168
7 Small Magnetic Black Hole
with the operator given by (7.54). With ˆ
a −1 , ˆ
b −1 given by (7.119) and making
use of (7.57) and (7.58) we have
P
2
0
∂
∂x
(P
−2
0 ˆ
a −1 ) +
∂
∂y
(P
−2
0
ˆ
b −1
= −2 g
∗ F ij k
i
j
= −4 g
∗ F ij k
i v
j .
(7.123)
Hence we conclude from (7.121)–(7.123) that
ˆ
c 0 = 1 + 1 + 2 Q 1 + 8 g
∗ F ij k
i v
j ,
(7.124)
neglecting O 2 -terms. We also have from (7.111) that
R (1)(1) − R (2)(2) = 2 (E (1)(1) − E (2)(2) ) = O
1
r
, R (1)(2) = 2 E (1)(2) = O
1
r
.
(7.125)
But
R (1)(1) − R (2)(2) =
1
r 2
∂ ˆ
a −1
∂x
−
∂ ˆ
b −1
∂y
+ O
1
r
,
(7.126)
and
R (1)(2) =
1
2 r 2
∂ ˆ
b −1
∂x
+
∂ ˆ
a −1
∂y
+ O
1
r
,
(7.127)
and thus (7.125) are satisfied on account of (7.120). Using the first of (7.117) we
have the field equation
R (3)(4) =
2 g
r 2
∗ F ij k
i v
j
+ O
1
r
.
(7.128)
But
R (3)(4) = −
P 2
0
2 r 2
∂
∂x
(P
−2
0 ˆ
a −1 ) +
∂
∂y
(P
−2
0
ˆ
b −1 )
+ O
1
r
=
2 g
r 2
∗ F ij k
i v
j
+ O
1
r
,
(7.129)