2.6 A Quick Comparison of the two Observers (Space-Time Diagrams)
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This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
2.6 A Quick Comparison of the two Observers (Space-Time Diagrams)
For a moment, I want to go back and compare the two observers in
Diagram 2-8. Consider how the O observer would explain the experiment done
by the O' observer. First note that in the coordinate system used by the O
observer, the point marked C' is above the x axis. This means that in the O
observer's frame of reference, C' happens after the origin (when the two
observers pass by one another). However, we concluded that for O' the C'
event happens at the same time as the two observers are passing one another.
What does that mean?
Look at the parts of the experiment O' did (including the actions of O'
and the events A', B', and C') as they appear in the O observer's frame. In
that frame, O' sends out a light signal when his own clock reads t' = -T,
but note also that he is moving along with that signal (according to O). The
distance between them changes slowly at the beginning according to O because
O' is moving along with the signal in the same direction. Then, according to
O, the two observers pass by one another. Next, the C' event happens and the
light bounces back toward the two observers. In the frame of the O observer,
the O' observer is now racing towards the light beam, and so the distance
between them is changing very quickly. Finally, the light beam reaches O' as
his clock is ticking t' = +T.
So, we see that in the O frame of reference, because O' is moving along
with the light before C' and is moving towards the light after C' that means
C' has to happen after the "half way point" (when the two observers pass one
another).
HOWEVER, relativity says that O' cannot agree with that analysis. In
the frame of O', it is the O observer who is moving. Further, O' cannot
agree that the distance between him and the light is changing slowly before
C' and quickly after C'. Why can't he agree? Well, because then he would
measure the speed of the light in his frame of reference and find it to be
different going away from him than it is coming back to him. As discussed in
Section 1.2, relativity dictates that for ANY inertial observer, when he
measures the speed of light he MUST find the speed to be c--ALWAYS, and in
ALL directions. If O' has to find that the light is traveling at the same
speed going and coming back, then O' also has to conclude that in his frame
C' really, truly happens at the same time as the origin (when and where the
two observers pass one another). O' thus has a different coordinate system
than O, and he measure space and time differently.
And so, in one frame of reference C' really, truly happens after the
two observers pass one another, but in another frame of reference C' really,
truly happens and the same time the two observer's pass. We find that the
notion of simultaneity is relative, and we will discuss this further in just
a bit.
Next, though, I want to address a possibility you might be thinking
right now. That is, why can't it simply be that O' is just wrong in
interpreting things as he does and that O is correct. One might want to
claim that the reason O' is confused is that he is moving while O is not.
But next we will see that we can interchange the two observers, and it
becomes obvious that there is no absolute way to claim that one of them is
the "correct" observer.
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