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Complications of finite space
Posted By: gremlinn, on host 204.210.33.178
Date: Sunday, March 25, 2001, at 03:06:11
In Reply To: Re: A Question of the Science Variety...Kind of posted by Den-Kara on Sunday, March 25, 2001, at 00:29:34:

> Like how space never ends...that gives me a headache.

Actually, I think a universe in which space never ends (and is also nice and flat, without weird gravitational distortions or anything like that) is pretty nice to deal with conceptually...much easier than a limited universe would be. In mathematical terms, the infinite universe would be R^3, and the finite universe might be some (convex)** subregion of [0,1]^3.

In reality, when describing anything that actually happens in a nice flat Newtonian universe (by which I mean a finite system of distinct point-sized particles, with definite positions and velocities, following certain determinate, time-independent laws of motion), you never really need to look at the whole space at once (assuming that all matter and energy was at one time contained in a bounded* region of space), because for any finite number of particles, moving at finite speeds, you can always bound them in a (possibly larger) finite volume of space at any future time.

That's not the case if at the very beginning, the particles were not in a bounded region of space, but in that case the infinite universe is still easier to deal with, as explained in the next paragraph. Since there's no real reason to believe that the mass of the universe is currently unbounded, we can just ignore this possibility.

Now if the universe DID have a boundary -- I imagine it would appear like an impenetrable mirror -- you'd have lots of complications for predicting the trajectory of particles when they bounce off the edge of the universe, especially if the boundary of the universe weren't a nice big box with flat walls. If the universe were a really weird, blob-shaped region, the equations for the evolution of the universe would be horrendous, and possibly impossible to describe in closed form (i.e. with a finite collection of the standard mathematical functions).

So for this reason, I'd definitely have to say that dealing with space that ends would be troublesome, to say the least.

Of course, like I said, all of the above only applies to nice determinstic Newtonian universes with nice flat Euclidean space, which is not at all what our universe is like. You could have a nice deterministic Newtonian system on a NON-flat space, like a torus, and that would actually be no worse than an infinite flat space. Or you could ruin the whole thing by taking away determinism (throwing in quantum mechanical effects?) and Newtonian laws of motion, and you get a complicated universe no matter what the underlying shape of space is. Still, I think the infinite universe would probably be easier to deal with.

It should be noted that I'm looking at this more from a philosophical point of view than a physically realistic point of view...that's why I had so many simplifying assumptions.

--gremlinn

(*) a bounded region in n-dimensional space is one which can be contained in a sufficiently large n-dimensional rectangular box
(**) convex seemed like a good assumption because that guarantees that your space won't have any weird holes or anything in the middle. A space is convex if for any two points A and B in the space, the whole line segment joining A to B is also in the space.