4.2 Bateman Electromagnetic Waves
81
from which we conclude that the geodesic and shear-free conditions (4.69) are the
integrability conditions for the differential equations (4.78). The general solution of
(4.78) is G = G(u + ¯
ζ Y, ζ − v Y), where G is an arbitrary analytic function of its
arguments. If we write X 1 = u + ¯
ζ Y, X 2 = ζ − v Y then
dX 1 ∧ dX 2 = (1 + ¯
ζ Y u − v Y ζ ) (du + Y d ¯
ζ ) ∧ (dζ − Y dv) ,
(4.80)
and now we can write the Maxwell 2-form (4.75) as
F =
1
2
F ij dx
i
∧ dx
j
=
1
2
(F ij + i
∗ F ij ) dx
i
∧ dx
j
= G(X 1 , X 2 ) dX 1 ∧ dX 2 .
(4.81)
Using (4.6) and (4.7), and the standard notation for Jacobian determinants, we find
from (4.81) that the electric and magnetic 3-vectors E, B are given by
E
1
+ i B
1
= F 01 = G(X 1 , X 2 )
∂(X 1 , X 2 )
∂(t, x)
,
(4.82)
E
2
+ i B
2
= F 02 = G(X 1 , X 2 )
∂(X 1 , X 2 )
∂(t, y)
,
(4.83)
E
3
+ i B
3
= F 03 = G(X 1 , X 2 )
∂(X 1 , X 2 )
∂(t, z)
.
(4.84)
We also have
i (E
3
+ i B
3 ) = F 12 = G(X 1 , X 2 )
∂(X 1 , X 2 )
∂(x, y)
,
(4.85)
−i (E
2
+ i B
2 ) = F 13 = G(X 1 , X 2 )
∂(X 1 , X 2 )
∂(x, z)
,
(4.86)
i (E
1
+ i B
1 ) = F 23 = G(X 1 , X 2 )
∂(X 1 , X 2 )
∂(y, z)
.
(4.87)
However (4.85)–(4.87) are consistent with (4.82)–(4.84) because, by direct calculation, we have
∂(X 1 , X 2 )
∂(x, y)
= 2 i (−Y + v Y Y ζ − ¯
ζ Y ¯
ζ ) = i
∂(X 1 , X 2 )
∂(t, z)
,
(4.88)
∂(X 1 , X 2 )
∂(t, y)
= i
Y
2
+ 1 − v (Y ζ − Y Y v ) + ζ (Y Y ¯
ζ + Y u )
= i
∂(X 1 , X 2 )
∂(x, z)
,
(4.89)
∂(X 1 , X 2 )
∂(y, z)
= i
Y
2
− 1 + v (Y ζ + Y Y v ) + ¯
ζ (Y Y ¯
ζ − Y u )
= i
∂(X 1 , X 2 )
∂(t, x)
.
(4.90)