Main Question or Discussion Point
When discussing the shape of the universe (flatness/curvature), I often hear of three possible examples; spherical, flat and hyperbolic. Presenters will often use a 2-D analogy of how a flat sheet can be curved or shaped, like a saddle, table, or surface of a ball, where triangles can be defined with different sums to their angles. This analogy makes perfect sense to me. In the same discussion they often talk about how the universe is expanding and depending on the shape, rate of expansion and matter density, it may rapidly expand into infinity, just slow down enough to never quite collapse, or turn around and collapse back down to a point. In GR I believe we are talking about space-time, which is a 4-D ‘space’. My question is this, when thinking of the 2-D analogy, do I extend this analog to a 3-D space, or a 4-D space in my imagination? Part of my confusion is whether to include the time-like aspect of space-time into the shape of the universe, or is it handled separately? In the spherical example, I understand the universe to be closed, so does this imply that time also goes in a circle and closes on itself? Can a spherically closed universe just be a 3-sphere and then expand forever without collapse?