76
4 Bateman Waves
Thus we have confirmed that the integral curves of the vector field k i constitute
a null geodesic congruence on Minkowskian space-time which is “shear-free". In
reviewing the derivation of Robinson’s equation above it is clear that, in addition
to the geodesic condition (4.30), the key further ingredients of the derivation are
Eqs. (4.24) and (4.25) (equivalently (4.35)) which ensure that the time evolution of
the electromagnetic field via Maxwell’s equations preserves the algebraic conditions
(4.10).
We have not made use of the electromagnetic energy tensor
E
ij
= F
i
k F
jk
−
1
4
η
ij F lk F
lk .
(4.45)
This has the algebraic symmetries E ij = E ji and η ij E ij = 0 and also, as a
consequence of Maxwell’s vacuum field equations (4.8) and (4.9), satisfies
E
ij
,j = 0 .
(4.46)
When F ij in (4.11) is substituted into (4.45) the result is
E
ij
= −|E|
2 k
i k
j .
(4.47)
Using this in (4.46) gives
k
i
,j k
j
= λ k
i with λ = −k
i
,i −
1
|E| 2 (|E|
2 ) ,j k
j .
(4.48)
With k i = (1, k) we have
λ = −
1
|E| 2
∂
∂t
|E|
2
− ∇ · k −
1
|E| 2 k · ∇|E|
2
= 0 ,
(4.49)
with the final equality following from (4.23). By this means we have the first
of (4.48) with λ = 0 and so we have recovered (4.30). When the derivation of
(4.46) from Maxwell’s vacuum field equations is examined carefully it reveals that
not all of the content of Maxwell’s equations has been utilised. Among the extra
information in Maxwell’s equations is the Eq. (4.44).
The discussion so far has been restricted to Minkowskian space-time in rectangular Cartesian coordinates and time because this is required in the next section.
However it can be presented in covariant form on a general space-time and for this
the reader may consult the Appendix B.
4 Bateman Waves
Thus we have confirmed that the integral curves of the vector field k i constitute
a null geodesic congruence on Minkowskian space-time which is “shear-free". In
reviewing the derivation of Robinson’s equation above it is clear that, in addition
to the geodesic condition (4.30), the key further ingredients of the derivation are
Eqs. (4.24) and (4.25) (equivalently (4.35)) which ensure that the time evolution of
the electromagnetic field via Maxwell’s equations preserves the algebraic conditions
(4.10).
We have not made use of the electromagnetic energy tensor
E
ij
= F
i
k F
jk
−
1
4
η
ij F lk F
lk .
(4.45)
This has the algebraic symmetries E ij = E ji and η ij E ij = 0 and also, as a
consequence of Maxwell’s vacuum field equations (4.8) and (4.9), satisfies
E
ij
,j = 0 .
(4.46)
When F ij in (4.11) is substituted into (4.45) the result is
E
ij
= −|E|
2 k
i k
j .
(4.47)
Using this in (4.46) gives
k
i
,j k
j
= λ k
i with λ = −k
i
,i −
1
|E| 2 (|E|
2 ) ,j k
j .
(4.48)
With k i = (1, k) we have
λ = −
1
|E| 2
∂
∂t
|E|
2
− ∇ · k −
1
|E| 2 k · ∇|E|
2
= 0 ,
(4.49)
with the final equality following from (4.23). By this means we have the first
of (4.48) with λ = 0 and so we have recovered (4.30). When the derivation of
(4.46) from Maxwell’s vacuum field equations is examined carefully it reveals that
not all of the content of Maxwell’s equations has been utilised. Among the extra
information in Maxwell’s equations is the Eq. (4.44).
The discussion so far has been restricted to Minkowskian space-time in rectangular Cartesian coordinates and time because this is required in the next section.
However it can be presented in covariant form on a general space-time and for this
the reader may consult the Appendix B.