The Dynamics of the Expanding Universe
We'll illustrate expansion dynamics using Newtonian
Happily, we will derive the same dynamical equations that come out of
relativity for a relativistic cosmology, with a few terms redefined.
Start with a test particle on the surface of an
sphere of radius R. Its equation of motion starts with F=ma and works out to be:
Since density is proportional to R-3, and we
"now" with a 0 subscript, and R0=1, we have
Which we can insert into the equation of motion to get
Note that if rho0
is nonzero, the Universe must be expanding or contracting. It cannot be
How do we integrate this? Multiply both side by Rdot
And remember that
Now, another rememberance:
So that we have
where -k is a constant of integration. Replacing rho0
and then dividing by R2, we finally get
What does this mean?
- If k=0,
always positive, and the expansion continues at an ever slowing pace
rho is dropping). This is called a critical
- 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.