226
C Gravitational (Clock) Compass
and
a 0
∂p (α)
∂x
+ b 0
∂p (α)
∂y
= P
2
0 ω (γ )(β) p
(β)
∂p (γ )
∂x
∂p (α)
∂x
+
∂p (γ )
∂y
∂p (α)
∂y
= ω (γ )(β) p
(β) (δ
γ α
− p
(γ ) p
(α) )
(C.23)
= −ω
(α)(β) p (β) .
(C.48)
Hence (C.46) simplifies to
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 .
(C.49)
The line element (C.30) can be written
ds
2
= −(ϑ
(1) )
2
− (ϑ
(2) )
2
− (ϑ
(3) )
2
+ (ϑ
(4) )
2 ,
(C.50)
with the basis 1-forms given by
ϑ
(1)
= r P
−1
0 (dx + a 0 du) = r P
−1
0 (x ,i + a 0 u ,i )dX
i
= −P 0
∂p i
∂x
dX
i
= −P 0
∂p (α)
∂x
λ (α)i dX
i ,
(C.51)
ϑ
(2)
= r P
−1
0 (dy + b 0 du) = r P
−1
0 (y ,i + b 0 u ,i )dX
i
= −P 0
∂p i
∂y
dX
i
= −P 0
∂p (α)
∂y
λ (α)i dX
i ,
(C.52)
ϑ
(3)
= dr = r ,i dX
i
= −p i dX
i
= −p
(α) λ (α)i dX
i ,
(C.53)
ϑ
(4)
= (1 − r h 0 ) du = (1 − r h 0 ) u ,i dX
i
= v i dX
i .
(C.54)
Finally we shall also require p i,j and we can obtain this by rewriting (C.40),
using (C.41) and (C.42), as
δ
i
j = v
i u ,j + p
i r ,j + r p
i
,j = (1 − r h 0 )
−1 v
i v j − p
i p j + r p
i
,j .
(C.55)