In fact what Engywook mentioned above is precisely what I was originally trying to get at at the beginning of this thread…
It’s been a long time since I looked at the geometry of special relativity, but I think you’ve got a problem right here in step 1: how are you defining the “distance” from a given point in space-time to the “edge”? The only way I can think of to define this distance is to look at the last point in time when the edge was “here” (i.e. the Big Bang), and multiply the intervening time by the speed at which the edge appears to be receding, namely c. But that calculation will return the same result independant of where I perform it. It other words, by this definition the edge is equally far away from every point in the universe.
green_bladder:
It is possible the universe is ‘connected’ in such a way that a line of sight through the universe from you in a straight line could impact you in the back of the head.
If the universe is expanding slowly enough that all parts of the universe (the balloon) are visible, then no part of the universe is ‘cut off’ to you, and you could actually (given sufficient lifespan, energy, resistance to acceleration, etc) perform the journey.
The cube universe was just a convenient way to try to give you a physical ‘thing’ to hold in your hands in some sense, and to connect up related parts of the universe.
Locally, like, anywhere you can see, there is ‘direction’. Surely you can point ‘over there’ and head that way. Locally space appears pretty flat. The fact that it may wrap around on itself ‘way out there’ is not easily visible here.
But it does mean, in a sense, there’s no universal direction. Just a local one. It’s a lot shorter to go from here to the drugstore by way of 11th street than, say, travelling across the universe in any number of ways to get there.
Interestingly, though, if the universe is not cut off to us in some way, it means there are an infinite number of ways to take what appears to be a dead straight line between any two points. An infinite number of ‘repetitions’ are available with sufficient distance.
You can get an even simpler example of this ‘connection’ of the universe I’m talking about. Take a long rectangular strip of paper. It obviously has four distinct ‘edges’. But it’s not that much of a leap of logic to tape the two short edges together and say ‘I mean this to be a seamless edge’. Just because we can’t easily manufacture such a loop of paper as a whole doesn’t mean we can’t make the leap of logic and math to connect these edges and treat the loop as now having only two ‘edges’.
(note, interestingly, that there are two ways to connect the paper in this manner, and this produces two distinct topologies, but if you’re on the surface and don’t consider the thickness or the undetectable ‘other side’ of the paper, a life form wouldn’t tell the difference)
In the cube example, we produce a hyper-toroid (a 3 dimensional equivalent of the 2 dimensional toroid/doughnut surface).
You don’t have to ‘bend’ the cube around, either, if you consider that face 1 just ‘connects’ to face 2 (you pass through the face, and reappear on the other side). But you, light, and everything else cross this boundary identically, so there’s no way of identifying any sort of seam or edge. A beam of light passing directly through face 1 would appear at face 2, and eventually (in a nice, flat universe) hit face 1 again eventually.
Producing a connected ‘balloon’ universe in 3 dimensions is tougher to visualize (hypersphere), so I won’t go into it, but mathematicians love this stuff.
Assuming the universe to be an approximate sphere with ~13 billion light year radius, is it at all possible that it is expanding into a boundless region of space, or simply one far, far greater than that? I know, I know, the universe is all of space/time, etc, but I was asked by a student a while back (it was off topic, in a math class) if there’s any reason to reject that model, including the possibility that universes could be flowering up all over the place, but so far away that none has colided with ours yet. What evidence is there that there isn’t another object like our expanding universe a few hundred billions light years thataway? (I’m pointing kind of eastward) What would be the consequences of two such objects expanding until they overlapped? Could such a model account for the observed acceleration of expansion in our own neck of the woods?
The question then becomes in how many dimensions are you talking about ‘overlapping’?
We can have, as I’ve been saying, an unbounded 3 dimensional universe, but have it bounded in 4 or more dimensions, where the shape and size of space is entirely defined by the matter within it…
…and yet, in a higher dimensional sense be just a finite bounded blob, maybe amongst a sea of other as yet undetectable blobs.
