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

5 Cosmological expansion
In this lecture we discuss Hubble’s observation of galactic redshifts in the
context of general relativity. The cosmological principle provides simplifying
assumptions, allowing to an “equation of motion” for the scale factor R
known as the Friedmann equation. This equation will be a starting point for
many of our discussion of physics of the early universe.
5.1 Spacetime in Relativity
In special relativity, a key concept is the invariant spacetime interval ds 2 :
ds 2 = (c dt) 2 − ( dx 2 + dy 2 + dz 2) ,
which separates two points in spacetime. The interval ds is the same for any
observer in an inertial reference frame. It can also be written equivalently as
ds 2 = ∑ g µν dx µ dx ν ,
where g µν is the metric tensor and dx µ = (ct, x, y, z) are 4-vectors. For the
flat spacetime of special relativity, g µν is the Minkowski metric, represented
by a 3 × 3 matrix with diagonal elements 1, −1, −1, −1, and all other
elements zero.
In spherical coordinates, then the interval in flat spacetime becomes:
ds 2 = (c dt) 2 − dr 2 − r 2 ( dθ 2 + sin 2 θdφ 2) .
This can be compared with the Schwarzschild interval which is a solution of
the equations of general relativity for the case of spherical symmetry at a
distance r from a mass M:
ds 2 =
[
1 − 2GM ] [
(c dt) 2 − 1 − 2GM ] −1
− r ( 2 dθ 2 + sin 2 θdφ 2) . (20)
rc 2 rc 2
The proper time dτ measured by a clock at rest at a distance r from the
mass is given by
[
(c dτ) 2 = 1 − 2GM ]
(c dt) 2 ,
rc 2
where dt is the time measured in a distant inertial frame. We see that as
one approaches the distance r = 2GM/c 2 ≡ R s , the proper time interval
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