The Rbtx Coordinate Transform

[Col. Rbtx]

The Rbtx Forward and Inverse Transform
by
Robt. William Brown


In my previous paper I have tried to show how to express the difference between Galilean distances and Einsteinian distances as a 'power of gamma' in the following equation:

H= gamma^{p} * [sqrt]{(x/[gamma])^2 + y^2 + z^2}...........eq.101

gamma^{p} = H / [sqrt]{(x/[gamma])^2 + y^2 + z^2}................eq.201

Recalling:
H= einsteinian distance
h= galilean distance

and:

h= [sqrt]{(x/[gamma])^2 + y^2 + z^2}
H= [sqrt]{(x^2 + y^2 + z^2}

WE can simplify equation 201 to be:

gamma^{p} = H / h

gamma^{p}= einsteinian distance / galilean distance...............!!

Then through this power of gamma ratio, I setforth this defintion of TIME:

T' = t' + {[gamma](T-VX/c^2) - [gamma](t -Vx/c^2)] / {H / [sqrt]{(x/[gamma])^2 + y^2 + z^2}................

We then couple the above defintion of TIME, to the Galilean transform and have the Rbtx-galilean Forward Transform:

T' = t' + {[gamma](T-VX/c^2) - [gamma](t -Vx/c^2)] / {H / [sqrt]{(x/[gamma])^2 + y^2 + z^2} eq. 501
X'= X-VT
Y'= Y
Z'= Z

where:
x = [gamma](X-VT)
y= Y
z= Z

t = previous time coordinate in 'rest frame'
x= previous distance coordinate in 'rest frame'.


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AT the same time I have also tried to show how this 'ratio of distances' can be used to adjust 'einteinian time intervals' to compute within 'galilean distances'. Showing that it is possible that light can also measure c, within the galilean framework (in a 'forward transformation'.)

I originally thought that it was feasable to use the 'galilean transform' in the inverse transformations. But I have since discovered that the Rbtx INVERSE TRANSFORM must take a more Fitzgerald formi.e.) But let's call them 'contra-galilean distances', since we are multiplying the Lorentz defintion by gamma, instead of dividing (as previously done):

X= [gamma]^2(X' - (-VT'/c^2)
Y= Y'
Z= Z'

This means that to arrive at the correct 'inverse time transformations' we must use this approach:

H= 1/gamma^{p} * [sqrt]{(x*[gamma])]^2 + y^2 + z^2}...........eq.101-A

gamma^{p} = 1/ {H / [sqrt]{(x * [gamma])^2 + y^2 + z^2}................eq.201-A

T' = t + {[gamma](T'-(-VX'/c^2) - [gamma](t' -(-Vx' /c^2)] / 1/ {H / [sqrt]{(x * [gamma])^2 + y^2 + z^2}....(eq.501-A)

Then we couple the above equation (501-A) to the distance equations given above, and we have:

Rbtx/contra-galilean Inverse Transform:

T' = t + {[gamma](T'-(-VX'/c^2) - [gamma](t' -(-Vx'/c^2)] / { 1/ {H / [sqrt]{(x * [gamma])^2 + y^2 + z^2}}
X= [gamma]^2(X'- (-VT'/c^2)
Y= Y'
Z= Z'

where:
x = [gamma](X'-VT')
y= Y'
z= Z'

t' = previous time coordinate in 'motion frame'
x' = previous distance coordinate in 'motion frame'.

In summation: The above equations are somewhat tricky to express but the concept that they are built on is
a simple and sound principle.
1.Determine the ratio between Einsteinian distances and Galilean (or contra-galilean) distances.
(We have the option of expressing this ratio as a 'power of gamma' and we can call it the Que factor.

Que = [gamma]^p ..............forward transform ................eq.601

Que = 1/ [gamma]^p' ...........inverse transform.................eq.601-A

2. We take an 'interval of einsteinian time' and adjust it by the Que factor so that it will fit into the 'galilean distance interval', so this will insure that the velocity of light remains constant!

When I first envinsioned this method of coordinate transformation I presumed that we would always 'decrease' the einsteinian time-interval by the que factor to keep distances absolute, but we find that this is not so! (Just compare equations 601 and 601-A to understand why.)

I also presumed that 'absolute distances' would be synonmous with 'galilean distances' but we find that since the Rbtx inverse transform is more akin to the Fitzgerald inverse transform' we must contend with contra-galilean distances. Which means that 'absolute distances' are a result of :

galilean distances...........coupled with...........contra-galilean distances

Where galilean distances are usually LESS than Einsteinian distances, while contra-galilean distances are usually GREATER than Einsteinian distances!

This indicates that the original Galilean concept of 'absolute distances' was a 'unilateral concept that had no real mathematical or bilateral basis, at least not in regard to 'light movement'.

I will further state that since we can arrive at various transforms (even the Lorentz) through the basic method I have used to arrive at the Rbtx transfomations above, the ideas described here might at least have some value in studying the derivation of the LORENTZ transform, from a unique perspective.

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Compliments of Col. Rbtx, the Barnyard Physicist of Texas.

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