242
E Small Magnetic Black Hole
+
ˆ
p ˆ
c
2 r
e
ˆ
α cosh ˆ
β
∂ ˆ
M
∂r
,
(E.5)
F (3)(4) =
∂ ˆ
K
∂r
− e
− ˆ
α ( ˆ
a cosh ˆ
β − ˆ
b sinh ˆ
β)
∂ ˆ
L
∂r
+e
ˆ
α ( ˆ
a sinh ˆ
β − ˆ
b cosh ˆ
β)
∂ ˆ
M
∂r
.
(E.6)
The exact Maxwell vacuum field equations read:
∂f 1
∂r
−
∂f 4
∂y
+
∂f 5
∂x
= 0 ,
(E.7)
∂f 1
∂u
−
∂f 2
∂y
+
∂f 3
∂x
= 0 ,
(E.8)
∂f 2
∂r
−
∂f 4
∂u
−
∂f 6
∂x
= 0 ,
(E.9)
∂f 3
∂r
−
∂f 5
∂u
−
∂f 6
∂y
= 0 ,
(E.10)
with
f 1 = −r
2
ˆ
p
−2 F (3)(4) ,
(E.11)
f 2 =
1
2
ˆ
c r ˆ
p
−1 e
ˆ
α (− sinh ˆ
β F (1)(3) + cosh ˆ
β F (2)(3) ) + r ˆ
p
−1 e
ˆ
α (sinh ˆ
β F (1)(4)
− cosh ˆ
β F (2)(4) ) − r
2
ˆ
p
−2 e
ˆ
α (− ˆ
a sinh ˆ
β + ˆ
b cosh ˆ
β) F (3)(4) ,
(E.12)
f 3 =
1
2
ˆ
c r ˆ
p
−1 e
− ˆ
α (− cosh ˆ
β F (1)(3) + sinh ˆ
β F (2)(3) ) + r ˆ
p
−1 e
− ˆ
α (cosh ˆ
β F (1)(4)
− sinh ˆ
β F (2)(4) ) − r
2
ˆ
p
−2 e
− ˆ
α (− ˆ
a cosh ˆ
β + ˆ
b sinh ˆ
β) F (3)(4) ,
(E.13)
f 4 = r ˆ
p
−1 e
ˆ
α (− sinh ˆ
β F (1)(3) + cosh ˆ
β F (2)(3) ) ,
(E.14)
f 5 = r ˆ
p
−1 e
− ˆ
α (− cosh ˆ
β F (1)(3) + sinh ˆ
β F (2)(3) ) ,
(E.15)
f 6 = F (1)(2) + r ˆ
p
−1 ( ˆ
b F (1)(3) − ˆ
a F (2)(3) ) .
(E.16)
With the Maxwell tensor F (a)(b) given in (7.144)–(7.149) these functions acquire
the approximate forms:
f 1 = −r
2 (P
−2
0 K 1 + O 1 ) + O(r
3 ) ,
(E.17)