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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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