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Re: SR examination question



At 12:02 PM 5/18/01 -0400, Michael Edmiston wrote:

I find it very helpful to draw space-time graphs,

Yes yes yes yes yes.

... a semantic argument about what "when" means.

Yes yes yes yes yes.

The ambiguity of what "when" means is precisely why this question is
utterly and hopelessly ambiguous.

To see exactly why, let's draw the diagram and look at some of the
spacetime events.
http://www.monmouth.com/~jsd/physics/sp-rel.gif

First of all, let me repeat my vehement objection to the notion posted
yesterday that the problem could be solved by using the "God's Eye View" to
observe things. Please remember what the "R" stands for in SR. There is
no God's Eye View. My God thinks any view is as good as another.

Being in an evenhanded mood today, I have chosen a laboratory frame that is
midway between Jane and Peter. Jane is propagating to the right with a
rapidity of 0.45 (velocity 0.42) while Peter is propagating to my left with
equal rapidity. They are separating from each other with a rapidity of
0.90 (velocity 0.72). This is not exactly the velocity specified in the
original question but it makes a nice illustration.

My laboratory X-axis is the horizontal heavy green line.
My laboratory T-axis is the vertical heavy green line.

Peter's personal world-line is the heavy blue line. The blue lines that
run parallel to this are the world-lines of reference points displaced 30
and 60 light-seconds (in his frame) to the right of his position.

The blue lines that cross Peter's world line are successive snapshots of
his X-axis, at 30-second intervals. They are contours of constant
Peter-time.

The world-lines and the constant-time contours form in Peter's frame a
number of _unit squares_ 30 seconds on a side. To the naive eye, these
squares appear as trapezoids in the diagram, but they really are squares if
you use the Lorentz metric.

Jane's personal world-line is the heavy red line. The red lines that run
parallel to this are the world lines of reference points displaced by
multiples of 30 light-seconds (in her frame) from her position.

The red lines that cross Jane's world line are successive snapshots of her
X-axis, at 30-second intervals. They are contours of constant Jane-time.

To the naive eye, Jane's unit squares appear as trapezoids on the diagram,
but they really are squares if you use the Lorentz metric.

The spacetime event where Peter's watch indicates one minute of elapsed
time is circled in blue.

The spacetime event where Jane's watch indicates one minute of elapsed time
is circled in red.

Originally at 12:27 PM 5/17/01 +0200, Mark Sylvester wrote:
When Peter observes the second hand on his
watch to have made one complete revolution, how many revolutions will
Jane observe the second hand of her watch to have made?

As foreshadowed above, that all depends on what "when" means. Jane-when
or Peter-when or lab-when or whoever-wherever-when?????

We start from the crucial event, circled in blue. Then...
1) If we identify with Peter, we follow Peter's constant-time contour,
sloping down and to the right in the diagram. This corresponds to a
Jane-time considerably less than 1.0, as you can see by noting where this
contour crosses Jane's world-line.
2) OTOH if we identify with Jane, we follow Jane's constant-time contour,
sloping up and to the right in the diagram. This corresponds to a
Jane-time considerably greater than 1.0, as you can see by noting where
this contour crosses Jane's world-line.
3) On the third hand, if we believe that God favors my (arbitrarily
chosen) lab frame, then we follow the contour of constant lab-time, which
is horizontal in the diagram. We conclude that the crucial event occurs at
Jane-time equal to 1.0.

Conclusions (1) and (2) are independent of the choice of lab
frame. Peter-when and Jane-when are perfectly well defined
concepts. Assuming our protagonists were paying attention when they
crossed paths at the (0,0,0,0) event, Jane knows what Peter-when means, and
Peter knows what Jane-when means.

==========

Bottom line: Unless the context has established whose constant-time
contours are to be used (or unless the events in question are colocated in
space), any question that begins
"When ...."
is an automatic loser. The rest of the question doesn't even
matter. Tilt! Game Over!

Teach your students:
-- The notion of simultaneity-at-a-distance is nonunique.
-- The notion of simultaneity-at-a-distance is nonunique.
-- The notion of simultaneity-at-a-distance is nonunique.

====================

The following is only tangentially related to the current question, but it
is interesting, and serves as a useful check on the accuracy of
construction of the diagram: The black line is the world line of a photon
that just happened to be passing through. Note that it travels diagonally
through the unit square in _all three_ frames. This means the speed of
light is the same in all three frames.