Paradoxes Galore

Paradoxes bring to light flaws in the logical structure of a theory. We have had the famous twin paradox of the Special Theory of Relativity. In our attempts to understand the Reciprocal System of theory some paradoxes seem to be coming up for consideration. One such paradox, which we will name the Quasar Paradox, has recently been mentioned by a student of the Reciprocal System, in a privately circulated communication. Since the correspondent opines therein that this paradox requires revision of Larson’s theory, it might be educative and worthwhile to discuss the issue with a hope to see the truth.

Simply stated the paradox is as follows.1 Larson establishes that the total redshift of a quasar is the sum of the recession redshift z, and that due to the explosion that created the quasar amounting to n*z(1/2), (where n is normally 3.5). As the distance increases and the recession redshift reaches the value 0.326 the explosion redshift arrives at the 2 unit limit. At this juncture, according to Larson, the gravitation inverts and ceases to be inward in space, resulting in the final disappearance of the quasar into the cosmic sector of the physical universe. Now, in the words of the correspondent,

    The problem is that an observer closer to the quasar would see the relation z + 3.5*z(1/2) as less than that seen by a more distant observer (z being less) and so the speed [in the explosion dimension] would be less than 2 for the closer observer. I feel... that a quasar cannot both fly apart and not fly apart at the same time depending [on] one’s point of observation.

The New Paradigm

We submit that before undertaking an analysis of the paradox it would be fruitful to draw attention to certain factors which act as preconditions for an unbiased appraisal of the Reciprocal System. The first thing to be recognized is that the Reciprocal System involves a fundamental change in our viewpoint concerning the basic constituent of the physical universe. Its principal tenet is that the universe is constituted entirely of motion. The previous viewpoint regards it as a universe of matter. The most important implication of this new viewpoint is that motion (space-time) is the content of the universe, whereas the concept of the universe of matter regards space and time as the background or setting on which matter plays. Throughout the ebb and tide of scientific thinking for the past 3000 years, Larson points out, the one unchanging element has been the ‘setting’ concept of space and time. This has become a thoroughly entrenched habit in the thinking of scientists and laymen alike.

Man’s endeavors to understand Nature have always been impaired by his limited and local viewpoints. He has tended to extrapolate what he perceived and experienced of the local and peculiar environs by merely enlarging their extent, without in the least suspecting that he might not be the centre of the universe. Only the increased power and scope of his observations have brought to light the fact that his global view is vitiated by his local viewpoint. He first thought that the earth was flat before realizing it is spherical. Then he went on thinking that his earth is the centre of the universe. The proliferation of epicycles eventually led to the discovery that it is the sun that is the real centre and so on. Every time such fundamental revolution in the view points had occurred it encountered bitter antagonism and a cold reception because the old and the new viewpoints were so disparate that the common man and the common scientist of the day could not grasp the truth.

We can now see history repeating itself as Larson discovers that our viewpoints about the most fundamental aspects of the physical universe, namely, space and time, have been, after all, local and limited. The view that space is stationary and three-dimensional and that time is one-dimensional and progressive is only apparently true and applicable only to the gravitationally bound system. Emancipating from this anthropocentric view and recognizing that both space and time are three-dimensional in their own right and progressive and that they are reciprocally related comprise the new revolution in human thought.

Larson states:

    Previous investigators have not realized that the “setting” concept is a creature of the “matter” concept; that it exists only because that basic concept envisions material “things” existing in a space-time setting. In attempting to construct a theoretical system on the basis of the concept of a universe of motion while still retaining the “setting” concept of space and time, these theorists have tried to combine two incompatible elements, and failure was inevitable. ... What is needed is to discard the “setting” concept of space and time along with the general concept of a universe of matter, to which it is intimately related, and to use the concept of space and time that is in harmony with the idea of a universe of motion.” (2)2

Realizing this, Larson has repeatedly cautioned in his works that the findings of the Reciprocal System ought to be adjudged from the standpoint of its basic premise(s) and that endeavors to evaluate the new concepts from the viewpoint of the untenable matter concept of the universe (and the associated “setting” concept) are going to lead to absurd results. We shall term the practice of using this old viewpoint in the context of the new theory the Fallacy of Incongruous Viewpoints.

