In the previous post below, we began our discussion of force and acceleration concepts, in the RST, noting that, although Larson redefined motion in terms of scalar magnitudes, he didn’t redefine force and acceleration in scalar terms. Yet, charge, gravitational as well as electrical and magnetic, is a scalar magnitude, not a vectorial magnitude. Force, on the other hand is not a scalar magnitude, but a vector magnitude; that is, a Coulomb force (charge), or a gravitational force (mass), affects another charge, or another mass, only in the direction defined by the distance between them.
Thus, the scalar magnitude of charge is manifest as a vectorial magnitude of force, whenever it interacts with another charge. We see the same thing with scalar motion in general. Galaxies A, B, and C, on a line, are moving away from each other, because of the scalar motion affecting them, but the scalar motion, expanding in all directions, is manifest as a vectorial motion of the galaxies, in a direction along the line between them.
However, in the LST concept, force is measured in units called Newtons, and given autonomous status, as if it were something independent of motion. Larson pointed out emphatically that this is a mistake (see especially The Neglected Facts of Science), but he didn’t go so far as to actually undertake to reexamine the force equation in light of this conceptual error.
Nevertheless, when we consider the so-called self-force concept, which is regarded as the force generated by a charge on itself, we get to the the crux of the major difficulty in LST theory that plagues it to this day. The electromagnetic theory of force is inconsistent. Because of this, classical electromagnetic theory was abandoned in favor of the quantum mechanical approach, but, in the end, this major change didn’t help solve the problem. As Richard Feynman explains it in his Lectures on Physics, page 28-4, the problem ultimately comes down to the idea of autonomous force, which Larson criticized, even though Feynman doesn’t use those words:
In deriving our equations of energy and momentum, we assumed the conservation laws. We assumed that all forces were taken into account and that any work done and any momentum carried by other “nonelectrical” machinery was included. Now, if we have a sphere of charge, the electrical forces are all repulsive and an electron would tend to fly apart. Because the system has unbalanced forces [read autonomous forces here], we can get all kinds of errors in the laws relating energy and momentum. To get a consistent picture, we must imagine that something holds the electron together. The charges [discrete fractions of charges distributed over the surface] must be held to the sphere by some kind of rubber bands - something that keeps the charges from flying off. It was first pointed out by Poincare that the rubber bands - or whatever it is that holds the electron together - must be included in the momentum and energy calculations. For this reason the extra nonelectrical forces are also known by the more elegant name “Poincare stresses.” If the extra forces are included in the calculations, [they] are consistent with relativity; i.e., the mass that comes out from the momentum calculation is the same as the one that comes from the energy calculation. Both of them contain two contributions; an electromagnetic mass and contribution from the Poincare stresses. Only when the two are added together do we get a consistent theory.
Thus, the LST’s autonomous forces, generated by the distributed electrical charges, on the surface of the sphere, have to be held together by some mechanism of restraint, a mechanical force of some kind, and if this is included in the calculations, then the mass of the electron, calculated from the energy of the electrical field from the charge, or from the mass component of the momentum associated with the velocity of the field, is the same. The bottom line is that the mass cannot arise solely from the electron’s charge; another, non-electrical, non-explainable, autonomous, force is required to hold the electron together against the strong electrical forces pulling it apart, and this leads to trouble. As Feynman observes:
Feynman goes on to explain in more detail why this is so, but it comes down to the same thing: if you allow the electron to be a point and, at the same time, act on itself, something Feynman characterizes as “perhaps a silly thing,” you have to modify Maxwell’s theory of electrodynamics. “Many attempts have been made,” writes Feynman, “but all of these theories have died.” Moreover, quantum mechanics doesn’t help, because there is no model of the electron in quantum mechanics. Quantum mechanics only describes a quantum state of the electron, not its structure. Feynman explains:
Clearly as soon as we have to put forces on the inside of the electron, the beauty of the whole idea [i.e. deriving mass from Maxwell’s equations], begins to disappear. Things get very complicated. You would want to ask: how strong are the stresses? How does the electron shake? Does it oscillate? What are all its internal properties? And so on. It might be possible that the electron does have some complicated internal properties. If we made a theory of an electron along those lines, it would predict odd properties, the modes of oscillation, which haven’t apparently been observed. We say “apparently” because we observe a lot of things in nature that still do not make sense. We may someday find out that one of the things that we don’t understand today (for example the muon), can, in fact, be explained as an oscillation of the Poincare stresses…there are so many things about fundamental particles that we still don’t understand. Anyway, the complex structure implied by this theory is undesirable, and the attempt to explain all mass in terms of electromagnetism…has led to a blind alley.
It turns out, however, that nobody has ever succeeded in making a self-consistent quantum theory out of any of the modified [classical] theories…We do not know how to make a consistent theory - including the quantum mechanics - which does not produce an infinity for the self-energy of the electron, or any point charge. And, at the same time, there is no satisfactory theory that describes a non-point charge. It is an unsolved problem.
