# Astronomy in a universe with two "straight" dimensions plus a highly curved dimension

So for lack of a better term (I'm sure there is a better term) imagine a universe without curvature as being like a sheet of paper. You have left, right, forwards, backwards, up, and down. Now take this sheet of paper and roll it. Specifically roll it so that it has a circumfrence of one meter. Go up 0.5 m and suddenly you're below where you started by 0.5 m. However this only works for up/down. You're still free to go as far left, right, straight, and back as you please. (NB: time still functions as normal.) That is the world in which I ask my question. (There is a similar idea present in string theory that features very small extra spatial dimensions of this phenomon, the difference here being this spacial dimension is large enough to notice and interact with.)

Now assuming that this universe experienced a big bang releasing lots of protons, neutrons, quarks, electrons, photons, etc, how would this universe advance astronomically? (Assume expansion takes place as normal, and that the one meter thing only applys in that universe's equivalent of today.) What versions of planets and stars would form and how would they interact?

• This is a really complicated question. I think in order to get a meaningful answer, you need to ask something super specific and basic. What version of planets and stars might form sounds to me like a topic you can burn multiple ph. d. students on. People sometimes overestimate the power of science. Some things can't be answered in some short SE answer Commented Aug 23, 2018 at 8:45
• Here are questions in a similar setting that may interest you: 1 2 Commented Aug 23, 2018 at 8:45
• It would be really interesting research paper because you would have to rewrite most of the field equations to fit your universe. And these equations form the basis for the particle pantheon. For one, magnetic fields would not form or be really strange having a preferred direction. But it would not be very interesting, astronomically, because there is not enough space in the up/down direction to get the stellar core densities for hydrogen fusion or the CNO cycle. Then again, general relativity would have to be rewritten as well for this curved space time. Commented Aug 23, 2018 at 8:55
• You do realize that in a universe like this planets and stars may not be possible at all, right? You only have 1 meter of space in one direction, meaning that nothing larger than 1 meter in that direction can exist, otherwise it will collide with itself. So you basically end up with a 2D universe with an additional coordinate where particles can move 1 meter before returning to the starting position. One would have to go through the equations in order to know how gravity works in this universe and if stars are even possible. Commented Aug 23, 2018 at 10:14
• Have you considered 3 normal dimensions plus 1 curled, instead of 2+1? That could give you something kind of like our universe but a little different, instead of something kind of like Flatland but a little different, which is both more interesting mathematically/physically and more likely to be relatable. Commented Aug 25, 2018 at 6:11

## Summary

You mentioned string theory, so you may know that compact dimensions with a small enough radius don't measurably alter the world's long-scale physics (otherwise string theory would already be falsified!). This is basically what happens with gravitational physics in the universe you propose. The result is that there is no gravity at all.

## Explanation

First, to build an intuition for what spaces like this would look like to an inside observer, you can download the app Curved Spaces. If you look at the "Torus Cubic" space within the "Flat" folder (keep the left arrow key pushed to remove the walls), you will see that any object seems to have infinite repeated copies across all three dimensions. Your proposed universe is like that, but objects only repeat across one dimension (up/down).

Now for the physics. Unless I'm missing something, five meters is a length scale big enough for electromagnetism, the strong and weak forces, and quantum mechanics in general to develop similarly to our 3D world; this would mean that hydrogen and helium atoms would probably form as usual.

However, the relevant length scale for gravity is much larger (think about the size of stars and planets), so gravitationally speaking your world would approximately behave like a 2D world. And it is known that general relativity in two dimensions is a purely topological theory without propagating degrees of freedom: modulo technical details, spacetime is always flat in this world.

This means that gravity as we know it wouldn't exist: no planets or stars would form, you would just have a primordial gas of particles without a force to condense them.

If you still want to impose the existence of stars and planets by hand, a moment of thought will convince you that they have to be cylindrically symmetric to the inside observer. My guess is that their external behavior would be that of the corresponding black hole-like solution, a so-called open cosmic string. As explained in that link under the "Gravitation" section, open cosmic strings don't produce a gravitational force either, only topological effects. So you would still have no orbits, the artificially-added cylindrical planets and stars would just move at constant speed without interacting unless they collide. If you are interested, this paper describes the interior solution of such a body and other possible configurations.

• When you say "relevant length" does that mean gravity just doesn't exist in this world or that gravity is present but its just irrelevant and doesn't do much of anything in the world Commented Aug 27, 2018 at 2:46
• @tox123 The latter. It still exists at scales on the order of meters and below (you could probably still perform some kind of Cavendish experiment), but it's far too weak to do anything meaningful, and at scales above 5 m it doesn't get any stronger. Commented Aug 27, 2018 at 6:35
• so you could hypothetically create an artificial black hole and it would function? Commented Aug 27, 2018 at 17:34
• @tox123 I think so, it would work provided its Schwarzschild radius is significantly less than 0.5 m. But if it accretes too much matter and grows past this limit, it will turn into a cosmic string. Commented Aug 27, 2018 at 18:04

So basically from what I can tell, Your universes is basically a cylinder shape and if you move through the center it's going to flip you around.

The only issue I see with the cylinder analogy, is that if I keep going down, or away from the center of the cylinder, it's going to get more and more linear. Eventually, I'll be so far away, that all the lines I see will appear straight. Basically, what I'm saying is that if you do have a curved axis, it's only going to be noticeable close to the center point of the curve. Draw any curve and then draw lines tangent to the curve going away from the center. There are going to be lines that will continue forever, away from the center and never cross over and that is where the large majority of your universe is going to exist and form (I assume that the big bang would occur in the center and push everything away from that centre).

I can't tell you how this universe would look or how fundamental particles will react, but you could represent it mathematically by defining a curved plane and setting that as the frame of reference. Basically you have new dimensions like i,j,k which when converted into our standard x,y,z are actually curved.

• If my hypothetical world is a 3-D manifold, then the cylinder analogy is a 2-D version. It's a hyper-cylinder if you will, a cylinder pushed along a fourth axis. Commented Aug 23, 2018 at 3:55
• If you want to redefine how particles work thats okay but I doubt anyone here as the expertises to help you. I'm trying to give you a way to represent how things in that world will look in ours. A circle on the X-Y axis is x^2+y^2=1. Lets that be your new axis, i. If you want it to keep looping over and over, you could better represent it using and Angle, and simply convert your X and Y into angles. you can then use i, to create two new tangential axis relative to it, so you have a cylindrical 3rd dimension. This gives you a way to represent your dimension in our dimension. Commented Aug 23, 2018 at 4:48
• I assume however that a particle/object in your world can have its movements split into 3 axis, or 3D. To it, those axis should always appear the same. To us, it could be moving and teleporting all over the place or exist in multiple areas at once because its axis aren't linear like the ones we have defined to represent our world. Commented Aug 23, 2018 at 4:50
• I am not sure this answer satisfies the requirement for the hard science tag. Can you maybe improve it?
– L.Dutch
Commented Aug 24, 2018 at 6:15