General Relativity and the Possible Geometries for Space

     The Friedmann-Robertson-Walker model allows for three possible structures for space: If the mass/energy density of the Universe is more than the critical value then space is a three-dimensional sphere; if the density is equal to the critical value then space is flat; if the density is less than the critical value then space is a hyperbolic three-dimensional sphere. The two-dimensional analogs of the above three spaces are respectively of the surface of a ball, the plane and the saddle surface:

sphere
Figure 1: Two-Dimensional Sphere
Curvature is positive, W > 1, and space is close.


plane
Figure 2: Two-Dimensional Plane
Curvature is zero, W = 1, and space is flat.


saddle
Figure 3: A Saddle Surface or Two-Dimensional Hyperbolic Sphere
Curvature is negative, W < 1, and space is open.

The ratio of the mass/energy density to its critical value is W; W determines the geometry of space.
     In the case of a sphere, the Universe is closed because it is of finite extension and it turns out that the expansion of the Universe will eventually stop, space will begin to contract instead of stretch and the Universe will finally collapse in a process resembling the opposite of the Big Bang. Hence, for this first structure, the Universe has a finite lifetime. When the Universe is open (the hyperbolic case) or flat, space is of infinite extent and the expansion continues forever.



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