7.5 Equations of Motion in Second Approximation
173
We can now write the tetrad components F (a)(b) of the perturbed Maxwell field
(7.144)–(7.149) more explicitly as
F (1)(2) =
1
r 2 (g + O 3 ) −
∗ F ij k
i v
j
+ P
2
0
∂m 2
∂x
−
∂l 2
∂y
−
1
3
g
∗ F
p
i
∗ F pj k
i k
j
+ O 2 + O(r) ,
(7.160)
F (1)(3) = −P 0
∗ F ij k
i ∂k j
∂y
− 2 P 0 l 2 + O 2 + O(r) ,
(7.161)
F (2)(3) = P 0
∗ F ij k
i ∂k j
∂x
− 2 P 0 m 2 + O 2 + O(r) ,
(7.162)
F (1)(4) =
1
r 2
5
2
g
2 P 0
∗ F ij k
i ∂k j
∂y
+ O 3
+
1
r
g P 0 a i
∂k i
∂y
− m P 0
∗ F ij k
i ∂k j
∂y
+ O 2
−
1
2
P 0
∗ F ij k
i ∂k j
∂y
− P 0
∗ F ij
∂k i
∂y
v
j
+ O 1 + O(r) , (7.163)
F (2)(4) =
1
r 2
−
5
2
g
2 P 0
∗ F ij k
i ∂k j
∂x
+ O 3
+
1
r
−g P 0 a i
∂k i
∂x
+ m P 0
∗ F ij k
i ∂k j
∂x
+ O 2
+
1
2
P 0
∗ F ij k
i ∂k j
∂x
+ P 0
∗ F ij
∂k i
∂x
v
j
+ O 1 + O(r) , (7.164)
and
F (3)(4) = K 1 + O 1 + O(r) .
(7.165)
Turning to the field equations R (A)(B) = 2 E (A)(B) with A, B = 1, 2 we first find,
using (7.76)–(7.84), that the tetrad components of the Ricci tensor here are given by
R (1)(2) =
1
r 2
1
2
∂ ˆ
a −1
∂y
+
∂ ˆ
b −1
∂x
+
1
2
P
−2
0 ˆ
a −1 ˆ
b −1 + 2 g
2 β 2
+
1
r
∂ ˆ
a 0
∂y
+
∂ ˆ
b 0
∂x
− 8 m β 2 + O 2
+ O(r
0 ) ,
(7.166)