Of course, in developing our RST-based theory, we escape the horns of this dilemma, by recognizing that electrical, magnetic, and gravitational forces, are properties of scalar, not vectorial, or M2, motions, and that they are certainly not autonomous entities that can exist independently on the surface of a sphere, as a consequence of interacting charges. In the new system, we can define force and acceleration (mass) in space|time terms in a way that eliminates the infinities of point particles and the need for Poincare stresses, in non-point particles.
That is to say, we can redefine force, just as we redefine motion: In the universe of motion, like everything else, force and acceleration must be motions, a combination of motions, or a relation between motions. Clearly, then, they are relations between motions, and/or relations between combinations of motions. As pointed out in the previous post below, we can see from their dimensions that force, dt/ds2, is one-dimensional energy (inverse speed), dt/ds, per unit of space, 1/ds, and acceleration, ds/dt2, is one-dimensional speed, ds/dt, per unit of time, 1/dt.
However, things are complicated quite a bit by the fact that numerical dimensions, are not necessarily geometric dimensions, that, while geometric dimensions are limited to three, numerical dimensions are not limited to three, even though they are related to the three geometric dimensions, in a special manner. Consequently, we have to be very careful in our expressions of multi-dimensional magnitudes, such as t/s2, or s/t2, in order not to confuse geometric dimensions with numerical dimensions.
For instance, while force, t/s2, can be interpreted as
F = ma,
because the dimensions are consistent, in this case, giving us
F = ma = (t3/s3)(s/t2) = t/s2,
it doesn’t necessarily follow that force should be defined this way in general. In the case of the electrical force equation,
F = Q1Q2/d2,
for instance, the force equation does not involve the dimensions of mass or acceleration:
F = (t/s)(t/s)/s2 = (t2/s2)(1/t) = t/s2 (substituting 1/t for s2, explained below), while,
in the gravitational equation, it does involve the dimensions of mass, but not of acceleration,
F = GM1M2/d2 = G(t3/s3)(t3/s3)/s2 = G(t6/s6)(1/t) = G(t5/s6).
Clearly, t5/s6, makes no sense geometrically, because there are only three geometric dimensions, not six (we don’t subscribe to a concept of hidden dimensions). Yet, when we recognize that the numerical dimensions, higher than three (four, counting zero) are geometrically mapped, so-to-speak, by the Bott periodicity theorem, into repeating groups of four dimensions, called tetrakti, in the reciprocal system of mathematics (RSM), we see that x5 corresponds to x5-4, and x6 corresponds to x6-4, geometrically. Therefore, t5/s6 is equivalent to t/s2, adjusted by a constant, presumably the observed gravitational constant G.
However, if this is true, then what is the 1/t term in the force equation, and where do the higher dimensions come from? Physically, 1/dt is a frequency, and it’s hard to see how time, t, can be equivalent to distance squared. Of course, this will take some more thought, but it’s possible that t is related to s2, because the temporal period of a cycle is related to the spatial area of a square, as the time it takes to complete one cycle defines a definite circumference and diameter, at unit speed. So, increasing/decreasing the distance between two locations, as the index, d, of a square area variable, might also be equvalently expressed as an increasing/decreasing period, as an index, t, of a circular area variable (π r2), even though this seems like a long shot at this point.
As far as where the higher dimensions come from, the separation of geometric from numerical dimensions makes this clear. The dimensions are actually the number of points involved; that is, two points (two S|T units), form the basis for a line, three an area and four a volume, but 5 is a point in the volume, 6 is a line in the volume, 7 is enough for an area, and 8 is enough for a second volume in the initial volume. Then 9 is a point in the volume of the volume, 10 is a line in the volume of the volume, etc, ad infinitum. Again, this is what Raul Bott proved as the periodicity theorem that bears his name.
I know this sounds crazy, but it’s where the path leads. The obvious conclusion is that what we have here, in order to make the dimensions come out right, is a new force equation, the self-force equation, if you will, except that the self-force is actually a scalar force. In other words, we can redefine the vector force equation, the equation that holds for the vectorial force between two charges, or two masses, in terms of the scalar motion equation that the vector force is part of. Just as the galaxies are moving apart scalarly, but a vector motion is part of that motion, as explained above in the beginning of this post, the charges/masses are moving scalarly, but the force property of that motion is directed along the line between them.
Thus, the equations,
F = Q1Q2/t, and F = GM1M2/t,
express the scalar force, because t, the period of oscillation, determines the radius of the sphere that in turn determines the 4 pi r2 area of the sphere, a part of which, is directed along a line between the charges, and is subject to the inverse square law along the line between them.
I might be all wet, but is there any other way to make the space|time dimensions come out right?