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The Question
(Submitted November 09, 1997)
I once heard a "story" about two twins, one of whom went on
a rocketship to a nearby star while the other stayed behind.
When the errant twin returned, his earth-bound twin had aged
considerably while he had not. What determines which of the
twins ages the most?
The Answer
Yes, you are correct, this is the classic "Twin Paradox" and it is a
product of special relativity.
The short answer is that one twin stays in a an inertial reference
frame, while the other doesn't. The twin that stays in an inertial
frame ages more.
Here are some more details:
Einstein's theory of relativity
assumes two things:
* The laws of physics are the same in all inertial reference frames
* The speed of light is always the same regardless of reference frame
This is counter-intuitive, but it has been verified over and over by experiment.
To see why it is counter-intuitive imagine two trains, each going 60 mph
one going North and the other one South.
Case 1: In the reference frame of the ground, each has a speed of 60 mph.
Case 2: In the reference frame of the northbound train, the northbound train
is still, but the southbound train is going 120 mph.
Case 2: In the reference frame of the southbound train, the southbound train
is still, but the northbound train is going 120 mph.
On the other hand, the light from each train's headlamp is moving at
exactly the speed of light, no matter who measures. Not c+60mph
or c-60mph, just plain c. (where c is the speed of light)
Once you make this assumption, then the other things that you usually
expect to be constant have to change. Quantities such as lengths and
times that are often constant independent of the observer's reference
frame must change to keep the speed of light the same in each reference
frame.
If the trains were moving close to the speed of light, the people in
each train would see their own train as normal, but looking out the
window at the other train, or the earth, they would see clocks moving
slow and everything a little shorter (in the direction of motion).
In the case of the twin paradox, the assumption is that the person gets in a
ship and then is in this different reference frame. At some point he turns around,
thus switching reference frames again, and when he gets back home he now
is back in reference frame of the Earth. Depending on how fast the ship went,
much less time elapsed for him then his twin brother who stayed at home.
This will be shorter by a factor of:
sqrt( 1 - (v/c)^2 ) : where v = speed and c = speed of light
There is one catch, how does his ship accelerate to this fast speed and
then turn around? It not only takes an enormous amount of energy, but
if it is going to be comfortable for the astronaut it cannot accelerate
faster than a "g" or so.
You can find even more details in the Physics FAQ at:
http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html
Jonathan Keohane
for Ask an Astrophysicist
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