6.3 Generalized Kerr–Schild Space–Times
139
which indicates that the Weyl tensor is Type N (pure gravitational radiation) in the
Petrov classification, with degenerate principal null direction k i . Solving (6.78) we
have
H = p
−1 q f (u) r + H(ζ, ¯
ζ , u) .
(6.80)
Substituting this into (6.77) and (6.79) we find that
p
2 ∂ 2 H
∂ζ ∂ ¯
ζ
+
3
H = 0 ,
(6.81)
and
0 = q
−2 p
2 ∂
∂ζ
q
2 ∂
∂ζ
(p q
−1
H)
,
(6.82)
which indicates that the arbitrary function f (u) in (6.80) is disposable. It can be
removed from the line element (6.73), without affecting the algebraic form of the
line element, by the coordinate transformation
u = γ (u
) , r =
dγ
du
−1
r
with
d 2 γ
du =
dγ
du
2
f (γ (u
)) .
(6.83)
Arguable the simplest geometrical construction is to have the null hyperplanes
u = constant nonintersecting. This requires κ = 0 and either Reβ = 0 or I mβ = 0
or Reβ = const. × I mβ. We shall take I mβ = 0. With κ = 0 we must have < 0
and so, writing
n =
−
6
=
β
α
,
(6.84)
we have
p = 1 − n
2 ζ ¯
ζ and q = α |1 + n ζ |
2 .
(6.85)
Now the line element (6.73) reads (noting that q −1 ˙
q = α −1 ˙
α)
− ds
2
= 2 p
−2 dζ d ¯
ζ + 2 p
−2
|1 + n ζ |
4 α du {d(α r) + p q
−1 H α du} .
(6.86)
A change of coordinates r, u replaced by r = r α and u = u(u ) given by du =
α du effectively reduces α to unity so that we may write (6.86) as
− ds
2
= 2 p
−2 dζ d ¯
ζ + 2 p
−2
|1 + n ζ |
4 du {dr + p |1 + n ζ |
−2 H du} . (6.87)
139
which indicates that the Weyl tensor is Type N (pure gravitational radiation) in the
Petrov classification, with degenerate principal null direction k i . Solving (6.78) we
have
H = p
−1 q f (u) r + H(ζ, ¯
ζ , u) .
(6.80)
Substituting this into (6.77) and (6.79) we find that
p
2 ∂ 2 H
∂ζ ∂ ¯
ζ
+
3
H = 0 ,
(6.81)
and
0 = q
−2 p
2 ∂
∂ζ
q
2 ∂
∂ζ
(p q
−1
H)
,
(6.82)
which indicates that the arbitrary function f (u) in (6.80) is disposable. It can be
removed from the line element (6.73), without affecting the algebraic form of the
line element, by the coordinate transformation
u = γ (u
) , r =
dγ
du
−1
r
with
d 2 γ
du =
dγ
du
2
f (γ (u
)) .
(6.83)
Arguable the simplest geometrical construction is to have the null hyperplanes
u = constant nonintersecting. This requires κ = 0 and either Reβ = 0 or I mβ = 0
or Reβ = const. × I mβ. We shall take I mβ = 0. With κ = 0 we must have < 0
and so, writing
n =
−
6
=
β
α
,
(6.84)
we have
p = 1 − n
2 ζ ¯
ζ and q = α |1 + n ζ |
2 .
(6.85)
Now the line element (6.73) reads (noting that q −1 ˙
q = α −1 ˙
α)
− ds
2
= 2 p
−2 dζ d ¯
ζ + 2 p
−2
|1 + n ζ |
4 α du {d(α r) + p q
−1 H α du} .
(6.86)
A change of coordinates r, u replaced by r = r α and u = u(u ) given by du =
α du effectively reduces α to unity so that we may write (6.86) as
− ds
2
= 2 p
−2 dζ d ¯
ζ + 2 p
−2
|1 + n ζ |
4 du {dr + p |1 + n ζ |
−2 H du} . (6.87)