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Aseyhe t1_j8eashy wrote

It might help to realize that the elapsed time for each object is just the length of its path in spacetime. This should make it clear that there is never any ambiguity about whose elapsed time is longer. Just compare path lengths.

In the traditional twin paradox, the question is whether a straight line (twin who stays behind) or a bent line (twin who travels) is longer. The straight line is longer, due to the particular form of the spacetime metric.

In your last paragraph you are asking, what happens if we bend spacetime into a cylinder-like shape, so that time goes along the cylinder and space goes around the cylinder? There is still no ambiguity: you are now simply comparing the length of a path drawn along the cylinder with a path that circles around the cylinder like a helix.

In particular, the two twins' situations are not symmetrical because when you bend the spacetime into a cylinder, you have to choose a special reference frame. That's the frame in which spatial surfaces exactly loop back on themselves and time points exactly along the cylinder. Try physically bending a sheet of paper into a cylinder so that the edges just meet. You will probably choose to make the corners meet as well, but that is just one option. You could also offset the corners as much as you want. These different possibilities correspond to different preferred frames.

So the two twins' elapsed times will depend on how fast they are moving with respect to this preferred frame.

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shimadon t1_j8f4fnn wrote

That's a very good answer. Everyone talks about acceleration all the time, but it's really the length of space time path. For example: you can have a situation in which one twin is in a space ship orbiting earth, and the other twin is in a space ship which is hovering above the surface, canceling gravity with a constant force upwards by its thrusters. In this case, the twin who is moving in orbit will be younger, but he is in constant free fall! It's the twin who hovering which will be older, but he was the one who was subjective constantly to an external force! So it's not acceleration, it's space time length...

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klawehtgod t1_j8la1mg wrote

Is it the overall length, or just how much of the movement is through the time dimension? Moving “faster” or “straighter” is just putting more of your movement through the time dimension and less through the space dimensions, right?

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Any-Broccoli-3911 t1_j8egw3r wrote

A spherical universe is one of the 3 solutions for a homogeneous and isotropic universe (flat and hyperbolic are the 2 other ones). There's no preferred frame in a spherical universe.

Still for the person who moves, the circumference of the universe is smaller. So he sees his twin on Earth crossing the universe in a shorter time than his twin sees him crossing the universe in the spaceship. So at the end of the travel, the spaceship twin will say that the total time passed is less than what the Earth twin will say.

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Aseyhe t1_j8ejbv8 wrote

I was referring to a toroidal universe, since its spacetime can be flat. Bringing in spacetime curvature complicates things unnecessarily!

However, the spherical universe does have a (local) preferred frame: the frame of a comoving observer.

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PoufPoal OP t1_j8hdnjc wrote

Thank you, that is a really usefull answer.

> So the two twins' elapsed times will depend on how fast they are moving with respect to this preferred frame.

In any case (any choosen frame), the one travelling would move faster, right?

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Aseyhe t1_j8hl9w3 wrote

Imagine the Earth is already moving with respect to the preferred frame imposed by the universe's geometry. Then depending on the direction of travel, the traveler could potentially be moving slower than the Earth, with respect to that preferred frame.

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PoufPoal OP t1_j8hr1v2 wrote

Ok, so, aren't we back in the paradox, where depending on the preferred frame we choose, the shower twin is not the same, then?

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Aseyhe t1_j8hrha5 wrote

But there's no paradox because there is an objective preferred frame in a universe that loops around, which is imposed by the choice of at what times you "connect the opposite edges" of the universe.

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WavingToWaves t1_j8htl4b wrote

Wdym by „straight line is longer than bent line”?

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Aseyhe t1_j8k0iwn wrote

In the metric of spacetime, spatial distances and temporal distances enter with the opposite sign. In ordinary Euclidean geometry, the distance between two points separated by x, y, and x along each cardinal direction is sqrt(x^(2)+y^(2)+z^(2)). In the Minkowski geometry of flat spacetime, on the other hand, the (proper time) separation between two points separated by x, y, and z along the cardinal directions and t in time is sqrt(t^(2)-x^(2)-y^(2)-z^(2)). As a result, a straight line turns out to be the longest path between two points. (Technically, it's the path with the longest proper time between two timelike-separated points.)

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WavingToWaves t1_j8le25s wrote

That’s very interesting! Thanks for the answer 👍 I will read about the reasoning behind that

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