What would an "open geometry" for the universe look like?
My high school Astronomy students and I are having a hard time conceptualizing a negative curvature universe. We can conceptualize a flat universe - no edge, infinite, open, with space-time expanding forever into time. We also do O.K. with positive curvature - closed, finite, spherical, and subject to collapse. Negative curvature...saddle-shaped? Is every point of the "saddle" expanding together? If so, it seems that it would violate the cosmological principle - different observers would see different phenomena depending on where they were on the saddle.
I agree that this is pretty confusing. The first thing to remember is that the sphere, the plane, and the saddle point are just the 3-dimensional analogues of the universe in 4 dimensions. That is, the three aforementioned shapes are the different ways you can represent a 2-d surface curved in the 3-d. But the universe is actually a 3-d "surface" curved in 4-d. Because we meager human beings can't visualize things in 4 dimensions, you can't *really* visualize the curvature of the universe. The sphere, plane, and saddle point are just meant to be analogies.
The situation you'd have in an actual open universe is one in which every point in space is like the saddle point (not like the entire saddle). This is impossible to visualize in 3-d. You just have to trust that it works out in 4-d.
Another way of thinking about it is that the sum of the angles of a triangle in a closed geometry is always greater than 180 degrees; in a flat geometry is always exactly 180 degrees; and in an open geometry is always less than 180 degrees. There's no way you can have something like that last case in 3-d (if it's to work out the same for all points in space). But it does work out in 4-d.
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