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In the example of two objects moving towards each other say at $0.5*c$ starting with synchronized clocks (edit: say synchronized by a third frame, like earth). it is understandable that each object observes the other object time being dilated relative to its clock (reciprocal dilation).

Am still struggling with one point though, when the objects pass each other and if they were able to exchange their clock information (e.g. number of ticks), what numbers will they observe?

  • Would both see the same number? if so how does SR explains the lack of dilation.
  • Or still observe dilation relative to own frame? How would one object send a value $y$ yet still observed by the other object as $y/\gamma$

Am trying to avoid introducing acceleration or direction change to avoid changing frames.

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    $\begingroup$ What does "starting with synchronized clocks" mean? If the clocks are synchronized in one object's frame they are not in synchronized in the other's. $\endgroup$ – WillO Dec 14 '20 at 17:46
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I'm going to assume that the clocks are initially synchronized in the earth frame (though you've left this ambiguous). So both clocks are set to 2:00 at the same moment, according to an observer on earth.

When the clocks pass each other, it's obvious from the symmetry that they both show the same time --- say 3:00.

A traveler with the first clock says: "Our clock was set to 2:00 an hour ago; the other guy's clock was set to 2:00 two hours ago; now they both show 3:00, so our clock was running at normal speed and his clock has been (and still is) running slow". A traveler with the second clock says exactly the same thing (with, of course, the clocks reversed).

If the clocks are initially synchronized in the frame of one of the travelers, then the story is a little different, but once again, the critical piece is that the travelers do not agree on how long ago each others' clocks were set.

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  • $\begingroup$ Thanks. So the disagreement on when the clock was synchronized is a result of information exchange speed? Or something else? $\endgroup$ – Bashar Gharaibeh Dec 14 '20 at 17:59
  • $\begingroup$ It's a result of the fact that the clocks are in motion relative to each other. $\endgroup$ – WillO Dec 14 '20 at 18:02
  • $\begingroup$ Got it. So if they synchronized at a point when they were stationary to each other then started moving, then acceleration would break the symmetry. $\endgroup$ – Bashar Gharaibeh Dec 14 '20 at 18:15
  • $\begingroup$ You don't need acceleration. Maybe they've both been moving forever. They synchronize their clocks at 2pm according to the earth observer, continue moving just as before, and all is as above. $\endgroup$ – WillO Dec 14 '20 at 18:24

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