Can someone explain the twin paradox to me

[stunner5000pt]

A mother and her son share the same birthday. Then say a mother is 24 years old when her son was born. Then when she was 50 (her birthday, and his) she went on an intergalactic flight and a return trip all at constant speed at (12 / 13) c. She returns back to find that her son and her are the same age. What age is that? How far away did she go? that is 1/2 the distance travelled.

So i know that t2 - t1 = gamma (t2' - t1')
but I'm a little confused on where do i go from here.

 

[yogi]

There are thousands of articles on the twin paradox - the paradox results from the fact that each observer according to the standard interpretation of the Lorentz transforms will measure the other clock running slow in accordance with the formula you cited - (neither has a privileged frame and therefore the situation is symmetrical at this point) but of course, both clocks can not be running slow relative to the other - but when the traveling entity returns, the general view is that a clock which accompanied the traveler that turns around will have accumulated less time than the clock carried by the person who stays home - this difference is explained in different ways by relativists - the usual explanation is that the traveler must decelerate and then re accelerate to return, and this acceleration places the traveler in two different frames (he is subjected to an acceleration force and therefore the symmetry between the two observers is broken. But the differece in age can also be explained w/o resorting to acceleration (this is the path integral approach used by Professor Resnic). Einstein first raised the issue in his 1905 paper - long before there were any experiments with high speed particles that verified the fact that clocks in motion appear to run slower when observed in another frame.

 

[Fredrik]

If you check some of the recent threads here, you will find that several of them discusses the twin paradox. For example, I wrote a little about it in "Am I thinking SR correctly?" (my last post in that thread). The most recent one is "SR paradox".

The key to understanding the twin "paradox" (which isn't really a paradox of course) is to realize two things:

1. Different observers don't agree on what events are simultaneous.
2. When the twin in the spaceship is on his way back to Earth he's in another frame than he was when he was moving away from Earth.

In the frame moving away from Earth, the twin on Earth is aging more slowly. In the frame going back to Earth, the twin on Earth is...still aging more slowly, but now (in the astronaout twin's frame) he's much older!

It's hard to explain this without spacetime diagrams, but I should at least mention a few basic facts. Every observer thinks of a certain set of 3-dimensional "slices" of spacetime as "space, at different times". But different observers are "slicing" spacetime in different ways.

You can't think of the twin moving away from Earth and the twin going back to Earth as "an observer". He's in a different inertial system on the way back. Because of this, he is now "slicing" spacetime in a different way. What this really means is that events that were simultanous to him when he was moving away from Earth are not simultaneous to him when he's going back to Earth.

The moment before he turned the spaceship around, a certain set of events on Earth were simultaneous (in his frame) with the events he was experiencing. Those are the events that are located on the same "slice" as the "press the return-to-earth button" event. One of those events involves a rather young twin on earth.

When the spaceship has reversed its course (even if it does so very quickly), a completely different set of events are simultaneous with the events he's experiencing. One of them involves a rather old twin on earth.

In other words, they both see each other aging more slowly through the whole trip, except at the turning point, when there's a huge jump in the age difference, from the astronaut's point of wiew. This is not a time dilation effect, but a consequence of the two things I mentioned above.

If you don't understand the stuff I said about "slices", I can only recommend a good introductory textbook on special relativitity, such as "A first course in general relativity" by Bernard Schutz. (I know the title says general relativity, but the chapter about special relativity is still the best introduction to SR I've read). Try to understand spacetime diagrams.

Don't let anyone tell you that you need general relativity here ("because one of the twins is accelerating"). That claim is wrong.