82
4 Bateman Waves
The electromagnetic waves described by (4.82)–(4.84) with (4.88)–(4.90) are
Bateman waves [3]. As a result we can say that every electromagnetic wave is a
Bateman wave [4]. Finally we observe that since G(X 1 , X 2 ) is the general solution
of (4.78) then the general solution of (4.69) is clearly any Y(ζ, ¯
ζ , u, v) given
implicitly by an equation of the form
W (Y, X 1 , X 2 ) = 0 ,
(4.91)
where W is an arbitrary complex analytic function of its arguments. This important
result in the study of geodesic and shear-free null congruences is originally due to
R. P. Kerr (unpublished) and is referred to in the literature as the Kerr theorem [5].
4.3
Some ‘Spherical’ Electromagnetic Waves
We now look for spherical electromagnetic waves propagating in the radial direction
in R 3 with respect to the origin r = 0 of R 3 . Thus the propagation direction in R 3
is
k =
x
r
,
y
r
,
z
r
with r =
x 2 + y 2 + z 2 .
(4.92)
Thus from (4.52) and (4.53) we have
Y =
k 1 − ik 2
1 − k 3 =
x − i y
r − z
=
2 ζ
2 r + u + v
with r
2
= ζ ¯
ζ +
1
4
(u + v)
2 .
(4.93)
Using these we can easily check that (4.69) are satisfied. Turning now to (4.81) we
first note that
X 1 = r − t and X 2 =
x − i y
r − z
(r − t) .
(4.94)
Then the Bateman waves (4.81) are given by the complex 2-form
F =
1
2
F ij dx
i
∧ dx
j
=
1
2
(F ij + i
∗ F ij ) dx
i
∧ dx
j
= ˆ
G
r − t,
x − i y
r − z
(dr − dt) ∧ d
x − i y
r − z
.
(4.95)
for some complex analytic function ˆ
G of its arguments. We will consider arguably
the simplest example of such waves by specialising to the case
ˆ
G = ˆ
G(r − t) = A(r − t) + i C(r − t) ,
(4.96)