Relativity/Time dilation with stop-watches.
Hello Uncle Ben,
Consider the following:
In frame S there is a stop-watch at point A, and a pole at point B (at a distance L from point A, as measured in frame S).
There is another stop-watch, in frame S', moving from right to left, with speed V relative to S. This stop-watch starts ticking when it passes the pole.
Suppose that, as observed in frame S, the stop-watch at point A starts ticking simultaneously with the starting of the moving stop-watch (that started when it passed the pole at pint B).
Let us suppose that the two stop-watches stop ticking when they coincide, ie when the S' stop-watch passes over point A where the S stop-watch resides.
When the two stop-watches stop ticking at point A, the stop-watch of S' will be found to have LESS ticks registered than the S stop-watch since, according to valid S-frame calculations based in SR, this S' stop-watch is slow and started ticking at the same time as the S stop-watch at point A.
How does the S' stop-watch interpret this objective fact since, in frame S', it is the S stop-watch that is running slow?
Is it the correct interpretation to say that:
In the rest-frame of the S' stop-watch, the stop-watch at point A must have started ticking BEFORE the S' stop-watch passed the pole at point B (and started ticking), because that is the only way the S' observer can interpret the fact that the stop-watch at point A shows more ticks than the stop-watch in S' even though it is determined to be slow in the S' frame.
Jim, Your question is a good one.
I think you understand the relativity correctly. But I find it helps me think straight if I avoid the idea of an object being IN a frame. I say it is AT REST in the frame, if that is what I mean to say. Even clearer, the object is at rest WRT that frame. We can also call it "the rest frame" of the object.
There are many frames in the world, and every obect is IN all of them at the same time, albeit that it may have a different velocity with respect to each. The phrase "with respect to" is so valuable but so long, I write WRT very often.
Also, I recommend that we avoid speaking of what an observer thinks. This is not a matter of psychology. It is physics. THE time is not at good concept; there are many times, even for one object. Instead of an observer thinking the time is so-and-so, the time IS (not APPEARS TO BE) different WRT different frames.
Finally, what a clock reads can be different than what it keeps. If it reads t, what it keeps is t only WRT its rest frame S. WRT another frame S' it keeps a time t'. In your question, the moving clock reads one thing as it passes the pole, but it has experienced another time WRT S. That is what you say when you say that it runs slow WRT the rest frame of the pole.
With all of that, I recommend that you restate the question. What I just went over solves most problems students have, but if not, I will try again.
PS. In your question you have the two clocks start ticking "št the same time." But they are far apart. There is no absolute simultaneity; so "at the same time" depends on a particular frame. This may be at the heart of your question.