28
2 Bivector Formalism
and, as a consequence of Maxwell’s equations (2.1), satisfies
E
ab ;b = 0 .
(2.60)
For the field F ab in (2.58) the electromagnetic energy-momentum tensor takes the
simple radiative form
E ab = (ξ c ξ
c ) k a k b ,
(2.61)
assuming of course that ξ c ξ c = 0. In this case (2.60) specialises to
k
a ;b k
b
= λ k
a ,
(2.62)
where λ is a real scalar function of the coordinates x a given by λ(x a ) =
−(ξ c ξ c ) −1 {(ξ d ξ d ) ,b k b + ξ d ξ d k b ;b }. Hence the integral curves of the vector field
k a are null geodesics as a consequence of Maxwell’s equations [1].
In 1956 Ivor Robinson made the important discovery that the integral curves of
k a , in addition to being geodesic satisfying (2.62), are shear-free (published later in
[2]) and in this natural way introduced into general relativity the concept of shear of
a congruence of null geodesics. With the present formalism, and with the benefit
of hindsight, we can readily obtain this additional information from Maxwell’s
equations
(f 1 L
ab ) ;b = 0 ⇔ f 1,b L
ab
+ f 1 L
ab ;b = 0 .
(2.63)
Using the first of (2.33) when multiplying this by L ac we find, since f 1 = 0, that
L ac L
ab ;b = 0 .
(2.64)
Writing this out explicitly using L ab given in (2.26), and the scalar products among
the tetrad vectors (see (2.13)), we have
(k a;b m
a k
b )m c − (k a;b m
a m
b )k c = 0 ,
(2.65)
from which it follows that
k a;b m
a k
b
= 0 and k a;b m
a m
b
= 0 .
(2.66)
Expressing k a;b k b in terms of the tetrad, and using the fact that this real covariant
vector is orthogonal to k a (identically since k a is null) and to m a (on account of the
first of (2.66)), we see that the first of (2.66) is equivalent to the geodesic equations
(2.62). To exhibit the significance of the second equation in (2.66) as a condition on
the congruence of null geodesic integral curves of k a , independently of the choice
2 Bivector Formalism
and, as a consequence of Maxwell’s equations (2.1), satisfies
E
ab ;b = 0 .
(2.60)
For the field F ab in (2.58) the electromagnetic energy-momentum tensor takes the
simple radiative form
E ab = (ξ c ξ
c ) k a k b ,
(2.61)
assuming of course that ξ c ξ c = 0. In this case (2.60) specialises to
k
a ;b k
b
= λ k
a ,
(2.62)
where λ is a real scalar function of the coordinates x a given by λ(x a ) =
−(ξ c ξ c ) −1 {(ξ d ξ d ) ,b k b + ξ d ξ d k b ;b }. Hence the integral curves of the vector field
k a are null geodesics as a consequence of Maxwell’s equations [1].
In 1956 Ivor Robinson made the important discovery that the integral curves of
k a , in addition to being geodesic satisfying (2.62), are shear-free (published later in
[2]) and in this natural way introduced into general relativity the concept of shear of
a congruence of null geodesics. With the present formalism, and with the benefit
of hindsight, we can readily obtain this additional information from Maxwell’s
equations
(f 1 L
ab ) ;b = 0 ⇔ f 1,b L
ab
+ f 1 L
ab ;b = 0 .
(2.63)
Using the first of (2.33) when multiplying this by L ac we find, since f 1 = 0, that
L ac L
ab ;b = 0 .
(2.64)
Writing this out explicitly using L ab given in (2.26), and the scalar products among
the tetrad vectors (see (2.13)), we have
(k a;b m
a k
b )m c − (k a;b m
a m
b )k c = 0 ,
(2.65)
from which it follows that
k a;b m
a k
b
= 0 and k a;b m
a m
b
= 0 .
(2.66)
Expressing k a;b k b in terms of the tetrad, and using the fact that this real covariant
vector is orthogonal to k a (identically since k a is null) and to m a (on account of the
first of (2.66)), we see that the first of (2.66) is equivalent to the geodesic equations
(2.62). To exhibit the significance of the second equation in (2.66) as a condition on
the congruence of null geodesic integral curves of k a , independently of the choice