7.4 Equations of Motion in First Approximation
169
by (7.123) and so (7.128) is automatically satisfied with an O(r −1 )-error. Next using
(7.115) and (7.116) we have the two field equations
R (1)(4) = −2
g
r 2 P 0
∗ F ij (u) (v
i
−
1
2
k
i )
∂k j
∂x
+ O
1
r
,
(7.130)
R (2)(4) = −2
g
r 2 P 0
∗ F ij (u) (v
i
−
1
2
k
i )
∂k j
∂y
+ O
1
r
.
(7.131)
Calculation of the tetrad components of the Ricci tensor here yields
R (1)(4) =
P 0
2 r 2
P
−2
0 ˆ
a −1 +
∂
∂y
P
2
0
∂
∂y
(P
−2
0 ˆ
a −1 ) −
∂
∂x
(P
−2
0
ˆ
b −1 )
+O
1
r
,
(7.132)
R (2)(4) =
P 0
2 r 2
P
−2
0
ˆ
b −1 −
∂
∂x
P
2
0
∂
∂y
(P
−2
0 ˆ
a −1 ) −
∂
∂x
(P
−2
0
ˆ
b −1 )
+O
1
r
.
(7.133)
With ˆ
a −1 , ˆ
b −1 given by (7.119) we have
∂
∂y
(P
−2
0 ˆ
a −1 ) −
∂
∂x
(P
−2
0
ˆ
b −1 ) = 4 g
∗ F ij
∂k i
∂x
∂k j
∂y
.
(7.134)
Then making use of the formulas (7.57)–(7.59) we find that
∂
∂y
P
2
0
∂
∂y
(P
−2
0 ˆ
a −1 ) −
∂
∂x
(P
−2
0
ˆ
b −1 )
= 4 g
∗ F ij (k
i
− v
i )
∂k j
∂x
,
(7.135)
∂
∂x
P
2
0
∂
∂y
(P
−2
0 ˆ
a −1 ) −
∂
∂x
(P
−2
0
ˆ
b −1 )
= −4 g
∗ F ij (k
i
− v
i )
∂k j
∂y
.
(7.136)
When these, along with (7.119), are substituted into (7.130) and (7.131) we see that
(7.130) and (7.131) are automatically satisfied. Finally we consider the second of
(7.117) and the field equation
R (4)(4) = 2 E (4)(4) = O
1
r
.
(7.137)