Isn’t math/topology fun? 
Either a ‘space we’re expanding into’ or a 'connected space we’re expanding from ‘inside’ manages to dodge the bullet of what is ‘outside’ in the physical sense that we detect.
The reason, I think, why the first case, where we’re expanding into predefined space, is rejected is that the universe appears to be on a large scale identical in all directions (isotropic) to the limit of our instruments. Unless we were by preposterously improbable accident at the dead center of an expanding universe that expands into existing space, we should see a bias one way or the other (more galaxies, more energy, etc).
I’m sure there are other clues to suggest why it is somewhat more likely that we’re in a topologically closed universe, which dodges the issue entirely by making there be no center or edge.
I suppose you could argue that we’re at the edge of the universe, but its expanding so fast and its so large that our visible universe doesn’t overlap with the edge of the whole universe and therefore we see similar things in all directions, but that seems more of a stretch to me, and I’m sure there are some cosmologists that could describe why this is also unlikely.
But we really don’t know the ‘shape’, and can’t yet actually prove the topologically closed physical universe. Neat mind puzzles though.
You’re again making the assumption that the universe is expanding into space, rather than creating new spacetime as it expands.
I should try looking at the SciAm article, but that window froze on me so I’m stumbling onward regardless.
From my reading, the general thought is that other universes could be popping up all over, but that they could never be detected from inside our universe (unless wormholes of some undefined sort might “connect” them in some undefined way.) There is no “outside” to our universe so that we are not going to be intersecting something else. There isn’t even any meaning to the phrase “a few hundred billions light years thataway”.
Now it may turn out that some enterprising cosmologist disagrees with these thoughts and that the SciAm article goes against this, but so it goes. Assumptions made for different models of the universe can turn up different results.
William_Ashbless said that:
and this is now thought to be untrue given the latest mapping of the observable universe. Notions about things we can’t measure yet have an extremely high uncertainty value.
Very interesting…
So if it is rejected that the universe is connected that a straight line could hit you in the head, what alternative is there? And why is it being rejected? Is it instead advocated that the universe IS expanding into unknown space? If it isn’t, but it isn’t topologically connected and closed, what is the alternative? I must read said SciAm article…
Oh well, doesn’t change the interesting discussion on topology, but it may definitely impact the OP.
I seem to recall a recent BBC News article about there being some idea that the universe was toroidal based on some measurements… high uncertainty, unverified, but it would still mean something like my statement you rejected.
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Quoted:
Rather than being infinite in all directions, as the most fashionable theory suggests, the universe could be radically smaller in one direction than the others. As a result it may be even be shaped like a doughnut.
“There’s a hint in the data that if you traveled far and fast in the direction of the constellation Virgo, you’d return to Earth from the opposite direction,” said Dr. Max Tegmark, a cosmologist at the University of Pennsylvania.
This is a very recent article.
My bet is the jury’s still out.
As I understood the article, there were two primary arguments for other universes:
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Matter, which would be predicted to get less and less densely spaced as one moved toward the edge of our universe, seems to be uniformly spaced. But all the matter in our universe is from our big bang, so I don’t follow this argument.
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The odds of an organized universe where atoms can exist for long periods of time is rare, so the fact that we’re in one might imply lots of others where things aren’t so good. I know that argument is bad statistics (taking all your data and forming a hypothesis, leaving you nothing to test your hypothesis on), so I don’t know what to make of the article overall.
Kind of northeastward of more southeastward?
Quoth William_Ashbless:
Not quite. We can’t actually observe anything happening right now 'way out there, but we can make an educated guess that things out there look pretty much like things around here. A being on a planet around a star 13.7 billion lightyears from here, observing at a time 13.7 billion years after the Big Bang, would probably see a night sky with stars scattered across it, and if he got out a small telescope, would see galaxies scattered around beyond the stars. We can’t say what constellations he would see, or anything like that, but we can guess that there are constellations.
