It should be noted that, the following are attempting to depict two coordinate systems that occupy the same space. So, although BR ( Bottom Ruler ) and TR (Top Ruler ) are physically depicted as being apart, one must use their imagination to understand that this is only for identification purposes.
GIVEN a point A at 2, and two coincident Cartesian coordinate systems BR and TR.
Diagram A the active version
axiom 1
Coincident Cartesian coordinate systems possess equally aligned points.
It would be impossible to have Point A and no corresponding point in TR.
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Diagram B the passive version
No transformation has taken place yet.
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TR and BR analogy with a rhombic dodecahedron.
Given: two coincident Cartesian coordinate systems
Blue called BR having the Rhombic dodecahedron as shown in yellow.
Tangerine called TR, also having the Rhombic dodecahedron in yellow.
( The coincident tangerine TR system, with its' rhombic dodecahedron nor its' point B are not shown. )
photo 1
We place a coincident point A on BR and a corresponding Point B on TR.
If I shift ( therefore translate ) the tangerine Cartesian coordinate system TR over + 3 units, ( along the x direction ) then
1 The yellow rhombic dodecahedron still exists on the table in the blue BR
Cartesian coordinate system.
2 Point A on the yellow rhombic dodecahedron still exists in the blue BR
Cartesian coordinate system.
3 The yellow rhombic dodecahedron still exists in the tangerine TR
Cartesian coordinate system.
4 Point B on the yellow rhombic dodecahedron exists in the tangerine TR
Cartesian coordinate system
5 NO transformation has taken place yet, simply the translation of the one coordinate system TR.
The Galilean would apply the unmoved yellow rhombic dodecahedron in BR to make a transformation of an identical yellow rhombic doecahedron in that same location ( coincident location that shares equal points ) ... as the unmoved one.
The Galilean would ignore the existence of the translated rhombic dodecahedron in the tangerine TR Cartesian coordinate system
( with its' own tangerine rhombic dodecahedron ) as soon as the tangerine TR Cartesian coordinate system began translation.
While this transformation finds a value for x' given x, it should be finding out what x is in the other frame. However, by solving for x in TR, the Galilean unwittingly manufactures a new value for x' in TR. This value of x' [ aka point B ] in TR is ALREADY known, as seen in Diagram B. It is 2. It is 2 when the frames are coincincent. It is 2 when the frames are translated. It is 2 also, should the frames be returned to coincidence after the transformations.
By analogy, Point B no longer exists, since the rhombic dodecahedron in the tangerine TR Cartesian coordinate system no longer exists in their transformation scheme. Thus, prior to any transformation,
the "Diagram D before translation" scenario below, depicts the proper existence of Point B, prior to any transformation.
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Diagram C the Galilean results
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Diagram D before TR translation
axiom 2
Origin translation does not alter point location with respect to the Cartesian coordinate system in which it resides.
Point B still exists prior to any transformation.
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Diagram D after TR translation
Depicting both Point A and Point B requiring a transformation.
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Diagram E Galilean after translation and replaced to coincidence.
Since point A equals 2 and BR is now coincident with TR...then point B = 2 in TR.
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Diagram F Galilean after translation and replaced to coincidence.
( with Point B )
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Diagram G The Galilean's two possibles values or a point A transformed to TR.
Violates axiom 3. Point A transformed has either a value of -1 or 2.
axiom 3
A Cartesian coordinate system cannot possess equally named points having unequal coordinate values.
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Conclusion - If physics is not allowed to use this one equation then time dilation and length contraction are mathematically invalid as well as conceptually invalid. To a mathematician, this is basically a simple if then relationship. Relativity itself has no supportive leg to stand upon. Indeed, this author believes the
arenas of both time and space should be returned to mathematical exactitudes. Without x' = x - vt...there is no Relativity.
Opinions, objections, questions, or comments regarding the above are invited. My address is swaterman@watermanpolyhedron.com
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