166
7 Small Magnetic Black Hole
pqkl k
p ∂k q
∂x
= −k k
∂k l
∂y
+ k l
∂k k
∂y
,
(7.102)
pqkl
∂k p
∂x
v
q
= −(v k − k k )
∂k l
∂y
+ (v l − k l )
∂k k
∂y
,
(7.103)
pqkl
∂k p
∂y
v
q
= −
∂k k
∂x
(v l − k l ) +
∂k l
∂x
(v k − k k ) .
(7.104)
These can be checked by direct substitution of k i from (7.16) or otherwise. Using
these we can rewrite (7.92)–(7.97) as
F (1)(2) =
1
r 2 (g + O 2 ) −
∗ F ij (u) k
i v
j
+ O 1 + O(r) ,
(7.105)
F (1)(3) = −P 0
∗ F ij (u) k
i ∂k i
∂y
+ O 1 + O(r) ,
(7.106)
F (2)(3) = P 0
∗ F ij (u) k
i ∂k j
∂x
+ O 1 + O(r) ,
(7.107)
F (1)(4) =
1
r 2 × O 2 +
1
r
× O 1 −
1
2
P 0
∗ F ij (u) k
i ∂k j
∂y
−P 0
∗ F ij
∂k i
∂y
v
j
+ O 1 + O(r)
(7.108)
F (2)(4) =
1
r 2 × O 2 +
1
r
× O 1 +
1
2
P 0
∗ F ij (u) k
i ∂k j
∂x
+P 0
∗ F ij
∂k i
∂x
v
j
+ O 1 + O(r) ,
(7.109)
F (3)(4) = P
2
0
∗ F ij (u)
∂k i
∂x
∂k j
∂y
+ O 1 + O(r) .
(7.110)
Neglecting O 2 -terms to simplify the presentation, since in this section we are
calculating the equations of motion with an O 2 -error (i.e. calculating the equations
of motion in first approximation), the electromagnetic energy-momentum tensor has
the following tetrad components:
E (1)(1) = E (2)(2) =
g
r 2
∗ F ij (u) k
i v
j
+ O
1
r
, E (1)(2) = O
1
r
,
(7.111)
E (1)(3) =
g
r 2 P 0
∗ F ij (u) k
i ∂k j
∂x
+ O
1
r
,
(7.112)