8.4 EOMs of a Reissner–Nordström Particle
197
with K 1 given by (8.46), and
1
4
0
K 2 = −3 A i
2
p
i
− 2 e
2
˙
w 1 + 6 e
2 h 0 w 1 − 3 m ˙
Q 1 − 6 e
2 h 0 ˙
Q 1
−2 e
2 P
2
0
∂h 0
∂x
∂w 1
∂x
+
∂h 0
∂y
∂w 1
∂y
− 2 e
2 Q 1 (a i a
i
+ h
2
0 )
−
1
2
Q 1
0
K 1 ,
(8.70)
with K 2 given by (8.47).
8.4
Equations of Motion of a Reissner–Nordström Particle
We begin by solving (8.69) for K 1 (x, y, u). We seek a solution which is well
behaved for −∞ < x, y < +∞. To find this we rewrite (8.69) in the form
1
4
0
K 1 = ˙
m + (−3 A i
1
+ 2 h
j
i ˙
a j ) p
i
− 9 e
2 V 1 .
(8.71)
The first term on the right hand side here is an l = 0 spherical harmonic. Hence to
have a solution K 1 which is non-singular for −∞ < x, y < +∞ we must have
˙
m = 0 ⇒ m = constant .
(8.72)
We have introduced the projection tensor h i
j = δ i
j − v i v j to emphasise that the
second term on the right hand side here is the Minkowskian scalar product of p i ,
which is an arbitrary unit space-like vector orthogonal to v i , with a fixed space-like
vector orthogonal to v i . We note that on account of (8.60) p i is an l = 1 spherical
harmonic. Also
V 1 = h
2
0 +
1
3
a i a
i ,
(8.73)
satisfies
0
V 1 + 6 V 1 = 0 ,
(8.74)
so that V 1 is an l = 2 spherical harmonic. With these observations we obtain the
required solution of (8.71):
K 1 =
0
Q 1 + 2 Q 1 = (6 A i
1
− 4 e
2 h
j
i ˙
a j ) p
i
+ 6 e
2 V 1 + C
1
(u) ,
(8.75)
197
with K 1 given by (8.46), and
1
4
0
K 2 = −3 A i
2
p
i
− 2 e
2
˙
w 1 + 6 e
2 h 0 w 1 − 3 m ˙
Q 1 − 6 e
2 h 0 ˙
Q 1
−2 e
2 P
2
0
∂h 0
∂x
∂w 1
∂x
+
∂h 0
∂y
∂w 1
∂y
− 2 e
2 Q 1 (a i a
i
+ h
2
0 )
−
1
2
Q 1
0
K 1 ,
(8.70)
with K 2 given by (8.47).
8.4
Equations of Motion of a Reissner–Nordström Particle
We begin by solving (8.69) for K 1 (x, y, u). We seek a solution which is well
behaved for −∞ < x, y < +∞. To find this we rewrite (8.69) in the form
1
4
0
K 1 = ˙
m + (−3 A i
1
+ 2 h
j
i ˙
a j ) p
i
− 9 e
2 V 1 .
(8.71)
The first term on the right hand side here is an l = 0 spherical harmonic. Hence to
have a solution K 1 which is non-singular for −∞ < x, y < +∞ we must have
˙
m = 0 ⇒ m = constant .
(8.72)
We have introduced the projection tensor h i
j = δ i
j − v i v j to emphasise that the
second term on the right hand side here is the Minkowskian scalar product of p i ,
which is an arbitrary unit space-like vector orthogonal to v i , with a fixed space-like
vector orthogonal to v i . We note that on account of (8.60) p i is an l = 1 spherical
harmonic. Also
V 1 = h
2
0 +
1
3
a i a
i ,
(8.73)
satisfies
0
V 1 + 6 V 1 = 0 ,
(8.74)
so that V 1 is an l = 2 spherical harmonic. With these observations we obtain the
required solution of (8.71):
K 1 =
0
Q 1 + 2 Q 1 = (6 A i
1
− 4 e
2 h
j
i ˙
a j ) p
i
+ 6 e
2 V 1 + C
1
(u) ,
(8.75)