108
5 Gravitational (Clock) Compass
in effect changing from the parameter s to the parameter u along the arbitrary world
line, we can write
dX i
ds
=
du
ds
[1 + (a · p) r] v
i
+ W
(α) λ
i
(α)
,
(5.40)
with
W
(α)
= p
(α) dr
du
+ r
∂p (α)
∂x
dx
du
+ r
∂p (α)
∂y
dy
du
+ r ω
(α)(β) p (β) .
(5.41)
Substituting (5.40) into (5.38) we obtain
ds
du
2
= (1 + (a · p) r)
2
− |W|
2 ,
(5.42)
with
W = u + r (ω × p) ,
(5.43)
and
u =
dr
du
p + r
dx
du
∂p
∂x
+ r
dy
du
∂p
∂y
.
(5.44)
We see that, since u is proper-time along the world line r = 0, u is the 3-velocity of
the observer with world line (5.37) relative to the observer with world line r = 0.
Hence (5.42) can finally be written
ds
du
2
= 1 − |u|
2
+ 2 {a · p − u · (ω × p)} r
+{(a · p)
2
− |ω × p|
2
} r
2 .
(5.45)
This formula is, of course, exact (in particular it does not have any restriction on r).
Equation (5.42), or equivalently the equation (5.45), can be compared directly to the
results in [49] and with eq. (22) in [17].
Frequency Ratio in Curved Space-Time
We now consider a general curved space-time, guided by Eqs. (C.42) and (C.45) we
choose a family of time-like hypersurfaces r(x i ) = constant in this space-time with
unit space-like normal
p
i
=
dx i
dr
= −g
ij r ,j and g ij p
i p
j
= −1 .
(5.46)