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(SR) Lorentz t', x' = Intervals
Section - 6. Relating two coordinate measures/systems.

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Taking care to not damage our brave little ant, I place
my yard-stick onto the table, zero end to the left, 36"
end to the right.

Now I place the 'just coordinate' meter-stick on the table
in the same orientation,  in a random location, and find
that the ant's coordinate on the meter-stick is 51.

The formula relating centimeters to inches is cm=i*2.54
but we want a formula similar to x'=(x-vt)/sqrt(1-vv/cc).
That would be c=i/.03937 approximately, but let's use x'
for the meter-stick reading, and x for the inch reading:

    x'=x/.3937.

    3.5/.3937 = 8.89  
    
Wait a minute.  It's not just science but definition 
that says c=i/.3937=8.89, so something is wrong.  8.89
is not 51.

We already knew that 51 cm was just an arbitrary coordinate. 
Arbitrary not because that point isn't 51 cm from the zero 
end of the meter-stick, but because the zero point was in an 
arbitrary position.

Let's put the meter-stick in a position where it's 
zero point is at the yard-stick zero point.

What is the centimeter coordinate now?  Hey. 8.89,
just like the formula says.

The only way for a 'transform' like x'=x/g to work, 
whatever g might be, is for both coordinate systems
to have their zero points aligned, in which case
saying the two measures are not intervals is pure
idiocy.

Noe that with both zero points at the same position
both x' and x are great measures for scientific
purposes, in any and every case where we were smart
enough to put those zero points at a useful location.

There is one extension of x'=x/g that will let us
use the meter-stick in arbitrary position. 

When the cm reading was 51, the zero point of the
yard-stick read (51-8.89=) 42.11 cm. If we call that
point x.z' we get 

     x' = x.z' + x/.3937.
 = 42.11 + 3.5/.3937
 = 42.11 + 8.89
 = 51.

Obviously, in this formula x/.3937 is the distance
from the x' coordinate of the location where x=0. 
An interval.

Just as obviously, the fact that we now have the
correct formula for relating an x interval to an
arbitrary x' coordinate, does not mean that x'
is anything more than nonsense for use in any
scientific formula.

Unless we were smart enough to put the x zero
point in a useful location, and use (x'-x.z') in
the scientific formula. (x'-x.z') equals the useful,
Ratio Scale value x/.3937.


So, we have discovered a basic fact: a transformation
formula like x'=x/g works only if the two zero points
of the coordinate systems coincide. That makes it non-
sense to say the two coodinates are only coordinates
and not intervals.  Both must be values that represent
distances from their respective zero points unless you
take the proper steps to adjust for the discrepancy.

Make sure you understand that although the inclusion
of x.z' made it possible to correctly calculate x',
the result is nonsense when it comes to use of x'
for general length/distance purposes; it is x'-x.z' 
that is a useful number in such cases. It could be
that we're measuring a sheet of paper with one end
at x=0 and the other at x=3.5; x'=51 is nonsense as
a centimeter measure of the paper.

But, you say, the Lorentz transform contain a -vt term.

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Top Document: (SR) Lorentz t', x' = Intervals
Previous Document: 5. Single-system, little-purpose ambiguity.
Next Document: 7. Distances and moving coordinate axes.

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Last Update March 27 2014 @ 02:12 PM