Quoth Dr. Zoidberg:
No. All modern models of the Universe are unbounded. The model favored by the MAP data is that the Universe is flat, and that it’s therefore either infinite, or topologically nontrivial. There is some circumstantial evidence in the data to support the belief that the Universe is finite in size (and therefore topologically nontrivial, possibly like the 3-torus you mentioned), a little bit bigger than the observable Universe, but there’s no “smoking gun” for that. A flat universe does not imply an edge, and since there is no edge, no points are closer to the edge than others.
Thanks, Chronos. I’m not the topologist, astronomer, or physicist. Just a dabbler in those dark arts. Glad I didn’t get too much abjectly wrong.
Quoth William_Ashbless:
Not quite. We can’t actually observe anything happening right now 'way out there, but we can make an educated guess that things out there look pretty much like things around here. A being on a planet around a star 13.7 billion lightyears from here, observing at a time 13.7 billion years after the Big Bang, would probably see a night sky with stars scattered across it, and if he got out a small telescope, would see galaxies scattered around beyond the stars. We can’t say what constellations he would see, or anything like that, but we can guess that there are constellations.
Quoth Dr. Zoidberg:
No. All modern models of the Universe are unbounded. The model favored by the MAP data is that the Universe is flat, and that it’s therefore either infinite, or topologically nontrivial. There is some circumstantial evidence in the data to support the belief that the Universe is finite in size (and therefore topologically nontrivial, possibly like the 3-torus you mentioned), a little bit bigger than the observable Universe, but there’s no “smoking gun” for that. A flat universe does not imply an edge, and since there is no edge, no points are closer to the edge than others.
No, in my case 1 we have a universe with no curvature (in a higher dimension) or with “negative” curvature like a saddle. This kind of universe would never curve back on itself so that if it was finite there would indeed be an edge somewhere. My case 2 is a finite but unbounded universe like a hypersphere, hypertoroid or other multiply connected topology. Your balloon universe is a example of case 2.
Dr. Zoidberg… okay, fine, I’m not well versed enough in either cosmology or topology to counter the point. I’ll let someone else (like Chronos) argue about it.
The problem I see with your case 1 is it still implies ‘what’s outside’ if there’s truly an edge. I can accept, possibly, a universe that has infinite ‘space’ that the matter is expanding into, but then there’s no edge.
So I have to ask you what, in your case 1, this edge is defined as.
Chronos, what do you mean by “topologically nontrivial”? Does that just mean “not flat”? What possible nontrivial topologies are possible? Anything other than the 3-torus? And what is the circumstantial evidence that the universe is finite in size and a little larger than the observable universe? I am uncomfortable with the idea of an infinite universe, so I’m looking for any straws to grasp at to retain my finite viewpoint now that the recently released MAP data has finally killed the possibility of a closed universe. Not that the universe gives crap what I’m comfortable with, but if I could hold on to my finite, unbounded viewpoint, I’d be very greatful.
PS- It occurs to me that one cool possible topology would be the Mobius version of the 3-torus – that is, if you gave a half-twist to one of the sets of opposing faces of the cube before bringing them together. That way, when you travel all the way around the universe and return to your starting point, you’d be you’re own mirror image. Of course, then you’d promptly starve to death, because all chiral biomolecules would then be wrong-handed.
If you read further into that article, like the very next paragraph, you’ll see that it immediates starts backpedaling. Articles like this are ubiquitous in science journalism. Almost any major finding will have a few scientists off to one side who say that the data shows something very different from what most think. Very occasionally the mavericks are right. Most often, however…
What’s fascinating to me is that it was in an article about the MAP data that I read the comment that the flatness it indcates disproves the back-of-the-head notion. Interpreting data is an art.
Of course the jury’s still out. But it looks to me that the best way to see the back of your head will be with a pair of mirrors.
