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(SR) Lorentz t', x' = Intervals
Section - 8. Time intervals.

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Top Document: (SR) Lorentz t', x' = Intervals
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Next Document: 9. Einstein's (1905) derivations.
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Instead of using our sticks, let's get out two clocks.

Mind you, we're not going to deal with different clock
rates here, just establish the same basics as for distance.

Your clock says 9:00 Eastern Standard Time (EST) and we 
note that t=540 minutes when we put down the clock.

Blindly, let's turn the setting knob of your twin's Pacific 
Standard Time clock and put it down before us.

According to what we see, EST's 540 minutes (9:00) corre-
sponds to PST's 14:30; t'=870.


We know the formula relating PST to EST is t' (pacific)
= t (eastern) - 180 (minutes). Thus, it is not correct 
that the second clock can have an arbitrary setting, 
because 870 <> 540-180. 

We know that the two clocks are related by t' = t/1 since 
both are using the same second, hour, etc units. But 870 
(14:30 in minutes) is not 540/1-180, so once again we know 
something is wrong. 

However, t'=t.z' + t/1 works. EST midnight equals PST 0.0 
(midnite)  - 180,  so t.z' = -180, and

     t' = -180 + 540/1  = 360.

Since EST-180=PST, 9:00 EST is 6:00 PST = 360 minutes.

We see thus that like distance measures/coordinates, time
axis origins (zero points) must either be 'lined up' or 
adjusted for. 

So, the Lorentz/Einstein t'=t/sqrt(1-vv/cc) must be the moving 
system elapsed time interval since the time axes were both at 
a common zero. There is no t.z' adjustment:

      t' = (t - vx/cc)/sqrt(1-vv/cc)   (Eq 1t)

Make sure you understand that in the clock case, if the
EST is showing a good number for elapsed time since the
travelling observer passed NYC, then the PST clock is
silliness. t.z' must be zero or must be taken out of
time lapse calculations for the PST clock to be used
intelligently, just as was true for x.z'.

What is lacking as yet for Lorentz t' is the -vx/cc term that
corresponds to the x' formula -vt term.

Break it up into two parts: v/c and x/c.  

v/c is a scaling factor that changes velocity from whatever 
kind of unit you are using over to fractions of c.

x/c is distance divided by velocity, which is time. x/c
is thus the time interval since the two time axes
had a common zero point - which they have to have in the
Lorentz transforms which do not have the t.z' term we
learned to use above.

Thus, (-vx/cc)/sqrt(1-vv/cc) is the interval amount the 
moving system clock has been changed - since the common/
adjusted time - over and beyond the elapsed time interval
represented by x/sqrt(1-vv/cc).

We have discovered that the only way for t' to be t/g
is for t' and t to have a common zero point, just as
for x' and x. It would be otherwise if the t' formula
contained an adjustment t.z' under some name or other,
but the necessity to include such a term correlates
100% with t' numbers that aren't directly usable.

As for x and x', our knowledge of how to setup a proper
formula relating t and t' is of no use unless we use
the knowledge in scientific formulas; (t'-t.z'+xv/gcc)
gives us the only directly useful value: t/g.


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Top Document: (SR) Lorentz t', x' = Intervals
Previous Document: 7. Distances and moving coordinate axes.
Next Document: 9. Einstein's (1905) derivations.

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Last Update March 27 2014 @ 02:12 PM