FURTHER MATHEMATICS OF THE RECIPROCAL SYSTEM This paper will present in
the most concrete, explicit manner the mathematics of space–time,
radiation, and matter of the Reciprocal System. Readers without
special knowledge of the Reciprocal System are first urged to study Larson’s
books, especially
Starting from
where c is the speed of light.
If we choose the reference point to be x
Now consider a second system, the 0´–system, moving translationally with respect to the 0´–system in the x–direction. What is the equation for the progression in the 0´–system? From the inverse Lortentz transformations,
Upon substitution, we obtain
But since the left side of the equation equals zero, so must the right side:
Thus the progression as determined by 0' is also spherical. And so the equation for the progression is invariant under a Lorentz transformation.
In polar coordinates the equation is simply
In
the Reciprocal System the speed of light is the natural unit of velocity
and so r and t must take equal natural values. The space–time
progression is thus
In the Reciprocal System radiation is the combined motion of a simple harmonic oscillation in one dimension and a uniform translation in a perpendicular direction.
The equation for a simple harmonic oscillation in one dimension (say the y direction) is
where A is the amplitude and
f
for The other variable to be determined
in eq.(10) is the frequency, f
The natural unit of frequency must occur when the average velocity is c.
the Rydberg frequency. (Actually, Larson derived the natural unit of time from the Rydberg frequency, but I think it was instructive to do the reverse, and this method will be used to calculate rotational and rotational vibration frequencies as well. Of course, this method assumes that the natural unit of time can be found by some other means.) Because of the discrete nature of the Reciprocal System, it is only possible to have integer multiples or reciprocal integer multiples of the Rydberg frequency. Putting the values of A and
f
where
Perpendicular to the oscillation is a translation at unit velocity (the speed of light). Let x be perpendicular to y. Then
From eq.(17) t can be found in terms of x and c and put in eq.(15). The result is
if x is given in meters. This is the equation for a monochromatic wave of radiation in the Reciprocal System.
Particles of matter consist of rotating photons. Subatoms have one rotating photon; atoms have two rotating photons (both photons rotate about the same central point). The rotational motion has a translational effect, which will be discussed after the mathematics of the rotation has been worked out.
1 A photon can rotate around
either of two horizontal axes passing through its midpoint, and also around
itself. In the Reciprocal System the true physical zero is motion
at unit speed. Anything physical must have a motion either greater
or less than unit speed. This deviation is called a speed displacement
by Larson. The first particle has 1 speed displacement around one
horizontal axis of the photon and is called the rotational base.
Actually there are two rotational bases: one with one speed displacement
above unity, the other with one speed displacement below unity.
As will be discussed later, the one displacement unit neutralizes the
translational motion of the photon in the original dimension, but the
progression now continues in the remaining dimension, so the
In the above table the speeds are calculated from the displacements as follows. For displace ments of n
The natural frequency of rotation must occur when the speed is c.
where R is the Rydberg frequency,
as before. In these terms, then, the cosmic rotational base is a
photon that has a vibrational oscillation of 1.6440288 x 10
Many more permutations appear to be possible, but the probability principles keep eccentricity to a minimum. Since none of the above particles has an effective displacement of 1 or more, they are all massless (aside from the mass contribution of an electric charge). The diameter of all the particles is one natural space unit, reduced by the (one–photon) interregional ratio, or 3.2054 Å. However, because these particles do not exert any force in the uncharged state, a particle–measuring probe would not be able to detect any size of these particles at all. 2. intermediate systems Intermediate
particles have two rotating photons, but one of the two sets has no effective
displacement and thus contributes no primary mass. The two intermediate
particles are the neutron and the mass one hydrogen isotope (and their
cosmic analogs). There are only two kinds of rotations that can
combine to form this kind of particle, the proton type and the neutrino
type We identify the combination of the material proton rotation
and the material neutrino rotation as the mass one hydrogen atom; the
combination of the material proton rotation and the cosmic neutrino rotation
as the neutron; the combination of the cosmic proton rotation and the
cosmic neutrino rotation as the mass one atom of cosmic hydrogen; and
the combination of the cosmic proton rotation and the material neutrino
rotation as the cosmic neutron. The proton is a single system with displacements
2–1–(1), effective displacements 1–1–(1), speeds
The new notation makes clear the two photons involved and the five rotations (to be discussed next). 3. Atomic cycles Atoms have two rotating photons, but here both systems have effective displacements and both systems ordinarily have the same velocities. Let the first photon be called A and the second be called B. A and B are mutually perpendicular. We have the following five rotations: (i) the rotation of A about B produces disk a; (ii) the rotation of B about A produces disk b; (iii) then disk a can be rotated about A; (iv) then disk b can be rotated about B; (v) finally the whole structure can be rotated in the electric dimension. This last rotation is in the scalar direction opposite to that of the previous rotations. Cosmic atoms have speeds above unity for the first four types of rotations, whereas material atoms have speeds below unity for the first four types. The electric rotation may be above or below unity for both cosmic and material atoms. The first particle with two effective rotating systems is deuterium, the second is helium, etc. A table similar to that for the intermediate particles can be made.
All other atoms can be given
appropriate values in the same manner. In the solid state, however,
the values that govern the physical properties are not the actual rotations,
but the 4. Electric charges and magnetic charges According to the Reciprocal
System an electric charge is a rotational vibration about the electric
axis, and the magnetic charge is a rotational vibration about one of the
magnetic axes. Both charges have the same natural frequency, calculated
as follows. In one cycle the motion covers a distance of ¹
* s
At the unit level, v
Solving for f
This frequency is one–half
that of a full rotation and can thus be considered to be effective in
one direction only half the time. One negative electric charge is
a rotational vibration of R/2p = 5.233106 x
10
The rotational motion of particles
has a translational effect. The maximum inward translation is two
full units, giving one net inward unit. In terms of rotation we
can have 2 For atoms with magnetic displacements
of less than 4 and electric displacements of less than 8, the frequency
of the rotating photons is normally one displacement above unity, or 2R
(the frequency of photons in cosmic atoms is (
And the speeds corresponding to electric displacements can be listed as follows:
In the
solid state, the values for electric rotation can be further altered.
Larson states that a combination of one atom of electric displacement
x with another atom of electric displacement 8–x results in a neutral
bond. This bond gives rise to an electric speed of
The reader can continue the table all the way to element 118. Again, one must first determine the kind of bond involved before the electric rotational speed can be determined. Since different
atoms have different rotational speeds and thus different rotational forces,
a particle probe of equal energy shot at atoms of different elements would
“penetrate” to different depths. Thus experimenters have
concluded that “nuclear” size is proportional to atomic weight.
Actually what they are measuring is atomic size, and according to the
Reciprocal System this is constant (2.914 Å diameter)–but
the force
1. Dewey B. Larson, |