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So we all know about time dilation, and how Alice travelling in a spaceship relative relative to Bob ages more slowly than him.

When Alice gets back to Earth, she has biologically aged time $t$, while Bob has biologically aged $t + \delta t$.
But this is fine since Alice has just made a quantitative gain lifespan, while she will still qualitatively live the same life as Bob. I.e., she was able to perform the same tasks that Bob did in time $t$.

So, what if Bob had a telescope and looked into Alice's spaceship (which has windows)? Would he see Alice going on with her slow-mo life?
And what would Alice see if she looked down at Bob?

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From the time they separate to when they meet, Alice has had less time than Bob. She has had less time to do stuff, like ageing, writing a diary or watch movies.

As to what would twins see if they looked at each other, the answer depends on the specifics of the trajectory that Alice takes.

Let’s assume that Alice only accelerates to turn around once she is far enough and she wants to come back. While she is coasting at constant velocity in interstellar space, Bob would see her moving in slo mo.

(To see this, consider Alice having a computer program that emits a laser flash towards Bob every second. Bon can observe that. You know that every second on the spaceship is more than a second long as seen from Bob’s point of view, by time dilation. So he receives a flash every 1.2 seconda for example. But say that computer is also playing music, and Alice is dancing to that music. If she is keeping rhythm, she is synchronised with the laser flash, so if Bob could see her, he would actually see her moving in slow mo!)

But by special relativity, Bob is moving at constant speed in Alice’s frame, so she would see him moving in slo mo.

If that sounds paradoxical, and that is why this is called the Twin Paradox. The reason its not contradictory, and why it’s not actually a paradox, is that when Alice turns on her thrusters to accelerate herself to come back to Earth, she will slow down even more from Bob’s point of view. And she would see Bob speed up.

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    $\begingroup$ Both of the last two sentences are incorrect. All through Alice's return journey, she sees Bob's clocks moving at exactly the same speed at which Bob sees Alice's clocks moving. Each sees the other speeded up but, after correcting for the doppler effect, says the other is moving slowly. $\endgroup$ – WillO Aug 27 '18 at 23:55
  • $\begingroup$ I think there is some ambiguity here. You use the phrase "speed up" and "moving in slow mo". I am confused at when you are talking about relative speeds and when you are talking about time intervals. Can you please edit the question to be more clear and objective in these things? $\endgroup$ – Aaron Stevens Aug 28 '18 at 0:00
  • $\begingroup$ It's popular to claim it is not a paradox as of late: it's a paradox, because each see each person sees the other aging differently--but they see the same thing, but A ages less than B. What they see is not just time dilated either, because they don't see each other "now", they're looking back into the past. They can't see "now" because they're space-like separated. $\endgroup$ – JEB Aug 28 '18 at 3:51

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