Edge of the universe?

[William_Ashbless]

Flat in 4 dimensions (3 dimensions of space and one dimension of time) should imply that gravitationally the universe is expanding fast enough that gravity just can’t slow it down to recollapse, but only barely.

But as long as we’re talking about magic walls, imagine one on one side of the universe that if you fly through you end up on the other side of the universe seamlessly.

Let’s take your piece of metal. If you’re living on that 2 dimensional sheet, you see everything as flat.

If the sheet were rolled into a tube, but the tube were really large, any local region would appear totally flat.

Now, connect the ends of the tube. If the tube’s large enough, any local region would appear totally flat.

So all that we can say about making any sort of measurement directly on space is that it might appear to be locally flat.

However, the gravitational expansion of the universe might be within the topology that exists and overall independant of it. This describes the growth of the sheet.

Its the curvature of this growth rate over time that determines the ‘closed’ ness of the universe in terms of whether we will end up in a big crunch.

So the doughnut grows or shrinks. At this point, the doughnut might be so large that everything is physically, in 3 dimensions, flat. But the universe could still be, over time, closed (it will collapse down into a teeny doughnut), or open (it will be forever growing as a massive doughnut), or just on the cusp.

In all cases its still a doughnut, and locally will always appear basically flat.

But there’s hints we can see some of the non-flatness at a truly massive scale.

Regardless, none of this changes the possibility that the universe is topologically nontrivial, and this dodges the notion of requiring a magic wall.

[William_Ashbless]

Excellent, thanks bryanmcc.

[Inoshiro]

*Originally posted by William_Ashbless *
**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’.
**]

In this case it wouldn’t be inevitable unless you stayed on course for the edge of reality. You could turn around if you wanted to unlike a black hole where you are doomed at some point. I don’t like this view myself since, as Chronos pointed out, it implies that there may be galaxies and stars situated near the edge such that in one direction they would see almost nothing while in the opposite direction they would see the rest of the universe.

*Originally posted by William_Ashbless *
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.
**

If the universe is flat I would tend to favor the idea of an infinite space and stuff expanding into it myself. But you always hear about how all space and time expanded out from a singularity at the big bang. So I was trying to reconcile a flat topology with that idea of all space and all time beginning at some point and expanding from there. If your universe is a sphere or some other connected shape you don’t have a problem. The thing just grows larger over time. It was the whole flatness issue that I was trying to work with and combining that with the idea of spacetime itself having a starting point I got my cone analogy. Of course you need to extend that into a hyper-cone for the real universe.

[William_Ashbless]

Again, I think the flatness issue goes away to some degree. Either you have the universe mathematically connected in such a way that you don’t see the curvature because everything (including your perception) is within the curvature you see, or the mathematical properties of the space is such that there is no curvature (the torus example, as bryanmcc pointed out doesn’t require any visible internal curvature mathematically), or the space is so large that we have no way of detecting the curvature short of seeing ‘repeats’ of the universe.

I still think we’re having a disconnect between gravitational expansion and spacial topology.

[Inoshiro]

*Originally posted by bryanmcc *
**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. **

OK, I think this is what topologists call a “non-embedded” torus. Meaning that it is not imagined to be part of a higher dimensional space but exists this way on its own. This is one possibility and you are correct. There is no curvature in this “Asteroids” type of universe. Of course, the Asteroids universe can’t be embedded in a real 3-spatial world without distorting it. So my problem is solved I suppose. Flat universe, unbounded and no curvature. I think my issue was that I kept trying to envision our 3-manifold embedded in a higher dimension with the resulting need for large scale curvature in finite, unbounded topologies.

[William_Ashbless]

Well, waitaminute, Zoidberg.

If you’re living on your distorted torus, is it not possible that your perceptions (your sensors) are distorted within the space along with the space itself such that it appears locally flat? (Assuming, again, that the torus is small enough that it doesn’t appear locally flat anyway, even to an outside observer).

It may not matter how the ‘outside dimensions’ see us. We’re inside. So we’re distorted along with the universe.

So the question is, what is ‘real curvature’ that could be measured inside the universe, and what is ‘apparent curvature’ that you’re inventing just to have a convenient shape in front of you?

[Inoshiro]

*Originally posted by William_Ashbless *
**Well, waitaminute, Zoidberg.

If you’re living on your distorted torus, is it not possible that your perceptions (your sensors) are distorted within the space along with the space itself such that it appears locally flat? (Assuming, again, that the torus is small enough that it doesn’t appear locally flat anyway, even to an outside observer).

It may not matter how the ‘outside dimensions’ see us. We’re inside. So we’re distorted along with the universe.

So the question is, what is ‘real curvature’ that could be measured inside the universe, and what is ‘apparent curvature’ that you’re inventing just to have a convenient shape in front of you? **

If you were a flatlander who lived on the surface of a large sphere you would notice that if you made a very small triangle the angles would sum to about 180 degrees. If you made one that was, say, a few light years across, the sum of angles would be noticeably larger. The value of PI would also change noticeably with scale. Larger circles would have a smaller value of PI associated with them as more of the curvature of space was enclosed by them.

[William_Ashbless]

Okay, fair enough. I can’t easily see a way around that analysis.

So we have a… non-embedded connected topology of some sort, or the curvature’s so small we can’t currently measure it, if you’re willing to buy a finite but unbounded universe.

[aeropl]

I guess the OP did a much better job at phrasing my question than I did when I asked this a while ago. This thread is answering my question quite well.

[Ring]

Keep in mind that if the universe is presently infinite then its always been infinite, even at a bazillionith of second after the big bang.

[Lobsang]

Space and ‘the universe’ are two different things.

Space is that in which the universe lives. The universe is not infinitely big. Space is.

