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## Main Question or Discussion Point

Since I am new to the forum here, I apologize in advance that in case similar example has been explained in another thread. If so, please kindly refer me to the appropriate place to read further.

In my example, there will be four reference objects, earth, E, distant star, S, spaceship A, and spaceship B. Initially, all four objects are at rest arrange in the following way, A is right next to E; S is ahead of A and E with a distance of 0.866 light year, and B is behind A and E also at a distance of 0.866 light year. (I chose this number just so that the time dilation fact will end up being roughly 0.5 later)

S--------------------A/E--------------------B

Now, let’s assume A starts to accelerate at a very high rate with respect to E, so that A reaches a speed of 0.866c almost instantly. B will also accelerate in a way such that the proper distance between A and B remain constant at 0.866 light year all the time. And right before A reaches S, A decelerates almost instantly again and stop at S. Again, B will also decelerates to keep the distant between A and B constant. Before A starts to move, observers on S,E,A and B all synchronize their clocks.

I would like to claim the event (E

Now, I will try to figure out what does each observer see during the event.

1 from an observer on Earth E

a. S is always at same distance away from E and the two clocks run the same.

b. A is at first rest next to E, then accelerate instantly to a speed of 0.866c, then move at that speed towards S for most of the distance, then A decelerates and stops at S. Assume the acceleration and deceleration are almost instant, the clock on E should show roughly 1 year has passed. And E should see that the clock on A shows roughly 0.5 year passed.

c. Due to the length contraction, E should B “jump” 0.433 light year towards E almost instantly, and the clock on B should also jump 0.5 year almost instantly. Then B moves at constant speed then decelerates to then stop and both clocks on B should show roughly 1 year has passed and the clock on E should show 0.5 year. (This is the part I think I might I have made some mistakes)

2 from an observer on spaceship A

a. B and A are always on the same inertial frame, as defined, so B is always at same distance away from A and the two clocks run the same. (Not entirely sure about this either)

b. Immediately after the acceleration, the distance between S and E contract by half, A should see S “jump” 0.433 light year towards A, and again the clock on S instantly advances by 0.5 year. Then A continues to move at a constant speed of 0.866c. When A comes to a rest right next to S, the clock on A should show 0.5 year passed and clock on S shows 1 year.

c. A should see E accelerate away from A and continue move towards B at a constant speed. Right before A decelerates to stop, while A should almost reach S, A should see E almost reach halfway between A and B, and the time on both clocks should be 0.5 year. (distance between A and B is 0.866 light year, while distance between S and E as seen by A is 0.433 light year). Again, when A decelerates almost instantly, A should see E “jump” instantly towards B and the time on E should “jump” forward 0.5 year. Thus, when A stops at S, clock on A shows 0.5 year and clock on E shows 1 year.

3 from an observer on spaceship B

a. B should see E first accelerate then move at constant speed until E reaches almost half way between A and B. Then E will appears to “jump” right in front of B. When E stops at B, clock on E shows 1 year and clock on B shows 0.5 year.

So far, I think most of observations about the events agree with each other, and indicates that when A comes to a stop at S, time on A/B has passed roughly 0.5 year and time on E/S has passed 1 year. However, I am confused about the events I stated in 1c and 3a. Somehow they shows contradicting results.

I think the contradiction could arise from two main issues, 1st, I did not handle the “time jump” and “distance jump” during length contraction correctly, 2nd, I assumed that the event of A stops at S and the event of B stops at E happens simultaneously in both reference frames

So my ultimate question would be where I was wrong and what time the clocks on each object should show, when A first stops at S and also when B first stops at E (if the two events are not simultaneous)

In my example, there will be four reference objects, earth, E, distant star, S, spaceship A, and spaceship B. Initially, all four objects are at rest arrange in the following way, A is right next to E; S is ahead of A and E with a distance of 0.866 light year, and B is behind A and E also at a distance of 0.866 light year. (I chose this number just so that the time dilation fact will end up being roughly 0.5 later)

S--------------------A/E--------------------B

Now, let’s assume A starts to accelerate at a very high rate with respect to E, so that A reaches a speed of 0.866c almost instantly. B will also accelerate in a way such that the proper distance between A and B remain constant at 0.866 light year all the time. And right before A reaches S, A decelerates almost instantly again and stop at S. Again, B will also decelerates to keep the distant between A and B constant. Before A starts to move, observers on S,E,A and B all synchronize their clocks.

I would like to claim the event (E

_{AS})where spaceship A reaches distant star S, and the event(E_{BE}) where spaceship B reaches Earth E happens simultaneously for both A/B and E/S. The reason I think is that the very moment A stops at S, A joins the reference frame of S/E. Since B moves in a way such that proper distant between A and B observed by A and B is always constant, the moment A stops, A should see B stops at E as well. Conversely, for observer on E/S, he should see E_{AS}and E_{BE}happens at same time as well.Now, I will try to figure out what does each observer see during the event.

1 from an observer on Earth E

a. S is always at same distance away from E and the two clocks run the same.

b. A is at first rest next to E, then accelerate instantly to a speed of 0.866c, then move at that speed towards S for most of the distance, then A decelerates and stops at S. Assume the acceleration and deceleration are almost instant, the clock on E should show roughly 1 year has passed. And E should see that the clock on A shows roughly 0.5 year passed.

c. Due to the length contraction, E should B “jump” 0.433 light year towards E almost instantly, and the clock on B should also jump 0.5 year almost instantly. Then B moves at constant speed then decelerates to then stop and both clocks on B should show roughly 1 year has passed and the clock on E should show 0.5 year. (This is the part I think I might I have made some mistakes)

2 from an observer on spaceship A

a. B and A are always on the same inertial frame, as defined, so B is always at same distance away from A and the two clocks run the same. (Not entirely sure about this either)

b. Immediately after the acceleration, the distance between S and E contract by half, A should see S “jump” 0.433 light year towards A, and again the clock on S instantly advances by 0.5 year. Then A continues to move at a constant speed of 0.866c. When A comes to a rest right next to S, the clock on A should show 0.5 year passed and clock on S shows 1 year.

c. A should see E accelerate away from A and continue move towards B at a constant speed. Right before A decelerates to stop, while A should almost reach S, A should see E almost reach halfway between A and B, and the time on both clocks should be 0.5 year. (distance between A and B is 0.866 light year, while distance between S and E as seen by A is 0.433 light year). Again, when A decelerates almost instantly, A should see E “jump” instantly towards B and the time on E should “jump” forward 0.5 year. Thus, when A stops at S, clock on A shows 0.5 year and clock on E shows 1 year.

3 from an observer on spaceship B

a. B should see E first accelerate then move at constant speed until E reaches almost half way between A and B. Then E will appears to “jump” right in front of B. When E stops at B, clock on E shows 1 year and clock on B shows 0.5 year.

So far, I think most of observations about the events agree with each other, and indicates that when A comes to a stop at S, time on A/B has passed roughly 0.5 year and time on E/S has passed 1 year. However, I am confused about the events I stated in 1c and 3a. Somehow they shows contradicting results.

I think the contradiction could arise from two main issues, 1st, I did not handle the “time jump” and “distance jump” during length contraction correctly, 2nd, I assumed that the event of A stops at S and the event of B stops at E happens simultaneously in both reference frames

So my ultimate question would be where I was wrong and what time the clocks on each object should show, when A first stops at S and also when B first stops at E (if the two events are not simultaneous)