1.2 Null Geodesic Congruences
9
Now k i , l i , m i , ¯
m i constitute a null tetrad at P which will continue to be a null tetrad
along C if we take m i to be parallel transported along C so that
m
i ;j k
j
= 0 = ¯
m
i ;j k
j .
(1.46)
In terms of this null tetrad we can write the components g ij of the metric tensor as
g ij = −m i ¯
m j − ¯
m i m j + k i l j + l i k j .
(1.47)
We now specialise the infinitesimal connecting vector ζ i by requiring it to satisfy
the following equations:
ζ i e
i
= 0 and (δ
i
j − e
i e j ) k
j ζ i = 0 .
(1.48)
Thus ζ i is orthogonal to e i and is also orthogonal to the projection of k i orthogonal
to e i . The physical interpretation of these conditions will appear below. As a
consequence of them ζ i is orthogonal to k i and to l i in (1.43) and so can be
expressed on the null tetrad as
ζ
i
= ¯
ζ m
i
+ ζ ¯
m
i ,
(1.49)
with ζ a complex valued function of the coordinates x i . Substituting this into the
transport law (1.42) results in
∂ ¯
ζ
∂r
m
i
+
∂ζ
∂r
¯
m
i
= ¯
ζ k
i ;j m
j
+ ζ k
i ;j ¯
m
j .
(1.50)
Taking the scalar product of this equation with m i leads to the propagation law for
ζ(x i ) along C:
∂ζ
∂r
= −σ ¯
ζ − ρ ζ ,
(1.51)
with
σ = k i;j m
i m
j and ρ = k i;j m
i
¯
m
j .
(1.52)
Although it appears that σ and ρ depend upon the choice of tetrad, in fact |σ | and
ρ are independent of the choice of tetrad and can be constructed solely from a
knowledge of g ij and k i . It is easy to see that the argument of the complex variable
σ is dependent upon the choice of tetrad since it is obviously not invariant under
the simple tetrad transformation m i → e i ψ m i , for some real constant ψ. To see
explicitly that |σ | and ρ can be derived from g ij and k i we first expand k i;j on the
null tetrad taking into consideration that k i;j is real, k i k i;j = 0 and k j k i;j = 0 and
9
Now k i , l i , m i , ¯
m i constitute a null tetrad at P which will continue to be a null tetrad
along C if we take m i to be parallel transported along C so that
m
i ;j k
j
= 0 = ¯
m
i ;j k
j .
(1.46)
In terms of this null tetrad we can write the components g ij of the metric tensor as
g ij = −m i ¯
m j − ¯
m i m j + k i l j + l i k j .
(1.47)
We now specialise the infinitesimal connecting vector ζ i by requiring it to satisfy
the following equations:
ζ i e
i
= 0 and (δ
i
j − e
i e j ) k
j ζ i = 0 .
(1.48)
Thus ζ i is orthogonal to e i and is also orthogonal to the projection of k i orthogonal
to e i . The physical interpretation of these conditions will appear below. As a
consequence of them ζ i is orthogonal to k i and to l i in (1.43) and so can be
expressed on the null tetrad as
ζ
i
= ¯
ζ m
i
+ ζ ¯
m
i ,
(1.49)
with ζ a complex valued function of the coordinates x i . Substituting this into the
transport law (1.42) results in
∂ ¯
ζ
∂r
m
i
+
∂ζ
∂r
¯
m
i
= ¯
ζ k
i ;j m
j
+ ζ k
i ;j ¯
m
j .
(1.50)
Taking the scalar product of this equation with m i leads to the propagation law for
ζ(x i ) along C:
∂ζ
∂r
= −σ ¯
ζ − ρ ζ ,
(1.51)
with
σ = k i;j m
i m
j and ρ = k i;j m
i
¯
m
j .
(1.52)
Although it appears that σ and ρ depend upon the choice of tetrad, in fact |σ | and
ρ are independent of the choice of tetrad and can be constructed solely from a
knowledge of g ij and k i . It is easy to see that the argument of the complex variable
σ is dependent upon the choice of tetrad since it is obviously not invariant under
the simple tetrad transformation m i → e i ψ m i , for some real constant ψ. To see
explicitly that |σ | and ρ can be derived from g ij and k i we first expand k i;j on the
null tetrad taking into consideration that k i;j is real, k i k i;j = 0 and k j k i;j = 0 and