/drawingfollowup.rel     from 1985    for jan98, sep98, sep99, feb01, jan,sep02

ASSIGNMENT on your drawing of a Lorentz Transformation, see /drawing.rel

1.  Label your CONTOURS.
Both x and x' values
increase towards
* the * right * ;
 both t and t'
values increase
 going  * up * .
See my own crude
 illustration,
to help get these
 straight.
     You now know
 the t x t' x'
values of ANY POINT
 (i.e. EVENT)
ON YOUR SHEET
 by simply reading
off the value
 on the APPROPRIATE
contour running through it,
or else, if, say,
 the point lies
*between* two  x  contours
 and you
wish its  x  value,
 by *interpolating*.  -->  DO EVERYTHING BY USING
*ONLY* YOUR CONTOURS AND LABELS---*never* take readings with a ruler!

2.  Determine your *speed*.  Method: Take two events, no. ``1'' and ``2'',
at the same (')-place---which means on the same  x'  contour of course---
read off their  x, t  values, I call them  x1 t1  and  x2 t2 , then
compute       x2 - x1
          v = ------- .  If you can follow these steps yet do not
              t2 - t1
understand *why* this is the speed  v  of Prime as seen by Unprime, ASK!
     The rest is finding your Lorentz contraction FROM THE PICTURE,
that is, without using the formula for that (except for a check, at step 4).
Also (step 5), you are to EXPLAIN AWAY THE PARADOX seemingly involved,
in each observer's finding the other's stick shorter than his own.
Steps for this:

3.  Color (')'s ``unit stick'', say, RED.  [If coloring between x'=0
and x'=1 gives an unpleasantly narrow strip, let your ``unit'' include
*several* steps of x'; thus, if \Delta x' = 1  means ``1 foot'' and you
prefer to use a unit of ``one yard'', you would color from x'=0  to x'=3,
                               for example.]
    Color ( )'s unit stick a different color, say, YELLOW.
...You needn't redraw everything just to match my suggested colors.
    To see what ( ) sees:  Rub against both sticks at ONE FIXED TIME
AS JUDGED BY ( ), MEANING, confine your attention to one value of  t :
to one  t  contour.  The red part is the red stick, then.  The yellow
part is the yellow stick.  You will see which is shorter, and you can
measure their ratio with a ruler TO GET THE RIGHT ANSWER  *BADLY* .  To
get it  *well* , find the  *length*  of each stick  *without*  a ruler, by
reading the  x  values of the two  RED  endpoints then subtracting one
from the other to get the RED LENGTH AS SEEN BY ( );  then the same,
i.e., x values, for the YELLOW endpoints, to get the YELLOW LENGTH.
You use x values, not x' values, because you are duplicating Unprime's
measurements.  Find the  SHORT LENGTH                        ^^
                         ------------  ratio.  Is the other guy's stick
                          LONG LENGTH                     the short one?
                                                         TURN OVER ----------->
                                                                          page2

     Next, do the OTHER POINT OF VIEW:  Get what Prime i.e. (') sees,
by taking a rubbing at a fixed  t' : this means that you choose one t' contour,
and now read off x' values for ends of the RED portion, then also x' values
for ends of the YELLOW portion, etc.

4. Of course,  SHORT
               ----- = sqrt{1 - v^2}  by *formula* for the Lorentz contraction.
               LONG                   [Of course ``v=1'' means, speed
                                      3*10^8 m/s ;  ``v=1/2'' means 1.5*10^8m/s
but the  m/s  unit  for speed is inconvenient here,
           and is  *not*  recommended.]
Note that  (SHORT/LONG)^2 = 1 - v^2 ,  so  (SHORT/LONG)^2 + SPEED^2
should come out *near* 1.  Most of you will use approximate readings, so don't
expect to get  *exactly*  1.

           ( But it is  *possible*  to keep to only ``nice''
points, through which all four types of contours run, for example, and
so to deal with exact answers only:  the use of integers p,q has biased
the situation, so that  v  and, perhaps surprisingly,  sqrt{1 - v^2} , are both
 RATIONAL! ) 

5.  Although you have now indeed found that each observer sees the
other's stick as shorter than his own, that may still seem paradoxical:
Two observers can  *not*  look at one snapshot, or at one rubbing, and
honestly disagree as to whether red is shorter than yellow, or yellow
shorter than red !

* Explain * in * what * way * your * work * avoids * this * objection. *

                              END