4.1 Electromagnetic Radiation
71
write P·P = |P| 2 . The gradient of a scalar ϕ will be denoted ∇ϕ = ((∇ϕ) α ) = (ϕ ,α )
with the comma denoting partial differentiation. We also have ∇ · P = P α ,α and
(∇ × P) α = αβγ P γ ,β . We shall not raise the indices on the permutation symbols.
In terms of the bivector field F ij on Minkowskian space-time, or the pair of vector
fields E, B on three dimensional Euclidean space, Maxwell’s vacuum field equations
read:
F
ij
,j = 0 ⇔ ∇ · E = 0 and
∂E
∂t
= ∇ × B ,
(4.8)
and
∗ F
ij
,j = 0 ⇔ ∇ · B = 0 and
∂B
∂t
= −∇ × E .
(4.9)
To illustrate the role in this context of a null geodesic congruence we consider
the Maxwell field E, B or F ij satisfying (4.8) and (4.9) subject to the algebraic
conditions
F ij F
ij
= 0 and
∗ F ij F
ij
= 0 ⇔ |E|
2
= |B|
2 and E · B = 0 .
(4.10)
Such Maxwell fields describe pure electromagnetic radiation (see Chap. 2). With
these conditions holding it is straightforward to verify that (4.6) and (4.7) can be
written in the forms
F ij = q i k j − q j k i and
∗ F ij = w i k j − w j k i ,
(4.11)
with
k
i
=
1,
E × B
|E| 2
, q
i
= (0, E) and w
i
= (0, B) .
(4.12)
We see that
k
i k i = 0 , q
i k i = 0 , w
i k i = 0 ,
(4.13)
and in addition
q
i w i = 0 , q
i q i = −|E|
2 , w
i w i = −|B|
2 .
(4.14)
with the first of (4.13) and the first of (4.14) following from (4.10). Hence k i is a
null vector field on Minkowskian space-time and q i , w i are space-like vector fields
71
write P·P = |P| 2 . The gradient of a scalar ϕ will be denoted ∇ϕ = ((∇ϕ) α ) = (ϕ ,α )
with the comma denoting partial differentiation. We also have ∇ · P = P α ,α and
(∇ × P) α = αβγ P γ ,β . We shall not raise the indices on the permutation symbols.
In terms of the bivector field F ij on Minkowskian space-time, or the pair of vector
fields E, B on three dimensional Euclidean space, Maxwell’s vacuum field equations
read:
F
ij
,j = 0 ⇔ ∇ · E = 0 and
∂E
∂t
= ∇ × B ,
(4.8)
and
∗ F
ij
,j = 0 ⇔ ∇ · B = 0 and
∂B
∂t
= −∇ × E .
(4.9)
To illustrate the role in this context of a null geodesic congruence we consider
the Maxwell field E, B or F ij satisfying (4.8) and (4.9) subject to the algebraic
conditions
F ij F
ij
= 0 and
∗ F ij F
ij
= 0 ⇔ |E|
2
= |B|
2 and E · B = 0 .
(4.10)
Such Maxwell fields describe pure electromagnetic radiation (see Chap. 2). With
these conditions holding it is straightforward to verify that (4.6) and (4.7) can be
written in the forms
F ij = q i k j − q j k i and
∗ F ij = w i k j − w j k i ,
(4.11)
with
k
i
=
1,
E × B
|E| 2
, q
i
= (0, E) and w
i
= (0, B) .
(4.12)
We see that
k
i k i = 0 , q
i k i = 0 , w
i k i = 0 ,
(4.13)
and in addition
q
i w i = 0 , q
i q i = −|E|
2 , w
i w i = −|B|
2 .
(4.14)
with the first of (4.13) and the first of (4.14) following from (4.10). Hence k i is a
null vector field on Minkowskian space-time and q i , w i are space-like vector fields