Start with a test particle on the surface of an expanding sphere of radius R. Its equation of motion starts with F=ma and works out to be:

Since density is proportional to R

Which we can insert into the equation of motion to get

*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, another rememberance:

So that we have

Or,

where -k is a constant of integration. Replacing rho_{0}
with rho*R^{3},
and then dividing by R^{2}, we finally get

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 goes to zero. Expansion stops, gravity wins, and the universe then starts to collapse. 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**.