180
7 Small Magnetic Black Hole
Hence in order to satisfy
R (A)(A) − 2 E (A)(A) =
1
r 4 × O 3 +
1
r 2 × O 3 +
1
r
× O 1 + O(r
0 ) ,
(7.206)
we must have ˆ
c 1 = −2 h 0 + O 1 (which will be sufficient accuracy for our purposes)
and
ˆ
c 0 = 1 + 8 g
∗ F ij k
i v
j
− 8 g P
2
0
∂m 2
∂x
−
∂l 2
∂y
+
5
3
g
2 ∗ F
p
i
∗ F pj k
i k
j
+ 2 + 2 Q 2
= 1 + 8 g
∗ F ij k
i v
j
−
5
9
g
2 ∗ F
ij ∗ F ij +
20
9
g
2 ∗ F
p
i
∗ F pj v
i v
j
+
10
3
g
2 L (1) −
139
3
g
2 S (3) + 96 g
2 S (2) − 4 g
2 S (1) + 16 g U(u)
∂
∂x
(log P 0 )
−16 g V (u)
∂
∂y
(log P 0 ) + 2 + 2 Q 2 + O 3 .
(7.207)
We note for use below that now
−
1
2
ˆ
c 0 = 8 g
∗ F ij k
i v
j
−
1
2
2 + 2 Q 2 ) +
10
3
g
2 L (1)
−139 g
2 S (3) + 288 g
2 S (2) − 12 g
2 S (1) + 16 g U(u)
∂
∂x
(log P 0 )
−16 g V (u)
∂
∂y
(log P 0 ) .
(7.208)
At this point in the derivation of the equations of motion in second approximation
of the magnetic black hole we have obtained all of the required perturbations in
the metric tensor and the potential 1-form appearing in (7.72)–(7.75) and (7.76)–
(7.83) with the exception of the functions ˆ
f −1 = O 2 and Q 2 = O 2 . The latter
function determines the behaviour of the geometry of the wave fronts of the radiation
generated by the motion of the magnetic black hole and thus is key to our derivation
of the equations of motion. However we need ˆ
f −1 in order to derive Q 2 . Both of
these functions are obtained from the field equation R (4)(4) = 2 E (4)(4) . The leading
term on both sides of this equation is O(r −3 ), neglecting O 3 -terms. We find that
R (4)(4) =
1
r 3
2 ˆ
f −1 + 4 g
2 h 0 − 2 m P
2
0
∂
∂x
(P
−2
0 ˆ
a −1 ) +
∂
∂y
(P
−2
0
ˆ
b −1 )
+
W
r 2 + O
1
r
=
1
r 3 (2 ˆ
f −1 + 4 g
2 h 0 + 8 m g
∗ F ij k
i v
j ) +
W
r 2 + O
1
r
, (7.209)
7 Small Magnetic Black Hole
Hence in order to satisfy
R (A)(A) − 2 E (A)(A) =
1
r 4 × O 3 +
1
r 2 × O 3 +
1
r
× O 1 + O(r
0 ) ,
(7.206)
we must have ˆ
c 1 = −2 h 0 + O 1 (which will be sufficient accuracy for our purposes)
and
ˆ
c 0 = 1 + 8 g
∗ F ij k
i v
j
− 8 g P
2
0
∂m 2
∂x
−
∂l 2
∂y
+
5
3
g
2 ∗ F
p
i
∗ F pj k
i k
j
+ 2 + 2 Q 2
= 1 + 8 g
∗ F ij k
i v
j
−
5
9
g
2 ∗ F
ij ∗ F ij +
20
9
g
2 ∗ F
p
i
∗ F pj v
i v
j
+
10
3
g
2 L (1) −
139
3
g
2 S (3) + 96 g
2 S (2) − 4 g
2 S (1) + 16 g U(u)
∂
∂x
(log P 0 )
−16 g V (u)
∂
∂y
(log P 0 ) + 2 + 2 Q 2 + O 3 .
(7.207)
We note for use below that now
−
1
2
ˆ
c 0 = 8 g
∗ F ij k
i v
j
−
1
2
2 + 2 Q 2 ) +
10
3
g
2 L (1)
−139 g
2 S (3) + 288 g
2 S (2) − 12 g
2 S (1) + 16 g U(u)
∂
∂x
(log P 0 )
−16 g V (u)
∂
∂y
(log P 0 ) .
(7.208)
At this point in the derivation of the equations of motion in second approximation
of the magnetic black hole we have obtained all of the required perturbations in
the metric tensor and the potential 1-form appearing in (7.72)–(7.75) and (7.76)–
(7.83) with the exception of the functions ˆ
f −1 = O 2 and Q 2 = O 2 . The latter
function determines the behaviour of the geometry of the wave fronts of the radiation
generated by the motion of the magnetic black hole and thus is key to our derivation
of the equations of motion. However we need ˆ
f −1 in order to derive Q 2 . Both of
these functions are obtained from the field equation R (4)(4) = 2 E (4)(4) . The leading
term on both sides of this equation is O(r −3 ), neglecting O 3 -terms. We find that
R (4)(4) =
1
r 3
2 ˆ
f −1 + 4 g
2 h 0 − 2 m P
2
0
∂
∂x
(P
−2
0 ˆ
a −1 ) +
∂
∂y
(P
−2
0
ˆ
b −1 )
+
W
r 2 + O
1
r
=
1
r 3 (2 ˆ
f −1 + 4 g
2 h 0 + 8 m g
∗ F ij k
i v
j ) +
W
r 2 + O
1
r
, (7.209)