The Twin Paradox

In what consists the Twin Paradox?

I consider twins A and B. 

At the age of 25 B sets off on a  prolonged space journey with \(v=12/13 \cdot c\), while A stays on earth.

After 26 years, earth time, A is precisely 51 years old, as his brother B returns to earth from his journey.

HOW OLD IS B?

I want ignore the short acceleration phases when starting, reversing flight direction and landing and I assume that B flew away for 13 years with velocity \(v\) and afterwards flew back 13 years with velocity \( -v\).

The term \( cos \theta = 5/13\). 

Therefore, for each of the out-bound flight and the return flight B aged only 5 years.

On his return to earth he is therefore only 35 years old, while his twin brother is celebrating his fifty-first.

From the point of view of the space traveller is everything reversed, and thus A must be younger than B!

Actually, however, the whole arrangement is asymmetrical: only A is the whole time at rest in an inertial frame, while B is exposed in different phases of his journey to accelerations. 

During the non-accelerated flight phases B actually has the impression that his brother A works somewhat slowly.

Anyway, the reasoning from A's frame is correct: twin B s younger.

Twin B, in order to leave the earth and travel to a distant galaxy, must accelerate to speed \(v\).

Then when B reaches the galaxy must slow down and eventually turn around and accelerate in the other direction.

Finally when B reaches the earth again he must decelerate from \(v\) to land once more on earth.
Since B's route involves acceleration, his frame cannot be considered an inertial reference frame and thus none of the reasoning like time dilation or length contraction can be applied.

To deal with this situation in B's frame I need a much more complicated analysis involving accelerating frames of reference and General Relativity.

It turns out that while B is moving with speed \(v\) A's clock does run comparatively slow, but when B is accelerating the A's clock run faster to such an extent that the overall elapsed time is measured as being shorter in B's frame.

Thus the reasoning in A's frame is correct and B is younger.

Reference:
Epstein Explains Einstein
An Introduction to both the Special and the General Theory of Relativity
“As simple as possible - but not simpler !”
by David Eckstein
&
http://www.sparknotes.com/physics/specialrelativity/applications/section2.rhtml
Image: http://www.sparknotes.com/physics/specialrelativity/applications/section2.rhtml