Learning Twin Paradox for Freely-Falling Observers

[vanhees71]

The “twin paradox” is often discussed in the introductory treatment of special relativity. Under “twin paradox” we understand the fact that if two twins start from the same place with synchronized clocks, traveling in an arbitrary way and then meet again at the same spacetime point, where they compare their clocks, in general, they find different times. According to the clock hypothesis, the time of a proper clock is independent of acceleration and given by the proper times of each twin,
$$\tau_j=\frac{1}{c} \int_{\lambda_1}^{\lambda_2} \mathrm{d} \lambda \sqrt{\eta_{\mu \nu} \dot{x}_j^{\mu} \dot{x}_j^{\nu}}.$$
Here the $x_{j}^{\mu}(\lambda)$ ($j \in \{1,2 \}$) are the world lines of the twins in terms of Galilean (Minkowskian) coordinates in an inertial reference frame with $(\eta_{\mu \nu})=\mathrm{diag}(1,-1,-1,-1)$. The clock hypothesis has been verified in various experiments, e.g., by comparing the lifetime of particles (like muons) or radioactive nuclei in particle...

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[timmdeeg]

Only one small thing, to my understanding "travelling in an arbitrary way and then meet again at the same spacetime point, where they compare their clocks, they find different times" requires different lengths of their wordlines, which is always true in case one twin stays "at home".

 

[vanhees71]

True, if the twins move on "symmetrical paths" their clocks can also read the same. I'll change the sentence to

"...where they compare their clocks, in general they find different times..."