Ratbay Myrzakulov

3.2 FRW case
From here we work with the spatially flat FRW metric
ds 2 = −dt 2 + a 2 (t)(dx 2 + dy 2 + dz 2 ), (3.18)
where a(t) is the scale factor. In this case, the non-vanishing components of the Levi-Civita
connection are
Γ 0 00 = Γ 0 0i = Γ 0 i0 = Γ i 00 = Γ i jk = 0,
Γ 0 ij = a 2 Hδ ij , (3.19)
Γ i jo = Γ i 0j = Hδ i j,
where H = (lna) t and i,j,k,... = 1,2,3. Now let us calculate the components of torsion tensor.
Its non-vanishing components are given by:
T 110 = T 220 = T 330 = a 2 h,
T 123 = T 231 = T 312 = 2a 3 f, (3.20)
where h and f are some real functions (see e.g. Refs. [12]). Note that the indices of the torsion
tensor are raised and lowered with respect to the metric, that is
Now we can find the contortion components. We get
T ijk = g kl T ij l . (3.21)
K 1 10 = K 2 20 = K 3 30 = 0,
K 1 01 = K 2 02 = K 3 03 = h,
K 0 11 = K 0 22 = K 0 22 = a 2 h, (3.22)
K 1 23
K 1 32
= K 2 31 = K 3 12 = −af,
= K 2 13 = K 3 21 = af.
The non-vanishing components of the curvature R ρ σµν are given by
R 0 101 = R 0 202 = R 0 303 = a 2 (Ḣ + H2 + Hh + ḣ),
R 0 123 = −R 0 213 = R 0 312 = 2a 3 f(H + h),
R 1 203 = −R 1 302 = R 2 301 = −a(Hf + f), ˙
R 1 212 = R 1 313 = R 2 323 = a 2 [(H + h) 2 − f 2 ]. (3.23)
Similarly, we write the non-vanishing components of the Ricci curvature tensor R µν as
R 00 = −3Ḣ − 3ḣ − 3H2 − 3Hh,
R 11 = R 22 = R 33 = a 2 (Ḣ + ḣ + 3H2 + 5Hh + 2h 2 − f 2 ). (3.24)
At the same time, the non-vanishing components of the tensor S µν
ρ
are given by
S1 10 = 1 (
K
10
1 + δ 1
2
1Tθ
θ0
S 10
1 = S 20
2 = S 30
S 23
1 = 1 2
− δ1T 0 θ
θν ) 1 = (h + 2h) = h, (3.25)
2
3 = 2h, (3.26)
(
K
23
1 + δ 2 1 + δ 3 1
S 23
1 = S 31
2 = S 21
3 = − f 2a
)
= −
f
2a , (3.27)
(3.28)
and
T = T 1 10S 10
1 + T 2 20S 20
2 + T 3 30S 30
3 + T 23
1 S 1 23 + T 2 31S 31
2 + T 3 12S 12
3 . (3.29)
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