Frame of references in time dilation

[J Goodrich]

Let's suppose that I am the Captain at a space dock sitting in interstellar space. My crew decides to go on a trip in our ship, so they board and set off at .9c.

t = t0 / sqrt( 1 - v^2/c^2 ), t0 = 1, v = .9c, therefore
t = 1 / sqrt( .19 ) = 2.3 (approximately)

So as they speed off, 1 second of the time that I experience corresponds to 2.3 seconds of the time that they experience, which suggests that if I log that they travel (assuming they say go in a big circle back to our dock, not stopping/altering their speed) a month they come back having experienced/aged 2.3 months.

I believe this is all correct.

Now here is my problem: since motion is relative, could the crew in the ship not have said that me, sitting in our dock, is instead in motion and that they were stationary? From their respective, shouldn't have I (and really the rest of the universe) have aged 2.3 seconds for their every second? What makes one perspective more correct than the other or how is this apparent contradiction otherwise solved?

 

[HallsofIvy]

Let's suppose that I am the Captain at a space dock sitting in interstellar space. My crew decides to go on a trip in our ship, so they board and set off at .9c.

t = t0 / sqrt( 1 - v^2/c^2 ), t0 = 1, v = .9c, therefore
t = 1 / sqrt( .19 ) = 2.3 (approximately)

So as they speed off, 1 second of the time that I experience corresponds to 2.3 seconds of the time that they experience, which suggests that if I log that they travel (assuming they say go in a big circle back to our dock, not stopping/altering their speed) a month they come back having experienced/aged 2.3 months.

From who's point of view? From your point of view, time moves more slowly in the crews frame of reference- you experience 2.3 seconds to their one second. From their point of view, your time has slowed and they experience 2.3 seconds to your one second.

I believe this is all correct.

Now here is my problem: since motion is relative, could the crew in the ship not have said that me, sitting in our dock, is instead in motion and that they were stationary? From their respective, shouldn't have I (and really the rest of the universe) have aged 2.3 seconds for their every second? What makes one perspective more correct than the other or how is this apparent contradiction otherwise solved?

No, they were not in an inertial frame- first they had to accelerate to .9c, then move in a circle, which, even though speed is constant, velocity is not so they were accelerating, and finally declerate to 0 relative to you. While velocity is relative, acceleration is not. They would have felt forces you did not and so were not in an inertial frame of reference.

This is really just a restatement of the "twin paradox" which has been done many times on this forum. As long as two observers are both in "inertial" frames of reference, they are moving at constant speed relative to each other and each observes the other as aging more slowly. There is no "paradox" until they are both stationary relative to each other and that requires breaking the "inertial" frame of reference- and the usual formulas of special relativity no longer apply.