A
Congruences of World Lines
A.1
Properties of η abcd
1. η abcd =
√ −g g abcd are the components of a tensor:
First define
a b c d =
∂x p
∂x a
∂x q
∂x b
∂x r
∂x c
∂x s
∂x d η pqrs .
(A.1)
This quantity is totally skew-symmetric. Its one independent component is
0 1 2 3 =
√
−g
∂x p
∂x 0
∂x q
∂x 1
∂x r
∂x 2
∂x s
∂x 3 pqrs =
√
−g det
∂x p
∂x a
.
(A.2)
But
g a b =
∂x p
∂x a
∂x q
∂x b g pq ⇒ g
=
det
∂x p
∂x a
2
g ⇒ det
∂x p
∂x a =
−g
√
−g
.
(A.3)
Hence
0 1 2 3 =
−g ,
(A.4)
and thus
a b c d =
−g a b c d = η a b c d .
(A.5)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. A. Hogan, D. Puetzfeld, Frontiers in General Relativity, Lecture Notes
in Physics 984, https://doi.org/10.1007/978-3-030-69370-1
209
Congruences of World Lines
A.1
Properties of η abcd
1. η abcd =
√ −g g abcd are the components of a tensor:
First define
a b c d =
∂x p
∂x a
∂x q
∂x b
∂x r
∂x c
∂x s
∂x d η pqrs .
(A.1)
This quantity is totally skew-symmetric. Its one independent component is
0 1 2 3 =
√
−g
∂x p
∂x 0
∂x q
∂x 1
∂x r
∂x 2
∂x s
∂x 3 pqrs =
√
−g det
∂x p
∂x a
.
(A.2)
But
g a b =
∂x p
∂x a
∂x q
∂x b g pq ⇒ g
=
det
∂x p
∂x a
2
g ⇒ det
∂x p
∂x a =
−g
√
−g
.
(A.3)
Hence
0 1 2 3 =
−g ,
(A.4)
and thus
a b c d =
−g a b c d = η a b c d .
(A.5)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. A. Hogan, D. Puetzfeld, Frontiers in General Relativity, Lecture Notes
in Physics 984, https://doi.org/10.1007/978-3-030-69370-1
209