4.1 Electromagnetic Radiation
75
and writing e α e β + b α b β = δ αβ − k α k β in the final term in (4.36) we finally arrive
at the expression
k α,β + k β,α = −k
μ
,μ δ αβ + ξ α k β + ξ β k α ,
(4.38)
with
ξ α =
1
2
k
μ
,μ k α + (k μ,λ e
μ k
λ ) e α + (k μ,λ b
μ k
λ ) b α .
(4.39)
We note that
ξ · k =
1
2
k
μ
,μ ,
(4.40)
and
|ξ |
2
=
1
4
(k
μ
,μ )
2
+ (k μ,λ e
μ k
λ )
2
+ (k μ,λ b
μ k
λ )
2
=
1
4
(k
μ
,μ )
2
+ k μ,λ k ρ,σ k
λ k
σ (δ μρ − k
μ k
ρ )
=
1
4
(k
μ
,μ )
2
+ |(k · ∇)k|
2
=
1
4
(k
μ
,μ )
2
+
∂k
∂t
2
,
(4.41)
with the final equality coming from (4.20). To obtain from (4.38) an equation
involving only k (and not e and b) we use (4.40) and (4.41) to arrive at
(k α,β +k β,α )(k
α,β
+k
β,α ) = 2 (k α,β +k β,α ) k
α,β
= 2 (k
λ
,λ )
2
+2
∂k
∂t
2
. (4.42)
To see what equation on Minkowskian space-time this is equivalent to, in terms of
k i = (1, k α ), we put k (i,j ) = (k i,j + k j,i )/2 and calculate
k (i,j ) k
i,j
−
1
2
(k
i
,i )
2
=
1
2
(k α,β + k β,α ) k
α,β
− (k
α
,α )
2
−
∂k
∂t
2
.
(4.43)
Hence (4.42) and (4.43) show that in Minkowskian space-time the propagation
direction k i of electromagnetic radiation is not only geodesic, as indicated above
by (4.30), but also satisfies Robinson’s [2] equation
k (i,j ) k
i,j
−
1
2
(k
i
,i )
2
= 0 .
(4.44)