# Length contraction in a doughnut shaped universe.

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Hey guys, I am back. I once heard a paradox where a finite universe loops on itself when you go near the speed of light due to length contraction. Unfortunately, I forgot the solution. All I know was it was on PBS space time. I heard from this video from kurzgesagt, that one possible model of the universe was a doughnut. Basically, you go in one direction, and it takes long to loop around, but in another direction, you get back quicker. My thought is what if you went at such a percentage of the speed of light, that the short loop of the doughnut loops around. Now, what would the other person see that goes around the long loop of the doughnut going at the same speed. One of the people would experience a little bit of length contraction (I am underexaggerating with that one, but you get the point), while the other one experiences an actual paradox, even though they go at the exact same speed. What would each person see in this paradox? Thank you for taking the time to read this, and have a good day.

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23 hours ago, grayson said:

I once heard a paradox where a finite universe loops on itself when you go near the speed of light due to length contraction.

PBS space time is a pop science source. They can say what they want.

Speed for one is relative, so for instance I am currently going very near light speed relative to a muon waiting for me to go by it. The universe is unaffected by my moving at this speed. The speed at which something goes relative to something else has zero effect on the geometry of the universe.

23 hours ago, grayson said:

I heard from this video from kurzgesagt, that one possible model of the universe was a doughnut.

It is possible, but a 3D torrid surface, sort of like the video game asteroids takes place in a 2D torrid surface. It means that there are preferred axis orientations, and that if you travel along any of these axes, you get back where you started. Any other direction and you don't. Your post seems to describe something like that.

Such a universe would be unbouned, but finite in volume, very much like the surface of Earth, except retaining the flat curvature.

23 hours ago, grayson said:

My thought is what if you went at such a percentage of the speed of light, that the short loop of the doughnut loops around.

The universe is much larger than the distance to the event horizon, and since one cannot reach the event horizon, one cannot traverse even one loop no matter the speed.

Presuming that expansion wasn't accelerating (or happening at all, for simplicity), there would be no event horizon, and thus one could go all the way around. So in asteroids, the universe isn't expanding at all, and if one goes vertical, you get back to the staring point after 1 lap, in say 5 seconds. If you go diagonal at the same speed, it might take a minute to get back approximately to the center of the screen, the starting point.

23 hours ago, grayson said:

Now, what would the other person see that goes around the long loop of the doughnut going at the same speed. One of the people would experience a little bit of length contraction

You seem to presume that one direction is longer than another, sort of like the asteroids screen not being square. That's fine, but it isn't necessarily the case.  If two depart from the center at the same speed and time, the vertical guy gets back before the one going the long way. If the aspect ratio is 3:4, then it will take 3 or 4 laps for them to meet each other again.

Length contraction plays no role in this. For one, it is a coordinate effect, not strictly a physical one. A length contracted ship going at 0.9c still gets 3.6 light years from here in 4 years (as measured by 'here', the frame in which he is contracted).

23 hours ago, grayson said:

One of the people would experience a little bit of length contraction

It being a coordinate effect, one does not experience length contraction. One is by definition always stationary relative to ones self, and thus there is no contraction to experience.

And a paradox isn't something experienced either. You've not identified any paradox. You just say that there is one.

What each person would see, if moving fast relative to stars, is a bunch of stars moving fast relative to him. They both see that. There is no paradox identified.

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That video you posted is not PBS Spacetime, which can be a little speculative, but does an excellent job of explaining mathematical theory in terms of Physics.

I do think you have gotten the wrong impression from the video, or others like it.
The universe's topology has nothing to do with time dilation/length contraction.
A toroidal universe is actually ( topologically ) flat around the short circumference, because it can be 'transformed' from a flat sheet.
The long circumference, however, is not, but you ( and the video, are only considering three dimensions.
Adding a fourth dimension doesn't allow for visualizing a 'doughnut' shape. What cosmologist use is a 'flat torus' which is best exemplified by the PacMan game, ( although I did own Atari computers and have played Asteroids ) and where going off one flat edge brings you to the opposite edge.

The video does explain that one could ( if the universe was small enough or you could travel fast enough ) notice a difference in travel time, to get back to the original position, when going around the two different circumferences, but that is certainly not time dilation.

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I thought there might be a paradox but I can't create one after all.

Suppose the universe "wraps around" 1 light year in distance, and assume it behaves the same as it it was flat. Then you could see what appears to be an infinite row of Earths, each subsequent one looking one year older than the last. One that looks n years older is "old light" from Earth that has made n loops around the universe before reaching you.

Lets say you can travel near enough the speed of light that it takes about a year Earth time to loop around the universe. If you leave Earth at the start of 2024, you'll return to Earth at the start of 2025 Earth time, even though the journey is almost instantaneous according to proper time of the traveler. Before you start, the clock on Earth as seen 1 LY away shows 2023, the one beyond it shows 2022. Thinking only of how things appear, there's no need to worry about relativity of simultaneity. As you travel one loop, you see 2 years pass on the "destination Earth" clock, so you see it showing 2023 when you start, and 2025 when you arrive. The next clock beyond it shows 2022 when you start, and also must have 2 years appear to pass during your journey, so it shows 2024 when you arrive. If you keep going, it shows 2026 when you get to it.

There's no paradox there. From the perspective of Earth, the ship and the image of Earth in 2024 travel around the donut in opposite directions and meet at the far end after half a year, and the ship returns at the start of 2025.

Now if you add another loop that's 2 light years long, it's the same thing, just double everything. You could have one ship travel the first loop twice, and meet a ship that travels the longer loop once, after 2 years Earth time. Negligible time would pass for both travelers. Or, you could have one traveler do the long loop in 2 years Earth time, and the other do the shorter loop in 2 years, ie. at a speed of c/2. One would see 4 years pass on their "destination Earth" and the other would see 3, where they would meet. Ie. they both start in 2024, and one sees Earth around the long loop looking like 2022, and arrive in 2026; the other sees Earth looking like 2023, and arrives in 2026. One would have aged a negligible time and the other would age 2x .866 years (according to Lorentz factor).

In terms of distances everything should be similar. Traveling at near c, the lengths would be negligible. At half c, traveling a proper light year would be measured as .866 light years traveled distance.

I can't see any paradoxes here. I think it would be equivalent to if you had a flat universe with a set of copies of Earth spaced a light year apart, all at relative rest and with synchronized clocks. Also add copies of the traveler so they could see "their distant selves". Creating that with copies wouldn't introduce any paradoxes.

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