C Gravitational (Clock) Compass
225
which implicitly determines x, y, r, u as scalar functions of X i on Minkowskian
space-time. We will need the gradients of these functions with respect to X i , denoted
by a comma in each case. To obtain these we start by differentiating (C.39) with
respect to X j giving
δ
i
j =
v
i
+ r
∂p i
∂u
u ,j + p
i r ,j + r
∂p i
∂x
x ,j + r
∂p i
∂y
y ,j .
(C.40)
Multiplying this by v i , p i , ∂p i /∂x and ∂p i /∂y yields successively
v j = (1 − h 0 r)u ,j ⇒ u ,j = (1 − h 0 r)
−1 v j ,
(C.41)
p j = −r ,j ⇒ r ,j = −p j ,
(C.42)
∂p j
∂x
= r
∂p i
∂x
∂p i
∂u
u ,j − r P
−2
0 x ,j
⇒ x ,j + a 0 u ,j = −
P 2
0
r
∂p j
∂x
,
(C.43)
∂p j
∂y
= r
∂p i
∂y
∂p i
∂u
u ,j − r P
−2
0 y ,j
⇒ y ,j + b 0 u ,j = −
P 2
0
r
∂p j
∂y
,
(C.44)
with the final two cases relying on (C.22) and (C.31). We first note from (C.41)–
(C.44) that
p
i ∂
∂X i = p
i
x ,i
∂
∂x
+ y ,i
∂
∂y
+ r ,i
∂
∂r
+ u ,i
∂
∂u
=
∂
∂r
.
(C.45)
Substituting (C.41)–(C.44) back into (C.40), using (C.18) and raising the covariant
index using η ij , we have
η
ij
= −P
2
0
∂p i
∂x
∂p j
∂x
+
∂p i
∂y
∂p j
∂y
− p
i p
j
+ v
i v
j
+r(1 − h 0 r)
−1
ω
ik p k − a 0
∂p i
∂x
− b 0
∂p i
∂y
v
j .
(C.46)
However the final term here vanishes since
ω
ik p k − a 0
∂p i
∂x
− b 0
∂p i
∂y
=
δ
βγ ω (α)(β) p (γ ) + a 0
∂p (α)
∂x
+ b 0
∂p (α)
∂y
λ
i(α) ,
(C.47)