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.

[removed]

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.

The one on Earth ages more.

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. Both will agree on the speed the other one was going. They disagree on the total distance and the time.

Thanks for all the answers. TBH, I never understood the concept of time slowing down with acceleration, because if you choose the right reference frame, it would seem that the return trip requires the twin to experience negative acceleration. Though writing this, that’s still acceleration I guess.

My take away is that I don’t understand the framework behind relativity at all.

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.

[removed]

[removed]

The one who hops into the rocket and leaves, has to accelerate (to gain speed). Because one frame experienced the acceleration, and the other frame didn't, the frames are not symmetrical. The U-turn is not the only element that requires acceleration (and deceleration upon landing)

The paradox itself was not that the twin's experiences are different, rather than the one who traveled more, aged less.

At least, this is what I remember from my university years.

[deleted]

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...

That's not the paradox. The paradox is that if each twin point of view is relative, then relative to the twin that is on the space ship, it's actually the earth twin that is traveling. So the spaceship twin should be older, not the earth twin. But it's not the case, and this is the paradox.

> if you choose the right reference frame, it would seem that the return trip requires the twin to experience negative acceleration.

While in relativity velocity is relative, accelerations are not. Accelerations can be measured and felt. Easiest example. You're driving along a perfectly flat road in the back of a windowless truck. There is no experiment you could do to determine what speed you're going. All speed feel the same. But if the truck accelerated- sped up, slowed down or even turned, you would notice.

Astronauts who went to the moon actually did live fractionally longer than people who stayed on Earth because of the speed they traveled at. Its not measurable in this case but it can, I believe, be observed in the behavior of relativistic jets from stars.

[removed]

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?

You're right, he has to accelerate to take off too, that's true. Thank you, I didn't even think of that.

> The paradox itself was not that the twin's experiences are different, rather than the one who traveled more, aged less.

No, sorry. The paradox is the fact that depending on which twin you take as a reference frame, each twin would age more than the other.

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.

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?

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.

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

It's only a paradox due to misunderstanding what is happening.

It's not the motion or velocity that causes the time dilation, it's the acceleration. The twin on the ship experiences acceleration (which is absolute, not relative) while the Earth-bound twin does not.

Once you take this into account, the paradox is easily solved.

Acceleration is still not the big picture. Only within the framework of special relativity it is true that an acceleration will always result in a shorter length of space time path. When you're talking about acceleration, you are restricted to special relativity only. But the twin paradox can be formulated in general relativity framework as well. In general relativity, you can have situations in which the accelerating twin is older, because in general relativity, even accelerating objects can have longer spacetime lengths...

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.)

[removed]

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?

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

[removed]

Only if the accelerating twin passes through regions of lower space-time curvature (lower gravitational field).

And unless he gets in a region of negative curvature (doesn't exist in our universe as far as we know) his speed must not be too fast.

[removed]