What Happens to the Twins' Ages in the Twin Paradox?

[Cruncher]

http://en.wikipedia.org/wiki/Twin_paradox

I read a little bit about this, and am a little bit confused.

I always learned about reference frames. When I jump, you can say that I am moving away from the earth, or the Earth is moving away from me. Why is this any different for near-light travel?

The proposed idea here is that it's "acceleration" which causes this(which I still don't understand why it matters).

This raised a thought question in me which I hope someone could answer. What if the traveling twin, first travels away from the stationary twin. After he gets there, the stationary twin does the same travel. Will they arrive the same age? Why or why not?

I'll add an edit ahead of time. I have a problem with either answer to my question.
Case 1) They do not arrive the same age. This seems absolutely impossible. They both made the same trip, at different times. How could something different have happened for each?

Case 2) They do arrive the same age. My problem here is that it implies that given 2 bodies A, and B, A accelerating toward B is different than B accelerating toward A, which goes against every intuition I have.

Please address these cases for your answer.

 

[mfb]

When I jump, you can say that I am moving away from the earth, or the Earth is moving away from me.

You can, but the descriptions are not equivalent. In one frame, every object in space "jumps" (without an external force), in the other, they do not (if you choose the center of mass system, which is basically equivalent to the system of earth). That is certainly a difference.

Why is this any different for near-light travel?

There is nothing different. Non-inertial frames see different physics.

After he gets there, the stationary twin does the same travel. Will they arrive the same age?

They will arrive at the same age.

Case 2) They do arrive the same age. My problem here is that it implies that given 2 bodies A, and B, A accelerating toward B is different than B accelerating toward A, which goes against every intuition I have.

See the jump-example. Non-inertial frames (where everything is accelerating magically) give different physics.

 

[stevendaryl]

Case 2) They do arrive the same age. My problem here is that it implies that given 2 bodies A, and B, A accelerating toward B is different than B accelerating toward A, which goes against every intuition I have.

According to Special Relativity, as well as classical physics, there is a big difference between A accelerating toward B and B accelerating toward A.

 

[Nugatory]

The proposed idea here is that it's "acceleration" which causes this(which I still don't understand why it matters).

You'll hear that said a lot, but it's not true - so not understanding it is quite understandable

The significance of the acceleration is just that is evidence that the twins have dfferent experiences. One twin feels acceleration, one twin doesn't, so their experiences are different; because their experiences are different it is not utterly illogical that they end up somehow different.

This raised a thought question in me which I hope someone could answer. What if the traveling twin, first travels away from the stationary twin. After he gets there, the stationary twin does the same travel. Will they arrive the same age? Why or why not?

It depends on the specific path they take through spacetime. If the stationary twin follows exactly the same acceleration profile except leaving later, they will both end up the same age at their meeting at the destination. That is, we'll have what you're calling Case 2:

Case 2) They do arrive the same age. My problem here is that it implies that given 2 bodies A, and B, A accelerating toward B is different than B accelerating toward A, which goes against every intuition I have.

The implication doesn't follow - fortunately.

It is easiest to analyze these problems from the standpoint of an inertial observer who is watching all this accelerating and flying back and forth. This observer can assign a position and time value to every interesting event: for example in the classic version of the twin paradox he would say that the traveler starts his journey at (t=0,x=0), travels for one year at .5c to reach his turnaround at (t=1,x=.5), and the twins meet back on Earth at (x=0,t=2) with all distances in light-years and all times in years.

The amount of time a twin will experience moving from one of these points to the next is given by $\tau=\sqrt{\Delta{t}^2-\Delta{x}^2}$ (and remarkably this value will come out the same for all inertial observers even though may have very different measurements of $t$ and $x$).

You use this to calculate the aging each twin experiences on his path through spacetime, and the accelerations are irrelevant except to the extent that different accelerations may send the twins on different paths.

 

[Dale]

This raised a thought question in me which I hope someone could answer. What if the traveling twin, first travels away from the stationary twin. After he gets there, the stationary twin does the same travel. Will they arrive the same age? Why or why not?
...
Case 2) They do arrive the same age. My problem here is that it implies that given 2 bodies A, and B, A accelerating toward B is different than B accelerating toward A, which goes against every intuition I have.

They arrive the same age. I don't understand your "problem", so unfortunately I cannot address it directly.

You can tell that A and B arrive the same age because of symmetry. Consider the reference frame where the beginning and ending are the same point. In this frame the twins travel along mirror-image paths. They are the same speed and acceleration profile, just rotated 180° and time reversed. Since the laws of SR are the same under a 180° rotation and time reversal, their ages are the same.