## 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:

**Figure 1: Two-Dimensional Sphere**

Curvature is positive, Ω > 1, and space is close.

**Figure 2: Two-Dimensional Plane**

Curvature is zero, Ω = 1, and space is flat.

**Figure 3: A Saddle Surface or Two-Dimensional Hyperbolic Sphere**

Curvature is negative, Ω < 1, and space is open.

The ratio of the mass/energy density to its critical value is Ω; Ω 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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