238
D de Sitter Cosmology
+ g
(0)
r2 g
(0)
s3 ( g
(0)
p0 k q k 1 + g
(0)
q1 k p k 0 )
pqrs
= det( g
(0)
ij ) + 2 H
k 0 k p g
(0)
r2 g
(0)
s3 g
(0)
q1 + k 1 k q g
(0)
r2 g
(0)
s3 g
(0)
p0
+k 2 k r g
(0)
p0 g
(0)
q1 g
(0)
s3 + k 3 k s g
(0)
p0 g
(0)
q1 g
(0)
r2
pqrs .
(D.4)
However, for example,
k p g
(0)
r2 g
(0)
s3 g
(0)
q1 pqrs = g
(0)
pl k
l g
(0)
r2 g
(0)
s3 g
(0)
q1 pqrs
= g
(0)
p0 k
0 g
(0)
r2 g
(0)
s3 g
(0)
q1 pqrs
= k
0 det( g
(0)
ij ) ,
(D.5)
and thus we can rewrite (D.4) neatly as
det(g ij ) = det( g
(0)
ij ) + 2 H det( g
(0)
ij ) k p k
p .
(D.6)
Hence we have from (D.2) that
det(g ij ) = det( g
(0)
ij ) .
(D.7)
We note that the inverse of (g ij ), namely (g ij ) such that g ij g jk = δ
i
k , is given
by
g
ij
= g
(0)
ij
− 2 H k
i k
j ,
(D.8)
with ( g
(0)
ij ) the inverse of ( g
(0)
ij ). Also writing det(g ij ) = g and det( g
(0)
ij ) = g 0
we have, in particular, the useful expression
k
i |i = k
i ;i ⇔
1
√
−g
∂
∂x i (
√
−g k
i ) =
1
√
−g 0
∂
∂x i (
√ −g 0 k
i ) ,
(D.9)
where a stroke denotes covariant differentiation with respect to the Riemannian
connection calculated with the metric tensor g ij and a semicolon denotes
covariant differentiation with respect to the Riemannian connection calculated
with the metric tensor g
(0)
ij . This last equation relies on a well known property of
the components of the Riemannian connection: