to test the validity of the new system by observation of the actual physical universe. If space-time actually progresses, as the new theory contends, then we should be able to recognize some phenomena in which identifiable objects without inherent motion of their own are being carried along in space by the progression of space-time.

In order to simplify the question of a reference system, let us assume that a large number of such objects originate at the same space-time location, which means that they originate at the same space location simultaneously. Due to the progression of space-time these objects immediately begin moving outward, but outward in space-time is a scalar direction, and the spatial motions of the individual objects will be distributed over all possible directions in accordance with the probability principles. Hence if there actually is a progression of space-time, we should observe objects of this kind originating at various spatial locations and moving away from the points of origin in all directions and at a constant velocity. We do not have to look very far to find physical entities which display exactly this behavior. Throughout the physical universe there are sources of light or other electromagnetic radiation from which photons emanate in all directions and recede from the points of emission at a constant velocity. This radiation phenomenon therefore furnishes the definite independent evidence that is necessary to demonstrate the reality of the space-time progression.

Additional confirmation is provided by the motions of the external galaxies. All galaxies except our immediate neighbors are receding from us in exactly the same manner as the photons of light that originate in our galaxy, except for the fact that the relative galactic velocity is a function of the distance, and has only reached about one-fourth of the velocity of light at the extreme range of our optical telescopes, and perhaps one-half of the velocity of light at the greatest distance accessible to radio observation. The lower velocities of the galaxies as compared to the velocity of the light photons are quite obviously due to the modifying effect of gravitation which, even at these enormous distances, still exerts a small force of attraction that operates against the progression. Thus the reality of the space progression, a basic feature of the new theory that has no counterpart in any other physical theory, is substantiated by two independent lines of evidence.

Space limitations preclude a detailed discussion of the development of the consequences of the Fundamental Postulates to the point where they require the existence of matter, but for present purposes it should be sufficient to say that this development indicates that the atoms of matter are rotating units in which the direction of rotation is opposite to that of the space-time progression; that is, irrespective of the *spatial* direction in which the atoms are moving, their *scalar space-time direction* is always *inward*, directly opposite to the *outward* motion of the space-time progression. Whereas the progression is continually carrying all physical objects outward away from each other, the inherent rotational motion of the atoms is carrying them inward toward each other. This is the phenomenon that we call gravitation.

As an aid in visualizing how gravitation operates, according to this theory, let us assume that a violent explosion has taken place and that we are looking at the results shortly thereafter without any knowledge of what has happened. We still see a cloud of flying particles apparently exerting a force of repulsion upon each other, and we will observe that this force has some peculiar characteristics: it acts instantaneously, without an intervening medium, and in such a way that it cannot be screened off or modified. According to the new development, gravitation is a force of the same general nature, except that it acts in the inverse direction: inward instead of outward. Like the apparent force which the particles of debris exert on each other, gravitation merely *appears* to be an action of one mass upon another; in reality each mass is *pursuing its own course independently of all others.*

Inasmuch as the motion of the progression originates everywhere and is constant regardless of location, whereas the gravitational motion originates at the location which the atom happens to occupy, and the component directed toward any other atom therefore decreases with distance in accordance with the inverse square relation, there is a point at which the two velocities are equal. Inside this equilibrium distance the gravitational motion is the greater, and there is a net gravitational effect. Beyond the equilibrium point the motion of the progression is the greater, and objects move away from each other, the net outward velocity increasing with the distance as the gravitational effect decreases. The actual behavior of the universe is exactly in accord with these predictions of the new theory.

Throughout the physical realm the new viewpoint as to the nature of space and time derived by the relatively straightforward and dependable process of extrapolation similarly resolves the dilemmas and difficulties which have resulted from basing physical theory on pure assumptions. It is evident from these results that space and time are actually entities of the same nature and that the great differences which we seem to see in them are merely the result of the gravitational motion of matter. Gravitation conceals the effect of the space progression in our immediate vicinity, and the progression is observable only at extreme distances, hence the most evident property of space is its three-dimensionality. The progression of time, on the other hand, is unchecked by gravitation, and the velocity of the progression is so high that any motion in three-dimensional time is negligible (relatively) except at extreme velocities. We therefore recognize only the progression. But science is now penetrating the regions of extreme distance and very high velocities, where the progression of space and the three-dimensionality of time play significant roles, and in order to remove serious obstacles to a clear understanding of phenomena in these regions it will be necessary to take heed of the salient point disclosed by the extrapolation process of the present investigation: the fact that *both* space and time actually have *all* of the properties that have hitherto been attributed to *either* of them individually.

References