Master Thesis

One concludes that the space of constant curvature is conformally flat. So according to (3.55) one can always change the
coordinates to those, where the space-like part of the metric is of the form e 2σ(x) g i j, where g i j is the euclidean metric.
For easier calculation this can also be rewritten as:
dσ 2 1
(3) =
Ψ2 (x)
ds 2 =−d t 2 + a2 (t)
Ψ 2 (x)
where we can again change into the orthonormal frame:
�
d x i ⊗ d x i
i
θ 0 = d t , θ i = a(t) i
d x
Ψ(x)
(4.11)
�
d x i ⊗ d x i , (4.12)
i
(4.13)
ds 2 =η µνθ µ ⊗θ ν . (4.14)
In order to get an expression for the functionΨ(x) the curvature will be calculated by using Cartan’s structure equations.
First, the Levi-Civita connection is obtained by using the first Cartan’s structure equation with vanishing torsionΘ i = 0
and an antisymmetric connection of the orthonormal frame d g i j= 0=ω i j+ω ji (see Eq. 3.48):
dΘ 0 = 0 (4.15)
dΘ i = a ,0
Ψ d x0 ∧ d x i + a(Ψ −1 ) ,kd x k ∧ d x i = (4.16)
From this one can read off the non-vanishing connection coefficients:
= a ,0
a θ 0 ∧θ i − Ψ ,k
a θ k ∧θ i =−ω i
0∧θ 0 −ω i
k∧θ k = (4.17)
= ω i
j0θ 0 ∧θ j +ω i
jkθ k ∧θ j = (4.18)
= ω il
j η l0θ 0 ∧θ j +ω ik
j η klθ l ∧θ j = (4.19)
= −ω i0
j θ 0 ∧θ j +ω ik
j θ k ∧θ j , (4.20)
ω i0
j =−ω0i j =−δi j
whereδ µ
ν =ηµλ η λν= diag(1, 1, 1, 1) . The connection one-forms are then:
˙a
a
(4.21)
ω ik
j =−ωki j =−δi
Ψ ,k
j a +δk
Ψ ,i
j , (4.22)
a
ω i0 =−ω 0i =− ˙a i
θ
a
(4.23)
ω ik =−ω ki = Ψ ,i
a θ k − Ψ ,k
a θ i , (4.24)
The space-like curvature form can now be calculated by using Cartan’s second structure equation (3.53):
Ω i j
= dωi j+ω m
i ∧ω mj= (4.25)
= −ΨΨ ,jmθ m ∧θ i +ΨΨ ,imθ m ∧θ j−Ψ ,mΨ ,m θi∧θ j
(4.26)
This space-like curvature should have the form of the constant curvature form (4.7). That is to say
K=Ψ(Ψ ,mj+Ψ ,mi)−Ψ ,mΨ ,m
(4.27)
and soΨ ,mj= 0,∀m�= j must hold. The termΨ ,mj+Ψ ,mi must also be independent of i and j, because the rest does so.
Now one can argue, thatΨmust be of the form:
The space-like connection simplifies to:
Ψ= Kρ2
4 + 1 , ρ2 :=
�
i
x 2
i . (4.28)
ω ik =−ω ki = K
2a x i θ k − K
2a x k θ i . (4.29)
18 4.2 Using symmetries