THE SPECIAL CASE OF THE THEORY OF CONTROPY
I.
The Contropic Theory (hereinafter called “The Theory”) offers promising philosophical possibilities on the Universal scale. If efficient, stabilizing structure can develop from within, then the fate of the physical Universe might be far different from that predicted by probabilism. The theory proposes the existence of a sub-quantum "Simplest Possible Particle" (called "SPPs", and differentiated as being either gravitons or magnetons depending upon the orientation of their axis of rotation with the axis of translation), and defines it to be the "elemental" particle.
Other physical existences such as Photons, Neutrinos, quarks, Electrons, Mesons, Protons, Neutrons, Atoms, Molecules, Cells, Man, etc., are seen as being physical constructions that are held together (at least for a time) in opposition to a statistical, thermodynamic tendency for energy to disperse. Their continued existence is dependent on internal structure and organization that produces forces of containment to oppose those of decay. Such existences can be regarded as self-contained microsystems that are simpler in their structure and organization than are the supersystems that include them into their structures. For instance, the Cell is simpler in its self and in its functions than is the Man (who includes the Cell and its functions) in his. At the level of the Photon, the system and its functions should be most uncomplicated.
Structural forms based on the Quark Theory have been proposed for the Proton and the Electron, but there seems to have been little real interest in models for the physical structure of Photons. Fritz Zwicky (1929) proposed that photons continuously lose energy as they travel through space, thereby shifting their light toward the red end of the spectrum. LaViolette (1986) has supported this theory, known as the "Tired Light" theory, with four different observational tests. If such losses are indigenous rather than due to external causes, they would infer that there are structural inefficiencies in the Photon and/or at least in the beam of Photons, so acceptance of that theory would probably have inspired more interest in the structure of Photons. However, it has been generally rejected (at least up until LaViolette's work) in favor of Hubble's explanation of "Red Shift" as an entirely Doppler effect caused by recession of the galaxies from earth, and the Photon is thought to be an elemental particle.
Since Contropy is proposed as a parameter that is independent of the current paradigm, its correspondence to physical reality must be positively demonstrated before its philosophical implications can be taken seriously. The construction of Contropic models for structural stability within the comparatively uncomplicated realities of the simplest physical existences offers opportunities to test for such correspondence.
At least in its scientific applications, the operative principle of Ockham's Razor1 may well be the fact that simplicity is inversely proportional to the number of specifications that are required for the full description of the entity or effect in question. Each specification introduces its own degree of uncertainty or randomness to the implications of other specifications. Therefore, when all else is equal, the probability of error is proportional to the number of specifications required by a given hypothesis. Especially when a hypothesis aims at Universal application should it be grounded upon a minimum number of resultant specifications, or else attempts to deal with specific interactions might become unmanageable in the general case.
One of the most effective techniques that have been developed for the isolation of resultant behavior is the use of statistical analysis. A sample, or model, is constructed to derive typical behavior, and it is then used to extend that behavior to the general case in order to further define resultants. An elegant example of the use of this technique can be seen in the derivation of the "Perfect Gas Law" (for example: Holton and Brush, 1973b). Although this law has proven adequate for the description of behavior in a complex variety of atomic and molecular gases, it is based on few specifications:
There must be contained enough constituent elemental particles to be sufficient for statistical representation.
The particles are in random motion.
The only interaction between the particles is "perfectly elastic" collision and rebound.
Observe that a "gas" composed of SPPs (such as within the Photon) would conform to these conditions. Since the third specification ignores field forces, the SPP gas should be expected to be even more "perfectly" gas-like in its behavior than molecular gases, and thus, be subject to the same type of mathematical formulation. Since increases in Contropy are proportional to decreases in randomness, a corollary to the general theory can be drawn:
A system is less perfectly "gas-like" according to its Contropy.
The term "gas-like" becomes less relevant as interactions besides collision and rebound become controlling. Deviations from "The Perfect Gas Law" are significant in more complex molecular gases (see Burrell and Robertson, 1963, as well as Fig. 1c and Table 1), and "Cells" are clearly beyond its purview. Thus, the corollary defines a Special Case of The Theory comprised of "SPP'S", "Photons", "Sub-Atomic Particles", "Atoms", and simpler "Molecular Gases".
