I have to admit I am still confused by the wording some are using in this
discussion. I am bothered by wording that says "what does Peter read when
Jane reads" because this implies they can simultaneously take readings.
There is no such thing as simultaneity in this situation. All they can do
is record position and time for an event they detect, and they can calculate
what the other person will or has recorded for their observation of the same
event. It is true that observers can each record the time and position of
one specific event, but they do not make their recordings simultaneously.
The only thing that is the same for the two is they recorded the same event.
I find it very helpful to draw space-time graphs, and I also find it helpful
to do the complete Lorentz transformation for each event. I do this both
ways for confirmation. I put her data through the transformation to see
what numbers she calculates he wrote down, and I put his data through the
transformation to see what numbers he calculates she wrote down. What she
wrote down and what he calculates she wrote down had better be the same
thing (and vice-versa).
If Jane is moving along Peter's positive x axis, Peter is moving along
Jane's negative x axis, the relative velocity is 0.9 c, we can calculate the
time and position each person will record for various events if we know what
they recorded for some reference event. People are talking about "elapsed
time" or "rotations of the clock hand." This language makes no sense unless
there are two events. Delta-t and delta-x require two events. That's why I
said we must assume some "synchronization" event. I suggested each set
their clocks to t=0 at the moment their coordinate origins coincide in
space. This would not have to be the event used for time-1, but there has
to be some event used for time-1 before we can speak of a time-2 and then
calculate a delta-t.
If we choose they both set t=0 for the event of their coordinate origins
overlapping, then we can pick some other event end calculate positions and
times relative to this synchronization.
Suppose the event we choose is Peter reading his clock at t=1.00 minute. If
desired, we can imagine a strobe light on his clock flashes when the clock
hits 1.00 minute. Assuming his clock or strobe light is located at his
origin, he records this event as x=0 meters, t=1.00 minutes. We can do the
Lorentz transformation on this to see what Jane records for this event. She
records x=-6.19x10^8 meters and t=2.29 minutes.
Jane is smart, she understands the Lorentz transformation, she can calculate
what Peter recorded for this event. If she puts her data through the
Lorentz transformation she calculates Peter recorded x=0, t=1.00.
I don't like to say Jane recorded t=2.29 when Peter recorded t=1.00. That's
a semantic argument about what "when" means. I would rather say Jane
recorded t=2.29 for the same event for which Peter recorded t=1.00. She
recorded t=2.29 from her clock when she observed that Peter's strobe
flashed. At this point in my example it is especially not true to say Jane
measured t=0.436 when Peter recorded t=1.00 because Jane records t=0.436 for
a different event. Let's examine that event.
For some reason Jane set her strobe light to flash when her clock hit
t=0.436 minutes. Therefore she records this event as x=0, t=0.436 minutes.
We can use the Lorentz transformation to calculate what Peter records for
this event. He records t=1.00 minute and x=2.70x10^8 meters. Peter is
smart, he understands the Lorentz transformation, he can calculate what Jane
recorded for this event. He puts his data through the transformation and
realizes she recorded x=0, t=0.436 for this event.
We now see the timing chosen for Jane's strobe light to flash just happened
to be the correct timing for Peter to record that event as t=1.00 minutes.
This means from Peter's perspective the two flashes (his flash and her
flash) were simultaneous at t=1.00 minutes. They did not occur at the same
place from Peter's perspective. His flash occurred at x=0 and her flash
occurred at x=2.70x10^8 meters.
We also see these events were neither simultaneous nor at the same place
from Jane's viewpoint. She recorded her flash first at x=0 meters and
t=0.436 minutes. She recorded his flash later at x=-6.19x10^8 meters and
t=2.29 minutes.
Barring computational errors, I believe these numbers are correct for the
situation as I have described it, and I think my wording is appropriate.
Please tell me if there are errors or ambiguities in what I have described.
Now we need to see if the original problem reasonably described either of
these situations. The question states: "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?"
I agree this is poorly worded, but I would choose this is describing Peter's
observation of t=1.00 as the event both people are observing. In that case
we have to answer that Jane records t=2.29 minutes for this event. We
cannot conclude Jane records t=0.436 minutes because that refers to a
different event... it refers to the event of Jane reading her clock at
t=0.436 minutes.
This is tricky. Peter observing t=1.00; Jane observing t=2.29... versus
Peter observing t=1.00; Jane observing t=0.436... refer to different
space-time events. Which space-time event is the original question
describing? Although the wording could be construed to be either of the two
events, I personally think it refers to the event of Peter's clock hitting
t=1.00.
I do not feel strong about that personal opinion, and that is why the
question is poor. But I do feel strong that the correct way to analyze this
is to calculate the times and locations each person records for specific
events they observe.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817