Lecture Notes for Astronomy 321, W 2004 1 Stellar Energy ...

5.3 The Friedmann equation and a useful analog
The field equations of general relativity can be written in tensor form
G µν = 8πGT µν + Λg µν , (26)
where T µν is the stress energy tensor, which contains the description of the
mass and energy densities, G µν is the Einstein tensor, which gives the resulting
spacetime curvature, and Λ is the (infamous) cosmological constant.
We will not use this equation directly, in which many ordinary equations are
embedded.
Envoking the cosmological principle, and assuming matter and radiation
fields can be described as ideal frictionless fluids (more on this part later),
Friedmann found the following solution to Eq. 26 in terms of the scale
parameter R:
(Ṙ ) 2 ( ) 8πG
H 2 = = ρ tot − Kc2
(27)
R 3 R 2
Here, ρ tot is the total energy density and K is the curvature parameter introduced
above.
We can gain significant intuition about the Friedmann equation by showing
its equivalence to a more familiar model. Let us assume that ρ is dominated
by ordinary non-relativistic matter and that this density is uniform.
We put a test mass m at the edge of a sphere of radius R(t). Only the
mass within the sphere, M = 4πR 3 ρ/3, contributes to the gravitational force
acting on m. (This Gauss’s Law for gravity also holds in general relativity.)
Hence, the potential energy of m is −GMm/R, and the total energy
E = E K + U is
1
2 mṘ2 − GMm
R = E .
Substituting M = 4πR 3 ρ/3 and multiplying through by 2/(mR 2 ), then the
left-hand side is identical to Eq. 27. And the right-hand side, the total
energy term, becomes 2E/(mR 2 ). Hence, we can make the identification
E = −Kc 2 /(2m), or
E K + U = E ∝ −K .
5.4 Synopsis of matter-dominated solution
Under our assumption of a matter dominated universe, the analogy presented
above can be related to the familiar classical mechanics problem of a mass
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