In Lucas's "Universal Force" theory, the expression ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] has an important influence over the derivation of the
"Thomas Barnes's relativistic correction factor, γ_TB = (1 - β^2)/(1 - β^2*sin(Aθpc(POIo(t),t=0))^2)^(3/2)" :
A key objective of equations 4-32 through 4-43 is to derive that factor.
Lucas sets ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] to zero, in which case :
- the correct functional forms of (β*sin(Aθpc(POIo(t),t=0)^n are obtained
- the coefficients of the series [are divergent, are not binomial series]
I personally cannot justify arbitrarily setting ∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)] to zero within the context of Lucas's theories. In other words, I have not been able to get the same result as Lucas. That's not a big deal for me, as I suspect that the relativistic correction factor may be more of an [instrumentation, measurement, observation] issue rather than a basic phenomena of electromagnetism.
My attempts up to 16Oct2019 were based on applying an [iterative, non-feedback] approach similar to Newton's method for solving implicit equations. Two variants were attempted :
1. Non-inclusion of expressions with (cos(Aθpc(POIo(t),t=0)) - 1)
- Once I corrected numerous mistakes I had made, these led to the proper functional result, but the coefficients of the infinite series were constantly expanding.
- see sub-directory "$d_Lucas""formulae Lucas/cos - 1 no, iterative, non-feedback/"
2. Non-inclusion of expressions with (cos(Aθpc(POIo(t),t=0)) - 1) - as recommended by Lucas
- The proper functional form was NOT obtained. Again, the coefficients of the infinite series were constantly expanding.
- I do NOT have any solid reason (certainly no proof) that allows the (cos - 1) terms to be dropped! Severely restrictive (sometimes inconcistent) can lead to this, but I am not at all comfortable with these!
- see sub-directory "$d_Lucas""formulae Lucas/cos - 1 yes, iterative, non-feedback/"
In neither case above were the binomial series coefficients obtained.
16Oct2019 Status
That's it. I give up on the relativistic correction factor, from an [iterative, non-feedback] perspective, even though it is quite likely that I have made [simple, fundamental] errors.
Only versions of 4-32 through 4-37 that drop expressions with (cos(Aθpc(POIo(t),t=0)) - 1) are shown below in this document (this could change in the future!). However, you can compare results by looking at :
"$d_Lucas""formulae Lucas/cos - 1 no, iterative, non-feedback/Lucas 4-32-37 no cos - 1.txt"
"$d_Lucas""formulae Lucas/cos - 1 yes, iterative, non-feedback/Lucas 4-32-37 with cos - 1.txt"
But as per the section "Multiple conflicting hypothesis" below, other approaches may also be considered.
I currently think that "∂[∂(t): Rpcs(POIo(t),t)^(-5)*t*(cos(Aθpc(POIo(t),t)) - 1)]" DEFINITELY should be included in derivations, but that :
- is based on a iterative solution, rather than an electromagnetic feedback effect as per Thomas Barnes (Lucas's source)
- ruins the functional form of results for the relativistic correction factor
- the coefficients of the series [are divergent, are not binomial series]
Note that [special, general] relativity does provide an explanation, but :
- in a non-phenomenological sense (more like data-fitting)
- "General relativity is a turkey" - see ?link to my web-page?
The concepts of [Thomas Barnes, Oleg Jefimenko, Ed Dowdye, Rami Ahmad El-Nabulsi, others] may provide a much more solid explanation for the "relativistic correction factor", but I have not verified these concepts step-by-step as of 24Oct2019.