An infinite universe must be flat?

[OldGuy]

@Apex_Rogers

The thing is that any constant curve will eventually meet itself, assumptions be damned. The place where assumptions factor in is where we postulate that the universe must be constantly curved, if it is curved at all. If in fact the universe is constantly curved, then unless it is flat, it must meet itself and thus be finite.

Doesn’t a helix have constant curvature? It never meets itself? Would it be possible to have a 3D object curved similarly in 5D space and not meet itself? Or are we again talking about the distinction between intrinsic and extrinsic? And if so isn’t the statement about “any curve” an extrinsic one? I’d think any 1-D manifold has no intrinsic curvature.

[by-tor]

Just FYI: “recent measurements (c. 2001) by a number of ground-based and balloon-based experiments, including MAT/TOCO, Boomerang, Maxima, and DASI, have shown that the brightest spots are about 1 degree across. Thus the universe was known to be flat to within about 15% accuracy prior to the WMAP results. WMAP has confirmed this result with very high accuracy and precision. We now know that the universe is flat with only a 2% margin of error.”

WMAP- Shape of the Universe

[MikeS]

@OldGuy

Doesn’t a helix have constant curvature? It never meets itself? Would it be possible to have a 3D object curved similarly in 5D space and not meet itself? Or are we again talking about the distinction between intrinsic and extrinsic? And if so isn’t the statement about “any curve” an extrinsic one? I’d think any 1-D manifold has no intrinsic curvature.

You’re spot-on here: a 1-D curve cannot have any intrinsic curvature. The proper statement is that any manifold with constant intrinsic curvature has a finite volume; more specifically, if I’m reading this Wikipedia page correctly, all manifolds of constant finite sectional curvature are obtained by taking some chunk of a sphere of appropriate dimension and gluing its edges together in a “smooth” way. (I’m oversimplifying here, but such is life on the SDMB.) This is a mathematical fact about intrinsic curvature, and follows from our definitions of what exactly we mean by “curvature”. The analog of a curve drawn on (embedded in) a flat plane is an OK one for gaining intuition about why the rigourous intrinsic-curvature statement above might be true, but as you’ve discovered the analogy breaks down if you look at it too closely.

[Chronos]

The thing is that any constant curve will eventually meet itself, assumptions be damned.

Again, this is not correct. A space of constant negative curvature will generally not meet itself.

[ashman165]

@Biggirl

This I understand. What got me going was the conjecture that an infinite universe must be flat because if it was curved, it would be finite because it would meet itself. Since we do not know the nature of what is outside of our observable universe, I wondered how this conclusion was reached.

If we can detect curvature (which, according to the show I watched, we haven’t) then that would mean the universe is finite. My question is-- why would you come to that conclusion? What’s to say that this curve does not persist for infinity?

The answer-- I thought-- was because of the assumptions we have about the nature of the time/space/place in which this universe exists. If we work with these assumptions, then a constant curve will eventually meet itself.

But now you’ve told me I’m wrong. Dang! I thought I had the universe figured out!

Oh okay. I think I see where you’re getting at. And please excuse my profoundly poor typing earlier; twas the day after my birthday and, um, some of my brain might have still been drinking. I think I’m better now.

Anyway, it isn’t necessary for a curve to sweep out some finite area. A constantly curving something needn’t come back on itself. It might, of course, but it equally might not.

And to say what’s going on beyond the points at which we can no longer see in the universe is entirely speculative. Granted, people will call it a “good” speculation, but all speculation is equal. If there were any evidence for either proposition, it wouldn’t be speculative anymore. =P

Essentially, if you think you understand how the universe works, then you’ve missed something. =P