The danger is especially strong for all of us who happen to live at the junction of the new paradigm of the universe of motion and the old untenable one of the universe of matter existing in a framework of space and time. Since none of us is in a position to maintain that we are absolutely infallible, it becomes imperative, whenever we encounter a difficulty or paradox in the Reciprocal System, to first establish that we have not unconsciously fallen prey to the Fallacy of Incongruous Viewpoints, before we can legitimately conclude that the logic of the theory is faulty.

Content vs. the Container

Now the crucial point to see is that it is not legitimate to imagine that the quasar is located ‘out there’ in our co-ordinate space. When we picture the large-scale universe we tend to imagine that the stationary co-ordinate reference frame--namely, the container space--as extending indefinitely in the three spatial dimensions and picture quasars and other distant galaxies as studded at specific locations in that stationary reference frame. This, of course, is an unconscious habit of thought carried over from the previous paradigm of ‘container space’ belonging to the concept of the universe of matter.

On examining we find that the stationary reference frame is an artifact applicable only to a gravitationally bound system of material aggregates. The very existence of the stationary reference frame requires unit inward motion to counter the ever-present unit outward motion of the natural reference system. Otherwise we cannot have a stationary reference frame. This needed inward motion is supplied only by a material system that is gravitationally bound. Therefore, whether we explicitly acknowledge or not, the stationary reference frame can exist only in conjunction with a gravitationally bound system3

Since the domain of the net inward motion of a gravitationally bound system ends at its gravitational limit, each such system has its own stationary reference frame. Beyond the gravitational limit the domain of the familiar three-dimensional space does not exist: it is, thereafter, a domain of equivalent space.4 The familiar three-dimensional space, the space adopted in all our picturizations, ends at the gravitational limit of the gravitationally bound system to which the stationary reference frame is anchored.

The truth that there cannot be one universal stationary co-ordinate reference frame and that each gravitationally bound system has its own stationary reference frame is not immediately and sufficiently recognized. Larson denies “that all spatial locations could be defined in terms of an absolute spatial reference system, and that time could be defined in terms of a universal uniform flow.”5 “In order to get the true picture,” Larson remarks, “it is necessary to realize that no single reference system is capable of representing the whole of physical reality.”6

The Two-Galaxy Paradox?

Imagine two galaxies A and B of unequal masses, each beyond the ambit of the gravitational limit of the other, and two stationary reference frames attached to each respectively. The spatial separation between the two stationary reference frames need not, in general, be the same as measured from A and B individually. This is because each such stationary reference frame is reckoned from the background of the gravitational motion of the gravitationally bound system to which it is anchored and the magnitude of this (the gravitational motion) is contingent on the mass of the gravitationally bound system. The estimate of the intervening distance according to the observers belonging respectively to either stationary reference frame need not be the same since the observer is observing from the background of the gravitational motion in which he is situated, and this differs for both of them.

In fact, this distance is proportional to the recession speed and the reciprocal of the Hubble constant. “... the astronomers have assumed that the Hubble constant is a fixed characteristic of the physical universe...” Larson explains, “The Hubble “constant”... like the gravitational limit... is a property of each individual mass aggregate. In application to the galactic recession this so-called constant is a function of the total galactic mass ...”7 More specifically, we have shown elsewhere8 that the Hubble constant is inversely proportional to the fourth-root of the galactic mass. We have shown there how the consideration of the mathematical relations that are applicable to the region beyond the gravitational limit directly leads us to the observed linear relationship between the recession speed and the distance that the Hubble law states. More recently we have also shown9 how the large-scale structure of the distribution of galaxies and voids that has emerged from the latest astronomical observations follows from the theory--both qualitatively and quantitatively--by considering the limitation of the conventional three-dimensional spatial reference frame and applying the mathematical relations appertaining to the region beyond.

