7.3 Magnetic Black Hole Perturbation of Background
163
with L 2 , M 2 , K 1 given by (7.51) and l 2 = O 1 , m 2 = O 1 . We have labelled the O 1 -
terms in ˆ
L 2 and ˆ
M 2 explicitly as l 2 , m 2 respectively because they will be required
in the sequel. In contrast we shall not require a knowledge of the O 1 -term in ˆ
K 1 .
The perturbed line element is given by (7.26)–(7.30) with p, α, β, a, b, c
replaced be their perturbed counterparts ˆ
p, ˆ
α, ˆ
β, ˆ
a, ˆ
b, ˆ
c having the forms
ˆ
p = ˆ
P 0 (1 + ˆ
q 2 r
2
+ . . . ) ,
(7.76)
with ˆ
q 2 = q 2 + O 1 , q 2 given by (7.68) and
ˆ
P 0 = P 0 (1 + Q 1 + Q 2 + O 3 ) with Q 1 = O 1 and Q 2 = O 2 .
(7.77)
Also
ˆ
α = ˆ
α 2 r
2
+ . . . , ˆ
β = ˆ
β 2 r
2
+ . . . with ˆ
α 2 = α 2 + O 1 and ˆ
β 2 = β 2 + O 1 ,
(7.78)
with α 2 , β 2 given by (7.63) and (7.64),
ˆ
a =
ˆ
a −1
r
+ ˆ
a 0 + ˆ
a 1 r + . . . ,
(7.79)
with
ˆ
a −1 = O 1 , ˆ
a 0 = O 1 , ˆ
a 1 = a 1 + O 1 ,
(7.80)
ˆ
b =
ˆ
b −1
r
+ ˆ
b 0 + ˆ
b 1 r + . . . ,
(7.81)
with
ˆ
b −1 = O 1 , ˆ
b 0 = O 1 , ˆ
b 1 = b 1 + O 1 ,
(7.82)
with a 1 , b 1 here given by (7.65) and (7.66) and finally
ˆ
c =
g 2
r 2 −
2 (m + 2 ˆ
f −1 )
r
+ ˆ
c 0 + ˆ
c 1 r + ˆ
c 2 r
2
+ . . . ,
(7.83)
with
ˆ
f −1 = O 2 , ˆ
c 0 = 1 + O 1 , ˆ
c 1 = −2 h 0 + O 1 , ˆ
c 2 = c 2 + O 1 .
(7.84)
Here h 0 and c 2 are given by (7.15) and (7.70) respectively.
With these expansions the null hypersurfaces u = constant in the perturbed
space-time have the potential to play the role of the histories of the wave-fronts