210
A Congruences of World Lines
2. How is η abcd related to η abcd ?:
η
abcd
= g
ar g
bs g
cp g
dq η rspq ,
(A.6)
is totally skew-symmetric. Hence it has only one independent component
η
0123
=
√
−gg
0r g
1s g
2p g
3q pqrs =
√
−g g
−1
= −
1
√
−g
,
(A.7)
and thus
η
abcd
= −
1
√
−g
abcd .
(A.8)
3. η abcd is covariantly constant:
Since η abcd has the important property, written as a determinant, that
η
apqr η clmn = −
δ a
c δ a
l δ a
m δ a
n
δ
p
c δ
p
l δ
p
m δ
p
n
δ
q
c δ
q
l δ
q
m δ
q
n
δ r
c δ r
l δ r
m δ r
n
,
(A.9)
and since δ a
b;c = 0 we have
η abcd;f η
pqrs
+ η abcd η
pqrs ;f = 0 .
(A.10)
It also follows from (A.9) that η abcd η abcd = −24 and so multiplying (A.10) by
η pqrs yields
η abcd;f = 0 .
(A.11)