C Gravitational (Clock) Compass
229
Using (C.63), (C.65) and (C.66) we see that
R A4B4 X
A X
B
= −2 H
(A,B = 1, 2),
(C.71)
with X A = (X, Y ). On account of the simplicity of the Riemann tensor (in particular
that it has only two independent components) all of the information contained in it
can be extracted using the observer with world line X = Y = Z = 0 and observers
with world lines Z = constant. The ratio of arc lengths or proper-times along such
world lines is, by (C.70) and (C.71),
ds
dT
2
= 1 − u
A u
A
− R A4B4 X
A X
B .
(C.72)
We note that in general the final term in (C.70) can be written
2 H
1 − u
3
2 = −R A4B4 X
A X
B
− 2 R A4B3 X
A X
B u
3
−R A3B3 X
A X
B (u
3 )
2 .
(C.73)
However this contains no more information on the Riemann tensor than the final
term in (C.72) since
R A4B3 X
A X
B
= −R A3B3 X
A X
B
= 2 H
= −R A4B4 X
A X
B .
(C.74)
C.3
Plane Gravitational Waves II
The function H in (C.62) and (C.63) has the property that it vanishes on the world
line X = Y = Z = 0. Its essential analytical properties are that the vacuum field
equations require it to be a harmonic function,
H XX + H Y Y = 0 ,
(C.75)
with the subscripts denoting partial derivatives, and the curvature tensor components
must be functions of T − Z so that
H XX − H Y Y = 4 a(T − Z) and H XY = 2 b(T − Z) .
(C.76)