4.1 Electromagnetic Radiation
73
The consistency of the second of Maxwell’s equations in vector form in (4.8) and
(4.9) with the vector form of the algebraic conditions (4.10) requires
∂
∂t
|E|
2
= −|E|
2 (∇ · k) − k · ∇|E|
2 ,
(4.23)
b · (∇ × b) − e · (∇ × e) = 0 ,
(4.24)
e · (∇ × b) + b · (∇ × e) = 0 .
(4.25)
For the derivation of (4.20) we will require (4.22) and (4.23). Using the second of
Maxwell’s equations in vector form in (4.8) and (4.9) and k given by (4.12) we have
∂k
∂t
= −
1
|E| 2
∂
∂t
(|E|
2 ) k +
1
|E| 2 ((∇ × B) × B + (∇ × E) × E)
= −
1
|E| 2
∂
∂t
(|E|
2 ) k +
1
|E| 2
−∇|E|
2
+ (B · ∇)B + (E · ∇)E
.
(4.26)
But using (4.15)
(B · ∇)B + (E · ∇)E =
1
2
∇|E|
2
− (k · ∇|E|
2 ) k
+ |E|
2 ((b · ∇) b + (e · ∇) e) ,
(4.27)
and now this and (4.22) with (4.23) in (4.26) yields
∂k
∂t
= (∇ · k) k + (∇ · e) e + (e · ∇) e + (∇ · b) b + (b · ∇) b .
(4.28)
However using (4.19) we have
(e · ∇) e + (b · ∇) b = −(∇ · e) e − (∇ · b) b − (k · ∇) k − (∇ · k) k ,
(4.29)
and when this is substituted into (4.28) the result is (4.20). In the context of
Minkowskian space-time (4.20) means that
k
i
,j k
j
=
∂k i
∂t
+ k
α k
i
,α =
0,
∂k
∂t
+ (k · ∇) k
= 0 .
(4.30)
Hence we see that the integral curves of the vector field k i constitute a null geodesic
congruence on Minkowskian space-time confirming the result of Mariot [1]. We
note that k is a unit 3-vector in the direction of the Poynting vector (energy flux
density) and k i is the light-like propagation direction in Minkowskian space-time of
pure electromagnetic radiation.