7.4 Equations of Motion in First Approximation
165
F (2)(4) =
1
r 2 × O 2 +
1
r
× O 1 + P 0
∂K 1
∂y
+ M 2
+ O 1 + O(r) ,
(7.90)
F (3)(4) = K 1 + O 1 + O(r) .
(7.91)
Substituting for L 2 , M 2 , K 1 into these components from (7.51) we can rewrite them
in the form
F (1)(2) =
1
r 2 (g + O 2 ) + P
2
0 F ij (u)
∂k i
∂x
∂k j
∂y
+ O 1 + O(r) ,
(7.92)
F (1)(3) = −P 0 F ij (u) k
i ∂k i
∂x
+ O 1 + O(r) ,
(7.93)
F (2)(3) = −P 0 F ij (u) k
i ∂k j
∂y
+ O 1 + O(r) ,
(7.94)
F (1)(4) =
1
r 2 × O 2 +
1
r
× O 1 + P 0 F ij (u)
1
2
k
i
− v
i
∂k j
∂x
+ O 1 + O(r) ,
(7.95)
F (2)(4) =
1
r 2 × O 2 +
1
r
× O 1 + P 0 F ij (u)
1
2
k
i
− v
i
∂k j
∂y
+ O 1 + O(r) ,
(7.96)
F (3)(4) = F ij (u) k
i v
j
+ O 1 + O(r) .
(7.97)
It is useful at this stage to rewrite these again in terms of the components, in
coordinates X i , of the dual of the tensor F ij given by
∗ F ij =
1
2
ij kl F
kl ,
(7.98)
where ij kl is the Levi–Civita permutation symbol in four dimensions with the
convention that 0123 = −1, and F kl = η ki η lj F ij . To facilitate this we note the
following useful formulas:
pqkl k
p ∂k q
∂y
= k k
∂k l
∂x
− k l
∂k k
∂x
,
(7.99)
pqkl k
p v
q
= P
2
0
−
∂k k
∂x
∂k l
∂y
+
∂k l
∂x
∂k k
∂y
,
(7.100)
pqkl
∂k p
∂x
∂k q
∂y
= P
−2
0 (k k v l − k l v k ) ,
(7.101)