184
7 Small Magnetic Black Hole
Now make the 1-parameter family of infinitesimal Lorentz transformations on the
tangent vector, v i → ¯
v i = v i − (4/3) g ∗ F ij v j − (1/m) ω i j v j . When this is done
and we then drop the bars we arrive at the final form of the equations of motion of the
small magnetic black hole in second approximation (re-introducing the designation
of the O 3 -error which we have been leaving understood for convenience):
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
+ T i + O 3 ,
(7.228)
with
T i =
g
m
{−ω ij
∗ F
j
k + ω kj
∗ F
j
i − 2 G
∗ F ik } v
k
= O 2 .
(7.229)
Since ω ij can be expressed as an integral with respect to u, whose range we naturally
take to be (−∞, u], we see that (7.229) is a “tail term" (depending on the past history
of the source) in the equations of motion (7.228). The first term on the right hand
side of (7.228) is the external 4-force in first approximation, the second term is the
radiation reaction due to the electromagnetic radiation emitted by the accelerating
magnetic black hole and the third term on the right hand side of (7.228) is a second
order correction to the external 4-force. All of these terms have analogues in the
case of a small charged black hole moving in an external field [9]. In particular it
is interesting to note that the second order contribution to the external 4-force in
(7.228) is
−
8
3
g
2 ∗ F
p
k
∗ F pj h
k
i v
j
= −
8
3
g
2 F
p
k F pj h
k
i v
j ,
(7.230)
on account of (7.158). The charged analogue of this term is [9]
+
4
3
e
2 F
p
k F pj h
k
i v
j ,
(7.231)
where e is the electric charge of the small black hole.
7.6
Review of Approximations
In the Appendix E the exact Maxwell field equations are given by (E.7)–(E.10) and
for our model these are satisfied approximately as indicated in (E.23)–(E.26). The
exact Einstein field equations are R (a)(b) − 2 E (a)(b) = 0 and again for our model
(i.e. to derive the equations of motion in second approximation of a small magnetic
black hole) these are satisfied approximately as follows:
R (3)(3) − 2 E (3)(3) = O 1 + O(r) ,
(7.232)
R (A)(3) − 2 E (A)(3) =
1
r 2 × O 3 +
1
r
× O 1 + O 1 + O(r) ,
(7.233)
7 Small Magnetic Black Hole
Now make the 1-parameter family of infinitesimal Lorentz transformations on the
tangent vector, v i → ¯
v i = v i − (4/3) g ∗ F ij v j − (1/m) ω i j v j . When this is done
and we then drop the bars we arrive at the final form of the equations of motion of the
small magnetic black hole in second approximation (re-introducing the designation
of the O 3 -error which we have been leaving understood for convenience):
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
+ T i + O 3 ,
(7.228)
with
T i =
g
m
{−ω ij
∗ F
j
k + ω kj
∗ F
j
i − 2 G
∗ F ik } v
k
= O 2 .
(7.229)
Since ω ij can be expressed as an integral with respect to u, whose range we naturally
take to be (−∞, u], we see that (7.229) is a “tail term" (depending on the past history
of the source) in the equations of motion (7.228). The first term on the right hand
side of (7.228) is the external 4-force in first approximation, the second term is the
radiation reaction due to the electromagnetic radiation emitted by the accelerating
magnetic black hole and the third term on the right hand side of (7.228) is a second
order correction to the external 4-force. All of these terms have analogues in the
case of a small charged black hole moving in an external field [9]. In particular it
is interesting to note that the second order contribution to the external 4-force in
(7.228) is
−
8
3
g
2 ∗ F
p
k
∗ F pj h
k
i v
j
= −
8
3
g
2 F
p
k F pj h
k
i v
j ,
(7.230)
on account of (7.158). The charged analogue of this term is [9]
+
4
3
e
2 F
p
k F pj h
k
i v
j ,
(7.231)
where e is the electric charge of the small black hole.
7.6
Review of Approximations
In the Appendix E the exact Maxwell field equations are given by (E.7)–(E.10) and
for our model these are satisfied approximately as indicated in (E.23)–(E.26). The
exact Einstein field equations are R (a)(b) − 2 E (a)(b) = 0 and again for our model
(i.e. to derive the equations of motion in second approximation of a small magnetic
black hole) these are satisfied approximately as follows:
R (3)(3) − 2 E (3)(3) = O 1 + O(r) ,
(7.232)
R (A)(3) − 2 E (A)(3) =
1
r 2 × O 3 +
1
r
× O 1 + O 1 + O(r) ,
(7.233)