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Consider a train moving with respect to ground. Then a observer on ground says that the train clock runs slow while a observer on the train claims that the ground clock runs slow. Who is right? In order to resolve this contradiction, in many books, two stationary clocks and one moving clock to compare time are used.

Why? Why not only one in each frame?

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    $\begingroup$ Both of them are right. Why does one of them have to be wrong? Everything is fine as long as the space-time interval is conserved between two events. $\endgroup$ Jun 16, 2017 at 6:39

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The famous two stationery clock experiment's that you are referring to is used so that you can compare the moving clock with the stationery clock when $\Delta x$ between them is 0 so that both the observers will read the same thing on them.

Different clocks which are synchronous for one observer in space are not synchronous for the other person who is travelling with some velocity because $\Delta t '$ depends on $\Delta x$.

All the main problem's in relativity arises because people have a hard time understanding relativity of simultaneity which means that two events occurring at the same time for one observer doesn't occur simultaneously for the second observer.

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Because it is the clearest way to show the underlying physical effect. If you try to compare the elapsed time on one clock in one frame with a single clock in another frame, you can only do it by passing signals between the two clocks, which introduces complications and obscures the principle involved. If you have two clocks in the second frame you can directly compare times as the clock in the first frame passes each of them in turn.

That all said, the effect that is explained by the set up of a single clock being passed by two others is a property of the geometry of spacetime, so you don't really need clocks at all to explain it.

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