What would the "shape" of a closed universe be?

Factual Questions
1

I was into this other thread about red-shifts. Then somebody asked if we were the center of the universe relative to us. (I’m only paraphrasing here) I was about to answer about the symmetry of the universe from our perspective and that the universe could be closed or open…

Then I realized that a closed universe might have a more complexd topology than simply heading in one direction and ending up in the same place. For example, the universe might be closed in one dimension but infinite in another. To add complexity to that, it could be sort of helical in one dimension, so that heading in one direction puts you X distance from where you started, but never actually intersecting with the origin.

What other topologies are possible?

2

First of all, “closed” and “infinite” are not synonymous in cosmology. A closed universe is one with a positive curvature, and is guaranteed to be finite. But you can also have a flat or open (negatively curved) universe which, due to various topological identifications, still wraps around.

If I remember correctly, there are 19 different flat 3-dimensional topologies. The simplest, of course, is just R[sup]3[/sup], a space infinite in all dimensions. For any of the others, the simplest way to describe them is by defining the basis cell and which of the faces are identified.

The next possibility is an infinite but finite-thickness slab, with the opposite faces identified. The slab may be rotated by an arbitrary angle on iteration, but we would not expect this in the actual Universe, since such a space would have a well-defined center. Each iteration might also be inverted, but we also wouldn’t expect this, since that would make the space non-orientable (like a Moebius strip), and certain properties of the weak nuclear force seem to imply that our Universe is orientable.

The next step after that is a “chimney”, which is infinite in one direction, but has a finite parallelogram cross section in the other two directions, and opposite faces identified with each other. Again, you could mathematically have an inversion in the identification, but it’s probably unphysical. You can also have a “chimney” shape which is hexagonal in cross section, rather than quadrilateral.

Of course, once you have those chimney spaces, you can chop up the infinite dimension, too, leaving your unit cells as prisms of the cross section shape (so a parallelopiped, or a hexagonal prism). And again, you can have rotations in your identifications, so long as the shapes match up. In the simplest case, here, your unit cell is a right rectangular prism (a box), with the opposite faces directly identified, in which case it’s referred to as the three-torus (since it’s analogous to the two-torus, topologically the surface of donut).

Finally, what I consider to be the most elegant one, your basis cell can be a rhombic dodecahedron, with opposite faces directly identified (no rotations are possible, on this one).

There are also a number of possible topologies for curved spaces, but I’m not too familiar with those. The only specific examples I know of are in a positively-curved space: You can have a basis cell shaped like a regular dodecahedron, with opposite faces identified with rotations of various angles.

3

What about a chiral universe? Where as I suggested, travelling in one direction never intersects with the origin, but leaves you X distance from it.

4

Okay, if you’re just interested in the spacelike slice (assuming there is such a thing), the list is given by Thurston’s Geometrization Conjecture, which was proved by Perelman. This gives eight building blocks: spherical, euclidean, hyperbolic, sphere-cylindrical, hyperbolic-cylindrical, SL(2,R), nil, and sol. Every three-manifold is a connected sum of quotients of these basic shapes.

In four dimensions, however, there’s actually no classification possible!

5

Something like a parking garage? The problem is that in GR everything in sight is a Riemannian manifold, and in the situation you describe in a Riemannian manifold there will be a different path that does get you back where you started.

Oh, and you’ll grok a lot better if you get rid of this silly artificial “origin” nonsense.

6

But if I start to travel in one direction I must have an origin. Forgive me, I am a simple chemist and know nothing of these things you describe. I thought most of what chronos said made sense, but I have no idea what a Riemannian manifold is.

“grok”?

7

I just searched the message boards for grok and I am not insulted. Sounds pretty accurate.

8

Okay, the quick-n-dirty: a manifold is a space that “close up” looks like regular n-dimensional space. Like the sphere (surface of the Earth) looks pretty much like a plane close up (thus the flat-Earthers). As you zoom out, you start to see that it curves around into a spherical shape.

Near any given point in spacetime, it looks like a 4-dimensional analogue of the plane, but it starts to get curved as you zoom out.

I’m still not sure whether you’re concerned with all of spacetime, or with the shape of “space” as a “constant-time” slice through spacetime, to whatever extent such a thing is even sensible. In the latter case, close up space looks like a 3-dimensional version of the plane, but it gets curved as you zoom out.