My guess is that the edge, if it exists and if you could reach it would be simply the end of your existence. Since we are talking about spacetime we have something like a black hole. When you pass the event horizon of a black hole you can’t avoid the singularity since it is quite simply in your future. If you got to the “edge” of spacetime you get to the edge of reality for you and the end of your existence. A flat 2-space, bounded disk universe could be visualized as a cone with the third axis as time. The big bang is the point of the cone. A spaceship that went fast enough to reach the edge just goes out of existence wherever its world-line intersects the surface of the cone.
If we imagine a hyperspherical or hypertoroidal universe we still have to wonder what is the something (nothing?) that it expands into. I think this can be avoided if we imagine that there is no higher dimension into which our three are “embedded” as the topologists say. Then the “edges” are just connected to each other somehow and we don’t need to worry about what’s “outside”
Ok, flat universe but no edge. I don’t get it. If I have a flat, finite piece of sheet metal, one that NEVER curves back on itself, I have a bounded piece of metal. If it won’t ever curve back it can’t link up with itself at some point. If it forms into a toroid or multi-toroid it still has a curvature. How can we combine (truly and consistently) flat with unbouded? I can see how a thing can be locally flat and then still be unbounded and finite. Cubes are like this. But at some point there’s a radical “curve” or discontinuity. I don’t understand.
Attaching a philosophical boundary to the edge of physical space?
That doesn’t sit well with me. I can’t see any justification for defining the edge of space as an inevitability. Smacks too much of sailing off the edge of the world. I can see you simply ‘turning around’.
I understand the ‘event horizon’ analogy, but there’s nothing that suggests there’s any sort of gravitational influence caused by this edge.
At best, I can see you defining the edge as ‘what can be seen by way of the distance light could have travelled’ but then we’re just back to the observable universe.
I’m not sure I follow your cone description. Indeed, if the universe is topologically connected as in your case 2, but is gravitationally capable of pulling itself back to a point, then indeed travelling far enough and long enough will end up with you arriving at the big crunch, inevitably (having possibly passed over a point you’ve already been at).
But I don’t think this is what you mean.
I’m not sure what to think about what you’ve said, but I simply don’t buy that there’s this magic wall that, when you hit it, you just go out of existence. Either its infinite in 3 dimensions and stuff is expanding into it, or its topologically nontrivial and its finite in 3 dimensions but not bounded.
Sure, you can ask what is outside the hyperdimensional object that is the universe, but since we work in 3 dimensions of space, and 1 dimension of time, from a physical point of view one has to ask what that even means. For most people, what you can see and where you can get to constitutes ‘the universe’.
Dr. Zoidberg, this specific issue was addressed by William_Ashbless above in this thread: a torus doesn’t have to be curved. Your standard donut shaped torus has curvature, yes. But it is possible to construct others (at least mathematically) that aren’t. Roll a square piece of paper into a cylinder. This object has no mathematical curvature – Euclidean geometry works the same on this surface as elsewhere (which can be seen in that you draw circles, triangles, parallel lines, etc. on the piece of paper before you roll it up, and they retain their properties; as opposed to a sphere or something curved, where triangles have the wrong number of degrees, for instance). Now, if you stretch the to ends of the tube around and attach them, you’ll have a torus without curvature. Obviously, you can’t do this in the real world – paper doesn’t behave that way and you’d have to use something like rubber where you’d introduce curvature and distort your geometric figures. But you can do it mathematically, and the result is the same as a Pac-Man or Asteroids video game, namely that when something moves off the surface on one “side,” it instantly reappears on the other. And there are no discontinuities, either – you can reposition the “screen” anywhere along the surface, and everything behaves the same. I believe this is called a “square torus,” although my mathematical vocabulary is limited. I believe the “3-torus” that Chronos mentions above is the same concept with another dimention: gluing opposing sides of a cube to each other without adding distorting curvature.