4.5 Bateman Gravitational Waves
89
Just as in the electromagnetic case the Eqs. (4.142) and (4.143) can be rewritten
using (4.137) in the forms
ˆ
k γ ,β ˆ
w
γ
ˆ
w
β
= ˆ
k γ ,β ˆ
q
γ
ˆ
q
β ,
(4.151)
and
ˆ
k γ ,β ˆ
q
γ
ˆ
w
β
+ ˆ
k γ ,β ˆ
w
γ
ˆ
q
β
= 0 ,
(4.152)
respectively. From these it follows that
ˆ
k α,β + ˆ
k β,α = − ˆ
k
γ
,γ δ αβ + ξ α ˆ
k β + ξ β ˆ
k α ,
(4.153)
with
ξ α = ( ˆ
k γ ,σ ˆ
q
γ ˆ
k
σ ) ˆ
q α + ( ˆ
k γ ,σ ˆ
w
γ ˆ
k
σ ) ˆ
w α +
1
2
ˆ
k
γ
,γ ˆ
k α .
(4.154)
It now follows from (4.150) and (4.154) that, as in the electromagnetic case,
ξ α ˆ
k α =
1
2
ˆ
k
α
,α and ξ
α ξ
α
=
1
4
( ˆ
k
α
,α )
2
+
∂ ˆ
k
∂t
2
.
(4.155)
These equations then lead again to Robinson’s shear-free condition on k i :
k (i,j ) k
i,j
−
1
2
(k
i
,i )
2
= 0 .
(4.156)
4.5
Bateman Gravitational Waves
From the Riemann tensor components given by (4.120) and (4.121) we form the
complex 2-form
+ ij =
1
2
(R ij kl + i
∗ R ij kl ) dx
k
∧ dx
l
= (E ik k j − E jk k i )k l dx
k
∧ dx
l
+ i (H ik k j − H jk k i )k l dx
k
∧ dx
l
= −
+ ji .
(4.157)
With k i in (4.136) we can write these more explicitly as
+ 0α = −(E
αβ
+ i H
αβ ) dx
β
∧ (dt − ˆ
k
γ dx
γ ) ,
(4.158)
+ αβ =
ˆ
k
α (E
βσ
+ i H
βσ ) − ˆ
k
β (E
ασ
+ i H
ασ )
dx
σ
∧ (dt − ˆ
k
γ dx
γ ) .
(4.159)