[Exapno_Mapcase]

This may be philosophy, but it is not physics.

[Beer_Penguin]

*Originally posted by Joe Random *
**And how do you propose traveling back in time? If you have a working time machine, I’d suggest that you patent it and make yourself filthy, stinkin’ rich instead of posting to the SDMB
**

Wouldn’t patenting a time machine be pretty futile? I could just buy one, copy down the schematics, go back in time a minute before the original patenter filed his patent and file my own. Then again, doing so would be futile on my part, as as soon as I did so someone else could do the same to me as I had done to the original patenter. Hrm.

Years ago, me and a friend had an idea for a ‘disinventor gun,’ that would ‘disinvent’ any object it was fired upon. Then, the owner of the disinventer gun could invent the object that had been ‘disenvented,’ and make a bunch of money. But we were never quite sure what would happen if you fired a ‘disinventor gun’ at another ‘disinventor gun,’ and fearing some type of nexus of the universe, mass hysteria, dogs and cats living together scenario, we stopped the plans to complete the device.

[bryanmcc]

*Originally posted by Ring *
**Keep in mind that if the universe is presently infinite then its always been infinite, even at a bazillionith of second after the big bang. **

Which is precisely why I’m uncomfortable with an infinite universe. Again, not that the universe necessarily gives a rat’s ass what I’m comfortable with.

[Ring]

*Originally posted by bryanmcc *
**Which is precisely why I’m uncomfortable with an infinite universe. Again, not that the universe necessarily gives a rat’s ass what I’m comfortable with. **

It’s like trying to imagine nothing- the mind reels.

[Inoshiro]

*Originally posted by Ring *
**Keep in mind that if the universe is presently infinite then its always been infinite, even at a bazillionith of second after the big bang. **

This is my case 3 above. The thing that bothered me about it was that you always have the cosmologists hammering away at this idea that “all time and space were once compressed into a singularity”. Then we got the big bang. So you go from a singularity to an infinity in a Planck second. I suppose it’s possible but the finite and bounded spacetimes always appealed more to me. Like I said in my first post, I spent most of my life convinced that the universe was a hyper-sphere. Then there was that W-Map data that got me thinking about this again.

[FriendRob]

*Originally posted by Dr. Zoidberg *
**So you go from a singularity to an infinity in a Planck second. **

Well, not really, since the singularity is telling us that our physics is wrong for times less than the Planck time. It makes more sense to say the universe has always been infinite, AFAWK. We just don’t “know” all the way back to t=0.

[William_Ashbless]

So just because our current physics don’t work nicely back before a certain point in time, you consider the universe to be infinite in spacial extent?

This doesn’t follow. Even if the universe is infinite in spacial extent, if you allow a big bang at all (beginning of the universe, mass-wise anyway), there’s still problems back around t=0.

[aubries]

*Originally posted by bup *
[B

Kind of northeastward of more southeastward? **

South. But only, like, five degrees, tops.

[Chronos]

Quoth brianmcc:

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?

A topologically trivial space would be one which is infinite in all directions. A nontrivial space is anything else. Assuming a uniform curvature, the curvature ends up dictating what topologies are possible.

There are 18 possible topologies for a flat, spatially 3-d Universe, and I might even be able to remember them all. You can have:
1: Space infinite in all three dimensions
2: Basic cell a slab, infinite in two dimensions, and repeating in the third, with an arbitrary twist
3: Basic cell a “chimney”, infinite in one dimension, reapeating in the other two with no twist
4: Basic cell a chimney again, infinite in one dimension, repeating in another with no twist, repeating in the third with 180[sup]o[/sup] twist
5: Basic cell a rectangular prism, finite in all dimensions, repeating with no twists (this is the 3-torus)
6: Basic cell a rectangular prism, repeating with no twists in two dimensions, repeating with a 90[sup]o[/sup] twist in the third
7: Basic cell a rectangular prism, repeating with no twists in two dimensions, repeating with a 180[sup]o[/sup] twist in the third
8: Basic cell a hexagonal prism, opposite sides of the prism glued together, top and bottom glued with a 60[sup]o[/sup] twist
9: Basic cell a hexagonal prism, opposite sides of the prism glued together, top and bottom glued with a 120[sup]o[/sup] twist
10: Basic cell a rhombic dodecahedron (a cube with pyramids on all of its sides, such that the slopes of the pyramids match), opposite faces glued together
11: Basic cell a slab, infinite in two dimensions, repeating with inversion in the third
12: Basic cell a chimney, infinite in one dimension, repeating directly in one, and repeating with inversion in one
(six more that I can’t remember, which involve inversions)

Cases 11-18 are probably out, since they’re non-orientable, and particle physics suggests that the Universe is orientable. Case 2 (with any non-zero rotation angle) and case 4 aren’t homogeneous, since they have a “center” about which you’re rotating. Cases 2, 3, and 4 have infinite volume, which sort of defeats the purpose of having a non-trivial topology: If space is infinite, it might as well be simple.

I’m not certain, but I understand there are an infinite number of possible topologies for positive or negative curvature universes.

As for that circumstantial evidence, MAP observed less Fourier power in very long wavelengths than was expected, in other words, that there’s slightly less large-scale variation than expected. One explanation is that the Universe is finite in size, and you can’t fit in scales larger than the Universe, but there are several other explanations, as well, and we don’t yet know which is correct.

And an infinite Universe is infinite at all times after T=0. It might be meaningful to say that at exactly T=0, it was a single point, but we’re far from even having a clue about that. However, it is meaningful to say that all of what is now the observable Universe was once contained in an extremely small region, far smaller than an atom. As noted before, the observable Universe is not the whole thing, and is an observer-dependent description.