8.3 Perturbations of a Reissner–Nordström Particle
195
Now the differential equation (8.50) for w 1 becomes
0
w 1 = 2 ˙
Q 1 − 4 h 0 Q 1 .
(8.55)
Turning now to the field equation (8.12) we have
M = m + 2 e
2 h 0 − 2 e
2 w 1 + O 3 ,
(8.56)
and
˙
M = ˙
m + 2 e
2 ˙
h 0 − 2 e
2 w 1 + O 3 .
(8.57)
With h 0 given by (8.28) and satisfying (8.30) we can write
˙
h 0 = ˙
a i k
i
− h
2
0 .
(8.58)
It will be useful to introduce the projection of k i orthogonal to v i using the
projection tensor h i j = δ i j − v i v j . This projection will be denoted
p
i
= h
i
j k
i
= k
i
− v
i
⇒ p i p
i
= −1 and p i v
i
= 0 .
(8.59)
It follows from (8.36) that each component p i of the projection is an l = 1 spherical
harmonic:
0
p
i
+ 2 p
i
= 0 .
(8.60)
Since ˙
a i v i = −a i a i , on account of the orthogonality of the 4-velocity and the
4-acceleration, we can write (8.57) as
˙
M = ˙
m + 2 e
2
˙
a i p
i
− 2 e
2 a i a
i
− 2 e
2 h
2
0 − 2 e
2
˙
w 1 + O 3 .
(8.61)
With H given by (8.44), M by (8.56) and ˙
M by (8.61) we have
˙
M − 3 H M = ˙
m − 3 m h 0 + 2 e
2
˙
a i p
i
− 2 e
2 a i a
i
− 8 e
2 h
2
0 − 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 .
(8.62)
The equations of motion for the world line r = 0 in second approximation will have
the general form
m a i = A i
1
+ A i
2
+ O 3 ,
(8.63)