Macroscopic perspectives of statistical populations within this case can neglect deviations from randomized behavior; thus, stochastic methods can be applied without impairment. In this, The Theory is in basic alignment with Classical as well as, to some extent, with Quantum Mechanical theory. (Possibly the proposed SPP might be used to define a limiting case of Quantum Mechanics, wherein the Hamiltonian could find its simplest expression in a reality of Newtonian collision and rebound, operating with invariant time within three coordinates of Euclidean space?) If The Theory has validity, the constants of dimensional relationship derived from the Contropic perspective should agree with those already accepted in the real world. Such derivations should be reasonably simple and straightforward because of the low degree of structure and organization effective within these magnitudes of Contropy. This paper undertakes such derivations as a test of the corollary, and thus, of The Theory.
It is supportive first to test the corollary in the most complicated of the included Contropic magnitudes, "Molecular Gases". It can be seen by inspection that the corollary is consistent with the observed behavior of molecules in general, in that their least ordered state is the most gas-like; and more, that the simpler molecular gases deviate less from "perfect" behavior than do the more complex gases. The corollary is presented in graphical form in Figure 1 (a, b, & c). It should be noted that the scales of Figures 1a and 1b are purely relative. In Figure 1c (with the exception of Nitrogen), the product of the number of interatomic bonds times the molecular weight of the gas has been taken as a measure of Contropy as indicated by complexity, and plotted against the data of Burrell and Robertson (U. S. Bureau of Mines Technical Papers 131 and 158). Nitrogen, which is multivalent, fits the curve when treated as if it had only one atomic bond. This is perhaps indicative of the fact that Nitrogen has a factor of simplicity, which is lacking in the other considered gases.
These data are presented in Table 3.
(**See discussion) (*Burrell and Robertson, U.S. Bureau of Mines Technical Papers 131 and 158.)
Although Figures 1a, 1b, and 1c are based on the corollary, they are also illustrative of the general concept of Contropy. In particular the parallel of The Theory with Teilhard de Chardin's "Rise of Consciousness" theory can be seen in Figure 1a, in which the special case that is under consideration here is shown in context with those larger concepts.
The Rise of Consciousness theory holds that increasing complexity and organization is equivalent to rising "Consciousness". The Theory, as previously developed, claims that the Evolutionary Process demonstrates increasing complexity within the Universe. Now, the corollary provides that the degree of deviation from perfect gas behavior can be taken as a measure of the Contropy that is effective within a system.
Taken together, these considerations indicate that the axes in Figure 1a might as well be labeled in the terms of the Rise of Consciousness; that is, increasing "Consciousness" as the ordinate and increasing "Time" as the abscissa. The point is that Figure 1a shows a profile of the Evolutionary Process as depicted by either theory.
However, as shown by Figure 1c, systems of Contropic magnitude less than that of Molecular gases can be expected to have a high degree of gas-like behavior. To that extent, statistically- based mathematical formulations within those levels should be facilitated.
It has been speculated here that SPPs are confined within the Photon of Electromagnetic Energy as a "standing wave". In that case, outside perspectives can resolve all the complexities of SPP motions into a single resultant identity that has an axis of translation and a singular location in time and space. The corollary provides a basis for derivations of the dimensional interrelationships "within" that identity, which itself defines a contained coordinate system that moves at (nearly) uniform velocity with respect to nearly all outside coordinate systems. But further elaboration of the Photon model is needed.
Within the Photon the motion of the wave (of SPPs) would be paramount. It is derivable (for example: Holton and Brush, 1973c) that the standing wave length, , is related to the length of the "vibrating medium", L, by Equation (2.):
where n is any positive integer. Because of Ockham's Razor and other considerations, the most likely specification for the elemental Photon model is the simplest relationship,= 2L, and that is here assumed as the base ("rest") case.
If the wave rotates about its axis in addition to its length-wise vibration, then the Photon will appear to occupy a volume of Space that is approximately spherical, depending upon the amplitude of the wave. This simplification is made more explicit by assumption that the amplitude of the wave in the base case is equal to /4 cm. The Photon model as seen by an outside observer at relative rest is shown in Sketch #2. Note that the axis of the wave is in the "y"direction, while the SPPs are displaced in the "x" and "z" directions.
