| The Friedmann equation governs the
dynamics of the universe.
From this equation the big bang is
derived so it's nice to have
an intuitive feel
for the type of physics that gives rise to it.
It
comes out of Einstein's theory of general relativity
but a classical derivation gets us most of the way there
and gives some physical insight into what the equation represents.
We can consider the Friedmann equation as
an equation of energy conservation.
Consider a test particle of unit mass on the surface of a sphere
enclosing mass M.
The potential energy of the particle is given by:
V = - GM / R(t)
The kinetic energy of the particle is given by
2T = (dR(t)/dt)2
From conservation of energy we know that
(d/dt)E = 0
Where E is the total energy of our system.
If the system is conservative
( i.e. if energy is conserved)*
then
it can be prooven that the total energy
is the sum of the kinetic enery T and
the potential energy V.
Luckly gravity is a conservative system
so
E = T + V
so
(d/dt) { (dR(t)/dt)2 - 2GM/R(t) } = 0
integrate wrt to time and let the constant of integration = B for the moment.
we then obtain the following equation:
(dR(t)/dt)2 - 2GM/R(t) = B
now the mass is just given by
M = 4 pi rho R(t)3 / 3
putting this in results in the following equation:
(dR(t)/dt)2 - 8G pi rho R(t)2 / 3 = B
which is the Friedmann equation.
In order to obtain the constant B you need the full machinery of
general relativity. It turns out that B = -kc 2
where k is the curvature index and c is the speed of light.
The essence of the equation is that it is an equation of energy conservation.
That a Newtonian treatment should give an answer so
close to the answer given by general relativity is
a little surprising. The two theories diverge
on small scales e.g. the perihelion advance of
mercury and the deflection of light by the sun are different
in the two theories.
Although the derivation of the Friedmann equation
gives an approximately
similar answer without general relativity
the Newtonian treatment would never have anything to say about
curvature.
This represents a fundamental difference
in the two worldviews.
I think the fact that the constants come out right
on the left hand side of the equation might have something
to do with the ubiquitousness of
Gauss's theorem but that is rather a speculation on my part.
For reference i suggest any introductory text on
cosmology. MV Berry "Principles
of Cosmology and Gravitation" is very light
and a nice technical introduction.
John Peacock's Cosmological Physics
gives some deep insight into these questions.
* an example of a system that is non conservative is
any system with friction. This is why we have no
perpetual motion devices on earth.
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