I'm reading about the twin paradox in special relativity - if there are two identical twins, one of whom who sets off in a high speed rocket to a planet, and then heads back, will find the twin who remained on Earth to have aged comparatively more due to time dilation. However, from the perspective of the travelling twin, it was the Earth that was moving and so the Earth twin should be the younger one. The 'paradox' can be resolved by realising that the travelling twin does not remain in an inertial frame for all time, since he has to turn around at the planet and hence accelerate.
Now, there are three possible curvatures the universe could have - positive, negative, or flat. As far as I'm aware, we're quite sure it's flat. However, if it was positive, does this mean that the travelling twin, if he travelled long enough, would loop back to Earth eventually? And therefore he would be able to return to Earth without having changed inertial reference frame, and then we would actually have a paradox?
I'm thinking perhaps that travelling through curved space itself constitutes acceleration, since in a way you're changing direction? In which case the travelling twin wouldn't remain in an inertial reference frame and we could solve the 'paradox' as before?
Answer
And therefore he would be able to return to Earth without having changed inertial reference frame, and then we would actually have a paradox?
No paradox. The equations of general relativity can be solved with a universe hat wraps around, even if it is flat.
What happens is that you see two earths, the one you are at, and an image of an older earth where light left long ago, wrapped all around the universe and just now got to you.
As you move towards it the image of the far away gets bluer an you see the images of the people on earth celebrating holidays arrive more often than you age.
As you move away from the nearby earth, you see the image of the nearby earth get redder and you see the images of the people on earth celebrating holidays arrive less often than you age.
Eventually the image of the earth that was initially farther away is an earth that is actually closer, and eventually you land on it. And then you could travel to the other side of the earth and sit and wait for the image of you landing to finally arrive from the long way around.
And you never see a paradox. If you focus on what actually happens there isn't a paradox.
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