7.5 Equations of Motion in Second Approximation
177
We also find that
T 1 = 2 (F
0
(1)(3) + i F
0
(2)(3) )(F
1
(1)(4) + i F
1
(2)(4) )
= 8 g P
2
0 a i
∂k i
∂ ¯
ζ
∗ F pq k
p ∂k q
∂ ¯
ζ
− 8 m
P 0
∗ F ij k
i ∂k j
∂ ¯
ζ
2
+ O 2
= 8
g 2
m
P
2
0
∗ F ij
∂k i
∂ ¯
ζ
v
j ∗ F pq k
p ∂k q
∂ ¯
ζ
− 8 m
P 0
∗ F ij k
i ∂k j
∂ ¯
ζ
2
+ O 2 ,
(7.188)
where we have made use of the equations of motion in first approximation (7.143).
We can rewrite (7.188) as
T 1 =
∂
∂ ¯
ζ
8 g 2
m
+ 16 m
P
2
0
∗ F ij k
i ∂k j
∂ ¯
ζ
∗ F pq k
p v
q
+8 m P
2
0
∗ F
p
i
∗ F pj k
i ∂k j
∂ ¯
ζ
+ O 2 .
(7.189)
Now with R (A)(B) = 2 E (A)(B) we obtain from (7.178), (7.186), (7.187) and (7.189):
m 2 − i l 2 =
1
3
g C ij kl k
i v
j k
k ∂k l
∂ ¯
ζ
− 4 g
∗ F
p
i
∗ F pj k
i ∂k j
∂ ¯
ζ
−8 g
∗ F ij k
i ∂k j
∂ ¯
ζ
∗ F pq k
p v
q
+ P
−2
0 (U (u) − i V (u)) + O 2 ,
= O 1 ,
(7.190)
where U(u) = O 1 and V (u) = O 1 are functions of integration, and
P
−2
0 ( ˆ
a 0 + i ˆ
b 0 ) = −
8
3
m C ij kl k
i v
j k
k ∂k l
∂ ¯
ζ
+ 4 m
∗ F
p
i
∗ F pj k
i ∂k j
∂ ¯
ζ
+
4 g 2
m
+ 8 m
∗ F ij k
i ∂k j
∂ ¯
ζ
∗ F pq k
p v
q
+P
−2
0 (A 0 (u) + i B 0 (u)) + O 2 ,
= O 1 ,
(7.191)
where A 0 (u) = O 1 and B 0 (u) = O 1 are functions of integration. We have chosen
functions of integration A 0 (u) and B 0 (u) so that P
−1
0 ˆ
a 0 and P
−1
0
ˆ
b 0 are guaranteed
to be free of singularities in x, y for −∞ < x,y < +∞. This ensures that
the components of the metric tensor of the perturbed space-time are free of such