Ambiguity of time dilation/twin paradox

[ghwellsjr]

There were some misunderstandings. In particular I believe you had misinterpreted purpose of adding observer C. If you would like to start from scratch - not referring specifically to any particular past exchange - I would be happy to reply. Otherwise I don't see the point.

You're right, I didn't realize that the OP knew that C would have to accelerate and so my second diagram does not represent his understanding. But I did realize that he wanted C to remain midway between A and B although I don't believe C can do that at a constant speed like the OP stated. But I have no idea how C must accelerate to maintain that goal, do you?

 

[ShreyasR]

You're right, I didn't realize that the OP knew that C would have to accelerate and so my second diagram does not represent his understanding. But I did realize that he wanted C to remain midway between A and B although I don't believe C can do that at a constant speed like the OP stated. But I have no idea how C must accelerate to maintain that goal, do you?

I did not state that C must maintain constant speed. If i have said so, i am sorry. C is free to accelerate such that the condition is met.

Just thinking of it in the probabilistic way, C is somewhere in between A and B. So there is a non-zero probability that C can remain exactly equidistant between A and B. But the problem is how must C accelerate in order to achieve this? I am not so good at calculations. And I am not free this week. I have internal assessment in college. I will try solving the problem mathematically and graphically ASAP and if i get stuck, i'll post it here.

 

[1977ub]

Initially B had a period of acceleration, which I wished to shrink to zero. likewise the deceleration and the accel/decl on the return leg. In A's frame, to stay midway between A & B, C can just halve the velocity of B. Anyhow I'm pretty sure the point was to ask about C's frame as if it was as adequate to measure time dilation as A's and we've discussed how it isn't. I don't think anything remains to be done.

 

[ghwellsjr]

You're right, I didn't realize that the OP knew that C would have to accelerate and so my second diagram does not represent his understanding. But I did realize that he wanted C to remain midway between A and B although I don't believe C can do that at a constant speed like the OP stated. But I have no idea how C must accelerate to maintain that goal, do you?

I did not state that C must maintain constant speed. If i have said so, i am sorry. C is free to accelerate such that the condition is met.

You did in post #61:

OK, so when does C change speed so as to remain midway between A & B?

C doesn't change speed, C turns back(towards earth) when C observes that B turns back...

I assumed you meant that just like B doesn't change speed (he just changes direction), that C also doesn't change speed, he just changes direction (both according to A's rest frame).

Just thinking of it in the probabilistic way, C is somewhere in between A and B. So there is a non-zero probability that C can remain exactly equidistant between A and B. But the problem is how must C accelerate in order to achieve this? I am not so good at calculations. And I am not free this week. I have internal assessment in college. I will try solving the problem mathematically and graphically ASAP and if i get stuck, i'll post it here.

You are correct, as long as you establish what you mean by "equidistant between A and B". As 1977ub pointed out in post #71:

He could catch wind of B's itinerary beforehand and then plan to do everything that B does, only traveling at half the velocity. I guess that makes sense from A's frame.

However, you originally stated that you not only wanted C to remain equidistant from A and B (trivial to define in any frame) but you wanted C to make the measurements:

Also, u can include a 3rd person C, who moves such that A and B are equidistant from him, So he'll measure the speeds and accelerations of A and B to be exactly the same wrt himself (but in opposite directions), throughout the whole trip. This should mean that The calculations of C should result in A and B being the same age after the trip isn't it?

I took this to mean that you also wanted C to measure that A and B were equidistant from him in his own rest frame. But since we all agree that his own frame is non-inertial we have the added problem that there is no standard way to establish a non-inertial frame. However, prior to my posting on this thread, the discussion centered around using radar methods to establish a frame and that's what I was focusing on when I drew the three diagrams in post #42.

With that in mind, I was using the second diagram to show how C's non-inertial frame must start out. In that frame, C will be inertial and "he'll measure the speeds and accelerations of A and B to be exactly the same wrt himself (but in opposite directions)" and he will measure the distances to A and B to be the same.

We also know that approaching the end of the scenario, an inverted situation applies where C is inertial and makes all the same measurements of A and B (but in opposite directions). However, somewhere in between, C has to accelerate in order to get from the first inertial position to the last inertial position and I believe he can do this in such a way that he will always measure A and B to be equidistant from himself but his measurements and observations of the speeds and accelerations and times of A and B cannot remain identical.

But I'm encouraging you to put this problem on the back burner until you learn how an observer uses radar to establish a reference frame. You need to feel real comfortable with that relatively easy task before tackling the very difficult task of establishing C's trajectory to maintain equal distance between A and B.

 

[ghwellsjr]

Initially B had a period of acceleration, which I wished to shrink to zero. likewise the deceleration and the accel/decl on the return leg.

Yes, we have all agreed to make B accelerate instantly at the start, midpoint and end of the scenario. To do otherwise would only add complication and not add understanding.

In A's frame, to stay midway between A & B, C can just halve the velocity of B.

Yes, since distance is frame dependent, we can use A's frame to establish a trajectory for C in which he remains midway between A & B. However, this is not the only thing that the OP wanted. He also wanted C to measure the speeds of A and B to be identical but in opposite directions. This won't ever be true for this trajectory defined by A's frame.

Anyhow I'm pretty sure the point was to ask about C's frame as if it was as adequate to measure time dilation as A's and we've discussed how it isn't.

True, because the standard definition of Time Dilation requires an Inertial Reference Frame (IRF) and A is the only observer who remains inertial in this scenario, but that doesn't mean that C's initial IRF (the second diagram in post #42) is not just as adequate to show Time Dilation, just like B's initial (and final) IRF is. Furthermore, we can use C's non-inertial reference frame to show how he can observe the Proper Time on the clocks of A and B during his travel, just like we can use B's non-inertial reference frame to show how he observes the Proper Time on the clocks of A and C, and how we can use A's inertial reference frame to show how he observes the Proper Time on the clocks of B and C.

I don't think anything remains to be done.

Well we could show a trajectory for C (in A's IRF) such that C measures himself to be equidistant from A & B at the beginning and end of the scenario and show how he's not equidistant in between. We could also show C's observations of A's and B's clocks and how they agree with the final outcome. We could show other trajectories for C, maybe even find one in which he measures himself to remain equidistant from A & B during the entire scenario and then see how he observes the others clocks and still comes to the correct conclusion at the end. There're lots of fun things left to do. I hate to leave anyone with the notion that only one frame (A's) is adequate to analyze or demonstrate what is happening in a scenario.