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I have been asked to retitle this question and I have been asked to edit the question to adhere to mainstream physics. And I simplified it. So here it is:

Two cars approach each other at the same constant speed relative to earth. Special theory of relativity (SPR) tells us that the clocks inside each car will run slower than an identical clock at rest. Most importantly, the clocks inside each car run at the same (slower) rate because they are traveling at the same speed. SPR also tells us that whenever relative motion exists between two inertial frames of reference, the approaching clocks will run at different rates. Yet in this case the approaching clocks run at the same rate (as demonstrated above). How can SPR explain this paradox?

dmckee --- ex-moderator kitten
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M. Pope
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    You're falling back onto the idea of absolute time by saying "the clocks run at the same rate". That happens to be true in the Earth's frame, but there's nothing special about that frame. Different observers can disagree on which clocks are running faster or slower. – knzhou Aug 07 '16 at 19:28
  • For more information, just search for 'time dilation'. I think a good half of the results will be about this same paradox. – knzhou Aug 07 '16 at 19:29
  • In SR all velocities are relative. You are free to assume which clock is at rest and which is moving. So I am allowed to say that the car on our right is at rest, the Earth's clock is approaching it at $v$, and the left car is approaching the right car at $2v$ and the Earth's clock at $v$. In this situation all assumptions concerning the speeds of the clocks will change. Which only adds to the paradox. That's Special Relativity ... – bright magus Aug 07 '16 at 19:33
  • Master question on time dilation: http://physics.stackexchange.com/q/241772/. And the next thing you're going to want to ask about is covered by the "twin paradox". Please read before asking. – dmckee --- ex-moderator kitten Aug 07 '16 at 19:34
  • The twin paradox is obvious, one twin accelerated and one twin did not. The car question is fundamentally different since both cars have identical acceleration/speed. That both cars experience the same slower clock rate has been proven experimentally. No disagreement there. That SR predicts a different clock rate for the two approaching cars is the unanswered paradox. Conclusion - relative motion does not always result in relativistic effects. – M. Pope Aug 07 '16 at 19:54
  • Any opinions that relativity is wrong posted as comments will be deleted. You are not allowed to dodge the judgement of the community by posting comments rather than answers. – dmckee --- ex-moderator kitten Aug 07 '16 at 21:06
  • @M.Pope: SR predicts different clock rates for the two approaching cars according to either of the two approaching drivers, and the same clock rate for the two approaching cars according to the earthbound observer. To say "SR predicts different clock rates...." without saying according to whom is to miss the whole point of relativity. – WillO Aug 08 '16 at 02:17
  • According to an observer at rest - thx willO. I thought that was understood. – M. Pope Aug 08 '16 at 15:36
  • "Any opinions that relativity is wrong posted as comments will be deleted" -dmckee. Be informed that mainstream physics is quite aware of the disparity of results prowedicted by SR. There's no hiding our heads in the sand - SR has it's limitations. Deleting comments (and I'm guessing you'll delete my account now) that are contrary to your personal understanding does help to further our understanding. – M. Pope Aug 08 '16 at 15:44
  • "According to an observer at rest" There is no unique "observer at rest". That's the whole point. No one in this scenario has a privileged point of view. Mainstream physics is well away of the discrepancy between "common sense" and the theory, but it is also well aware that the experimental evidence favors the theory. And, this being a science, that's the end of it. Common sense can push off. – dmckee --- ex-moderator kitten Aug 09 '16 at 02:05
  • This is a simple scenario that has two conflicting results. According to the observer at rest (i.e., stationary wrt earth), the clock in car1 (approaching from left) and car2 (approaching from right) run at the same slower rates since they have the same speed. True? But since they are traveling in opposite directions (i.e., towards each other), the diver of car1 should see a different clock rate than that in car2 (and vice versa), because of the relative motion between the two cars. True? – M. Pope Aug 09 '16 at 15:26
  • When the cars stop either the clocks in both cars agree (as predicted by the stationary observer) or they read differently (as predicted by the drivers of both cars). But both can't be true. This is not complicated and has a simple explanation common to most SR paradox challenges....... – M. Pope Aug 09 '16 at 15:49

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