E Small Magnetic Black Hole
243
f 2 =
1
r
−g P
−2
0 ˆ
a −1 − 2 g
2 M 2 + O 3
+ g
∂h 0
∂x
+4 m M 2 + O 2 − r
∂K 1
∂y
+ 2 M 2 + O 1
+ O(r
2 ) ,
(E.18)
f 3 =
1
r
−g P
−2
0
ˆ
b −1 + 2 g
2 L 2 + O 3
+ g
∂h 0
∂y
−4 m L 2 + O 2 + r
∂K 1
∂x
+ 2 L 2 + O 1
+ O(r
2 ) ,
(E.19)
f 4 = r(−2 M 2 + O 1 ) + O(r
3 ) ,
(E.20)
f 5 = r(2 L 2 + O 1 ) + O(r
3 ) ,
(E.21)
f 6 =
1
r 2 {g + O 3 } + P
2
0
∂M 2
∂x
−
∂L 2
∂y
+ O 1 + O(r) .
(E.22)
Substituting these into the left hand sides of Maxwell’s equations (E.7)–(E.10) we
arrive at
∂f 1
∂r
−
∂f 4
∂y
+
∂f 5
∂x
= −2 r P
−2
0
K 1 − P
2
0
∂L 2
∂x
+
∂M 2
∂y
+ O 1
+ O(r
2 )
= r × O 1 + O ( r
2 ) ,
(E.23)
using (7.53),
∂f 1
∂u
−
∂f 2
∂y
+
∂f 3
∂x
=
1
r
g
∂
∂y
(P
−2
0 ˆ
a −1 ) −
∂
∂x
(P
−2
0
ˆ
b −1 )
+2 g
2
∂L 2
∂x
+
∂M 2
∂y
+ O 3
− 4 m
∂L 2
∂x
+
∂M 2
∂y
+O 2 + r
∂ 2 K 1
∂x 2 +
∂ 2 K 2
∂y 2 + 2
∂L 2
∂x
+
∂M 2
∂y
+ O 1
+O(r
2 ) ,
=
1
r
6 g
2 P
−2
0 F ij k
i v
j
+ O 3
− 4 m P
−2
0 F ij k
i v
j
+O 2 + O 1 × r + O(r
2 ) ,
= O 2 ×
1
r
+ O 1 + O 1 × r + O(r
2 ) ,
(E.24)