This "perennial" is almost as old
as Einstein's special theory of relativity itself. Einstein did not invent the
paradox, although it was stimulated by his 1905 paper, where he spoke about the
fact that two clocks that were separated, one staying inertial and the other one
being moved away from the first one and then brought back again, will not read
the same time.
The popular press explained it in terms of twins of closely the same age, where
one twin sets off on a long, fast journey and eventually returns home. Special
relativity then predicts that the "away twin" will be younger than
the "at home" twin.
The "paradox" arises out of an abuse of special relativity's freedom
to choose any inertial frame as the reference frame and make all calculations
relative to it. The "abuse" on it's part arises out of choosing a
non-inertial frame (the away twin) as reference and then making wrong
conclusions---like that the home twin can just as well be considered as in
motion relative to the away twin. This then means that the home twin should
therefore suffer the same amount of time dilation as has been calculated for
the away twin and could thus be considered to end up the younger one (or at
least both twins still being the same age).
To confuse the issue even further, many explanations of the difference between
the two reference frames are very confusing and unconvincing---even some given
in reputable technical books. Search the web for "Too Many Explanations: a
Meta-Objection" (the author found it on "http://math.ucr.edu")
and see for yourself.
None of the one's discussed there is fully convincing either---which may be
just another "meta-objection" (what does "fully convincing"
mean anyway?) In my eBook "Relativity 4 Engineers" I give three
reasonable explanations: (i) a simple "hand waving" argument; (ii) a
very relativistic calculation and (iii) a more engineering-like representation
and calculation. I will concentrate on the "engineering" solution
here, using electromagnetic signals and relativistic Doppler shift.
Pam and Jim are twins that decided to put Einstein to the test. On January 1st,
2007 Pam quickly accelerated her spacecraft to a speed of 0.6c and flew away
inertially for 4 years, when she will quickly turn around and head back to
Earth at 0.6c again. Jim stays at home and the twins agreed to send each other
a New Years greeting on every January 1st until Pam returns safely.
This scenario is illustrated in the Minkowski spacetime diagram, Figure 1, a
split image for clarity. This shows a crucial non-symmetry in the signals that
the home twin and the away twin receive. Although the amount of stretch and
shrinkage of the received periods are the same, the amount of time that the
signals are stretched and shrunk is very different between the respective
twins.
Figure 1
The Doppler shift ratio (period of received signal (Tr) to period of
transmitted signal (T) for the outbound leg (opening velocity) is
Tr/T = sqrt[(1+0.6)/(1-0.6)] = 2.0
and the same ratio for the inbound leg (closing velocity) is
Tr/T = sqrt[(1-0.6)/(1+0.6)] = 0.5,
as can be clearly seen in figure 1. The non-symmetry comes from the fact that
the away twin receives compressed period signals immediately after turnaround,
while the home twin has to wait until the first signal after turnaround arrives
at home before noticing the event.
In the four years that Pam heads away from home, she will receive only two New
Year's messages from Jim. This is because Jim here represents the T period of 1
year. Pam is the receiver, with the Tr period of 2 years. On her calendar she
will receive "happy New Year 2008" only on Jan 1, 2009, and the next
one ("happy New Year 2009") on Jan 1, 2011. Weird, but this is due to
the increasing distance between them and the time light takes to cross the gap.
During her return trip, the situation is reversed, so in the last four
years, she will receive eight New Year's messages from Jim, one every six
months! She will receive the last (tenth) message, on New Year's Day 2015 on
her calendar, as she makes a close fly-by of Earth.
Does this solve the twin paradox? Not quite, yet.
That was Pam receiving Jim's messages. At what rate will Jim receive New
Year's messages from his sister? For the first eight years, he will also have
to wait two years for every 'happy New Year' message. This means that it will
be 2015 on Earth before Jim gets the message that his sister has sent on New
Year's Day 2011, with a note that she has just turned around for the home leg
of her trip.
Then, for the last two years, Jim will receive a New Year's message every
six months - four of them. Add them up and Jim will receive only eight messages
from Pam in the decade that he waited for her return. Conclusion: Pam recorded
only eight years during her voyage, while Jim recorded ten years.
I hope this clears up at least some of the confusion that surrounds the
"twin paradox".
Burt Jordaan (http://www.relativity-4-engineers.com)