88
4 Bateman Waves
and so the algebraic conditions (4.138) and (4.139) are preserved under this time
evolution provided ˆ
q, ˆ
w satisfy
ˆ
q · (∇ × ˆ
w) + ˆ
w · (∇ × ˆ
q) = 0 ,
(4.142)
and
ˆ
w · (∇ × ˆ
w) − ˆ
q · (∇ × ˆ
q) = 0 .
(4.143)
Also we see from (4.117) that E αβ E βα = 2 |q | 4 and thus we find from (4.118) that
∂
∂t
|q |
4
+ ∇ · (|q |
4 ˆ
k) = 0 .
(4.144)
To interpret the Eqs. (4.142)–(4.144) we first note that the Bel–Robinson tensor
(with vanishing Ricci tensor) is given by Penrose and Rindler [7]
T ij kl =
1
4
R i
p
j
q R kplq +
∗ R i
p
j
q ∗ R kplq
.
(4.145)
This symmetric tensor is, as a result of the Bianchi identities, divergence-free which,
in the linear approximation, reads
T
ij kl
,i = 0 .
(4.146)
For the case of (4.120) the Bel–Robinson tensor is given by
T
ij kl
= |q |
4 k
i k
j k
k k
l .
(4.147)
Since (4.144) is equivalent to
|q |
4 k
i
,i
= 0 ,
(4.148)
we have from (4.146) and (4.147) that
(k
i k
j
,i ) k
k k
l
+ k
j (k
i k
k
,i ) k
l
+ k
j k
k (k
i k
l
,i ) = 0 ⇒ k
j
,i k
i
= 0 ,
(4.149)
and so the integral curves of k i are null geodesics on account of (4.148). We note
that with k i given by (4.136) we can write (4.149) in the form
∂ ˆ
k α
∂t
= − ˆ
k
α
,β ˆ
k
β .
(4.150)