172
7 Small Magnetic Black Hole
Calculating R (1)(3) and R (2)(3) using (7.76)–(7.84) we find that
R (1)(3) = −
1
r 2 P
−1
0 ˆ
a −1 + O
1
r
,
(7.152)
R (2)(3) = −
1
r 2 P
−1
0
ˆ
b −1 + O
1
r
,
(7.153)
Hence Einstein’s field equations
R (1)(3) − 2 E (1)(3) =
1
r 2 × O 3 + O
1
r
,
(7.154)
and
R (2)(3) − 2 E (2)(3) =
1
r 2 × O 3 + O
1
r
,
(7.155)
yield
ˆ
a −1 = 4 g P
2
0 M 2 + 4 g P
2
0 m 2 + O 3 = 2 g P
2
0 F ij k
i ∂k j
∂y
+ 4 g P
2
0 m 2
= −2 g P
2
0
∗ F ij k
i ∂k j
∂x
+ 4 g P
2
0 m 2 + O 3 ,
(7.156)
and
ˆ
b −1 = −4 g P
2
0 M 2 − 4 g P
2
0 l 2 + O 3 = −2 g P
2
0 F ij k
i ∂k j
∂y
− 4 g P
2
0 m 2
= −2 g P
2
0
∗ F ij k
i ∂k j
∂y
− 4 g P
2
0 l 2 + O 3 ,
(7.157)
For convenience we note, using (7.51), (7.68), (7.99), (7.102) and
F
p
i F pj =
∗ F
p
i
∗ F pj −
1
2
η ij
∗ F
pq ∗ F pq ,
(7.158)
that
L 2 =
1
2
∗ F ij k
i ∂k j
∂y
, M 2 = −
1
2
∗ F ij k
i ∂k j
∂x
and q 2 = −
1
6
∗ F
p
i
∗ F pj k
i k
j .
(7.159)