The Dynamics of the Expanding Universe
We'll illustrate expansion dynamics
using Newtonian gravitational
Happily, the same dynamical equations 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 is
Since density is proportional to R-3,
"now" with a 0 subscript, and R0=1, we have
Which we can insert into the equation of
motion to get
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, also remember:
So that we have
Replacing rho0 with rho,
and dividing by R2,
What does this mean?
- If k=0, then
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 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!