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If I'm correct, in the Twin Paradox, an explorer travelling to a star in a spaceship moving with 0.95c and coming back to Earth after 25 years of his subjective time, will have gone 12.5 light years back and forth, while if you're on earth then for someone travelling 25 light years means 80 years of wait?

Now, if two twins establish communication, it will be taking more and more time for the signal to travel, and, if I'm correct, the signal will travel with net velocity of 0.05c (since the signal will always have to be reaching the receding spaceship). And the twins will see each other as if they are aging simultaneously? But when A travels towards the earth, and keeps the communication session with his earth twin uninterrupted during all time, he will have arrived back on Earth and now he should be catching current earth signal from his twin. I'm wondering what I'm not accounting for, because 80-25=55 years seem to have disappeared. Because when the exploring twin arrives, he suddenly sees his brother aged by an additional 55 years.

I guess one could probably set up an integral to account for all time it takes for the signal to travel back and forth between the two frames. The question is why/when the explorer sees his twin suddenly aged by the additional years. The ODE $$dt = \frac{ds}{0.95c}, s(0) = 0$$ will give about t = 82.24 years if we integrate ds from 0 to 12.5c.

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  • $\begingroup$ Time dilation and retardation (that is the light-speed signal delay) are distinct effects. The former is not a consequence of the latter. $\endgroup$ – dmckee Dec 15 '16 at 0:42
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Do you actually need so complex way to compare clock rate? Contrary to widespread opinion, you can measure ANY clock rate this way.

It is better to do as Einstein proposed. Brother E (Earthman) places synchronized clocks along the path of travel of his brother A (Astronaut). Then he sends a light beam to synchronize them. Now the clocks are perfectly synchronized and show the same time. Now A can compare his own clock readings with any clock on his way. For example A travels in a train and look at a platform, at each every 100 feet there is synchronized clock.

It is clear, the brother E will measure dilation of his brother A clocks. 99 % of people believe that travelling A twin will “see” or better to say will “measure” dilation of his twin brother E clock too. It is not quite correct. Looking at clocks on the platform successively he will see, that clock hands spinning at a breakneck speed.

Please look at the article in Wikipedia “Time Dilation”. The article itself is jumble of nonsense. There is a diagram in the bottom

https://en.wikipedia.org/wiki/Time_dilation#/media/File:Time_dilation02.gif

Now compare how many oscillations the travelling clock and any synchronized clock at rest makes during time of travel. Travelling one makes 3 and the stationary makes 7. It is 100% obvious that from the point of view of a moving clock, time in the reference frame runs much faster.

Contrary to widespread opinion, moving observer measures acceleration of time in the reference system and elongation of length of resting measuring rods.

Then - hocus pocus! Moving clock forgets that it was in motion just a second ago. Now it is at rest and introduces its own Einsteinian frame. Hooray! Another clock now dilates. The only problem is that now from the point of view of this clock (which now is in motion) time in the reference frame accelerates.

Since we cannot detect universal preferred frame, each of us (according to SR) may think of himself as he is “at rest”. We proudly measure dilation of every moving in our reference frame clock. However, in this reference frame every other moving observer detects acceleration of ours clock.

According to the SR both you and I can ascribe ourselves states of proper rest.

Imagine that we move relatively to each other with relative velocity 100 mph. I believe that we can describe our motion by means of reference frame, in which yours velocity is 0 and mine is 100. Or by means in reference frame in which yours velocity is 100 and mine is 0. Or by means in reference frame in which my velocity is 50 and yours is 50. Do you think we can use a frame, in which yours velocity is 0 and mine is 0 too?

Everything will be all right if the brothers will not be jumping like fleas from one frame to another.

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