The
Twin Paradox using Simultaneous Events
Simultaneity
(or lack thereof) is a terrific tool for understanding many of the
paradoxes associated with SR. And, if I am to be thorough, simultaneity
must be considered for all SR events between separate frames of reference.
Let's re-visit the twin paradox (John travels out 12 hours at 60% the
speed of light and returns at the same speed). Basically, there are three
frames of reference to consider. First, the twins are on the earth with no
relative velocity between them. Second, John embarks on the outgoing leg
of his trip. Thirdly, John (after instantaneously turning around) embarks
on his return leg of his trip. I am using the same example as before,
except I am using numbers from the Lorentz Transforms as opposed to the
Relativistic Doppler Shift to explain the observed phenomena.
1st
frame:
Hunter
and John each agree on everything they observe. This should be easy to
understand since there is no relative velocity between the two twins. They
are in motion together.
2nd
frame:
John
travels out 12 hours by his clock. With the two postulates in mind, we
realize that Hunter observes time dilation for John's outgoing trip. Thus,
if John records 12 hours, Hunter will record 15 hours. Remember that at
60% the speed of light, the time dilation will be 80%. Therefore, if John
records his time to be 12 hours, this is 80% of what Hunter records - 15
hours. But what does John observe for Hunter's time? He observes the time
dilation as effecting Hunter; therefore, he measures his trip to be 12
hours, but he observes 9.6 hours (80% of his clock's time) for Hunter's
time.
2nd
frame totals:
Hunter
measures his time to be 15 hours, but John's time to be 12 hours. John
measures his time to be 12 hours, but Hunter's time to be 9.6 hours.
Obviously,
the event, which is the end of the outgoing trip, is not simultaneous.
John thinks Hunter's time is 9.6 hours but Hunter thinks his time is 15
hours. On top of that, they both think that John's time is 12 hours, which
doesn't agree with either of the first two times.
3rd
frame:
From
Hunter's perspective, nothing new has happened. He remained in his initial
frame of reference and John returned at the same velocity he left with.
Therefore, Hunter measured the return trip to take 15 hours for his frame
(same as the outgoing trip) and observes the trip to take 12 hours for
John. From John's perspective, he encountered a major change. He actually
changed frames from one of traveling out to one of traveling back. Now, at
the start of the return trip, when John looks at his clocks, he observes
his clock to read 12 hours and Hunter's clock to read 20.4 hours. Think
about this. John now shows that Hunter's clock has jumped ahead from 9.6
hours to 20.4 hours. How can this be???? When John changed from the 2nd
frame to the 3rd frame, the established symmetry between Hunter and John
was broken. Thus, each views their own time as having no change. And since
John was the one that actually changed frames, he showed more elapsed time
for Hunter. From here on out, it is business as usual. The return trip is
clocked at 12 hours by John, but he observes 9.6 hours for Hunter. Again,
let's clean this up…
3rd
frame totals:
Hunter
measures his time to be 15 hours, but he measures John's time to be 12
hours. John measures his time to be 12 hours, but he measures Hunter's
time to be 9.6 hours. Remember, this 9.6 is only for the return trip after
the frame change.
Trip
totals:
Hunter
measured his time to be 15 hours for the outgoing trip + 15 hours for the
return trip…30 hours.
Hunter observed John's time to be 12 hours outgoing + 12 hours return
…24 hours.
John measured his time to be 12 hours outgoing + 12 hours return…24
hours.
John observed Hunter's time to be 20.4 hours (after outgoing trip and
frame change) + 9.6 hours for the return trip…20.4 + 9.6 = 30 hours.
Can
you find any events in which both John and Hunter agree on the time for
both themselves and the other? No, you can't. The lack of simultaneity is
the key to the paradox. Both twins are measuring and observing.
Unfortunately, they are not measuring and observing the same events. It is
impossible for them to consider something like the end of the first leg as
simultaneous when they each view it occurring at different times for
Hunter. It's interesting to note that the results are the same as the
Relativistic Doppler shift results. Is there a pattern here? SR allows for
various methods to be employed to resolve the problems. For this case, use
of space-time diagrams (there's those words again) would clearly show
every point that we have talked about. I have merely used the Lorentz
transforms in combination with the Relativistic Doppler effect.
Many
people have trouble with the twin paradox because of the way in which the
frame change is handled. In this case, the jump on John's clock for Hunter
after the frame change (9.6 to 20.4 hours) is the problem. There really is
no problem here. If you want to integrate the acceleration to use various
inertial frames during the turn around, it can be done (with the same
results). Another common approach is to imagine someone else in space that
passes John just when he reaches the point of his turnaround. This person
is heading towards Hunter at the same speed that John was travelling, so
there is no need to consider John any further. The key fact is that if we
then went back in the substitute's frame and looked at his clock for
Hunter, it would show that some amount of time had already been recorded
when the substitute began his trip towards Hunter. How far back should we
go? Since John traveled out 12 hours on the outgoing trip, we should go
back 12 hours in the substitute's frame. At this starting point for the
substitute, his clock for Hunter would read 10.8 hours. This is extremely
important. It clearly shows that both twins or the twin and the substitute
observe the other as having slower times. The big shift occurs when the
frame of reference is changed. This means that both observe the other to
have a slower time during the actual outgoing and return trips, but there
is a shift during the frame change that more than makes up for John's
account of Hunter's slowly running clock. After the frame change, the
damage has been done. John will still observe Hunter's clock to run slow,
but it will never slow down enough to compensate for the 10.8 hours that
were perceived during the frame change. Is this time jump a physical
occurrence? No. The time jump occurs because when John changes frames, he
is no longer using the same event as a reference. When John made his
turnaround, the event in Hunter's frame that John thought was simultaneous
with his turnaround changed. John's frame change caused this confusion
because his new frame uses a different time for the event in Hunter's
frame. More clearly, the turnaround event in Hunter's frame has a
different time value for the outgoing leg and the return leg, as perceived
by John. Keep in mind that in the above references to Hunter's frame, I'm
really talking about what John thinks Hunter's frame time would be. This
time difference is only apparent to John because it is his frame change
that causes the discrepancy. In Hunter's frame, nothing changes for Hunter
when John changes frames. Here again, by realizing that the two events are
not simultaneous, the paradox is resolved. The point I am trying to
emphasis is that there are a variety of ways to handle the paradox. All of
the methods yield the same result, but if you actually consider the
simultaneity of the situation, then the how's and why's become more clear.