244
E Small Magnetic Black Hole
using (7.51), (7.53), (7.52), (7.101) and (7.119),
∂f 2
∂r
−
∂f 4
∂u
−
∂f 6
∂x
=
1
r 2
g P
−2
0 ˆ
a −1 + 2 g
2 M 2 + O 3
−
∂K 1
∂y
− 2 M 2 −
∂
∂x
P
2
0
∂M 2
∂x
−
∂L 2
∂y
+O 1 + O(r) ,
=
1
r 2
3 g
2 F ij k
i ∂k j
∂y
+ O 3
+ O 1 + O(r) ,
= O 2 ×
1
r 2 + O 1 + O(r) ,
(E.25)
using (7.56), (7.102) and (7.119), and
∂f 3
∂r
−
∂f 5
∂u
−
∂f 6
∂y
=
1
r 2
g P
−2
0
ˆ
b −1 − 2 g
2 L 2 + O 3
+
∂K 1
∂x
+ 2 L 2 −
∂
∂y
P
2
0
∂M 2
∂x
−
∂L 2
∂y
+O 1 + O(r) ,
=
1
r 2
− 3 g
2 F ij k
i ∂k j
∂x
+ O 3
+ O 1 + O(r) ,
= O 2 ×
1
r 2 + O 1 + O(r) ,
(E.26)
using (7.55), (7.99) and (7.119). Satisfying Maxwell’s vacuum field equations
approximately, with the degree of smallness of the coefficients of the various powers
of r indicated here, is sufficient for us to determine the equations of motion of the
small magnetic black hole in second approximation.