Explanation requested for a relativistic scenario wherein two seemingly valid measurement procedures lead to different results for the same observable

[DemKal]

Hi all,
Consider the following scenario.
Two poles are 100km apart in frame S; also, in frame S, a stopwatch A moves from left to right, at 0.999c, and starts to tick at the left pole and a stopwatch B moves from right to left, at 0.999c, and starts to tick at the right pole; S perceives these two events as simultaneous. Conclusions drawn so far: A and B will meet at the center and display the same reading of approximately 7.5microsec.

At the meeting point B says: "I will deduce the remaining distance to the left pole based on A's reading and speed. I know that even though A shows 7.5microsec like me, it started ticking way before that because A is slow and the associated gamma factor, based on its speed as it comes at me, is about 1000; therefore I calculate that it has been ticking for approximately 7.5millisec and, multiplying this time with B's speed, gives me a distance to the left pole of about 2,200km!!"

But if B measured the distance between the poles simply by stopping his/her stopwatch when arriving at the left pole, noting the time interval elapsed on the stopwatch, and multiplying with 0.999c, then B would get the 'normal' Lorentz contracted (wrt to the 100km measured in S) distance of only 4.5km. It seems that two perfectly valid methods, used by B and B alone, lead to wildly conflicting results.

All thoughts and comments welcome.

 

[Ibix]

Remember that the poles are moving at 0.999c in B's rest frame. The calculation B makes based on A's clock reading is not the distance between the poles, it's the distance between where A (going at more than 0.999c) passed a pole (going at 0.999c), which hapened a long time before the other pole passed B, and the place where A and B meet.

 

[Dale]

First, don’t use $v=0.999 c$. It is an annoying value. Use $v=0.6 c$ so that $\gamma = 1.25$. That is fast enough that relativistic effects are easily obtained, but not so many decimal points are required.

Second, use units where $c=1$. If you want times in microseconds then use units of light-microseconds for distance. One foot is approximately one light nanosecond.

As @Ibix said, the quantity you are describing is not the distance between the poles. The same thing happens classically. If an ant crawls 5 m from the back of your car to the front while you drive 100 km, the distance between the front and back is 5 m not 100 km.

 

[DemKal]

thanks for your replies; yes, my wording is wrong concerning the distance between the poles and what I am concerned about. since i already got the values for these parameters, lets keep them. OK, rephrasing: at the meeting point B will calculate that A has been ticking for about 7.5millisec, which is about 1000 times longer than B's reading of about 7.5microsec at that point. the left pole will reach B when another 7.5microsec has elapsed on B's stopwatch; the starting of the ticking of A at the left pole is an objective event; how can A, according to B's own calculation, have been ticking for a time 1000 times longer than B at the meeting point when it only took another 7.5microsec on B's stopwatch for the left pole to arrive. ??

 

[Ibix]

thanks for your replies; yes, my wording is wrong concerning the distance between the poles and what I am concerned about. since i already got the values for these parameters, lets keep them. OK, rephrasing: at the meeting point B will calculate that A has been ticking for about 7.5millisec, which is about 1000 times longer than B's reading of about 7.5microsec at that point. the left pole will reach B when another 7.5microsec has elapsed on B's stopwatch; the starting of the ticking of A at the left pole is an objective event; how can A, according to B's own calculation, have been ticking for a time 1000 times longer than B at the meeting point when it only took another 7.5microsec on B's stopwatch for the left pole to arrive. ??

Because, as we said, A passed the left pole a long time before B passed the right pole as measured in this frame. A and B only pass the poles simultaneously in the rest frame of the poles.

This is easy to calculate. Let the poles lie at $x=\pm L/2$ in their rest frame, and let A and B pass their respective first poles at $t=0$ in this frame, with A doing speed $u$ and B doing speed $-u$. Clearly A and B meet at $x=0$ and $t=L/(2u)$ in this frame. Apply the Lorentz transforms to get the times and places at which A and B pass their respective first poles and the time and place where A and B meet, all using the frame where B is at rest. What do you get? Then plug in your values. Do you see where the difference comes from?

 

[Ibix]

Just to check: you are aware that events that are simultaneous in one frame are not simultaneous in another? Your wording in the OP lead me to think you were, but your subsequent confusion seems consistent with you not realising it.

 

[Dale]

how can A, according to B's own calculation, have been ticking for a time 1000 times longer than B at the meeting point when it only took another 7.5microsec on B's stopwatch for the left pole to arrive.

In B's frame A passed the left pole and started its clock a long time before B passed the right pole. So in B's frame the sequence is something like this:
1) A reaches the left pole when A and both poles are a long distance away and starts his clock
2) a long time passes and they (A and both poles) get closer
3) the right pole reaches B and finally B starts their clock
4) a short time passes
5) A reaches B
6) a short time passes
7) the left pole reaches B

The short times in 4) and 6) are equal, but both are smaller than the long time in 2). The long time in 2) has very little to do with the distance between the poles and is rather about the distance away that A was when it started its clock.

 

[DemKal]

yes, thanks much for this clarification