3.2 Non-Geodesic Null World Line
55
with h 0 given by (3.20). The form of P 0 in (3.21) suggests a coordinate transformation from x, y to ξ, η given by
ξ = P
−1
0
x +
2 v 1
v 0 − v 3
,
(3.23)
η = P
−1
0
y +
2 v 2
v 0 − v 3
.
(3.24)
These transformations result in the line element (3.22) taking the form
ds
2
= −r
2
dξ +
∂q
∂η
du
2
+
dη +
∂q
∂ξ
du
2
+ 2 du dr − 2 h 0 r du
2 ,
(3.25)
with q(ξ, η, u) given by
q =
1
2
A
1 η (ξ
2
−
1
3
η
2 ) +
1
2
A
2 ξ (η
2
−
1
3
ξ
2 ) +
a 0 − a 3
v 0 − v 3
ξη ,
(3.26)
with
A
1
= a
1
−
a 0 − a 3
v 0 − v 3
v
1 and A
2
= a
2
−
a 0 − a 3
v 0 − v 3
v
2 .
(3.27)
Writing the components of k i given by (3.16) and (3.17) in terms of ξ, η we arrive
at
k
i
= ζ
i
−
1
2
ζ j ζ
j v
i ,
(3.28)
with
ζ
i
=
1 − ξ v 1 − η v 2
v 0 − v 3
, −ξ , −η ,
1 − ξ v 1 − η v 2
v 0 − v 3
.
(3.29)
This simplifies the calculation of h 0 (in (3.20)) in terms of ξ, η since a i v i = 0. The
result is
h 0 = a i ζ
i
= A
1 ξ + A
2 η +
a 0 − a 3
v 0 − v 3
=
∂ 2 q
∂ξ∂η
.
(3.30)
We also note that q in (3.26) is a harmonic function and thus
q :=
∂ 2
∂ξ 2 +
∂ 2
∂η 2
q = 0 .
(3.31)