7.5 Equations of Motion in Second Approximation
183
We note that the addition of l = 0 and/or l = 1 terms to Q 2 are trivial, as explained
following (7.85) above. The equation (1) = 0 will give us the equations of motion
of the small magnetic black hole. To see this we first define
p
i
= h
i
j k
j with h
i
j = δ
i
j − v
i v j .
(7.223)
Here p i is the unit, space-like projection of the null vector k i orthogonal to v i .
Since k i is arbitrary it follows that p i points in any direction orthogonal to v i . For
substitution into (7.218) we can write the following quantities in terms of p i =
k i − v i :
L (2) = h
j
i ˙
a j p
i ,
∗ ˙
F ij k
i v
j
=
∗ ˙
F ij p
i v
j , L (1) =
∗ F
p
k
∗ F pj h
k
i v
j p
i ,
h 0 = a i p
i ,
∗ F ij k
i v
j
=
∗ F ij p
i v
j ,
∂
∂x
(log P 0 ) = c i p
i ,
∂
∂y
(log P 0 ) = d i p
i ,
(7.224)
with
c i =
1
2
(v
3
− v
4 ), 0, −
1
2
v
1 ,
1
2
v
1
and d i =
0,
1
2
(v
3
− v
4 ), −
1
2
v
2 ,
1
2
v
2
.
(7.225)
Now (7.218) becomes
(1) = 6
m a i − g
∗ F ij v
j
−
2
3
g
2 h
j
i ˙
a j +
8
3
g
2 ∗ F
p
k
∗ F pj h
k
i v
j
−
4
3
m g
∗ ˙
F ij v
j
+ 2 G a i − ˆ
U(u) c i − ˆ
V (u) d i
p
i .
(7.226)
This is the scalar product of p i , which is any unit space-like vector orthogonal to
v i , with a vector orthogonal to v i and it is required to vanish for all p i . Hence we
must have
m a i = g
∗ F ij v
j
+
2
3
g
2 h
j
i ˙
a j −
8
3
g
2 ∗ F
p
k
∗ F pj h
k
i v
j
+
4
3
m g
∗ ˙
F ij v
j
− 2 G a i + ˆ
U(u) c i + ˆ
V (u) d i .
(7.227)
We note that G(u) is given by (7.220) and thus G(u) can be written as an integral
with respect to u for which we would naturally choose as range of integration
(−∞, u] which results in −2 G a i in (7.227) contributing to a “tail term". We can
write ˆ
U c i + ˆ
V d i = ij (u) v j with ij = − ji vanishing except for 13 =
− 14 = ˆ
U/2 and 23 = − 24 = ˆ
V /2. Define ω ij (u) = −ω ji (u) by ˙
ω ij = ij .
183
We note that the addition of l = 0 and/or l = 1 terms to Q 2 are trivial, as explained
following (7.85) above. The equation (1) = 0 will give us the equations of motion
of the small magnetic black hole. To see this we first define
p
i
= h
i
j k
j with h
i
j = δ
i
j − v
i v j .
(7.223)
Here p i is the unit, space-like projection of the null vector k i orthogonal to v i .
Since k i is arbitrary it follows that p i points in any direction orthogonal to v i . For
substitution into (7.218) we can write the following quantities in terms of p i =
k i − v i :
L (2) = h
j
i ˙
a j p
i ,
∗ ˙
F ij k
i v
j
=
∗ ˙
F ij p
i v
j , L (1) =
∗ F
p
k
∗ F pj h
k
i v
j p
i ,
h 0 = a i p
i ,
∗ F ij k
i v
j
=
∗ F ij p
i v
j ,
∂
∂x
(log P 0 ) = c i p
i ,
∂
∂y
(log P 0 ) = d i p
i ,
(7.224)
with
c i =
1
2
(v
3
− v
4 ), 0, −
1
2
v
1 ,
1
2
v
1
and d i =
0,
1
2
(v
3
− v
4 ), −
1
2
v
2 ,
1
2
v
2
.
(7.225)
Now (7.218) becomes
(1) = 6
m a i − g
∗ F ij v
j
−
2
3
g
2 h
j
i ˙
a j +
8
3
g
2 ∗ F
p
k
∗ F pj h
k
i v
j
−
4
3
m g
∗ ˙
F ij v
j
+ 2 G a i − ˆ
U(u) c i − ˆ
V (u) d i
p
i .
(7.226)
This is the scalar product of p i , which is any unit space-like vector orthogonal to
v i , with a vector orthogonal to v i and it is required to vanish for all p i . Hence we
must have
m a i = g
∗ F ij v
j
+
2
3
g
2 h
j
i ˙
a j −
8
3
g
2 ∗ F
p
k
∗ F pj h
k
i v
j
+
4
3
m g
∗ ˙
F ij v
j
− 2 G a i + ˆ
U(u) c i + ˆ
V (u) d i .
(7.227)
We note that G(u) is given by (7.220) and thus G(u) can be written as an integral
with respect to u for which we would naturally choose as range of integration
(−∞, u] which results in −2 G a i in (7.227) contributing to a “tail term". We can
write ˆ
U c i + ˆ
V d i = ij (u) v j with ij = − ji vanishing except for 13 =
− 14 = ˆ
U/2 and 23 = − 24 = ˆ
V /2. Define ω ij (u) = −ω ji (u) by ˙
ω ij = ij .