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(SR) Lorentz t', x' = Intervals
Section - 7. Distances and moving coordinate axes.

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We discovered x'=x.z' + x/g as the correct formula
for relating one coordinate to another system's.

But the Lorentz transform contains another term, 
-vt/sqrt(1-vv/cc). What is it?

Let's start with our x'=51 cm, x=3.5", x.z'=42.11 example.

Every minute, let's move the meter-stick one inch to our
right.

At minute 0, the cm reading  was   51 cm.
At minute 1, the cm reading is now 50 cm.
At minute 2, the cm reading is now 49 cm.

In this instance, v=1 inch/minute. And t was 0, 1, 2.

What has happened is that we have made our x.z' a lie,
and increasingly so.  -vt/.3937 is the change in x.z'.


   x' = (x.z - vt/.3937) + x/.3937.

Obviously, vt/.3937 is not a coordinate; even most SRians
wouldn't imagine it was. It is an interval, the distance
over which the moving system has moved since t=0.


And, of course, x/.3937 is the distance of our brave
little ant from the point where x=0 and the centimeter
reading is x.z'-vt/.3937. Yes, every minute the meter-
stick moves to the right and the meter-stick coordinate
of the spot where x=0 gets less and less - and eventually
negative.

Make sure you understand that every minute the x' 
coordinate, because of -vt/g, becomes a better measure 
of, say, the  3.5"  paper we might be measuring with 
the yard-stick, given that 51 was too big a number and
-vt is negative.  That is, until the two origins coincide 
at x'=x=0,  and then it gets worse and worse.

With -vt positive (because v<0) the situation is different.

With 51 and -vt positive, x' just gets worse and worse
over time.

Quite obviously, the fact that we now have the
correct formula for relating an x interval to an
arbitrary x' coordinate even when the x' axis is
moving, 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'+vt/.3937) in
the scientific formula. (x'-x.z'+vt/.3937) equals the 
useful, Ratio Scale value x/.3937.

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Top Document: (SR) Lorentz t', x' = Intervals
Previous Document: 6. Relating two coordinate measures/systems.
Next Document: 8. Time intervals.

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Last Update March 27 2014 @ 02:12 PM