234
C Gravitational (Clock) Compass
Substituting (C.95)–(C.97) into (5.87) we find
ds
du
2
= 1 − |u|
2
+ 2 {(a · p) − u · (ω × p)} r
+
(a · p)
2
− |ω × p|
2
− R (A)(4)(B)(4) p
(A) p
(B)
+
4
3
R (A)(4)(B)(4) {u
(3) p
(A)
− u
(A) p
(3)
} p
(B)
−
1
3
R (A)(4)(B)(4) {u
(A) p
(3)
− u
(3) p
(A)
}
×{u
(B) p
(3)
− u
(3) p
(B)
}
r
2
+ O(r
3 ) .
(C.98)
It is interesting to note that while k i given by (C.90) when r = 0 is the propagation
direction of the radial gravitational waves relative to the observer with world
line r = 0 it cannot be the propagation direction of gravitational waves in the
neighbourhood of r = 0 (i.e. for small, non-zero, values of r). The reason for
this is because the Goldberg–Sachs [1] theorem requires the propagation direction
in space-time of gravitational waves propagating in a vacuum to be geodesic and
shear-free. Using (C.41) and (C.56) we have
k i,j =
1
r
(η ij + p i p j − v i v j ) − h 0 v i v j + a i v j + O(r) ,
(C.99)
from which we conclude that
k i,j k
j
= −h 0 v i + O(r) ,
(C.100)
and so k i is not even approximately geodesic for small r if a i = 0 (i.e. if r = 0 is
not a time-like geodesic). However we can construct an approximately null vector
field K i in the neighbourhood of r = 0, which coincides with k i on r = 0, and
which is approximately geodesic and shear-free. Such a vector field is given by
K
i
= p
i
+ v
i
−
1
2
{a
i
+ h 0 p
i
} r + O(r
2 ) ⇒ K
i K i = O(r
2 ) .
(C.101)
When differentiating this with respect to X i using (C.41), (C.42) and (C.56) it is
useful to note that the partial derivative of h 0 = a i p i reads
∂h 0
∂X i =
1
r
(a i + h 0 p i ) + O(r
0 ) .
(C.102)