178
7 Small Magnetic Black Hole
singularities since ˆ
a 0 , ˆ
b 0 appear in the metric tensor components in the form P
−1
0 ˆ
a 0
and P
−1
0
ˆ
b 0 . The same applies to the choice of the functions of integration U(u) and
V (u) in (7.190) which will ensure that
P
2
0
∂m 2
∂x
−
∂l 2
∂y
,
which appears in the metric tensor component involving ˆ
c 0 (see below) is free of
singularities in x, y for −∞ < x, y < +∞.
For later use we need to derive from (7.190) and (7.191) the expressions
P
2
0
∂m 2
∂x
−
∂l 2
∂y
= P
2
0
∂
∂ζ
(m 2 − i l 2 ) +
∂
∂ ¯
ζ
(m 2 + i l 2 )
,
(7.192)
and
P
2
0
∂
∂x
(P
−2
0 ˆ
a 0 ) +
∂
∂y
(P
−2
0
ˆ
b 0 )
= P
2
0
∂
∂ζ
(P
−2
0 ( ˆ
a 0 + i ˆ
b 0 ))
+
∂
∂ ¯
ζ
(P
−2
0 ( ˆ
a 0 − i ˆ
b 0 ))
.
(7.193)
We can express the results neatly by making use of the following quantities:
S (1) = C ij kl k
i v
j k
k v
l ,
(7.194)
S (2) = (
∗ F ij k
i v
j )
2
+
1
3
∗ F
p
i
∗ F pj v
i v
j ,
(7.195)
S (3) =
∗ F
p
i
∗ F pj k
i k
j
− 2
∗ F
p
i
∗ F pj k
i v
j
+
1
3
∗ F
ij ∗ F ij
+
2
3
∗ F
p
i
∗ F pj v
i v
j ,
(7.196)
S (4) = h
2
0 +
1
3
a i a
i ,
(7.197)
L (1) =
∗ F
p
i
∗ F pj k
i v
j
−
∗ F
p
i
∗ F pj v
i v
j ,
(7.198)
L (2) = ˙
a i k
i
+ a i a
i .
(7.199)
A dot, as in (7.199), indicates differentiation with respect to u. The significance of
these quantities is that, using the useful formulas (7.57)–(7.59) the S’s are l = 2
spherical harmonics (i.e. each S satisfies + 6 S = 0) while the L’s are l = 1