D de Sitter Cosmology
239
2. The components
i
ij of the Riemannian connection calculated with the metric
tensor g ij can be written in the form
i
ij =
i
ji =
1
2
g
ik g ik,j =
1
2 g
g ,j ,
(D.10)
with g = det(g ij ) and consequently if g < 0 then we can write
i
ij =
(
√
−g) ,j
√
−g
.
(D.11)
To see this we first note that the determinant of the metric tensor g ij can be
written
g = det(g ij ) = g p0 g q1 g r2 g s3 pqrs .
(D.12)
Then given the matrix (g ij ) = (g ji ) its cofactor matrix is (G ij ) with
G 0p = g q1 g r2 g s3 pqrs = G p0 ,
(D.13)
G 1q = g p0 g r2 g s3 pqrs = G q1 ,
(D.14)
G 2r = g p0 g q1 g s3 pqrs = G r2 ,
(D.15)
G 3s = g p0 g q1 g r2 pqrs = G s3 .
(D.16)
But
g
ik g ik,j = g
0p g 0p,j + g
1q g 1q,j + g
2r g 2r,j + g
3s g 3s,j .
(D.17)
Substituting here for the inverse (g ij ) = (g ij ) −1 of the metric tensor using the
cofactor matrix with components (D.13)–(D.16) results in
g g
ik g ik,j = (g p0,j g q1 g r2 g s3 + g q1,j g p0 g r2 g s3
+g r2,j g p0 g q1 g s3 + g s3,j g p0 g q1 g r2 ) ) pqrs
= (g p0 g q1 g r2 g s3 ) ,j pqrs
= g ,j ,
(D.18)
and thus (D.10) is established.
Reference
1. A. Trautman, Recent Developments in General Relativity (Pergamon Press Inc., New York,
1962), p. 459