In the twin paradox, what happens if the travelling twin never U-turn to get back to earth? (explanation in the post)

Submitted by PoufPoal t3_1118du3 in askscience

Hi, all.

First of all, apologies if my question is kind of naïve, or even dumb. I'm not that smart of a person, and I just know the basics in most science fields. Just think of me as a 5 year old trying to understand the world around him.

I'm trying to understand the twin paradox, and I've stumbled upon this video (in french, but automatic English translation is quite good), where he explains that the paradox is resolved by the fact that the situations of the two twins are not really symmetrical, despite of what we think, because the traveling twin has to U-turn at some point, and doing so subjects him to an acceleration (at 16:11). So, in short, it's because he has to accelerate at some point (to U-turn) that their situations are not symmetrical, and that he is indeed the one that age less.

After that, in relativity, there is no more any notion of simultaneity, which is why the traveling twin is not getting younger (or the Earth-one older) suddenly at the exact moment of the U-turn. This part I'm having a lot of trouble grasping, but that's not the point of my question.

So, I have two question about all this:

  • Is his explanation "correct", or at least does it give the right insight about the whole thing? If not, how does this really work?
  • If he's right, and the acceleration is the root of the age difference at the end, what would happen in this hypothesis:

> We are in an infinite universe without borders, where if you travel long enough straight forward, you come back to your starting point from the other side. The traveling twin hops in his rocket, take off and travel straight forward until he comes back to Earth (which has magically stayed in the same place and still exists, don't question it too far) from the other side. He has not made any U-turn, so in theory, I guess their situations are still symmetrical.

Which one would have age more, and why?

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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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Interesting-Month-56 t1_j8ei6vc wrote

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.

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Weed_O_Whirler t1_j8f80cw wrote

> 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.

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Any-Broccoli-3911 t1_j8eh54w wrote

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.

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peselev t1_j8esdxt wrote

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.

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shimadon t1_j8f5l6w wrote

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.

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wedgebert t1_j8i9ukt wrote

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.

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shimadon t1_j8iufiu wrote

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...

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Any-Broccoli-3911 t1_j8mpxqk wrote

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.

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PoufPoal OP t1_j8hdt64 wrote

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.

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michaelrohansmith t1_j8fuw15 wrote

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.

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