Thus, if vAB is the velocity of recession of galaxy A, as measured by the observer belonging to the stationary reference frame anchored to B; and xand H represent the distance and Hubble’s constant respectively, we have:

vAB = HB * xAB and vBA = HA * xBA   


(xAB/xBA)*(vBA/vAB) = HA/HB

Then all that we can say about the intervening distance is that XAB need not, in general, equal xBA. Seeing that the basic constituent of the universe is motion, a plausible *assumption* is that these two galaxies are connected by the common speed of recession, rather than by a common intervening distance (which would probably be more appropriate to assume in the context of the concept of container space). Then:

xAB/xBA = HA/HB = (MB/MA)¼   

where M represents the mass of the galaxy. This might seem an absurd result, but only if looked at from the unacceptable viewpoint of the “setting” concept.

Resolution of the Quasar Paradox

Eq.[2] gives enough clue to resolve the quasar paradox. Whatever happens to the quasar due to the 2-unit explosion speed happens to it in an objectively real manner. But since each observer is making his observation from within his gravitationally bound system, the phenomena pertaining to the excess speed components that cannot be directly represented in his ‘absolute space’ and ‘absolute time’ frames manifest variously depending upon the local gravitational motion. It must be realized that the inversion of gravity that marks the entry of the quasar into the cosmic sector is relative to the local spatial co-ordinate frame--not relative to the natural reference frame.

An analogy might help. Imagine an object of mass m situated on the surface of the earth and two scientists located vertically below the object ad depths d1 and d2 and estimating its potential energy. They would measure it to be respectively mgd1 and mgd2. Then it would seem that the object has a potential energy mgd1 and at the same time a different potential energy mgd2, depending on which observer’s standpoint one takes. The paradox disappears as soon as it is recognized that the datum levels from which the potential energy is regarded are different.

The term “at the same time” occurring in our statement of the quasar paradox can now be seen to be referring to a concept that is not applicable to the quasar situation under consideration, since this turns on the assumption of the existence of a universal, unique co-ordinate frame. What has given rise to the quasar paradox is the committing of the Fallacy of Incongruous Viewpoints: it is not, after all, as contended, due to any flaw in the structure of the Reciprocal System. The quasar paradox typically demonstrates (i) how difficult it is for us to cast off our allegiance to the concept of a universal container space and uniform absolute time that pertain to the concept of universe of matter that no longer is admissible in the context of the concept of the universe of motion; and (ii) consequently, how we might be misled to wrong evaluations of the Reciprocal System.

Perhaps it is not out of place here to note how rashly and caustically the correspondent condemns Larson’s theory, in the communication in which he refers to the quasar paradox, and discredits his monumental work. It would seem that one tends not only to underestimate Larson’s calibre but also to overestimate one’s own infallibility. Sure and faster progress in the study and research of the Reciprocal System will be accomplished only if we are seeking truth and thoroughly understand the pitfalls a student might encounter. Therefore it might be of some value to dwell on a few more items, in the context of the Reciprocal System, and examine how the unconscious slipping into the old habit of positing everything in the container space might confound our thinking.

Other Conceptual Difficulties

The Time Region. One concept that led some students astray has been that of the time region, the region inside unit space. Since the Reciprocal System asserts that less than a (natural) unit of space does not exist, some tended to interpret that the time region is some kind of pseudo-space. The principle mistake, however, has been to conceptually locate the time region in the frame of the container space. It is not realized that what it really alludes to is a region (or domain) of physical action. The inside of a unit of space, to which the concept refers, is still a region of space, not of any pseudo-space. This example typically illustrates how desperately we tend to hang on to the known concept of space by some stratagem of using such terms as ‘pseudo-space.’

Travel through Time. Another concept that might mislead students is the manner in which the radiation from an object moving away from us at a speed greater than unity reaches us. Larson points out that such radiation reaches us through time rather that through space. Any conclusion that might be reached by inadvertently and unwarrantedly assuming universal reference frames is bound to create mischief.

The Spatialization Syndrome. In the speakers of English ( and of languages of kinder grammatical structure), unfortunately, there is a strong tendency to spatialize everything, even those items that have no noticeable spatial structure (such as thoughts and emotions, say). This, therefore, predisposes the speaker of such a language to picturize/localize all items of knowledge in the container space. This language habit is so thorough that it requires the utmost detachment and awareness to recognize its illegitimacy whenever such is the case.

Moreover, our sentence structure divides reality into ‘actors’ and their ‘actions,’ largely due to the occurrence of the grammatical categories of substantive and verb. This practice is so ingrained that we are assisted to imagine that there is necessarily an ‘actor’ in each fact which, in reality, is only the necessity of a substantive in the sentence structure. For example, our sentence structure requires us to say ‘it is raining’ while, in truth, ‘raining’ is sufficient. Another example: we seem to be regarding ‘the thinker’ as distinct from ‘thinking,’ while in reality there is no separate thinker disassociated with thinking.

Larson explains at length in his Beyond Space and Time that we divide reality into two categories, answering respectively to the two questions ‘what it is’ and ‘what it does’ . In the context of the old paradigm of the universe of matter suppose one asks these questions, say, about the earth, one would answer ‘what it is’ by ‘matter’ and ‘what it does’ by ‘moves.’ However, from the point of view of the Reciprocal System, the answer to both these questions is ‘motion.’ Therefore one has to be wary not to fall prey to the attitude of misreading what in truth are only the necessities of English grammar as the characteristics of reality. We may call this attitude the Fallacy of Misplaced Categories.

The Space-Time Progression. Another source of potential misconception is the space-time progression, the background or datum of physical action. it would be educative to inquire as to how we visualize the uniform ubiquitous space-time progression. Do we visualize it as empty space spread out to infinity and ever expanding? do we tend to miss (or misunderstand) the significance of the concomitant expansion of time? Since space and time are reciprocally related, the expansion in space is nullified by the expansion in time, and each unit of space is not separated by other units of space but all are connected by the unit speed. The space-time progression is a speed manifold--not a space manifold, as is commonly visualized. The entire background of the space-time progression without displacements is a ‘point’ in the speed manifold--not an expanse in the space manifold (the container space).

In the stationary reference frame all spatial locations are at the same time. But this frame is not valid for the entire universe. Larson states:

    ”... It follows that the motions can be represented in the conventional fixed system of reference only by the use of multiple reference points... further elaboration of this point is necessary in order to avoid misunderstandings. The principle stumbling block seems to be a widespread impression that there must be some kind of a conceptually identifiable universal reference system to which the motions of photons and other objects that remain in the same absolute locations can be related. The expression “natural reference system” probably contributes to this impression, but the fact that a natural reference system exists does not necessarily imply that it must be related in any direct way to the conventional three-dimensional stationary frame of reference.”10


Paradoxes in a theory point to defects in its logical structure. However, paradoxes and difficulties encountered in the Reciprocal System might arise out of another source. One of such is the unconscious practice of looking at the concepts of the Reciprocal System from the standpoint of regarding space and time as setting or background for physical action rather than regarding them as the contents of the universe. This has been termed the Fallacy of Incongruous Viewpoints. The quasar paradox is seen to belong to this category. Other such unconscious factors identified are e spatialization syndrome and what has been termed the Fallacy of Misplaced categories, both of which were seen to have their origin in the linguistic habits. Before branding the Reciprocal System as fallacious on the strength of any paradoxes that might be proffered, it is mandatory to establish that the proponent is not guilty of committing any of the above fallacies. This danger is especially so with our generation since we are at the change over point and are still steeped in the old frame of thinking.


  1. The Universe of Motion, (Portland, 1984), p. 210.
  2. Nothing But Motion, pp. 18-19.
  3. Ibid., pp. 66-67.
  4. The Universe of Motion, p. 197.
  5. Nothing But Motion, p. 40.
  6. Ibid., p. 41.
  7. The Universe of Motion, p. 200.
  8. K.V.K. Nehru, “The Gravitational Limit and Hubble’s Law”, Reciprocity, XVI (2), Winter 1987-88, pp. 11-16.
  9. The Large-Scale Structure of the Physical Universe,” Reciprocity, XX (2), Summer 1991, pp. 5-8 and XX(3), Autumn 1991, pp 23-28.
  10. D. B. Larson, Nothing But Motion, p. 34

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