I think, perhaps, that a little judicious use of "tensors or differential geometry or some other crazy math that makes the minds of experimental solid state physicists expode" is called for in this case.
For one thing, I think we need to begin with a clarification. When physicists describe the universe as being like a flat plane, they are obviously talking in higher dimensions than 2. When we say that the universe is "flat" or "curved," lots of people think of the model of gravity they've seen before: a rubber sheet with heavy balls sinking into it. The concept is fine, but the dimensions are higher. What we are actually claiming is that it is 4-dimensional spacetime that is either "flat" or "curved."
Personally, when talking about the curvature of spacetime, I find it more useful to come back to the ideas of Euclid. If a space is "flat," then its geometry is Euclidean. If a space is "curved," then its geometry is non-Euclidean. An example of 2-dimensional non-Euclidean geometry is the surface of a sphere (hence the term "curved"?).
Now, in a nutshell, general relativity assumes that space may be curved (non-Euclidean), and it gives an expression for how to find the curvature of the space (these expressions are called Einstein's equations).
So my response to your question, NeoArcadian, is to tell you what research into the geometry of the universe has discovered:
1) Mass distorts (bends/curves) spacetime, resulting in gravity (this is a part of Einstein's equations)
2) The part of space not curved by mass is BASICALLY Euclidean.
So, in other words, until you put mass in the universe, it's almost nice and Euclidean.
I say "almost" and "BASICALLY" because of the discoveries made that the universe is expanding. The current best model into the inherent geometry of the universe is that it is an expanding Euclidean universe (not in the presence of mass).
So that just about takes care of tube-like universes and coffee-mug shaped wormholes; the experimental evidence just doesn't support it.
Wormholes themselves are actually incredibly interesting distortions that happen in otherwise FLAT space. We've only recently gone over this in my gravitation class, so I can't tell you with 100% certainty, but I think it is the case that what happens in a wormhole is this: Imagine a sphere of radius R. If there is a wormhole at the center of this sphere, then, for large R, if you decrease R, then the surface area of the sphere decreases (like you'd expect.) At some critical point, however, the area of the sphere hits a minimum, and, if you decrease R further, the area of the sphere begins to INCREASE. This increase is unbounded, so you can basically fit an infinitely large sphere either outside (
) or inside (
) the wormhole. So there's basically an entirely separate space on the "inside" of the wormhole.
I hope that was helpful at least a little bit. Please say so if I've gone too in depth in my discussion.
Last edited by M@Man (2005-03-04 13:39:36)
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