We'll illustrate expansion dynamics using Newtonian gravitational dynamics. Happily, the same dynamical equations come out of general relativity for a relativistic cosmology, with a few terms redefined.

Start with a test particle on the surface of an expanding sphere of radius R. Its equation of motion is

*Note that
if rho _{0}
is nonzero, the Universe must be expanding or contracting. It cannot be
static.*

How do we integrate this? Multiply both side by Rdot to get

Now, also remember:

So that we have

Or,

Replacing rho_{0} with rho,
and dividing by R^{2},

What does this mean?

- If
**k=0**, then Rdot is always positive, and the expansion continues at an ever slowing pace (since rho is dropping). This is called a**critical or flat universe**. - If
**k>0**, Rdot is initially positive, but will reach a point where it changes sign. Expansion turns into contraction. This is a**closed universe**. - If
**k<0**, Rdot is always positive, and never goes to zero -- expansion always continues. This is an**open universe**.

Note: we are ignoring any cosmological constant here!