2.1 Bivectors and Electromagnetic Fields
21
with
g
ab ϕ ,a ϕ ,b = 0 ,
(2.9)
and with the comma, as always, denoting partial differentiation with respect to the
coordinates x a . Hence the normal to each hypersurface ϕ ,a is orthogonal to itself
and is therefore tangent to the null hypersurface and is a null vector field whose
direction is the propagation direction in space-time of the histories of the waves.
This null vector field is determined by the Maxwell bivector F ab at each point of
space-time when the two invariants (2.7) vanish and further properties of this null
vector field will follow from Maxwell’s equations (2.5). More generally, without
assuming the vanishing of the two invariants, a Maxwell bivector determines two
null vectors at each point of space-time. To demonstrate this we will use a basis of
four null vector fields, a null tetrad, at each point of space-time.
Let λ a
(0) , λ a
(1) , λ a
(2) , λ a
(3) be an orthonormal tetrad at a point of space-time, with
λ a
(0) a unit time-like vector, and thus g ab λ a
(0) λ b
(0) = 1, and λ a
(1) , λ a
(2) , λ a
(3) mutually
orthogonal unit space-like vectors, each orthogonal to λ a
(0) , and thus
g ab λ
a
(α) λ
b
(β) = −δ αβ and g ab λ
a
(0) λ
b
(α) = 0 ,
(2.10)
with Greek indices taking values 1, 2, 3. These orthonormality conditions can be
inverted to express the metric tensor components in terms of the components of the
vectors in the orthonormal tetrad as
g
ab
= λ
a
(0) λ
b
(0) − λ
a
(1) λ
b
(1) − λ
a
(2) λ
b
(2) − λ
a
(3) λ
b
(3) .
(2.11)
This piece of algebra is an interesting challenge to the reader. In place of the
orthonormal tetrad we will work with an equivalent null tetrad consisting of two
real null vectors and a complex null vector and its complex conjugate defined by
k
a
=
1
√
2
(λ
a
(0) + λ
a
(1) ) , l
a
=
1
√
2
(λ
a
(0) − λ
a
(1) ) ,
m
a
=
1
√
2
(λ
a
(2) + iλ
a
(3) ) , ¯
m
a
=
1
√
2
(λ
a
(2) − iλ
a
(3) ) ,
(2.12)
with complex conjugation denoted by a bar. All of the scalar products involving
these vectors vanish except
k
a l a = 1 and m
a
¯
m a = −1 .
(2.13)
Writing the orthonormal tetrad vectors in terms of the null tetrad vectors
k a , l a , m a , ¯
m a and substituting into (2.11) results in
g
ab
= k
a l
b
+ k
b l
a
− m
a
¯
m
b
− ¯
m
a m
b .
(2.14)
21
with
g
ab ϕ ,a ϕ ,b = 0 ,
(2.9)
and with the comma, as always, denoting partial differentiation with respect to the
coordinates x a . Hence the normal to each hypersurface ϕ ,a is orthogonal to itself
and is therefore tangent to the null hypersurface and is a null vector field whose
direction is the propagation direction in space-time of the histories of the waves.
This null vector field is determined by the Maxwell bivector F ab at each point of
space-time when the two invariants (2.7) vanish and further properties of this null
vector field will follow from Maxwell’s equations (2.5). More generally, without
assuming the vanishing of the two invariants, a Maxwell bivector determines two
null vectors at each point of space-time. To demonstrate this we will use a basis of
four null vector fields, a null tetrad, at each point of space-time.
Let λ a
(0) , λ a
(1) , λ a
(2) , λ a
(3) be an orthonormal tetrad at a point of space-time, with
λ a
(0) a unit time-like vector, and thus g ab λ a
(0) λ b
(0) = 1, and λ a
(1) , λ a
(2) , λ a
(3) mutually
orthogonal unit space-like vectors, each orthogonal to λ a
(0) , and thus
g ab λ
a
(α) λ
b
(β) = −δ αβ and g ab λ
a
(0) λ
b
(α) = 0 ,
(2.10)
with Greek indices taking values 1, 2, 3. These orthonormality conditions can be
inverted to express the metric tensor components in terms of the components of the
vectors in the orthonormal tetrad as
g
ab
= λ
a
(0) λ
b
(0) − λ
a
(1) λ
b
(1) − λ
a
(2) λ
b
(2) − λ
a
(3) λ
b
(3) .
(2.11)
This piece of algebra is an interesting challenge to the reader. In place of the
orthonormal tetrad we will work with an equivalent null tetrad consisting of two
real null vectors and a complex null vector and its complex conjugate defined by
k
a
=
1
√
2
(λ
a
(0) + λ
a
(1) ) , l
a
=
1
√
2
(λ
a
(0) − λ
a
(1) ) ,
m
a
=
1
√
2
(λ
a
(2) + iλ
a
(3) ) , ¯
m
a
=
1
√
2
(λ
a
(2) − iλ
a
(3) ) ,
(2.12)
with complex conjugation denoted by a bar. All of the scalar products involving
these vectors vanish except
k
a l a = 1 and m
a
¯
m a = −1 .
(2.13)
Writing the orthonormal tetrad vectors in terms of the null tetrad vectors
k a , l a , m a , ¯
m a and substituting into (2.11) results in
g
ab
= k
a l
b
+ k
b l
a
− m
a
¯
m
b
− ¯
m
a m
b .
(2.14)