86
4 Bateman Waves
The null vector field is the degenerate principal null direction of the Riemann tensor.
We now choose the time-like u i to be orthogonal to each of the space-like vectors q i
and w i and we normalise k i with k i u i = +1. Then with respect to this unit time-like
vector field the electric and magnetic parts of the Riemann tensor are given by
E ik + i H ik = (R ij kl + i
∗ R ij kl )u
j u
l
= (q i + i w i )(q k + i w k ) .
(4.123)
Hence
E ik = q i q k − w i w k and H ik = q i w k + q k w i ,
(4.124)
and since both of these quantities must be trace-free we have
q i q
i
= w i w
i and q i w
i
= 0 .
(4.125)
The vector fields u i , k i , q i , w i are vector fields on Minkowskian space-time. With
u i = (1, 0, 0, 0) we have
q
i
= (0, q ) , w
i
= (0, w) , k
i
= (1, k) ,
(4.126)
with
q = (q
α ) = (−q α ) , w = (w
α ) = (−w α ) , k = (k
α ) = (−k α ) ,
(4.127)
and
|q |
2
= q
α q
α
= |w |
2 , |k|
2
= 1 ,
(4.128)
and the 3-vectors q, w, k are mutually orthogonal. We shall write
q = |q | ˆ
q , w = |q | ˆ
w with ˆ
q · ˆ
q = ˆ
w · ˆ
w = 1 and ˆ
q · ˆ
w = 0 .
(4.129)
Hence, from (4.124),
E αβ = |q |
2 ( ˆ
q α ˆ
q β − ˆ
w α ˆ
w β ) = E
αβ , H αβ = |q |
2 ( ˆ
q α ˆ
w β + ˆ
q β ˆ
w α ) = H
αβ .
(4.130)
Using (4.120) and (4.124) we have
R ij kl u
l
= E ki k j − E kj k i .
(4.131)
However in general [6]
R ij kl u
l
= E ki u j − E kj u i − η pij q H
p
k u
q .
(4.132)