@S1-A.ARPA,@MIT-MC.ARPA:FRIEDRICH%GAV@LLL-MFE.ARPA (06/22/85)
From: FRIEDRICH%GAV@LLL-MFE.ARPA
Quite a bit of discussion about the twin paradox has gone on. Let me
throw some more wood onto the fire by doing a small bit of algebra.
First, please allow me to use the capital B to denote beta, which repre-
sents the term v/c, where v is the relative velocity involved at the
moment, and c is the speed of light. Also grant me the use of sqrt(x)
to mean the square root of x.
Now, if you'll promise not to go to sleep ...
Give each twin a bright flashing light, which flashes at a rate f. Now
send twin R on the rocket, and give twin E a chair.
During the outbound trip of the rocket, each twin sees the other's light
flashing at a rate
f' = f * sqrt( (1-B)/(1+B) )
where the v buried in B is the rocket's velocity. Note that since B < 1,
f' is lower than f; in fact, at v = c, f' is zero.
(Let me stop a minute to show this; without special relativity,
f' = f / (1 - B)
but y (gamma, if you please) = sqrt(1 - B^2) is the time-dilation factor,
so
f' = (f / (1 - B)) * y
from which you can easily get the first equation.)
Now comes the asymmetry of the situation. Note that I will carefully
stay out of general relativity; it is not necessary to explain the twin
paradox. Immediately after the turnaround, twin R begins seeing twin E's
light flashing at a much higher rate:
f'' = f * sqrt( (1+B)/(1-B) )
Now this means that f'' is GREATER than f. But twin E, back on Earth, does
not see this enhanced rate immediately. If L is the distance of the trip,
twin E does not see the enhanced rate until
t = L/v - L/c
before the end of the trip. (Note that as v -> c, twin E sees the increased
rate for less and less time.) This enhanced flash rate is simply none other
than the Doppler effect; we are NOT comparing the speeds of the two twins'
clocks. Do not be misled into believing that we have shown that moving
clocks run FAST sometimes; realize that the light travel time is decreasing.
To summarize: during the outbound leg, each twin sees a reduced flash rate
from the other twin. Immediately after reversal, the rocket twin sees a
higher flash rate from the Earth twin's light. However, the Earth twin does
not see this high flash rate for quite a while yet; the light must get to
him from the turnaround point first. Herein lies the asymmetry of the sit-
uation. The rocket twin is the one that actually went through the turn-
around event; NOT the Earthbound twin.
The Earthbound twin sees the lower flash rate for a much longer period
of time, and NEVER sees a flash rate higher than the rocket twin sees.
Reference: Special Relativity, by A. P. French, which is part of the
MIT Introductory Physics Series, published by W. W. Norton and Company.
My edition is copyright 1966, and the relevant pages are 149 - 159.
Terry