Now when this Photon is seen in relative motion in the +x direction, the wave pulse which leaves +y en route to -y appears to traverse a diagonal path, reaching the x axis at a point which is vt centimeters out from x = 0 (where v is velocity of the Photon relative to the observer and t is the time elapsed between start of the pulse and its arrival at the x axis). Since the pulse must be carried by SPPs known to be at constant speed, it may be derived that there is a distortion of Time and Space between observers "within" and "without" the Photon, equivalent to the Fitzgerald-Lorentz equation. (See Sketch #3)
where tr represents time as seen be perspectives at relative rest and tm is time as seen by perspectives in relative motion. Squaring and rearranging both sides of Equation (4.):
Now, extracting the square root of both sides, and substituting the identity of Equation (3.):
The derived time dilation is in accord with Fitzgerald-Lorentz. The displacement of SPPs from the "y" axis, as seen by observers at relative rest, is given by Equation (5.):
Relative motion of the Photon, as above, causes an apparent foreshortening of the paths of SPP displacement in the "x" dimension due to the dilation of time:
The substitution of Equations (4b.) and (5.) now gives:
Squaring both sides:
The "contraction of length" is also in accord with Fitzgerald and Lorentz. Equation (6a.) indicates that observers in relative motion will perceive the Photon as an "oblate spheroid" with eccentricity equal to v/c. As the relative velocity approaches c the Photon presents less and less surface area to all perspectives except those along its path, becoming entirely focused at v = c. It is as if the Photon "sheds" a dimension due to its motion, so that it has height and width, but no length. Because SPPs that would escape from the Photon must do so through the surface, it seems that the Photon gains a measure of containment because of its motion. Sketch #4 illustrates these effects.
Consider now this waveform model from an "internal" perspective. The total Energy contained within the space occupied by the Photon, EP, may be stored within only a limited number of modes. Some energy must be absorbed in the organization of the wave from the otherwise random motion of the SPPs. If this energy is understood as “Energy of Structure," Es, which is separate from the "Energy of State", Epv, then by extension of Universal experience it can be assumed that Es is a relatively small fraction of EP, and its neglection should not cause any serious error in reasonable approximations.
The properties "angular momentum" and "kinetic energy of rotation" are not detectable from within, and so energy may not be stored within those modes. Such properties are negligible with regard to the individual SPP, and may not serve as a repository of energy. There are only three modes left for consideration:
1. epv = Intrinsic Energy of SPPs
2. eke = Kinetic Energy of Translation of SPPs
3. Epv = Energy of State of the Photon
Before examination of the distribution of EP among these modes, it is in order to define the dimensions of Space and Time "within" the Photon. Since the SPPs are each confined within three spatial dimensions--"x", "y", and "z"--along a path which is cm. in length per cycle, and because their velocity is c cm/sec along that path, then "Space" and "Time" can be defined in terms of and those factors of confinement:
The relationship of space with time within the Photon is found by dividing Equation (7.) by Equation (8.):
Each SPP within the Photon collides with the "wall of containment" twice per cycle, and it is reversed in its direction of travel each time. These collisions produce an expansive force that acts through the radius to perform work against the containment. If the total Energy of the Photon, EP, is exerted evenly over its surface, then the expansive force, F, is given by Equation (9.):
and the Pressure, Pp, within the Photon can be calculated.
Now, in view of the small degree of Contropy in this system, the corollary allows EPV to be given by Equation (11.):
Equation (11.) confirms gas-like behavior by the model, since the product of its Pressure times its Volume is a constant.
The universality of the model may be further tested by the equation of Pressure within the Photon, PP , as specified by the model with the expression for Pressure within molecular gases, PM,, as derived by Herapath in terms of mass density (m/V) of the gas and the velocity of translation, v, of its molecules:
This straightforward derivation of Einstein's Equation is encouraging to the model and to the corollary upon which it is based.
Returning to the implications of Equation (11.), it can be seen that two thirds of EP is distributed between ePV and eKE. If there are n SPP's contained within the Photon, and es is used to represent the Total Contained Energy associated with each SPP, then Equation (13.) summarizes these relationships: