10
1 Congruences of World Lines
also the definitions (1.52) of σ and ρ. The result is
k i;j = a k i k j + b k i m j + ¯
b k i ¯
m j + c m i k j + ¯
c ¯
m i k j
+σ ¯
m i ¯
m j + ¯
σ m i m j + ρ ¯
m i m j + ¯
ρ m i ¯
m j ,
(1.53)
with a a real-valued function of the coordinates and b, c complex-valued functions
of the coordinates with complex conjugation denoted by a bar as usual. Making use
of the scalar products among the tetrad vectors we derive from (1.53) the following
equations:
k
i ;i = −ρ − ¯
ρ ,
(1.54)
k i;j k
i;j
= 2 σ ¯
σ + 2 ρ ¯
ρ ,
(1.55)
k j ;i k
i;j
= 2 σ ¯
σ + ρ
2
+ ¯
ρ
2 .
(1.56)
From these we conclude that ρ = −θ − i ω with
θ =
1
2
k
i ;i and ω
2
=
1
2
k [i;j ] k
i;j
≥ 0 ,
(1.57)
with k [i;j ] = (k i;j − k j ;i )/2, and
|σ | =
1
2
k (i;j) k i;j − θ 2 ,
(1.58)
with k (i;j) = (k i;j + k j ;i )/2. We see explicitly from (1.57) and (1.58) that ρ and
|σ | are determined from a knowledge of g ij and k i and are therefore independent
of the choice of null tetrad. It therefore makes sense to use the local geometrical
construction leading to (1.51) to provide a geometrical interpretation of ρ and |σ |.
At a point P on the null geodesic C consider e i to be the 4-velocity of a small
plane circular opaque disk located in the path of a small bundle of photons having
world lines members of the null geodesic congruence in the neighbourhood of C.
Since ζ i k i = 0 we have ζ i in the rest-frame of the disk and we can take ζ i to be
the position vector of points on the boundary of the disk relative to the centre of the
disk. The second condition on ζ i in (1.48) requires the disk to be oriented relative
to the paths of the photons so that the photons strike the disk at right angles when
observed in the rest-frame of the disk. If Q is a second point on C to the future of
the point P and a small affine distance dr from P then, at Q we consider e i to be
the 4-velocity of a small plane screen on which a shadow of the disk is projected.
The position vector of points on the boundary of the shadow, relative to the centre
of the shadow, is given by ζ i which lies in the rest-frame of the screen on account
of the first equation in (1.48) and the second equation in (1.42). Again following
from the second equation in (1.48) the photons strike the screen at right angles to
the screen as observed in the rest-frame of the screen. The passage from the disk
1 Congruences of World Lines
also the definitions (1.52) of σ and ρ. The result is
k i;j = a k i k j + b k i m j + ¯
b k i ¯
m j + c m i k j + ¯
c ¯
m i k j
+σ ¯
m i ¯
m j + ¯
σ m i m j + ρ ¯
m i m j + ¯
ρ m i ¯
m j ,
(1.53)
with a a real-valued function of the coordinates and b, c complex-valued functions
of the coordinates with complex conjugation denoted by a bar as usual. Making use
of the scalar products among the tetrad vectors we derive from (1.53) the following
equations:
k
i ;i = −ρ − ¯
ρ ,
(1.54)
k i;j k
i;j
= 2 σ ¯
σ + 2 ρ ¯
ρ ,
(1.55)
k j ;i k
i;j
= 2 σ ¯
σ + ρ
2
+ ¯
ρ
2 .
(1.56)
From these we conclude that ρ = −θ − i ω with
θ =
1
2
k
i ;i and ω
2
=
1
2
k [i;j ] k
i;j
≥ 0 ,
(1.57)
with k [i;j ] = (k i;j − k j ;i )/2, and
|σ | =
1
2
k (i;j) k i;j − θ 2 ,
(1.58)
with k (i;j) = (k i;j + k j ;i )/2. We see explicitly from (1.57) and (1.58) that ρ and
|σ | are determined from a knowledge of g ij and k i and are therefore independent
of the choice of null tetrad. It therefore makes sense to use the local geometrical
construction leading to (1.51) to provide a geometrical interpretation of ρ and |σ |.
At a point P on the null geodesic C consider e i to be the 4-velocity of a small
plane circular opaque disk located in the path of a small bundle of photons having
world lines members of the null geodesic congruence in the neighbourhood of C.
Since ζ i k i = 0 we have ζ i in the rest-frame of the disk and we can take ζ i to be
the position vector of points on the boundary of the disk relative to the centre of the
disk. The second condition on ζ i in (1.48) requires the disk to be oriented relative
to the paths of the photons so that the photons strike the disk at right angles when
observed in the rest-frame of the disk. If Q is a second point on C to the future of
the point P and a small affine distance dr from P then, at Q we consider e i to be
the 4-velocity of a small plane screen on which a shadow of the disk is projected.
The position vector of points on the boundary of the shadow, relative to the centre
of the shadow, is given by ζ i which lies in the rest-frame of the screen on account
of the first equation in (1.48) and the second equation in (1.42). Again following
from the second equation in (1.48) the photons strike the screen at right angles to
the screen as observed in the rest-frame of the screen. The passage from the disk