Now, for space we get a “Riemannian” structure on this 3-manifold. What this means is that if you give me two directions from a point I can tell you the angle between them. And it’s an incredibly deep theorem that there are really only so many ways you can do this.

As for “grok”, you might also like to supplement your knowledge of the term with the Wikipedia article.

9

If we’re talking about the real Universe, we can probably dispense with any topology which would contain closed time-like curves. And a spacetime with nontrivial spatial topology is actually not globally Lorentz invarient, so there is a preferred reference frame to use for your constant-time slices.

10

Me and my damn linear brain. I’m using simple 3d cartesian coordinates because that’s what I grok.(am I using that right?) In a simple closed universe that I might describe, along any axis repeated origins (dammit I know you told me not to use origin, but it is an arbitrary place I call (0,0,0) because I need an origin even though all positions are equivalent.) occure. I think this would be a simple cube spacem in your definitions. In fact we could define the cube space with a second set of coordinates that define the distance at which the coordinates repeat say (x,y,z).

For a spherical space I guess that polar coordinates would be better, but they could be defined in terms of cartesian coordinates with some knowledge of the conversion that I long forgot.

Of course there could be fundamental flaws with my thinking all over the place so please point them out and maybe I can correct my model.

Anyway, round about my third glass of wine yesterday, I got to thinking what if the universe didn’t actually repeat, but repeated with a “screw” pitch. I don’t know how to describe this except to explain the result of traveling in one direction (Lets say using the cubic space I set up along X), rather than ending up at (x,0,0) = (0,0,0) you end up at (0,s,0). I see no reason that a closed universe might not exhibit this property, because the resulting space is also closed. Even more interesting, is that if you travel the opposite direction along x you end up at (-x,-s,0). In otherwords the universe is assymetric.

Of course, I just realized this doesn’t work because if I end up at (x,s,0) then I wasn’t traveling along the x axis.

Dammit and I was about to call stevie collect!(Why yes Prof. Hawkings said I could call him that.)

11

Stacking the cubes somewhat shifted from the regular lattice, yeah? The problem is that you’ve just changed which paths loop around. In fact the space you’ve made is still the first one you described, just with a change of coordinates. And one of the fundamental principles of general relativity is that coordinates don’t mean squat. Everything that’s true shouldn’t depend on your choice of coordinates at all. Alternately, any two models that differ by a change of coordinates are “really the same”.

12

Well, it’s topologically the same, but if you allow angle measurements, they’re distinguishable. Basically, if you skew your unit cells, you end up with a unit cell shaped like (for instance) a paralllelopiped rather than a rectangular prism. So the lines towards equivalent points in neighboring cells wouldn’t all be at right angles.

13

Apparently I am understanding something. Is there any way to create an assymetric universe? As in a “right” and “left” version? (The terms right and left may have presumptions that aren’t necesary for assymetry.)

The reason this intrigues me, is of course the assymetry of life. Don’t misunderstand me. As a chemist, autocatalytic enhancement of assymetric synthesis is right up my ally and I get it. In fact, my first cum was on it. Yet, I can’t help but wonder if there might have been some push that helped left win over right. If the universe itself were assymetric, then that would be an interesting addition to the equation.

14

Hm, for most guys, that would be a smuggled copy of Playboy, but to each their own, I suppose.

As for a handed universe, it’s quite possible, and our very own seems to be slightly handed. This is why I said I was discarding the non-orientable, Moebius-like topologies, since you can’t have handedness in one of those spaces. But to see the handedness in our Universe, you can’t look at the topological structure, but rather at the subatomic level. If the Universe were perfectly symmetric, then it would be equally likely for a long-lived neutral kaon to decay to a positive pion, an electron, and an antineutrino, or to a negative pion, a positron, and a neutrino. However, experiments show that the decay goes to the first of those approximately 49.835% of the time, and to the second 50.165% of the time.

But this is a very tiny effect, which has only ever been observed for kaons, which have no significance to biological processes. Almost certainly, it was a coin flip which way life developed, and left just happened to win the toss.

15

cummulative exam dammit! Arrrrrrggggghhhh!

I understand how it could be a coin flip. In every sense that is very plausible. My intuition still says one had some unseen advantage over the other.

16

[nitpick] B mesons as well kaons these days. Not that they’re any more relevant to biology.[/nitpick]