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I have looked for other answers, but most confused me more than clarified and did not provide specific calculations. So I am still struggling with this simple question.

Consider two persons, $P_a$ and $P_b$.

From $P_a$'s reference, $P_b$ is orbiting around him at speed $v$. But from $P_b$'s reference it is the other way around, and $P_a$ is orbiting around him at speed $v$

As I understand, from $P_a$'s reference point, $P_b$'s clock is ticking slower, more specifically, $t_b = t_a \sqrt{1 - v^2/c^2}$.

However, from $P_b$'s reference it is the other way around, and he measures $t_a = t_b \sqrt{1 - v^2/c^2}$.

Assume that both clocks started at 0.

Now here is my question:

After X years from $P_a$'s perspective, they decide that they will be both at the same speed.

There are two ways to do it. Either $P_a$ will accelerate to reach $P_b$ or $P_b$ will accelerate to reach $P_a$.

Both of them expect to find that the time has passed slower for the other one. What will happen when they meet? Does it make a difference who reaches the other person? Please, provide specific calculations for the answer!

John Doe
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2 Answers2

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Only uniform transalatory motion(moving with constant speed in a fixed direction) is relative. If you change the magnitude of your velocity or your direction then you're accelerating as dictated by Newton's first law. A body in circular motion with constant speed is an accelerating object since it's constantly changing its direction. Therefore the principle of relativity cannot be applied to it, that is, a body in constant circular motion cannot assume a state of rest and it's the other frame that is moving around.

Omar Nagib
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From As point of view (which may or may not be an inertial frame), it looks like B is going in a circle, thus A concludes that either 1) they (A) is not moving inertially or else 2) they (A) is moving inertially and hence the laws of physics apply and so B must be accelerating and hence feeling a nonzero force. In which case B is not moving inertially (since B is accelerating).

From Bs point of view(which may or may not be an inertial frame), it looks like A is going in a circle, thus B concludes that either 1) they (B) is not moving inertially or else 2) they (B) is moving inertially and hence the laws of physics apply and so A must be accelerating and hence feeling a nonzero force. In which case A is not moving inertially (since A is accelerating).

So we conclude that at most one is moving inertially.

If neither is moving inertially we don't have enough information.

If one is moving inertially, the laws of physics hold in that frame and hence the other one has its clock run slower.

Done.

Timaeus
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