C Gravitational (Clock) Compass
231
Hence if v A = 0, so that the time-like world line r = 0 is the history of an
observer accelerating in the direction of propagation of the gravitational waves (the
Z-direction), then in this case (5.87) simplifies to
ds
du
2
= 1 − |u|
2
+ 2 {(a · p) − u · (ω × p)} r
+
(a · p)
2
− |ω × p|
2
− R (α)(4)(β)(4) p
(α) p
(β)
r
2
+ O(r
3 ) .
(C.82)
The origin of the coordinate transformation (C.78) is to start with the line element
ds
2
= η ij dX
i dX
j
+ 2 H (dT − dZ)
2
= η ij dX
i dX
j
− R A4B4 (T − Z)
X
A
− w
A (T − Z)
×
X
B
− w
B (T − Z)
(dT − dZ)
2 .
(C.83)
Now in the final term here make the transformation (C.39). This involves
T − Z = w
4
− w
3
+ r (p
4
− p
3 ) ⇒
dT − dZ = (v
4
− v
3 )du + (p
4
− p
3 ) dr + O(r),
(C.84)
and
R A4B4 (T − Z) (X
A
− w
A (T − Z)) (X
B
− w
B (T − Z))
= r
2 R A4B4 (u) p
A p
B
+ O(r
3 ) .
(C.85)
Hence the final term in the line element (C.83) reads
−r
2 R A4B4 p
A p
B
(v
4
− v
3 )
2 du
2
+2 (v
4
− v
3 )(p
4
− p
3 ) du dr
+ O(r
3 ) .
(C.86)