4.1.4 Some Additional Notes (Relativity and FTL Travel - Paradoxes and Solutions)
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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.
4.1.4 Some Additional Notes (Relativity and FTL Travel - Paradoxes and Solutions)
There are four specific things I want to make note of concerning the
twin paradox as I have explained it.
First, we should note that the outcome of the above thought experiment
(i.e. the fact that Sam ended up younger than Ed) is completely dependent on
the fact that Sam turned around and headed back to Ed. If instead Ed had
done the acceleration when he saw his own clock tick 4 years and had headed
over to meet Sam, then Ed would be the one who had aged a total of 8 years
while Sam had aged 10 years. Notice that the twin who undergoes the
acceleration must actually have a physical force applied to him to cause
that acceleration. During the acceleration he is no longer an inertial
observer (this is why his frame of reference shifts while the other twin's
frame does not shift). That differentiates his situation from the twin who
does not accelerate, and that breaks the symmetry between the two observers.
Unless one of them goes through an acceleration, their situations are
completely symmetric, and there is no absolute answer to the question "which
twin is younger?"
Second, I want to note something particular about the acceleration Sam
went through. Look back at the lines of simultaneity drawn for Sam's frame
before and after he accelerated. As we noted, the point where his "t' = 4"
line of simultaneity cross the t axis (Ed's position) shifts upward when Sam
turns around. Notice, however, that if Sam had taken a longer trip, then he
would have done the acceleration when he was further from Ed. Then that
"shift" would have been even larger, and after the acceleration, Sam's new
frame of reference would be one in which Ed's clock had "jumped" ahead an
even greater number of years. So, for Sam, the longer the trip he takes, the
bigger the change will be when he switches his frame of reference, and that
will make him an even greater number of years younger than Ed when they get
back together. Of course, for Ed, the longer the trip is for Sam, the longer
Sam's clock will be running slowly. So, Ed too agrees (with a different
explanation) that Sam will be more years younger than Ed in the end if the
trip is longer. As a final point on this, note that when Sam first
accelerates to start his trip, he is right next to Ed, so the acceleration
doesn't have much effect at all (as is true for his final acceleration at
the end of the trip). That is why we basically ignored those accelerations.
Third, I want to note something about Sam's explanation of the events.
Recall that when he changed frames of reference, his clock read 4 years
while (in his new frame) Ed's clock read 6.8 years. One may think that Sam
has thus changed to a frame where Ed's clock has been running faster;
however, we know that in Sam's new frame, Ed is still moving with respect to
Sam. Thus, in Sam's new frame Ed's clock has still been running slowly the
whole time. To understand how this can be, consider a third observer (Tim)
who has always been in the frame of reference which Sam has during the last
part of the trip. Let's say that Tim was traveling along (going to Earth)
when he saw Sam headed towards him, and to Tim's surprise, Sam turns around
and joins Tim in Tim's frame of reference as the two come together. Thus,
after Sam turns around, he and Tim are moving together, side by side. Now,
Tim notices that right after Sam turns around, Sam's clock reads 4 years.
Regardless of what Tim's clock reads, he can reset his clock to 4 years, and
we can backtrack 4 years along Tim's path to identify the origin of Tim's
frame (Sam's new frame). In Diagram 4-3 I have drawn (along with everything
in Diagram 4-2 Tim's path, the origin (o') of Sam's new frame of reference,
and a line of simultaneity for Tim's and Sam's frame at that origin.
Diagram 4-3
t
^
|
t=10 + t'=8
|*
| *
| *
| *
| *
t=6.8\ *
| *
| \ *
| *
t=5+ - - - - * - - -> (t = 5
| \ line)
| \ \
t=3.6|\ \
| \ \
| \ \ (t'= 4
| \ line)
| \ \
| \
| \ \
-----o-------------------o'----->x
| \
| (Tim's path)
Notice that for Sam's new frame (the frame Tim has always been in) if
t'= 4 when Sam turns around, then the event at Ed's position which is
simultaneous with the origin in this frame (o') is the event "Ed's clock
reads 3.6 years". And there you have it. In Sam's new frame, while it is
true that Ed's clock is always been running slow, at the "beginning" for
this frame (i.e. at its origin) Ed's clock started at 3.6 years. In this new
frame of Sam's, Ed's clock had a "head start" (so to speak) when compared to
Tim's clock. That is why Ed's clock already reads 6.8 years while Sam's
clock reads only 4 years in Sam's new frame. In the end, we can describe the
events in whatever frame of reference we wish, and though they may each have
different explanation for what actually happens, they must all agree with
the final outcome when the two twins come back together.
The final note I want to make is, again, about Sam's "view" of the
events. When we say that before Sam's turn-around he is in a frame of
reference in which Ed's clock reads 3.2 years, and after the turn-around Sam
is in a frame of reference in which Ed's clock reads 6.8 years, one might be
tempted to say that as Sam accelerates, Ed's clock speeds up in Sam's frame
of reference. Of course, this doesn't change the way Ed sees his clock
running, but it is only the way things occur in Sam's changing frame of
reference. However, think about what would happen if Sam quickly changed his
mind after the turn-around and immediately turned BACK around to his
original heading. Then, in this new acceleration, Sam went from a frame
where Ed's clock read 6.8 years to a frame where Ed's clock reads 3.2 years
again. One would thus argue that Ed's clock went backwards in Sam's changing
frame of reference. Again, this doesn't have any real significance to the
way Ed is reading his own clock, but we have to come to terms with the fact
that Sam's new acceleration caused Ed's clock to go backwards in Sam's
changing frame. Perhaps the best way to think about this is simply to
realize that Sam is not in an inertial frame since he is accelerating.
Rather, Sam is simply changing into various inertial frames, and in each of
these inertial frames, moving clocks do tick slowly, time does goes forward
in all frames, etc. Either way you like to think about it, in the end, we
can explain the outcomes as needed.
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