In Larson’s work, all physical constants disappear except for the constant c, and the constant R. For example, the constants G and h, are derived in the RSt, in principle. In practice, however, it has not worked out as well as might be hoped. Both Nehru and Satz have attempted to improve on Larson’s derivation of the Planck constant, though no one has attempted to rederive G, as far as I know.1,2
It appears that, In the work of the LRC, however, these calculations will have to be redone, because here it must be recognized that the dimensional property of the space|time progression has been misunderstood in Larson’s work; that is, mathematically speaking, the geometric values of the physical dimensions of the progression, s/t (t/s), in Larson’s development, are s1/t1 (t1/s1), but they are s3/t0 (t3/s0) in our development. This means that the function of space per unit of time is non-linear: At time t1, one unit removed from time t0, space has progressed (1x2)3 = 8 cubic units; At time, t2, the space progression is (2x2)3 = 64 cubic units, and so on.
This does not effect the constant c, because it is measured in only one direction from the source, as the spherical expansion, when measured in polar coordinates, proceeds according to the equation,
r - r0 = c(t1 - t0).
Clearly, when
r0 = t0 = 0, then
r/t = c,
where r is the radius of the expanding sphere, regardless of the direction measured from the origin to the surface of the sphere. In other words, the volume of the expanding ball of space (the algebraic pseudoscalar) increases non-linearly, even though the radius of the expansion increases linearly.
At first glance, this might seem to be an immaterial observation (no pun intended), but it becomes relevant the moment we introduce space (time) direction reversals in the progression. When space (time) direction reversals are introduced in Larson’s development, the s/t progression oscillates in only one dimension, but, in the LRC development, this is impossible, since it would reduce the dimensions of the pseudoscalar and violate the conservation of motion. The space (time) reversals in the s3/t0 progression must be a 3D oscillation, expanding by a magnitude of (1x2)3 = 8, 3D, units in the outward direction, then contracting by a magnitude of (1x2)3 = 8, 3D, units in the inward direction, for each cycle.
Since, unlike c, the G and h constants necessarily arise in the context of the scalar vibrations, the 16, 3D, units of change per cycle must be taken into account. We can’t just focus on one dimension, ignoring the dynamics of the 3D, pseudoscalar, oscillation. Moreover, in the case of the gravitational motion, evidently only a portion of the total change is directed inward, giving rise to the gravitational constant of proportion, G. If this is the case with the Planck constant, as well, it follows that the proper physical dimensions of the vibrational motion of radiation are actually the dimensions of velocity, s/t, not the artificial dimensions of frequency, 1/t, just as Larson deduced.
The reason that using the dimensions of frequency, 1/t, as a parameter in the energy equation, works in spite of this is that the dimensions of Planck’s constant, similar to the dimensions of the constant G, which is, aside from the higher dimensions involved, similar to spatial distance, or motion multiplied by some interval of time, are actually inverse motion multiplied by some interval of space, yielding a value analogous to temporal distance. The difference, in this case, is that the constant h is an analog of spatial distance; that is, in the gravitational force equation,
Gm/s2 = ((s3/t3 * t) * t3/s3)/s2 = t/s2,
G is motion times time (a form of spatial distance), and in the energy of radiation equation,
hv = ((t2/s2 * s) * 1/t = t/s,
h is inverse motion times space ( a form of temporal distance), because inverse motion times space is time.
Since, in our development, unlike in Larson’s development, the initial combinations of speed displacement units are combinations of 3D motion, it appears that matter (3D time speed-displacement) precedes radiation, if radiation is taken to be motion in less than three dimensions. However, we have concluded that the SUDR|TUDR combination, 3D time speed-displacement plus 3D space speed-displacement, constitutes radiation, because it is this combination that propagates at unit speed relative to oscillating space and oscillating time (SUDRs & TUDRs). Yet, it is clear from observation that all radiation originates from matter, so the implication seems to be that the original conclusion was not correct, and that radiation forms as dimensionally reduced matter.
However, this line of thought also raises several other issues, especially issues with particle pair production and particle annihilation, where nothing is left behind in the conversion of radiation into matter and matter into radiation. If, in the latter case, the 3D scalar motions, constituting the mass of the electron and the positron, are transformed into the dimensionally reduced motion of radiation, and the motion is conserved according to the equation,
E = mc2,
the question becomes how, or why, the collision of the two particles changes the dimensions of the motion from 3D to 1D? But in the former case, the opposite occurs, so the question becomes how, or why, the 1D motion of the radiation is transformed into the 3D motion of mass, and the motion conserved according to the equation,
m = E/c2.
