Assume our universe is curved (even though evidence says it’s probably not) such that if you traveled in a straight line for a distance equal to the diameter of the universe in any direction, you’ll be where you started. Now suppose you had highly advanced technology or magic or whatever and you built a rod that went all the way across the universe in a straight line without ends, like if you built a rubberband around the equator of our planet. This rod can be bent in any direction, but cannot be stretched at all. Would you be able to basically unwrap this rod from around the universe and wad it up in one section of the universe, similar to pulling that rubberband southward and wadding it up in Antarctica?
Absolutely. :dubious:
I think your question is better rephrased into “how would the tension in the rod change as it is moved accross the universe”. That would depend on the shape of your universe, wouldn’t it?
In other words, given a point you can go in any direction defined by a vector [X,Y,Z] and after going the distance L you wind up where you started. You are asking if for a given point any direction vector would yield the same L and if that L depends on the point inside the universe. This depends on the shape of your universe, doesn’t it? How would you like to define it?
No need to raise that eyebrow, EvilJoe; this is actually a question that cosmologists are interested in.
The short answer is that we don’t know. The mathematical question you’re asking is whether or not the Universe is simply connected or not. There is currently no real consensus either way, although there have been serious attempts to answer this question.
For example, suppose we lived in a space that was compact in extent. There would then be multiple directions in which we could see the same distant galaxy: the most direct one, and some other directions where the light “loops around the Universe” to get to the distant Galaxy. (This is like being able to walk between two points on the same longitude line either directly or via both poles — one path is obviously shorter, but they’re both straight lines.)
Cosmologists have actually looked in for these kinds of repetitions in the sky, most notably by examining the cosmic microwave background, the so-called “afterglow of the Big Bang” (specifically, data from the WMAP satellite, which was designed to study the CMB.) What’s more, some scientists have claimed to find periodicities compatible with space having the topology of the Poincaré dodecahedral space, which is not simply connected; thus, your infinite rod couldn’t get bunched up in a corner of the Universe like you propose, because it’s wrapped around the Universe. However, there have been claims that the data are also consistent with other topologies, and more data from the WMAP satellite has been released since that group made their proposal.
I got ya. 
The crunching the rod down to Antarctica is where I got lost. In all seriousness though I think that the tentative fact that the universe is expanding would cause your rod to have to to be continually stretching to keep up with L. Since you said it can’t be stretched I’m not sure how you would do that.
The dodecahedral results are very shaky, and look remarkably like just finding patterns in noise. More rigorous tests for matching circles have failed completely (that is, not only are there not the matching circles corresponding to the Poincaré dodecahedron, there are no matching circles at all), so if the Universe is topologically nontrivial, it’s on scales larger than we can observe. Which, frankly, is really disappointing: If the Universe were measurably topologically nontrivial, we’d be able to tell all sorts of cool things.
Oh, I forgot to mention: A flat Universe does not rule out the possibility of the Universe wrapping around. A simple two-dimensional example would be the world of many video games, where if you travel off the left edge you end up on the right, and if you travel off the top edge, you end up on the bottom. This is not the same as the surface of a sphere, since the North Pole doesn’t connect directly to the South, and it is in fact truly flat, since you can project it onto a flat TV screen or computer monitor without distortion. If I remember correctly, there are 17 different three-dimensional topologies which wrap around in some way, yet are completely flat, plus of course many more curved ones which are almost-flat (that is, the curvature is on scales too large to be noticeable).
I should also mention that the failure of the matching-circles search doesn’t actually say anything about whether our Universe wraps around or not, just the scale on which it occurs. I don’t think there’s any way we can rule out the possibility that there’s some interesting topology going on somewhere past the limits of the observable portion of the Universe.
Especially if it the theorized hypersphere…
That’s the topology I had in mind. And let’s pretend that the universe’s size is static for the sake of this question.
If the Universe had that topology (specifically, S[sup]3[/sup]), then yes, your hypothetical magic rod could be scrunched up as you describe. However, there’s no evidence that the Universe actually does have this topology; indeed, if the Universe is flat, which all current experimental data seems to agree with, then the Universe can’t have this topology.
I thought the Asteroids universe was a toroid - that’s still curved, isn’t it? (just curved in a dimension not accessible by the inhabitants).
Of course bending any object with a diameter greater than zero involves compressing it on one side, and stretching it on the other.
Ah, but if the universe as a whole is expanding, wouldn’t the space between and within individual atoms proportionally expand at the same rate?
Various points brought up here:
and here:
http://bbs.stardestroyer.net/viewtopic.php?t=99675&highlight=
When mathematicians and relativists say “curved”, they almost always mean “intrinsically curved”, or curved in dimensions “accessible by the inhabitants.” The canonical example of this is the surface of a cylinder: it’s “extrinsically curved” because of the way it’s embedded in three-dimensional space, but it’s not “intrinsically curved” because you could take a chunk of it and flatten it out without wrinkling or distorting it. For comparison, the surface of a sphere is intrinsically curved: you couldn’t completely flatten out a chunk of it without wrinkling or distorting said chunk. The Asteroids Universe is intrinsically flat, too – it’s on your computer screen, after all.
As far as I know, nobody has ever come up with an experiment to test the extrinsic curvature of the Universe (if that notion even has any sense), so the intrinsic curvature is all we care about.
And as a nitpick, relativity says that you can’t make an unstretchable rod. If it was completely unstretchable, then pulling on one end would make the other end move at exactly the same time, even if the rod was millions of miles long. Relativity says (in essence) that you can’t affect something far away from you instantly, so relativity says that an unstretchable rod is impossible.
Very true!
That’s what I wanted to hear. I said in my first post that evidence suggests the world is probably flat, but I was curious if my rubberband-around-the-equator scenario could be expanded into another dimension. I kind of thought it would, but couldn’t imagine how it would work in my mind.
I realized my hypothetical rod isn’t actually possible but that wasn’t the important part of the question. Okay, let’s say it’s damn near unstretchable, like a cable.
(bolding mine)
I find your ideas intriguing and wish to subscribe to your newsletter.
Thanks for that explanation - it was useful.
I was just about to start a thread to ask something along this line. Could you please elaborate on this? The way I was going was with spinning a very long rod, so pretty much the same.