160
7 Small Magnetic Black Hole
where F ij (u) = −F ji (u) are the components in coordinates X i of the external
Maxwell field calculated on r = 0. The leading, in powers of r, Maxwell equations
d ∗ F = 0 require (7.51) to satisfy
K 1 = P
2
0
∂L 2
∂x
+
∂M 2
∂y
,
(7.52)
1 + 2 P
2
0
∂L 2
∂x
+
∂M 2
∂y
= 0 ,
(7.53)
with
= P
2
0
∂ 2
∂x 2 +
∂ 2
∂y 2
,
(7.54)
and
∂K 1
∂x
+ 2 L 2 −
∂
∂y
P
2
0
∂M 2
∂x
−
∂L 2
∂y
= 0 ,
(7.55)
∂K 1
∂y
+ 2 M 2 +
∂
∂x
P
2
0
∂M 2
∂x
−
∂L 2
∂y
= 0 .
(7.56)
Clearly (7.53) is a consequence of (7.55) and (7.56). The dependence of L 2 , M 2 , K 1
in (7.51) on u is arbitrary (since the world line r = 0 is an arbitrary time-like world
line), but their dependence on x, y is explicitly known because the dependence of k i
and P 0 on x, y is explicit in (7.14) and (7.16). Hence one can check that L 2 , M 2 , K 1
in (7.51) satisfy (7.52), (7.53), (7.55) and (7.56). For carrying this out we note the
following useful formulas [9]:
∂ 2 k i
∂x 2 = P
−2
0 (v
i
− k
i ) −
∂
∂x
(log P 0 )
∂k i
∂x
+
∂
∂y
(log P 0 )
∂k i
∂y
,
(7.57)
∂ 2 k i
∂y 2 = P
−2
0 (v
i
− k
i ) +
∂
∂x
(log P 0 )
∂k i
∂x
−
∂
∂y
(log P 0 )
∂k i
∂y
,
(7.58)
∂ 2 k i
∂x∂y
= −
∂
∂y
(log P 0 )
∂k i
∂x
−
∂
∂x
(log P 0 )
∂k i
∂y
.
(7.59)
We also note (7.52) and (7.53) imply that K 1 satisfies
1 + 2 K 1 = 0 ,
(7.60)
indicating that K 1 is an l = 1 spherical harmonic, since the operator in (7.54) is
the Laplacian on the unit sphere.