Of course, the particle pair production cannot occur spontaneously, that is, photons don’t decay into particle pairs spontaneously. For the phenomenon to occur, there must be an interaction of radiation with matter, and even then it cannot occur unless the energy of the radiation is at least twice the equivalent energy of the mass of the particle produced. These are our clues as to the physics involved, along with the fact that the properties of the produced pair of matter particles are always opposite, except for their masses and spins.
For example the mass and spin of the electron and positron are the same, but they have opposite charge. In LST theory, the reason why particle pair production cannot occur spontaneously has to do with conservation of momentum. Photons have the equivalent energy of the particle pair, according to E = mc2, but not the momentum; that is, the photon is massless, and consequently restless (it cannot be brought to rest in a spatial reference system), while the particles are massive and therefore restful (they can be brought to rest in a spatial reference system). How, then, can the created pair’s quantity of 1D translational motion, or momentum (t3/s3 * s/t = t2/s2), be created from the photon’s 1D vibrational motion, or ν (reckoned as 1/t)?
The LST answer is that the photon has to interact with matter in a way to bring the photon to rest, which is to say it has to interact in a way that doesn’t result in mere Compton scattering, reducing the energy of the photon through absorption of some of its energy by the scattered electron, but in a way that converts all of its energy into the moving mass of the particle pair.
Since, in the LST, the photon has angular momentum of spin 1, it is presumably conserved in the angular momentum of the spin 1/2 of the particle pair, but the photon’s motion of propagation, relative to matter, at light speed must be converted into linear momentum, or rest mass times velocity, according to the m = E/c2 equation.
Even so, before we can even begin to make these calculations, there is the immediate problem of the geometric versus algebraic disconnect, affecting the size of the units involved; that is, though the number of volume and area units is algebraically attainable, from the dimensional considerations, the actual size of the units must be geometrically determined.
As long as one-dimensional, linear, units are considered, the number and size of the units is straightforward, as only the radius of the expansion/contraction is relevant, but when the oscillation is a three-dimensional expansion/contraction, the number of orthogonal directions (i.e. the mathematical dimensions) is eight units (the eight diagonals from the origin to each corner of a 2x2x2 stack of cubes), not the one linear unit of the radius. Moreover, the size of these volume units is not the 1x1x1 algebraic unit, in each of the eight directions, but something less than this due to the geometric expansion from the origin in all directions simultaneously.
The change in the physical situation is shown in figure 1 below.
Figure 1. Squaring the Circle
Referring to figure 1 above, the algebraic value of one 1x1x1 cube is calculated using the formula of a prism (Vp = a x b x e), while the geometric value of the unit radius expansion within this cube is less and must be determined by the formula of a ball (Vb = 4/3π x r3). But in three dimensions there are eight 1x1x1 algebraic cubes, so the total volume of the ball is contained in the algebraic volume and is consequently something less than the 2x2x2 algebraic volume, with the radius d. For a unit volume, this turns out to be V2, which is equal to 4/3π, as shown. The area of the surface of the ball, the area of the sphere, is A2, which is equal to 4π.
However, given that the measure of the speed of the expansion, from the origin to a point selected on the surface of the sphere, is a one-dimensional line of one unit, the radius d, the question arises as to how to properly calculate this value in terms of scalar motion, rather than vector motion, if its proportional area and volume are relevant physical factors, which they must be.
For instance, in the case of a material explosion, the momentum of a particle, traveling from the origin to the surface in a one-dimensional line, would represent only a fraction of the total momentum and energy contained in the expanding volume, and, in fact, only a fraction of the momentum and energy contained in the fraction of the volume of the section contained within one of the eight 1x1x1 algebraic cubes of the total volume.
Yet, while the total scalar motion of the actual 3D unit of space|time has no associated mass or momentum, per se, it definitely has a three and two-dimensional component, not just the one-dimensional motion of one radius, and clearly, therefore, it must be taken into account in any calculation.
However, a definitive understanding as to how to proceed in this manner is not available at this point in the development, even though it seems clear enough that previous treatments, such as Nehru’s, which relies on a one-to-one identity between natural units of energy E and natural units of speed S/T, might have to be re-examined in the light of the three-dimensional units discussed here.
References:
1. K.V.K. Nehru, “Theoretical Evaluation of Planck’s Constant,” http://www.reciprocalsystem.com/rs/cwkvk/planck.htm
2. R. W. Satz, “A New Derivation of Planck’s Constant,” http://www.reciprocalsystem.com/rs/satz/constant.htm