THE LAW OF CONSERVATION OF DIRECTION Introduction Some students of the Reciprocal Syetem (RS) have been disputing the explanation of the intrinsic structure of the photons, given by Larson, the originator of the R.S. No amount of discussion, so far, seemed to throw additional light in overcoming the logical objections raised. An examination of the situation undertaken by the present investigator revealed that a crucial fact of fundamental nature is being missed hitherto, both by the originator and the other students. It is found that a recognition of this fact not only clarifies the photon situation entirely but also throws light on many collateral issues where gaps in the logical development of the theory exist, thus rendering the theory more cogent. Some of these new developments are reported in this Paper. The Difficulties with Larson’s Account of the SHM In the outline of the deductive development of his theory
Larson states: Since there is nothing like more outward (inward) or less outward (inward) the question arises as to the meaning of the statement “a continuous and uniform change from outward to inward”? Outward and inward, as applied to scalar motion, are discrete directions: the acalar motion could be either outward or inward. There are no intermediate possibilities. Larson is quite clear about this, at another juncture: “When the progression within a unit of motion reaches the end of the unit it either reverses or does not reverse. There are no intermediate posaibility.”² As such, the idea of a “constant and uniform change” is logically incompatible with, this concept of “outwardinward motion.” It muat be remembered that the mavgnitude of the motion is constant, being unity. Since simple harmonic motion (SHM) does seem to underlie the structure of photons, the crux of the problem of understanding the nature of the photon is the explanation of the genesis of the SHM given only uniform scalar speed. If a vibration of the type Larson prnposes is to exist, it can not be a SHM. The speed has to be a square wave. SHM seems possible only if one of the components (space or time) progressea nonuniformly while the other progresses uniformly. In fact, SHM will be the result under the two circumstances: (i) when a constant magnitude is continuously and uniformly changing its direction in the conventional reference system (as in rotation) and its projection in a constant direction is being considered; or (ii) when there is a constant direction and the magnitude is continuously and nonuniformly changing. The second alternative is precluded by definition (see item D of the Basic Principles in the “Outline”³). Is Rotation Primary ? A scalar motion has magnitude only, and no inherent spatial direction. It therefore has to be given a direction in order to be represented in a spatial ref erence system. Now a “constant and uniform change” envisaged by Larson can only happen if the representation in the reference system changes the vectorial direction uniformly as in rotation. As a matter of fact, SHM will be the result if two such rotations, mutually opposite in direction, are vectorially combined. But Larson does not pasit the possibility of rotation prior to the existence of photons. Thus he states: “While motion is possible without anything moving, rotation is not possible unless some physical object is available to be rotated.”^{4} The logical basis for this conclusion is hard to find. Rotation is as much a motion as translation is, and logically it must be as much possible without any “thing” rotating, as far as the primary motions are concerned. One wonders, in this context, if the author is completely free of the unconscious leanings to the frame of mind that underlies the view of the universe of matter as against the univerae of motion! Rotation is precluded only if space is onedimensional. Juat as soon as it is established that the stationary spatial reference system is threedimensional, rotation becomea a possibility. Larson himaelf, while discussing the status of the uncharged electron, refers to the general nature of space which includes rotation as much as linear translation. “Thus the electron is essentially nothing more than a rotating unit of space. This is a concept that is rather difficult for most of us when it is first encountered, because it conflicts with the idea of the nature of space that we have gained from a longcontinued, but uncritical, examination of our surroundings. ... The “space” of our ordinary experience, extension space, ... is merely one manifestation of space in general ...”^{5}. Therefore, what is not being explicitly recognized is that, in general, space has two intrinsic traits: translational and rotational. The translational trait manifests to us as the familiar “extension space,” whereas the rotational traitwhich manifests as difference in directionsia not so readily regarded by common experience as manifestation of space. Hence the representation of a uniform scalar motion in the conventional reference system can take either the form a uniform and continuous change of linear magnitude with a constant direction, or the form of a uniform and continuous change of direction, with a constant linear magnitude, that is, a rotation. Conservation of Direction As already pointed out, a scalar motion does not have a vectorial direction. The latter is a property acquired due to the coupling with the conventional stationary threedimensional spatial reference system, which involves also the identification of a reference point. A point of universal significance that needs to be recognized is that the representation of a scalar motion in the conventional reference system conserves direction. This is accomplished by the representation by substituting two opposite directionswhat we will call a ‘bidirection’ for the original lack of direction. For example, consider the motion of a point 0 that is made the reference point. Consider two locations, A and B, on a straight line passing through 0, and situated on opposite sides of 0 (Fig. 1). In the csse of an outward scalar motion we find both A and B receding from 0 (Fig. 1(a)). On the other hand, if O’s motion is vectorial we find B (or A) receding from 0, and A (or B) approaching it (Fig. 1 (b)). Thus a scalar motion gets represented as a ‘bivector’ and not merely as a vector. The appearance of a bivectorial motion in the conventional reference system, therefore, serves to distinguish an intrinsically scalar motion from vectorial motion. Representation of linear motion
An analogy might help to demonstrate the universality of the Law of Conservation of Direction. Imagine a long solid cylinder with a crosssectional area of an arbitrary shape. If the cylinder is now divided into two by cutting with a plane, two new surfaces, S_{1} and S_{2}, will be generated as the ends of the two halves of the cylinder where there were none prior to the cutting. Adopting the right hand corkscrew representation of areas, we can see that the two intersection surfaces, S_{1} and S_{2}, will be of equal area but opposite directions (one being the mirror image of the other). The original lack of (exposed) area is substituted by two equal areas of opposite vectorial directions. It is simply not possible to carry out the intersection such that only one new surface is generated. In an identical manner, the representation in the conventional apatial reference frame of a scalar motion, with its inherent lack of direction, is not possible with the ascription of only a single directionit requires the imputation of two mutually opposite directions, in other words, a bidirection. Photon: an Intrinsic SHM ? In case the representation of a scalar motion in the stationary threedimensional reference system is rotational motion instead of translational motion, the requirement of the conservation of direction still holds good, the reprssentation taking the form of a birotation. The birotation is a vectorial combination of two equal and opposite rotations, clockwise (CW) and counterclockwise (CCW). Some students of the RS have argued that the CW or CCW direction of rotation is the algebraic sign, the sense, of the rotation and not really a direction. Therefore they concluded that rotation has no true direction. But they are missing the point. What their conclusion means is that rotation does not have a direction in the sense of a direction of linear motion. The CW or CCW sense of rotation is relative to the axis of rotation, but the axis itself can be oriented in any direction in the threedimensional spatial reference system. Adopting the right hand corkscrew representation of rotation, the latter can be vectorially depicted . Because of the discrete unit limitation a mere change of direction (as in rotation) without any magnitude is not possible. Hence a unit of birotation involves half a unit of onedimensional space element in each of its component rotations. As shown in Fig. 2, let one component rotation be CW, and the refer ence point for this rotation be O, OA being the radius of rotation with the axis of rotation perpendicular to the plane of the paper. The reference point for the second component rotation, which is CCW, will be A, with AB as radius and axis parallel to that of the first rotation. Since the angular speeds of the two rotations are of equal magnitude, the visible result of this birotation is a SHM, with location B oscillating in the XX’ direction. This, therefore, is how the SHM is engendered by uniform motionthe basis of photon structure. Figure 2
Simple Harmonic Motion as Birotation At this juncture it might be mentioned that, in this Paper, we are endeavoring to discuss some logical difficulties occurring in the present development of the RS and to clarify them in the light of the discovery of the Law of Conservation of Direction. It is not possible, however, to undertake here the full development of the aspects we discuss beyond supplying the missing links in the logic. It is assumed that the reader is sufficiently familiar with the account of the development of RS as given by Larson in his works. Polarization Suppose now that a light beam is passed through a polarizer and one of the component rotations of the photons is filtered out. The outcoming photons will be constituted of a continuous uniform rotation, with the axis of rotation lying parallel to the direction of propagation. If a blackened disk is suspended by a fine filament and ia irradiated by such a circularly polarized beam of radiation such that the beam travels parallel to the suspenaion and atrikea the diak normally, a torque should appear. This, of course, is experimentally verified. It might be noted that in Laraon's account of the structure of the photon there is no explanation of thia fact. Vibration vs. Translation Since each unit of motion, by the reciprocal poatulate, consists of one unit of space in association with one unit of time, all motion takes place at unit speed. However, by a sequence of reversals of the progreasion of either time or space, while the other component (space or time) continues progressing unidirectionally, an efiective speed other than unity can result. Explaining this, Larson gives a tabulation² for the example of an effective apeed of 1/3 (see Table I).
It may be seen that in the case of the translational situation the
vectorial direction reverses in unison with the scalar direction. But
in the case of the vectorial vibration it is not so: it is perplexing
why the scalar and vectorial directions do not maintain a constant relationship
in the case of the vibrational motion (cp., for example, the third and
the fourth units in the tabulation). Larson comes up with an explanation of a sort, which sounds more like an apology: “...in order to maintain continuity in the relation of the vectorial motion to the fixed reference system the vectorial direction continues the regular reversals at the points where the scalar motion advances to a new unit of space (or time).”^{6} On the principles of probability, the alternative possibility, namely, the vectorial directional reversals occurring in unison with the scalar directional reversals appears more logical. The present recognition of the fact that the linear vectorial vibration is really the manifested result of a birotation now clarifies the situation. Both in the case of the vibration and in the case of the translation the vectorial directional reversal is in unison with the scalar directional reversal. In the vibrational case, the two component rotations involved in the birotation do promptly reverse their respective directions at the time of the reversal of the scalar direction. However, this does not produce any effect on their vector resultant, which continues uninterrupted as the SHM. Referring to Fig. 2, let A reach the poaition A’ and B the poaition B'. Thia is one extreme position of the oscillation of B. From this position whether OA’ continues rotate in the original CW direction or reversea and rotatea in the CCW direction (with the sense of rotation of the aecond rotation always being opposite to that of the rotation of OA) hardly mattersin either case the observable result ia the same oscillatory motion of B. Conclusions Summarizing some of the concluaions reached:
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