196
8 Reissner–Nordström Particle
with
A i
1
= O 1 , A i
2
= O 2 and A i
1
v
i
= 0 = A i
2
v
i .
(8.64)
Hence
m h 0 = m a i k
i
= m a i p
i
= A i
1
p
i
+ A i
2
p
i
+ O 3 .
(8.65)
Now (8.62) reads
˙
M − 3 H M = ˙
m − 3 A i
1
p
i
+ 2 e
2
˙
a i p
i
− 2 e
2 a i a
i
− 8 e
2 h
2
0
−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
+O 3 .
(8.66)
We note that the first line on the right hand side here consists only of O 1 -terms
while the second line consists only of O 2 -terms. With w given by (8.40), p 2 given
by (8.41) and using (8.32) to show that
P
2
0
∂h 0
∂x
2
+
∂h 0
∂y
2
= −a i a
i
− h
2
0 ,
(8.67)
we find that
e
2 p
2 (w
2
x + w
2
y ) = −e
2 a i a
i
− e
2 h
2
0
−2 e
2 P
2
0
∂h 0
∂x
∂w 1
∂x
+
∂h 0
∂x
∂w 1
∂y
− 2 e
2 Q 1 (a i a
i
+ h
2
0 )
+O 3 .
(8.68)
In similar fashion to (8.66) the first line on the right hand side in (8.68) consists only
of O 1 -terms while the terms on the second line are all O 2 -terms. Now substituting
(8.48), (8.66) and (8.68) into the field equation (8.12) results in the following
equations:
1
4
0
K 1 = ˙
m − 3 A i
1
p
i
+ 2 e
2
˙
a i p
i
− 3 e
2 a i a
i
− 9 e
2 h
2
0 ,
(8.69)