How can 3D space expand and contract without a fourth dimension?

[Feeble Wonk]

Please help enlighten an ignorant layperson. I am having great difficulty understanding how 3D space can bend, expand or contract in a finite universe in the absence of at least a 4th dimension.

The classic example offered is the balloon analogy, wherein the balloon expands - with all points diverging - but without a specific center from which expansion occurs. My objection to that analogy has always been that the center is "INSIDE" the balloon, not on it.

The good people in cosmology suggest that this is flawed thinking, and insist that the center of the balloon DOES NOT EXIST. They refer me to the topological concept of a 2D torus existing in three dimensions, in a universe that DOESN'T HAVE THREE DIMENSIONS!

It seems to me that this is an arbitrary set selection of dimensional space... simply defining a limited area of 3D space with 2D specificity. Can anyone explain this in terms that I can understand?

 

[holy_toaster]

The notion of expansion and contraction in cosmology is usually defined with respect to a set of flow lines of the matter. Assume the cosmic matter (galaxies or clusters of galaxies if you want) to be dust moving through space and time, like for example the particles in a fluid flow. Then contraction means that these flow lines get closer to each other over time and expansion the opposite.

Assume for the moment the universe were 3-dimensional, that is one time dimension and 2 space dimensions. Then "space" at each instant of time is a 2-dimensional surface and the flow lines of matter cross these surfaces at every point. Time moves forward along every flow line. So if you sit on one galaxy moving along one of these flow lines you see the other galaxies on the flow lines around you either coming nearer to you or moving away from you in the transfer from one 2-dimensional surface to another one.

This works similarly in the case of 3-space dimensions and one time dimension. Only the instants of time are now 3-dimensional spaces instead of 2-dimensional surfaces. So there is no need for additional dimensions to make sense of the notion of expansion and contraction.

Did that explanation help?

Furthermore I do not know what you mean with space "bending". Do you mean curvature? That is bit more complicated than expansion, but only slightly...